Tosin, Freeman, Toyin, Moshood, Bayo, Seun Joshua Rose

Dr Muideen

Dr Augustina

Kunle

This thesis is devoted to the mathematical and statistical modeling of epidemic data and It is divided into two broad parts, which are subdivided into different sections. The modeling of infectious diseases has been a subject of interest to researchers, policy makers, and medical practitioners, most especially during the recent global COVID-19 pandemic, which It has been devastating to the health infrastructure and socio-economic status of many nations. It has affected mobility and interaction among citizens due to the many daily new cases and deaths. Hence, the need to contribute to understanding the mechanisms of virulence and spread using different mathematical and statistical modeling approaches. The first part is dedicated to the mathematical modeling aspect, which consists of the deterministic and discrete approaches to epidemiology modeling, which in this case is mainly focused on the COVID-19 pandemic. The daily reproduction number of the COVID-19 outbreak calculation is approached by discretization using the idea of deconvolution and a unique biphasic pattern is observed that is more prevalent during the contagiousness period across various countries. Furthermore, a discrete model is formulated from Usher's model in order to calculate the life span loss due to COVID-19 disease and to also explain the role of comorbidities, which are very essential in the disease spread and its dynamics at an individual level. Also, the formulation of Susceptible-Infectious-Geneanewsusceptible-Recovered (SIGR) age-dependent modeling is proposed in order to perform some mathematical analysis and present the role of different epidemiology parameters, most especially vaccination, and finally, a new technique to identify the point of inflection on the smoothed curves of the new infected pandemic cases using the Bernoulli equation is presented. This procedure is important because not all countries have reached the turning point (maximum number of daily cases) in the epidemic curve. The approach is used to calculate the transmission rate and the maximum reproduction number for various countries. The statistical modeling of the COVID-19 pandemic using various data analysis models (namely machine and deep learning models) is presented in the second part in order to understand the dynamics of the pandemic in different countries and also predict and forecast the daily new cases and deaths due to the disease alongside some socio-economic parameters. It is observed that the prediction and forecasting are consistent with the disease evolution at different waves in these countries and that there are socio-economic determinants of the disease depending on whether the country is developed or developing. Also, the study of the shapes and peaks of the COVID-19 disease is presented. The peaks of the curves of the daily new cases and deaths are identified using the spectral analysis method, which enables the weekly peak patterns to be visible. Finally, the clustering of different regions in France due to the spread of the disease is modeled using functional data analysis. The study shows clear differences between the periods when vaccination has not been introduced (but only non-pharmaceutical mitigation measures) and when it was introduced. The results presented in this thesis are useful to better understand the modeling of a viral disease, the COVID-19 virus. i I thank God for assisting me in starting this journey and seeing it through to completion because without him, I would not have been able to complete this doctoral journey. Prof. Mustapha Rachdi and Prof. Jacques Demongeot, my wonderful supervisors, deserve my heartfelt thanks. Your help has been invaluable throughout this journey. Despite meeting you for supervision during the most difficult period of my life, you made the journey interesting for me. I truly appreciate it.

Résumé

Cette thèse est consacrée à la modélisation des épidémies et elle est divisée en deux parties qui se subdivisent elles-mêmes en différentes sections. La modélisation des maladies infectieuses a suscité depuis de nombreuses années (la première modélisation de la propagation de la variole de D. Bernoulli date de 1760)) l'intérêt des décideurs et des médecins, plus particulièrement lors de la récente pandémie de COVID-19, qui a déstabilisé les infrastructures de santé, altéré le statut socio-économique et affecté les interactions entre citoyens dans de nombreux pays, en raison de multiples cas et décès, d'où la nécessité de comprendre les mécanismes de virulence et de propagation, à l'aide de diverses approches de modélisation mathématique et statistique. La première partie est consacrée à l'aspect modélisation mathématique d'une épidémie, qui consiste en des approches déterministes et discrètes de la propagation épidémique, principalement axée sur la pandémie de COVID-19. Le nombre quotidien de reproduction durant la période de contagiosité est approché par discrétisation, fondée sur l'idée de dé-convolution, et la forme bi-phasique a été la plus souvent calculée dans divers pays, ceci étant possible, si on suppose que la maladie a commencé par un unique patient infectieux « zéro » au démarrage de l'épidémie et que les taux de transmission et la taille de la population susceptible sont constants en phase de croissance exponentielle des nouveaux cas observés. Ensuite, un modèle discret été formulé à partir du modèle d'Usher, afin de calculer la perte de durée de vie des infectés, due à la maladie COVID-19 et afin d'expliquer également le rôle des comorbidités, qui est très essentiel dans la propagation de la maladie et sa dynamique au niveau individuel. En outre, une formulation de la modélisation de la dynamique épidémique dépendante de l'âge des infectés, de type Susceptible-Infectious-Geneanewsusceptible-Recovered (SIGR), a été proposée, afin d'analyser le rôle de différents paramètres épidémiologiques, plus particulièrement celui de la vaccination. Enfin, une nouvelle technique a été proposée pour identifier le point d'inflexion sur les courbes lissées des nouveaux cas infectés, cela à l'aide de l'équation princeps de Bernoulli. Cette procédure est importante, car les vagues de nombreux pays n'ont pas atteint le point de retournement (maximum de cas quotidiens) de leur courbe épidémique, en particulier pour des vagues proches, dans lesquelles on peut avoir une succession d'épaulements, mais pas de maximas. La modélisation statistique de la pandémie de COVID-19, à l'aide de divers modèles d'apprentissage automatique et profond, est présentée dans la deuxième partie, afin de comprendre la dynamique de la pandémie dans différents pays, ainsi que de prédire et de prévoir les nouveaux cas quotidiens et les décès dus à la maladie, ainsi que certains paramètres socio-économiques. On observe que la prédiction et la prévision sont cohérentes avec l'évolution de la maladie durant différentes vagues dans ces différents pays, et qu'il existe des déterminants socio-économiques de la maladie, selon que le pays est développé ou en voie de développement. De plus, l'étude du nombre et des formes des pics de la maladie COVID-19 est présentée. Les pics des courbes des nouveaux cas et des décès quotidiens sont identifiés à l'aide d'une méthode d'analyse spectrale, qui permet de visualiser les modèles de pics hebdomadaires. Enfin, la classification des départements ii français vis-à-vis de la propagation de la maladie est effectuée par analyse fonctionnelle des données. Cette dernière étude montre des différences nettes entre la période où la vaccination n'a pas encore été introduite (seulement des mesures d'atténuation non pharmaceutiques) et celle où elle l'a été. Les résultats de cette thèse seront particulièrement utiles dans le futur pour mieux comprendre les mécanismes de propagation d'une maladie virale, ici la maladie due au virus SARS-CoV-2.

iii

Dedication

The thesis is dedicated to God Almighty, the Giver of Knowledge. To my mother -Mrs B. Oshinubi, who has been a source of inspiration despite her lack of a university education. iv Chapter 1 Introduction

General Background

Epidemiology and infectious disease modeling and its application to actual viral disease data has been envisioned by many researchers over the years. The modeling of infectious diseases started as early as 1760 by Daniel Bernoulli [START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF] who proposed the famous deterministic mathematical model Susceptible-Infected (SI) to solve the epidemic wave of smallpox, but it is a known fact that the stages of infectious diseases are the stages endemic, epidemic and eradicated while the model he presented defined but not modeled the endemic state. He said: "Le retour d'une épidémie longtemps suspendue fait un ravage plus terrible dans une seule année qu'une endémie uniforme ne pourrait faire pendant un nombre d'années considérable.", which means in english: "The return of a long-suspended epidemic wreaks more terrible havoc in a single year than a uniform endemic could do for a considerable number of years." This shortcoming has now been taken into account because we have had many models of diseases whose endemic state has been considered due to their endemicity. The improvement of the SI model is now extended to models such as Susceptible-Infected-Recovered (SIR), Susceptible-Exposed-Infected-Recovered (SEIR), Vaccinated-Susceptible-Infected-Recovered-Deaths (VSIRD), Susceptible-Infected-Recovered-Deaths (SIRD), etc. which provide an indepth understanding of the dynamics of viral diseases. Figure 1.1 is a simple SEIR model where α is the contact rate and γ is the recovery rate. The problem of the initial conditions of an epidemy has been already evoked by Bernoulli [START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF], but in general it is at best only through the hypothesis of a unique "patient zero" allowing the definition on him of his initial daily reproduction number [START_REF] Nishiura | Time variations in the transmissibility of pandemic influenza in prussia, germany, from 1918-19[END_REF]. The patient zero, also called index or primary case is the first patient identified in a given population during an epidemiological investigation. Despite the considerable means of current investigation, in particular those of the World Health Organization (WHO) and the member governments of the WHO, this patient is rarely identified, despite some cases as the H1N1 epidemic in Mexico [3]. In general, the sources of an infectious disease spread in a given reservoir indicates with their subsequent clusters the possible existence of an outbreak, but this search is in general very long and difficult as for HIV in North America [4]. Concerning these initial clusters, from observations made during investigations of the start of the outbreak in some countries [START_REF] Yong | Connecting clusters of covid-19: an epidemiological and serological investigation[END_REF][START_REF] Adam | Clustering and superspreading potential of sarscov-2 infections in hong kong[END_REF][START_REF] Desjardins | Rapid surveillance of covid-19 in the united states using a prospective space-time scan statistic: Detecting and evaluating emerging clusters[END_REF][START_REF] Pung | Investigation of three clusters of covid-19 in singapore: implications for surveillance and response measures[END_REF][START_REF] Ganyani | Estimating the generation interval for coronavirus disease (covid-19) based on symptom onset data, march 2020[END_REF][START_REF] Kremer | Authors' response: Estimating the generation interval for covid-19 based on symptom onset data[END_REF][START_REF] Jing | Household secondary attack rate of covid-19 and associated determinants in guangzhou, china: a retrospective cohort study[END_REF][START_REF] Böhmer | Investigation of a covid-19 outbreak in germany resulting from a single travel-associated primary case: a case series[END_REF][START_REF] Guttmann | Performance map of a cluster detection test using extended power[END_REF][START_REF] Shams | A death, infection, and recovery (dir) model to forecast the covid-19 spread[END_REF][START_REF] Chan | Surveillance and epidemiology of infectious diseases using spatial and temporal lustering methods[END_REF][START_REF] Hisada | Surveillance of early stage covid-19 clusters using search query logs and mobile device-based location information[END_REF][START_REF] Han | Exploring the clinical characteristics of covid-19 clusters identified using factor analysis of mixed data-based cluster analysis[END_REF][START_REF] Ladoy | Size and duration of covid-19 clusters go along with a high sars-cov-2 viral load: A spatio-temporal investigation in vaud state, switzerland[END_REF][START_REF] Tariq | Real-time monitoring the transmission potential of covid-19 in singapore, march 2020[END_REF], it is sometime possible to get spatial and temporal information on the start of the epidemy, but rarely these studies allows estimating the parameters S(0) and τ (0) in the concerned population and worse, they give no indication of how long they remain constant. This is the reason why we assumed that they remained constant only during the period of exponential growth of new cases observed. A way to solve this problem could be to use the probability distribution given in some of these articles for the length r of the contagiousness period (in general a not-symmetrical lognormal distribution) considered as a random variable and then, for a sample of values of r, simulate the case by adding another noise on the new observed cases X i at day i of the pandemic, e.g., a Gaussian noise concentrated on the mean values given by different databases devoted to the new cases counting. The world is recently facing with a new viral disease named COVID-19, whose modeling is the main focus of this thesis. The virus was first discovered in Wuhan China in December, 2019 and since then it has spread to more than 230 countries. There are different symptoms the host of this highly contagious virus do experience which vary in different ways, they are fever, shortness of breath, fatigue, coughing, loss of smell and taste, difficult in breathing and other lower respiratory effects. It is also observed that we have more asymptomatic than those who are symptomatic and it takes about 5-6 days on the average for symptoms to show, and however, it can take up to 14 days in some cases. As published by World Health Organization (WHO) [START_REF]World health organization[END_REF] in March 2020, there are four types of COVID-19 transmission: no cases or no reported cases for at least 28 days, sporadic cases (one or more cases, imported or locally acquired, detected in the last 14 days), clusters of cases (most cases of local transmission linked in chains of transmission detected in the last 14 days are predominantly limited to well-defined clusters), and community transmission (one or more cases, imported or locally acquired, detected in the last 14 days). The number of people infected per unit time is referred to as the incidence in epidemiology. This incidence increases as the disease spreads from persons to persons. The spread of the disease from patient zero to other susceptible part of the population during the contagiousness period without public health interventions is measured by the basic reproduction rate. This parameter is important in epidemiology modeling and it will be critically discussed and estimated in this thesis. This spread is more severe in people with comorbidities (kidney failure, obesity, cancer, complicated hypertension, chronic liver disease, psychiatric disorders, dementia, respiratory and heart failure) and severely immunocompromised persons (people with rare and serious pre-existing conditions or severe disabilities, history of stroke, patients with autoimmune diseases or aggressive immunosuppressive therapy, dialysis patients, and hematopoietic stem cell, solid organ or recent bone marrow transplant recipients) due to their high risk factor and vulnerability to infectious diseases. It has been on the top of the agenda of various stakeholders like researchers, scientists, politicians and medical practitioners to find a way to solve this global menace which has cripple health facilities and economy globally due to its high rate of incidence and fatalities of the disease which make it to be a major concern and significant disruption has been caused to the socio-economic structure. At the beginning of the pandemic, delays in reporting COVID-19 cases and incubation period (the number of days before an infected individual starts showing symptoms) affect policy making such as the introduction of mitigation measures. Because many countries most especially developing countries were not prepared even though it appears no countries was prepared for it so, there were many efforts to scale up Intensive Care Unit (ICU) beds, ventilators, awareness on sneezing and coughing in elbow, discouraging large gatherings, setting up new isolation centers, encouraging awareness on washing of hands, provision and use of sanitizers and masks, as well as various medical interventions and equipment necessary to help those who are critically ill depending on their symptoms and age because their was a rapid increase in the number of hospitalization cases. Also, place of work and schools started running their activities online in order to encourage e-learning and teleworking so as to avoid physical contacts and interactions like handshakes and greetings with hugs and kisses, and due to isolation, many were affected psychologically with little or no support from stakeholders. Till date, all efforts to mitigate its spread which are ensuring facilities that are open put hygiene concepts (well ventilated environment, sanitizers and nose masks) in place, lock-downs (partial or total), curfew, face-masks, limit in the number of people gathering, restriction of non-essential businesses, stoppage of social, recreation and sporting activities, aggressive contact tracing, massive testing and its prerequisite for travels, social distancing, and administration of different doses of vaccine to the population especially the most vulnerable (elderly) has not yield major result and we have seen many reinfections even in fully vaccinated people but we can't deny the fact that it has helped to reduce hospitalization and severe illness because the virus keep mutating most especially the Delta and Omicron variants which are known for their high contagiousness. There are a lot of concerns when this outbreak will be endemic because it can't continue to cripple activities all over the world because mobility, businesses and tourism must continue and if possible to live with it as normal flu however, the aggressive vaccination campaign its still ongoing whereby more doses is prescribed by medical scientist so as to proper manage the situation we have found ourselves. We have also observed that there are some demographic determinants of the spread of the disease, since we do have more incidence most especially when lock-downs are eased, during festive and winter period and less cases in other season. The spread and fatality also depends on age (severe in elderly and less severe in young and children) but due to its mutation, the recent Omicron variant is highly contagious, then the epidemic curve shows that children and adult closer to children (like teachers and parents) are more infected, reason why there is a call for vaccination of children which was not the case at the beginning of the pandemic. Developing an infectious disease model appropriate for the current COVID-19 epidemic that can predict the epidemic's development trend and can also fit an epidemic that has already occurred hence, our goal in this thesis is to contribute to the modeling of COVID-19 outbreak in two directions namely (i) the mathematical modeling and (ii) statistical modeling. In the next section, we present an overview of some literature work in these two directions.

Systematic Review of Literature

People across the globe are looking for a way to put an end to the COVID-19 pandemic even though on medical level, scientists are currently working on drugs to combat the disease spread and there were different approaches and suggestions by many researchers, some of these contributions in literature will be discussed in this section based on the direction of this thesis in terms of mathematical modeling, statistical modeling, public health policies and there medical implications. A lot of mathematical and statistical models have been proposed by researchers in order to suggest best way to combat the spread of the disease and to understand the disease population dynamics. On searching the key word 'Mathematical and Statistical modeling of COVID-19' on cross-reference, there are thousands of articles on this subject which only few of them will be discussed and the summary of models that have been developed and applied to the study of COVID-19 outbreak ranges from mathematical to statistical and theoretical ones, these models employing insight from mathematics, physics, statistics, medicine, epidemiology and infectious disease. The models summarized in this section are of the forms: difference equation, ordinary differential equation, partial differential equation, matrix model, numerical model, individualbased model, population-based model, regression model, survival function, hazard function, stochastic process, and deep learning model. Discrete-time dynamics are used for stochastic and deterministic population growth models while continuous-time dynamics can be used for host-virus models, stochastic population growth models and advection-diffusion models. Also, spatially homogeneous, deterministic and delay formulations are mostly used for advection-diffusion models, host-virus models and population growth models. Age-structured population models are characterized by linear system of difference equations and can also include stochastic effect and agent-based models that simulate actions and interactions among individuals within the population. They are computer model and can be deterministic or stochastic. Numerical models solve complex dynamical systems and realistic models. Diffusion equation can be used to study the evolution of these models. The idealistic assumption of semi-mechanistic model is that the country is viewed as a closed environment, with all infections assumed to occur within the population. Lastly, statistical tools help to formulate models based on artificial intelligence, survival analysis, machine learning, logistic regression, random effects and sometimes these models include spatial effects. The pandemic started at a time when many countries were not prepared for the impact the spread would have on the daily activities of the people and scientists are trying to understand if this strain will be like other Coronaviruses strains. Since the pandemic started in China, a model to study the country preparedness and management was proposed in [START_REF] Altakarli | China's response to the covid-19 outbreak: A model for epidemic preparedness and management[END_REF]. In [START_REF] Mackolil | Logistic growth and sir modelling of coronavirus disease (covid-19) outbreak in india: Models based on real-time data[END_REF] authors used logistic growth model and SIR model to study the outbreak in India which is one of the country that was badly hit by the pandemic and it is a country with a very high population so, there was a lot of interactions among people and movement of people from one place to another enabling contacts which lead to high rate of the spread of the disease [START_REF] Khanday | Mathematical analysis on the dynamics of covid-19 in india using sir epidemic model[END_REF]. There are different waves or exponential phases of the pandemic, and most countries are in their fourth wave while others are already in their fifth wave. A non-linear mathematical model was proposed in [START_REF] Yaghoby | A new nonlinear mathematical model for forecasting the waves of the COVID-19 epidemic[END_REF] to forecast different waves of COVID-19 pandemic. Also, authors in [START_REF] Kaxiras | Multiple epidemic wave model of the covid-19 pandemic: Modeling study[END_REF] proposed a model for the multiple epidemic waves to study different exponential growth phases of the pandemic. In [START_REF] Meiksin | Using the seir model to constrain the role of contaminated fomites in spreading an epidemic: An application to covid-19 in the uk[END_REF] authors formulated a SEIR compartment population dynamics model with an added formite term. The formite term is analyzed as a means of statistically quantifying the contribution of contaminated fomites to the spread of COVID-19 pandemic. The model helps to guide health policy makers to know the contribution of some fomites to the spread of infection in COVID-19 virus. The use of artificial intelligence techniques is important in infectious disease modeling. In [START_REF] Cámara | An epidemic model for economical impact predicting and spatiotemporal spreading of COVID-19[END_REF], authors developed an artificial intelligence tool which combines epidemiology model and medical data. This approach helped to understand Spanish population movements based on mobile phone geographically located information to determine their economy activities hence, an economic impact is predicted alongside spatio-temporal spread of COVID-19. Travel restrictions is one of the measures implemented which affected many countries whose national income depends heavily on tourism. [START_REF] Paunovic | Mathematical modeling and statistical analysis in post -covid-19 tourism recovery[END_REF] presented and discussed mathematical and statistical analysis of this measures and its effects in tourism sector. The author in [START_REF] Duan | Simulations of the covid-19 epidemic in nigeria using sir model[END_REF] simulated COVID-19 cases in Nigeria to better understand the peaks in the evolution of the disease recorded using SIR model with certain parameters to be optimized by using the Nelder-Mead algorithm with maximum likelihood estimator. The use of fractional derivative to model physical problems has played important role in epidemics modeling. It is known to understand the physical meaning of real data fielded into the model even though the model is old, and there has been some improvements in recent time. Some of those improvements include applying it to COVID-19 outbreak modeling. In [START_REF] Padmapriya | Fuzzy fractional mathematical model of covid-19 epidemic[END_REF] authors proposed a fuzzy fractional model for COVID-19 epidemic and due to high transmission rate of the disease, in [START_REF] Kozioł | Fractional-order sir epidemic model for transmission prediction of covid-19 disease[END_REF], fractional order of SIR model was used to understand and predict the transmission of COVID-19 disease. Age dependent modeling helps to better understand the dynamics of the disease at subpopulation level. Some articles have been devoted to study the role of vaccination at subpopulation level and the spread of the disease based on age whose fatality rate is higher in elderly class but for the young class, the transmission rate is high. Age structured compartment model was proposed to study the spread of COVID-19 in Belgium [START_REF] Deforche | An age-structured epidemiological model of the Belgian COVID-19 epidemic[END_REF] and in Switzerland [START_REF] Balabdaoui | Age-stratified discrete compartment model of the covid-19 epidemic with application to switzerland[END_REF]. To see the effect of vaccination in age modeling, authors in [START_REF] Verrelli | Two-age-structured covid-19 epidemic model: Estimation of virulence parameters to interpret effects of national and regional feedback interventions and vaccination[END_REF] used available data at national level in Italy to validate their proposed model. The effect of the pandemic on mental health of the people cannot be overemphasized as many fell into depression and anxiety of the unknown and of what will happen next most especially health workers, teachers, workers, students and lecturers due to isolation when various control measures were put in place. As a result, many were separated from families and loved ones, people could not interact because of social distancing, workers were forced to do teleworking and students to start taking classes online to reduce physical interactions. There were reports that psychiatric patients at home were increasing because there was little or no support on mental health during the lockdown. Some studies in [START_REF] Liu | Focus on the mental health of pediatric medical workers in china after the covid-19 epidemic[END_REF][START_REF] Li | Epidemic risk perception, perceived stress, and mental health during covid-19 pandemic: A moderated mediating model[END_REF] focus on mental health during the COVID-19 pandemic. The results presented show that epidemic risk perception is positively correlated with perceived stress, depression and anxiety. There is a delay effect between the contact date of a carrier of the virus and the time when the contact is infectious or starts showing symptoms and also delay in testing and reporting during quarantine or isolation. This delay can be modeled in a stochastic [START_REF] Rihan | Stochastic sirc epidemic model with time-delay for covid-19[END_REF] or a deterministic approach [START_REF] Kang | Epidemic spreading and global stability of a new sis model with delay on heterogeneous networks[END_REF]. Studies have shown that many carriers of the virus do not show symptoms (they are called asymptomatic) but they are infected. This observation and that of many other factors have led researchers to model this kind of phenomenon using different approaches but adding a term that takes into consideration delay. [START_REF] Dell'anna | Solvable delay model for epidemic spreading: the case of covid-19 in italy[END_REF] proposed a delay model to better understand the wide spread of the disease in Italy because at the beginning of the global pandemic there was a collapse of their health infrastructure and many deaths were recorded. The result presented by authors in this article present an insight for this delay and how it enhances the spread of the disease in Italy. The COVID-19 pandemic has spread all over the world and it is important to study the propagation of information related to the spread of the disease, that is why in [START_REF] Bianconi | Epidemic spreading[END_REF], the author presented a SIR model characterized by multi-layer networks which is a framework that incorporated the study of the interaction and awareness during epidemic spreading specifi-cally. A related study proposed this type of framework in the case of COVID-19 outbreak in terms of misinformation of the people in relation to the virus, spread of rumors and fake news, and also the role of media (social and print) in the epidemic spread [START_REF] Nian | A new spreading model in the environment of epidemicrelated online rumors[END_REF][START_REF] Zhao | Epidemic-related sites in covid-19 media reports[END_REF][START_REF] Süt | Epigenetic regulation mechanisms in viral infections: A special focus on covid-19[END_REF][START_REF] Jayaraj | Covid-19: When the misinformation is spreading faster than the virus[END_REF]. They proposed that all these factors can affect the spread of the disease and the population opinion based on the information it accesses. As the virus spread increases, government, health policy makers and experts suggested different prevention strategies which have helped to mitigate the spread of the disease and reduce the basic reproduction rate per time. Models incorporating these strategies have been developed, and in [START_REF] Zihady | Multiple variable social distancing model in reducing covid 19 spreading: A long term social distancing model in developing nations like bangladesh[END_REF][START_REF] Gounane | An adaptive social distancing sir model for covid-19 disease spreading and forecasting[END_REF] authors formulated a SIR model which was based on social distancing, a measure introduced by government in many countries to control COVID-19 disease spread. Recently, vaccines to effectively tackle the pandemic were produced after many trials by researchers. Vaccines such as Moderna, Pfizer, AstraZeneca, Novavax, Johnson and Johnson are many authorized in Europe which have been administered to people based on age and the number of doses depends on the efficacy of the vaccine because research proves that immunity decreases after some time reason why the third dose was introduced to better tackle the virus. Mathematical and statistical modeling of the pandemic taking into account vaccination has been studied. Author in [START_REF] Webb | A covid-19 epidemic model predicting the effectiveness of vaccination[END_REF] predicts the effectiveness of the vaccination, [START_REF] Temerev | Geospatial model of covid-19 spreading and vaccination with event gillespie algorithm[END_REF] used Geo-spatial model with Gillespie algorithm to investigate COVID-19 spread and vaccination, [START_REF] Alqahtani | Development and analysis of a seir model for covid-19 epidemic with vaccination and non-singular kernel[END_REF] analyzed and developed SEIR model for COVID-19 pandemic with vaccination and non singular kernel, [START_REF] Webb | A covid-19 epidemic model predicting the effectiveness of vaccination in the us[END_REF] investigates the effectiveness of vaccination in the United States using predictive epidemic model. Authors were able to demonstrate that the implementation of vaccination depends on other social interventions introduced to reduce transmission rate at the population level. There are other models that have been used to study COVID-19 disease. They are SEIR model with quarantine and fatality compartment [START_REF] Hidayat | Epidemic model analysis of covid-19[END_REF], lattice model [START_REF] Gavalas | A Lattice Model for COVID-19 Epidemic[END_REF], a recursive model [START_REF] Ilyin | COVID-19 spreading: a recursive model[END_REF], computational model [START_REF] Below | Comparative analysis of the spread of the covid 19 epidemic in berlin and new york city based on a computational model[END_REF], networks based model [START_REF] Pizzuti | Network-based prediction of covid-19 epidemic spreading in italy[END_REF][START_REF] Zuo | A new coupled awareness-epidemic spreading model with neighbor behavior on multiplex networks[END_REF], tailored model [START_REF] Arenas | Epidemic spreading: Tailored models for covid-19[END_REF], growth rate model [START_REF] Bianconi | Ostwald growth rate in controlled covid-19 epidemic spreading as in arrested growth in quantum complex matter[END_REF], parameter estimation model [START_REF] Prodanov | Analytical parameter estimation of the sir epidemic model. applications to the covid-19 pandemic[END_REF], Weibull distribution model [START_REF] Moreau | Using the weibull distribution to model covid-19 epidemic data[END_REF], Markovchain model [START_REF] Xu | A continuous markov-chain model for the simulation of covid-19 epidemic dynamics[END_REF], in-homogeneous spatial model [START_REF] Tsori | Epidemiological model for the inhomogeneous spatial spreading of covid-19 and other diseases[END_REF], fuzzy model [START_REF] Tsaur | Fuzzy portfolio selection in covid-19 spreading period using fuzzy goal programming model[END_REF], hybrid intelligent model [START_REF] Zhang | Prediction and control of covid-19 spreading based on a hybrid intelligent model[END_REF], case based rate reasoning model [START_REF] Zakharov | Cbrr model for predicting the dynamics of the covid-19 epidemic in real time[END_REF], forecasting model [START_REF] Bereketoglu | Forecasting model to predict the spreading of the covid-19 outbreak in turkey[END_REF], complex mathematical model [START_REF] Akyildiz | Complex mathematical sir model for spreading of covid-19 virus with mittag-leffler kernel[END_REF], time series model [START_REF] Jamshidi | A new model for epidemic spreading with a focus on covid-19[END_REF], and hierarchical epidemic risk model [START_REF] Cherkashin | Hierarchical epidemic risk modeling of spreading new covid-19 coronavirus[END_REF]. All these models were used to predict, forecast and understand COVID-19 spread dynamics. To conclude this section, it has been shown in this section that a lot of models have been developed and applied to predict and forecast the pandemic at different waves, by taking into account different control measures and interventions (quarantine, use of face-mask, social distancing and vaccination) introduced to avoid escalation of the disease. The estimation of the basic reproduction number, demographic models and socio-economic impacts of the pandemic, statistical and mathematical approaches were discussed in some articles. The goal of this thesis is to contribute significantly to the modeling of COVID-19 pandemic by developing, applying and suggesting new models using different approaches.

Thesis Outline

The first chapter is based on the general introduction. Chapter 2 deals with the state of art and the methodologies used in the thesis. Chapter 3 and Chapter 4 deals with the results obtained from the mathematical and statistical modeling of the COVID-19 outbreak which are from the scientific publications (published and in preparation) produced from the thesis. In the last chapter, the perspectives and conclusions derived from the thesis are discussed Chapter 2

State of the Art

Mathematical Methodology

In this section, we present the mathematical methodologies used in the thesis for the modeling of the COVID-19 disease. These methodologies include Bernoulli's epidemiological model, D'Alembert's epidemiological model, Ross and McKendrick SIR model, Leslie Model and Usher's Model.

Bernoulli's Epidemiological Model

Daniel Bernoulli [START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF] developed a mathematical model by taking into consideration the dynamics of smallpox infection. In the model, the whole population was divided into susceptibles and immunized where the probability of a newborn to be susceptible is denoted by p and the probability of being immunized to the smallpox infection is denoted as q. At age x, the natural death rate is denoted by µ(x), and the transmission rate is denoted by β(x). Only a fraction of susceptibles z(x) survives to become immune while the rest b(x) = 1 -z(x) dies due to the infection which means that b(x) is the fatality rate. We will refer to it as case fatality because it is not a rate (with dimension per unit of time) but a probability, i.e. a dimensionless quantity. Let p(x) denote the likelihood of a newborn being alive and susceptible at age x. The probability q(x) to be immunized for life and be alive is represented by the differential equation below:

dq dx = (1 -b(x))β(x)p(x) -µ(x)q,
with the initial condition q(0) = 0. Daniel Bernoulli used the logistic model as a phenomenological approach to fit data to the model he developed above corresponding to the daily new cases of the smallpox infection. He described the disease's endemic state following a wave, but he did not propose a model for describing the endemic dynamic that is now considered to be a stationary stochastic process with constant average and variance, nor did he use a precise method to detect critical boundary times corresponding to ruptures between epidemic and endemic phases.

D'Alembert's Epidemiological Model

D'Alembert [START_REF]Opuscules mathématiques[END_REF] developed in 1761 an alternative method for dealing with competing risks of death, which is also applicable to non-infectious diseases. D'Alembert developed an alternative solution in approaching epidemiological modeling developed by Daniel Bernoulli which nowadays one would call a non-parametric approach in contrast to the Bernoulli parametric model [START_REF] Dietz | Daniel bernoulli's epidemiological model revisited[END_REF]. Figure 2.2 shows the structure of D'Alembert's epidemiological model. D'Alembert approach is quite general and is not restricted to an immunizing infectious disease. Suppose λ(x) denote the death rate at age x due to a particular infectious disease and the natural death rate at age x is denoted as µ(x). Let ψ(x) denote the rate at which deaths due to a specific cause are recorded for people who die at age x. Then we have:

ψ(x) = λ(x)l(x),
where the survival function at age x, l(x), is defined by:

log l(x) = y≤x log(1 -µ(y)).
If the survival function l(x) is known, the mortality due to the disease can be calculated by dividing ψ(x) by l(x). The d'Alembert's epidemiological model is unquestionably more general than the Bernoulli epidemiological model, which is limited to immunizing against infections. The Bernoulli's epidemiological model, on the other hand, provides significantly more insight for the interpretation of infectious disease data.

Ross and McKendrick SIR Model

The most common SIR model which is an improvement of Bernoulli epidemiological model is the SIR Model by Ross and then by McKendrick [START_REF] Ross | An application of the theory of probabilities to the study of a priori pathometry.-part i[END_REF][START_REF] Kendrick | Applications of mathematics to medical problems[END_REF], which was assisted by W.O. Kermack. The Kermack-McKendrick model is a SIR model for the number of people infected by an infectious disease in a closed population over time. It was proposed to explain the rapid rise and fall in the number of infected cases observed in disease outbreaks. This model represents diseases against which people have a lifetime immunity and it does not consider the possibility of re-infection with the disease which is also an important factor that should be considered. The model is expressed as:

dS dt = δS + δI + (δ + γ)R -βSI -µS, dI dt = βSI -(µ + ν)I, dR dt = νI -(µ + γ)R
where S is the size of the susceptible, I the size of the infectious, and R the size of the recovered, with the total population N defined as N = S + I + R. β is the transmission rate, δ the birth rate (supposed to be equal to the death rate µ), γ is the loss of immune resistance, and ν is the immunization rate. Determining the epidemiological threshold is a critical component in determining whether or not to implement lockdown. This decision would also be based on the disease's fatality, which is beyond the scope of the SIR models. The basic reproduction number which is the epidemiological threshold is expressed as: R 0 = βN ν + δ The model can also be used for age dependent modeling by defining age classes between 1 and m and by denoting S j , I j , R j , j = 1, 2, . . . , m, each subpopulation of S, I, and R at age j, we can define at any stationary state (S * , I * , R * ) the probabilities for a newborn individual of being alive and either susceptible, infectious or immune at age j by the following formula:

(S i + I i + R i )
These formula make the link between the Bernoulli and the Ross and Kermack-McKendrick epidemiological models.

Leslie Model

Leslie model which is sometimes called Leslie matrix is the first discrete population growth model used for age structured modeling and it is well known for projecting how age is distributed in a particular population. The model was proposed by Leslie in 1945 [START_REF] Leslie | Some further notes on the use of matrices in population mathematics[END_REF][START_REF] Leslie | On the use of matrices in certain population mathematics[END_REF]. It majorly considers one sex (female), fecundity and deaths among the sub-population (different age classes) with a population of closed migration. Leslie model uses the age pyramid vector p(t) = (p j (t)) j=1,2,...,m , where p j (t) represents the size of the age class j at time t, with j ranging from the birth age 1 to the maximal death age m, whose discrete dynamics is governed by the matrix equation given as: where ∀ j = 1, . . . , m, b j = 1 -µ j ≤ 1 (where µ j is the death rate at age j) is the survival probability between ages j and j + 1 and f 1 is the birth rate at age j (i.e., the mean number of offsprings from an individual at age j). The dynamic stability of the Leslie system for a distance d is quantified by the tendency to return to its stationary pyramid p * after a perturbation p * + q, such as d(p * , L m (p * + q)) < Ke -mD tends exponentially to 0, when m goes to infinity, the parameter D being the stability module. For the L 2 distance to the stationary pyramid p * , D = |λ -λ , |, absolute value of the difference between the dominant and sub-dominant eigenvalues of L, i.e., λ and λ , (λ = e r , where r is the Malthusian growth rate, and p * is the left eigenvector of L corresponding to λ). For the distance (known as symmetrized divergence) of Kullback-Leibler, D = kH, where H is the entropy of p * and k is a constant [START_REF] Demetrius | Boltzmann, darwin and directionality theory[END_REF][START_REF] Rhodes | Evolutionary entropy determines invasion success in emergent epidemics[END_REF].

p (t) = Lp (t -1) with L = (l jk ) =         
The Leslie model has the advantage of not requiring a stable age distribution for valid population projections. It also allows for sensitivity analysis to see how changing certain age classes vital rates affects population size and age structure. It can also incorporate densitydependence, i.e., dampen values in the matrix to account for density-dependent factors limiting population growth, and it can derive useful mathematical properties from matrix formulas, such as stable-age distribution and finite rate of population change (i.e., the growth parameter λ). The model's disadvantages are that it requires a large amount of data (i.e., age-specific data on survival, fecundity, and population structure) and estimating fecundity is difficult atimes.

Usher's Model

An improvement of the Leslie matrix is the Usher's model which is sometimes refers to as Usher's matrix [START_REF] Usher | A matrix model for forest management[END_REF] which is widely used to predict population dynamics and to predict the dynamics of size-structured or age-structured populations. The model is formulated in which it is possible to remain in the same biological age corresponding to an increase of the longevity (a rejuvenation) or to pass over a biological age state corresponding to an acceleration of ageing between t -1 and t as modeled by Usher using the vector of age class sizes u(t) = [u 1 (t), • • • , u m (t)], whose discrete dynamics is ruled by matrix equation u(t) = U u(t -1), where:

U = (u ij ) =          f 1 + v 1 f 2 f 3 • • • f m-1 f m α 1 v 2 0 • • • • • • 0 0 α 2 v 3 • • • • • • 0 . . . . . . . . . • • . . . . . . . . . . . . . . . v m-1 . . . 0 0 0 • • • α m-1 v m         
and where f 1 , f 2 , . . . , f m are the fertility rates in age classes, v i is the probability to remain in state i, α i is the probability to go from state i to state i+1, v

i +α i = 1-µ i ≤ 1, ∀ i = 1, . . . , m,
where µ i is the death rate at age i. and v j . The probabilities P i belong to the interval [0,1]:

0 ≤ P i ≤ 1 ∀ i = 1, . . . , m.
We can also say that (v i + α i ) is the general probability of survival. The dynamical stability modulus for the L 2 distance between the current age pyramid [u i (t)/ j=1,...,m u j (t)] and the stationary age pyramid w is given by e -|λ-λ , | , where λ( resp. λ , ) is the dominant (resp. sub-dominant) eigenvalue of the matrix U. In general, the explicit calculation of the eigenvalues of the Usher matrix is not possible. Many theoretical properties of the Usher model are known, particularly its asymptotic behavior: the evolution to the stationary state is exponential and is characterized by the growth rate and the stationary distribution [START_REF] Mortier | Population dynamics of species-rich ecosystems: the mixture of matrix population models approach[END_REF][START_REF] Zetlaoui | Asymptotic distribution of stage-grouped population models[END_REF]. , where p i is the probability for an individual to stay in class i, q i is the probability to move up from class i to i + 1, m i is the probability of dying and f is the average fertility rate supposed to be constant (after [START_REF] Mortier | Population dynamics of species-rich ecosystems: the mixture of matrix population models approach[END_REF]).

Statistical Methodology

This section deals with the statistical methodologies used in the thesis for the modeling of the COVID-19 disease. These methodologies include time series, machine learning (supervised and unsupervised), deep structured learning (feed-forward neural network and recurrent neural networks), spectral analysis and functional data analysis. In Figure 2.4, we present the block diagram for the statistical methodology applications study. The data used throughout this study were collected from public database for all epidemiological, demographic, and socio-economic data and references were provided as appropriate in various sections and subsections. The data-set used raw file was overflowed by a large quantity of information collected. The raw file processing ensured the right attributes to be mined for this study. The data-set was pre-processed and cleaned on Microsoft Excel while some of the data-set was converted to Comma Separated Values (CSV) files. The data cleaning processes ensure the removal of unwanted attributes and the quality data-set creation for an excellent analysis. The data were normalized to generate a cleaned data-set. Figure 2.5 presents the steps involved in data preparation and cleaning before applying the statistical techniques. In subsequent section, we present the mathematical formulation of all the statistical methodologies used in this thesis having in mind that most of these models are already inbuilt in well known software like Python ® and R which were employed for the experimental work.

Time Series Modeling

Most of the data-sets used in this research are time series data and we shall give details about some of the time series modeling used to predict and forecast our data-sets.

p(t) = i=1,...,r a(t)p(t -i) + ε(t), (2.1)
where p(t) is the time series real value at a time t and ε is a random residue also known as noise, whose variance is to minimize. The auto-correlation curve is obtained by calculating the correlation A(k) between p(t) and the p(t -k)'s (t belonging to a moving time window) by using the formula:

A(k) = E[p(t)p(t -k)] -E(p(t))E(p(t -k)) σ(p(t)σ(p(t -k)) . (2.2)
where E denotes the empirical expectation calculated on a chosen moving time window of length m and σ is the standard deviation.The auto-correlation function A allows examining the serial dependence of the p(t)'s.

The noise (white if it is Gaussian) ε is a stochastic process and must obey the following rules:

E(ε(t)) = 0 V (ε(t)) = δ 2 ε < +∞ Cov(ε(r), ε(r -k)) = 0, ∀ r and k ̸ = 0
where V denotes the variance and Cov the auto-covariance of the stochastic process ε. In this thesis, we used the classical ARIMA (r, d, m) model, where r means the order of auto-regression, d the degree of trend difference (related to the polynomial character of the trend and discrete analog to the order of a differential equation) and m the order of moving average.

Exponential smoothing modeling

There are fifteen types of exponential smoothing modeling as stated in [START_REF] Hyndman | A state space framework for automatic forecasting using exponential smoothing methods[END_REF] which depend strictly on underlying error (Multiplicative -M or Additive -A), type of trend (None -N, Additive -A or Damping -D) and type of seasonality (None -N, Additive -A or Multiplicative -M). The simple exponential smoothing model which has no trend nor seasonality but a slowly varying mean for a time series p t is given as follows:

p t+h = l t (2.
3)

l t = αp t + (1 -α)l t-1
where p t+h the h-step ahead forecast, l t is the level of the time series and α is the parameter which determines the degree of the smoothing and is always estimated by minimizing the sum of the squared prediction errors.

It is important to first describe the underlying error models which can be described for simple exponential smoothing model with an additive error as:

p t+h = l t-1 + ε t (2.4) l t = l t-1 + αε t
and for the simple exponential smoothing model with a multiplicative error, is given as:

p t+h = l t-1 (1 + ε t ) (2.5)
l t = l t-1 (1 + αε t )
The improvement of Equation (2.3) with a trend behavior of the time series which can give rise to a model like MAN (exponential smoothing with additive trend without seasonal component and with a multiplicative underlying model) is expressed as:

p t+h = l t + m i=1 Φ i a t l t = αp t + (1 -α)(l t-1 + Φa t-1 ) (2.6)
a t = β(l t -l t-1 ) + (1 -β)Φa t-1
where β is the parameter that determines the rate of change in trend and Φ is the parameter that takes into consideration the damping of the trend. Other parameters are defined as previously. If we set Φ = 0, we will have a simple exponential model without trend (e.g. ANN or MNN) but if Φ = 1, it gives an exponential model with trend (e.g. AAN). Another improvement of Equation 2.3 is to add seasonality effects to the model. For additive seasonality we have:

p t+h = l t + m i=1 Φ i a t + s t-n+hn l t = α(p t -s t-n ) + (1 -α)(l t-1 + Φa t-1 ) (2.7)
a t = β(l t -l t-1 ) + (1 -β)Φa t-1 s t = γ(p t -l t-1 -Φa t-1 ) + (1 -γ)s t-n
for multiplicative we have:

p t+h = (l t + m i=1 Φ i a t )s t-n+hn l t = α(p t /s t-n ) + (1 -α)(l t-1 + Φa t-1 ) (2.8) a t = β(l t -l t-1 ) + (1 -β)Φa t-1 s t = γ(p t /(l t-1 -Φa t-1 )) + (1 -γ)s t-n
. where h n = ⌊(h -1) mod n⌋ + 1. The seasonality parameter is γ and the number of season is given as n which for instance for monthly data-set verifies: n = 12. Other parameters are defined in the same way as before. Table 2.1 gives a summary of the symbols we have described earlier for all the exponential smoothing models.

y(t) = g(t) + s(t) + h(t) + ε(t), (2.9)
where g(t) is the trend, a piece-wise linear or logistic growth curve for modeling the nonperiodic changes in time series defined for the logistic case by:

g(t) = C/(1 + exp(-k(t -z))),
where C is carrying capacity, z is the growth rate and m is the offset parameter.

Similarly, for the piece-wise linear case, we have:

g(t) = (s 0 + s 1 + • • • + s j ) × t + (C 0 + C 1 + • • • + C j ),
where j is the last change point before t.

Other parameters in Equation (2.9) are as follows: s(t) represents the seasonality, h(t) is the effect of holidays with irregular schedules and ε(t) is the error term which accounts for any unusual changes or noise not accommodated by the model [START_REF] Furtado | Epidemiology sir with regression, arima, and prophet in forecasting covid-19[END_REF].

Prophet model forecasts seasonality effects using the Fourier series, and the seasonality models are specified as periodic functions of t. The arbitrary smoothing of seasonal effects with a scaling time variable using Fourier series is of the form:

s(t) = a 0 2 + ∞ m=1 a n cos 2mπt P + b n sin 2mπt P ,
where P is the period for fitting the seasonality and the parameters a m and b m can be estimated from the series. The seasonality is multipliable by the trend if the argument "multiplicative" is specified then we have:

s * (t) = g(t) × s(t) if multiplicative s(t), otherwise.
If we take the period P = 1 and 9, by considering additive seasonality (as opposed to multiplicative), we have:

s(t) = s * 1 (t)+s * 9 (t) = s 1 (t)+s 9 (t) = 9 m=1 a n cos 2mπt 1 +b n sin 2mπt 1 + 5 m=1 a n cos 2mπt 9 +b n sin 2mπt 9
Also, h(t) can be a multiplicative seasonality and if we introduce a trainable coefficient σ m then it can be defined as follows:

h(t) = h * m (t) where, h m (t) = σ m h(t) h * m (t) = g(t) × h m (t) if multiplicative h m (t), otherwise.
for each m.

Spectral Analysis

The decomposition of a time series into underlying sine and cosine functions of different frequencies allowing us to identify those frequencies that appear to be strong or very important is called spectral analysis. Another way to think about spectral analysis is as a linear multiple regression problem, with the dependent variable being the observed time series and the independent variables being the sine and cosine functions of all possible frequencies. Spectral analysis always identifies the correlation of sine and cosine functions of different frequencies with the observed data, and if a large correlation is found, one can conclude that the respective function has a strong periodicity in the real data. Spectral analysis is useful for analyzing stationary time series and identifying noise-corrupted periodic signals [START_REF] Ray | Spectral analysis and time series. vol. 1: Univariate series[END_REF].

The periodogram is one of the most important tools in spectral analysis because it is an estimation called spectral density. It quantifies the contributions of individual frequencies to time series regression and is denoted as

P k = a 2 k + b 2 k
, where P k is the periodogram, a k cosine parameter, and b k sine parameter values at frequency k (for k = 1, 2, . . . , n/2). The periodogram values can be interpreted in terms of the variances of the real data at the respective frequencies. A Fourier line spectrum is a plot of P k (as spikes) versus k. The periodogram is created by joining the tips of spikes in the Fourier line spectrum to form a continuous plot and scaling it to an area equal to the variance. The periodogram can also be smoothed by using different Daniell kernel which can be log or sine or cosine. Another important tool in spectral analysis is multitapering which is an extension of single taper approaches that involves dividing the data into overlapping subsets that are individually tapered and then Fourier transformed. The individual spectral coefficients of each subset are averaged to reduce variance. The method of subdividing the data is defined by a set of tapers indexed by m = 1 • • • M ; the estimated spectral density at frequency f is expressed as follows:

Z(f ) = M -1 m=0 λ m Z m (f ) M -1 m=0 λ m ,
where Z m (f ) is the estimated spectral density using taper m and λ m are weights for each tapered spectral density estimate. Also, harmonic regression analysis of the frequencies of the time series is important. Suppose we have a time series p t which contains a periodic component, a phenomenon which spectral density is known for, the model will be of the form:

p t = µ + Zcos(f t + d) + ε t ,
which can further be expanded to be of the form:

p t = µ + a cos(f t) + b sin(f t) + ε t ,
where µ is the mean of the series, Z is the amplitude of variation and d is the horizontal offset. All other parameters remain defined as earlier stated. This nonlinear model can be fitted as a linear regression model with two independent variables, P = cos(f t) and Q = sin(f t). The regression coefficients are J = a and K = b. In practice, the variation in a time series can be modeled as the sum of several different individual sinusoidal terms occurring at different frequencies. The generalization of this model to the sum of k frequencies, to which we can optionally add a trend term to obtain the forecasting equation, can be expressed as follows:

p t = µ + zt + k m=1 a m cos(f m t) + k m=1 b m sin(f m t) + ε t .

Supervised Machine Learning Modeling

Supervised machine learning modeling involves classification and regression supervised learning problems and processes. In most cases, you give specific instructions to what the machine will do. Figure 2.6 shows the flowchart of used for supervised machine learning modeling, analysis and the process involved while Figure 2.7 shows the flowchart for k-fold crossvalidation strategy which is a re-sampling method to evaluate supervised machine learning models on a limited dataset sample.

Regression modeling

Here we describe the linear regression model, log-linear regression model, generalized linear models, polynomial regression (order ≥ 2) model, regression splines, multivariate ordinary least square model, ridge regression model and lasso regression model. Linear regression model uses some historic data of independent and dependent variables and consider a linear relationship between both. The model is of the form:

y = bx + a,
where a is the intercept, b is the slope, x is the independent variable and y is the dependent variable. Log-linear regression model is coming from the common exponential model which is given as:

y = ae bx , (2.10)
where y is the daily number of incidence in the case of modeling COVID-19 outbreak which is the subject of the thesis, x is the number of days, b is the slope and a is a constant, and the log-linear model is derived by taking the log of both sides of Equation (2.10) which can be written as:

log y = log a + bx.
It means that the log of the daily number of incidence will be used as the dependent variable. Generalized linear models are models that fit a linear response function y i as linear combination of the predictors x pi to a data that is likely to have errors ε i that are normally distributed [START_REF] Wood | Generalized Additive Models[END_REF]. It can be written as follows:

y i = b 0 + p b p x pi + ε i , (2.11)
where y i can be any exponential family distribution i.e., Binomial, Bernoulli, Gamma, Normal, Poisson etc. and ε i ∼ N (0, σ 2 ) which means that the mean is equal to 0 and constant variance is σ 2 . The intercept is b 0 and b p is commonly estimated using ordinary least squares model. Polynomial regression models use a similar approach as linear regression model but the dependent variable is modeled as a degree n (n ≥ 2) polynomial in x. We can describe a cubic polynomial (n = 3) regression as follows:

y = b 1 x + b 2 x 2 + b 3 x 3 + b 0 ,
where b 1 , b 2 , b 3 are the weights of the coefficients of x. Other parameters are defined in the same way as before. To model other type of polynomial regression, the power of x will just be increased.

A spline is a smooth piecewise polynomial function with a fixed degree z defined and constructed by joining polynomials smoothly at values τ i = 1, . . . , m. The values τ i are known as knots or breakpoints, and they are given as follows:

t 1 = τ 1 < • • • < τ m = t n .
Regression splines of order m with knots ε k , k = 1, . . . , K is a piecewise-polynomial of order m -1, which has a continuous derivatives up to order m -2 whose basis functions is as follows:

y = K+m n=1 β n h n (x) + ε.
The general formula for the truncated-power basis set is

h k (x) = x k-1 , k = 1, . . . , m h k+m (x) = (x -ε k ) m-1 + , k = 1, . . . , K
The regression spline can be linear or cubic but it is claimed that cubic splines are the lowestorder splines in which the knot-discontinuity is not visible to the naked eye. The set of linear or cubic splines with fixed knots is a vector space. There is almost never a good reason to go beyond cubic-splines. In practice the most widely used orders are m = 1, 2 and 4.

Multivariate ordinary least squares model allows us to test much more complex relations between variables. It can be can be represented as follows:

y = β 1 x 1 + β 2 x 2 + • • • + ε, (2.12)
where β 1 , β 2 , . . . are coefficients or weights, ε is the residual noise, y is the dependent variable and x 1 , x 2 , . . . are the independent variables. Ridge regression is a modeling approach that can be used to reduce model complexity and false positive which may result from the classical linear regression model. The cost function for ridge regression is given as follows:

m i=1 y i - p j=0 β j × x ij 2 + α p j=0 β 2 j ,
for all c > 0, p j=0 β 2 j < c, while α is the penalty term that regularizes the coefficients. If the coefficients take large values, the optimization function is penalized. Ridge regression puts constraints on the coefficients β. In data modeling, Lasso (Least absolute shrinkage and selection operator) regression model helps to regularise data models and to select the best feature for the model. The cost function for Lasso regression is obtained in the same way as that of Ridge regression with little modification which is given as:

m i=1 y i - p j=0 β j × x ij 2 + α p j=0 |β j |.

Support vector machine

Support vector machine is a supervised machine learning model that draws a hyperplane relating to the data points and creates a boundary of possible data points (high and low) for prediction. The goal of the model is to find the maximum margin, which means to minimize error. A tolerance margin (ϵ) is set to approximate the model and minimize error, individualizing the hyperplane that maximizes the margin while keeping in mind that some of the error is tolerated. The mathematical formulation of the support vector regression where b and w are the coefficients is given as:

y = f (x, w) =< w, x >= m j=1 w j x j + b, (2.13)
with w j > 0. The support points minimize prediction error where ∥w∥ is the magnitude of the vector w : min

w 1 2 ∥w∥ 2 , (2.14)
with constraints | y j -w j x j |≤ ε.

(2.15)

The error term is instead handled in the constraints, where we set the absolute error less than or equal to a specified margin, called the maximum error (ε). Parameters used for support vector machine are shown in Table 2.2.

Canonical correlation analysis

Canonical correlation is an aspect of multivariate statistical modeling that is used to simultaneously correlate several metric dependent variables and several metric independent variables measured on or observed with similar experimental units. It finds linear combinations within a data set with the goal of maximizing the correlation between these linear combinations [START_REF] Krzyśko | Canonical correlation analysis for functional data[END_REF]. Mathematically, it can be expressed as two groups of n-dimensional variables

X = [x i ] = [x 1 , x 2 , . . . , x p ] and Y = [y i ] = [y 1 , y 2 , . . . , y q ],
where

x i =      x i1 x i2 . . . x ip      , y i =      y i1 y i2 . . . y iq      .
The purpose of canonical correlation analysis is to find coefficient vectors

U n×r = X n×p A p×r , V n×r = Y n×q B q×r
If X and Y are both centered, we can concatenate them and calculate the covariance matrix given as:

C = Cov([XY ]) = 1 n -1 [XY ][XY ] T = C xx C xy C yx C yy ,
where C xx and C yy are within the set of covariance matrices and C xy = [C yx ] T is between the set of covariance matrices. The first canonical response a and b maximize the equation:

β = a T C xy b √ aa T C xx bb T C yy
The generalized formulae to calculate other canonical responses a i and b i ∀ i ≥ 2 can be expressed as follows:

β i = a T i C xy b i a i a T i C xx b i b T i C yy ,
subject to the constraint:

a j a T i C xx = 0 ∀ j < i, b j b T i C yy = 0 ∀ j < i.

Unsupervised Machine Learning Modeling

Principal component analysis

Principal component analysis is an exploratory data analysis technique which uses the observations in a data set say p for each m individuals or quantities [START_REF] Jolliffe | Principal component analysis: a review and recent developments[END_REF]. A p m-dimension vector x i of observations on the ith variable and a linear combination and the variance of the data set matrix denoted as X with minimum variance is defined as follows:

p i=1 b i x i = Xb i , var(Xb i ) = b T Sb i ;
where b is a vector whose components are constants which are commonly called eigenvectors, S is the sample covariance matrix associated with the data set. The linear combinations Xb i are called principal components of the data set. The elements of the eigenvectors b i are called principal components loadings and elements of the linear combinations Xb i are called the principal components scores which are values each m of the individuals or quantities will score on principal components. The first principal component gives the most information in principal component analysis.

Clustering analysis

Clustering is a data analysis technique that divides objects into different clusters by measuring distances and identifying individual clusters [START_REF] Lee | Clustering of time series water quality data using dynamic time warping: A case study from the bukhan river water quality monitoring network[END_REF]. K-means and Hierarchical techniques are two main types of clustering analysis that will be discussed in this section. Figure 2.8 shows the flowchart of clustering techniques algorithm. K-means clustering technique is a centroid-based or distance-based algorithm that uses distances to assign a point to a cluster. Each cluster is associated with a centroid in this clustering technique. The main goal of this technique is to maximize the sum of distances between the points and their respective cluster centroid. The steps involve selecting the number of clusters (k), randomly selecting the centroid for each cluster, assigning each point to the closest cluster centroid, computing the centroids of newly formed clusters, and then repeating the process of assigning each point to the closest cluster centroid to form another new cluster. Hierarchical Clustering is a popular clustering technique for estimating patterns in multidimensional data-sets. Examining groups of data with a similar pattern can lead to a better understanding of the functions and state of the data set used in this thesis. The steps involved are to determine the distance between each data point, consider all of the data points to be individual clusters, determine the number of cluster groups required, calculate the proximity of the new clusters and merge the two closest clusters to form new clusters and finally, merge all available clusters to form a new cluster. Ward's method was used in the hierarchical clustering method to determine the similarity between two clusters. This method was chosen due to its excellent performance in separating clusters even when there is noise between clusters. The ward's linkage method was used to compute the distance matrix between the closest features considered in the hierarchical clustering technique [START_REF] Hadi | Finding groups in data: An introduction to cluster analysis[END_REF]. It can be expressed mathematically using Equation (2.16).

D(c 1 , c 2 ) = ∥c 1 ∥∥c 2 ∥ ∥c 1 ∥ + ∥c 2 ∥ ∥c 1 -c 2 ∥ 2 , (2.16)
where c 1 and c 2 are the two closest clusters and the norm is defined on the set of subsets of the data set (e.g., the norm of their barycenter). The number of clusters to form and centroids to generate was set at three. The affinity used for this technique was Euclidean distance, which refers to the distance between all points in the selected data. The ward linkage method estimates the proximity between two clusters.

Deep Structured Learning Modeling

Deep structured learning modeling involves the use of artificial intelligence or artificial neural network in modeling different phenomena; however, in this case it will be used to model COVID-19 outbreak from several of its epidemiological, socio-economic and demographic determinants. Artificial neural network modeling was developed to study interconnected networks which can be in form of nodes or neurons with many layers involved. It can be applied to biological systems, stochastic processes, computing among several others. It is divided into two major classes which are feed-forward artificial neural network and recurrent artificial neural networks. These classes are based on several improvements over the years which show that one model is performed better than the others when applied to an experimental work. In Table 2.4, the list of hyper-parameters for artificial neural network modeling, their definitions and values are presented while in Figure 2.11, the work flow of how the deep structure learning modeling was carried out using France, India and Turkey as an example in the flow chart.

Recurrent artificial neural networks

Recurrent artificial neural network is a type of artificial neural network in which node connections form a directed or not directed graph along a temporal sequence. This enables it to exhibit temporal dynamic behavior [START_REF] Salem | Gated rnn: The long short-term memory (lstm) rnn[END_REF][START_REF] Hochreiter | Long short-term memory[END_REF]. There are two major types of recurrent artificial neural networks which are Long Short-Term Neural Network (LSTM) and Gated Recurrent Unit (GRU). The most advanced types of Recurrent Neural Networks are LSTM and GRU (RNN). The LSTM addresses an inherent problem of recurrent neural networks, the gradient vanishing problem. It is intended to model long-term dependencies and determine the optimal time lag for the time series by granting the memory unit the ability to decide, remember, or forget some information. This complex positioning among the recurrent neural networks allows the LSTM network to recall past data, making it easier to create connections between current and past data points, allowing the network to find patterns that play out over time.

The GRU is similar to the LSTM in that it uses gates to control data set flow, but unlike the LSTM, it lacks a separate cell state and instead has only a hidden state. The GRU has some parameters that help to accelerate data set training. Figure 2.9 depicts the LSTM and GRU architectures.

Feed-forward artificial neural networks

Feed-forward artificial neural network is sometimes called deep feedforward artificial neural network are models that enable information to flow through the function being evaluated, the intermediate computations used to define the function, and finally give an output. There are no feedback connections where the model's outputs are fed back into itself. The hidden layer of an artificial neural network is used to store and evaluate how significant one of its inputs is for the output. Figure 3b depicts an artificial neural network with only one hidden layer. A CNN (see Figure 2.10(a)) is a type of neural network composed of neurons with learnable weights and biases and it requires significantly less pre-processing than other classification algorithms. It is made up of filters that are applied to the input data set, effectively condensing the data set into a smaller resolution. CNN is well-positioned to remove noise in the inputed data set so that it can be fed into basic neural networks. The advantage of this method is that it approaches the data set spatially [START_REF] Miotto | Deep learning for healthcare: review, opportunities and challenges[END_REF].

A DNN is a multi-layer artificial neural network in which data flows from the input layer to the output layer without going backward, and the links between the layers only go forward and never touch another node again. It is also referred to as a feed forward network. A DNN is shown in Figure 2.10(b). A MLP is a feed-forward ANN with at least three layers of nodes: an input layer, a hidden layer, and an output layer, as shown in Figure 2.10(b), whereas ELM is used in the training algorithm for a feed-forward artificial neural network, which converges much faster than traditional methods and yields promising results [START_REF] Ravi | Deep learning for health informatics[END_REF].

Neural prophet forecasting modeling

Neural prophet forecasting model is an extension of Equation (2.9) with neural network components (auto-regression and covariates modules) and it is as expressed as follows:

y(t) = g(t) + s(t) + h(t) + f (t) + a(t) + l(t) + ε(t). (2.17)
For all m ∈ [1, h], h is the step length, and we have:

y(t+m-1) = g(t+m-1)+s(t+m-1)+h(t+m-1)+f (t+m-1)+a(t+m-1)+l(t+m-1),
where f (t) is the forecast covariates or future known variables, a(t) auto-regression of the form Equation (2.1) and l(t) is the lagging current covariate by m time period or lagged observations of the known variable. Other parameters are defined in the same way as before. f (t) can be multiplicative like seasonality which was earlier defined with a trainable coefficient σ m then it can be defined as follows:

f (t) = f * m (t)
where, for each m:

f m (t) = σ m f (t) f * m (t) = g(t) × f m (t) if multiplicative f m (t), otherwise.
Given a set of covariates x and if we create a separate lagged regressor module for each of m covariates x of length T , l(t) can be expressed as follows:

l(t) = p l x (x t-1 , . . . , x t-p ).
The p last observations of the covariate x are the inputs to the module (instead of the series y itself as in auto-regressor) [START_REF] Triebe | Neuralprophet: Explainable forecasting at scale[END_REF].

Functional Data Analysis (FDA)

Functional data analysis deals with data from curves, surfaces, or anything else that varies along a continuum (e.g. time, spatial location, wavelength, probability, etc.). Data is only sampled at certain points along the continuum, either very precisely or with intricate noise structure, because the observed data are generally filled with observational errors (or noise) that are superimposed on the underlying signal. The main assumption is that the underlying random processes are intrinsically regular. In the real world, we could see a scenario with M processes being observed at the same time [START_REF] Ramsay | Introduction to functional data analysis[END_REF]. Let t be a one-dimensional argument sometimes referred as time. Functions of t are observed over a discrete grid {t 1 , . . . , t K } ∈ ∆ at sampling values t k , which may or may not be equally spaced. Let Y be a vector of

M functional data Y = [Y T 1 , . . . , Y T M ] T
, each functional data is expressed as follows:

Y ik = X i (t k ) + ε ik , 1 ≤ k ≤ K, 1 ≤ i ≤ M,
where Y ik is a noisy observation of the stochastic process X i (t k ) and ε ik is a random error with zero mean and variance function σ 2 i associated with the i th functional raw data. In vector notation, the X i (t) is of the form:

X i (t) ≈ c T i .α(t), ∀t ∈ ∆, i = 1, . . . , M
, where c i and α(t) are n-vectors.

Smoothing techniques

The idea behind smoothing procedures is to estimate a curve that fits the data set well while not becoming too 'wiggly' or locally variable. This is accomplished by allowing the curve estimate at a point t to be heavily influenced by observations near t. B-spline technique is one of the tools used in smoothing (curve estimation modeling) a functional data and this can be done by changing the number of basis functions (n = 2, 3, 4, . . . , p). The most commonly used basis functions are Fourier, polynomial, splines and wavelets. A mathematical expression for the basis representation for the curve of the functional data set is of the form:

f (t) = n j=1 b j α j (t), ∀ t ∈ ∆, (2.18)
where b j 's are basic coefficients and α j 's are the basis functions for j = 1, . . . , n.

Sometimes one can use the Fourier basis for the functions to further see the variations in the curves, which is the most appropriate basis for periodic functions defined on an interval ∆ where α j 's take the following form:

α 0 (t) = 1/ |∆|, α 2r-1 (t) = sin(rωt) |∆|/2 and α 2r (t) = cos(rωt) |∆|/2 , for r = 1, . . . , n-1 2 ,
where n is the number of basis functions and n must be odd to compute a Fourier basis. The frequency ω determines the period and the length of the interval |∆| = 2π/ω. The non-parametric method (kernel smoothing) of smoothing for functional data is based on the smoothing matrix M given:

m ij = 1 h K( t i -t j h ), (2.19
)

m(h) = (s j (t i )) = K( t i -t j h ) T k=1 K( t k -t j h ) , (2.20)
where K(.) is the Kernel function and h is smoothing parameter.

Functional principal component analysis

Principal component analysis (PCA) is a dimension reduction analysis tool in unsupervised machine learning modeling, whereas functional principal component analysis (FPCA) is a dimension reduction with high correlation in functional data analysis that completes the statistical tools used in biomedical data modeling, particularly epidemiologic data. Let {x i (t)} i=1,m be a given set of function and let α be a weight. To compute functional principal components, the following steps are involved:

-We first find the principal component weight function α 1 (t) for which the principal component score is given as follows:

f i1 = α 1 (t)x i (t)dt
while maximizing i=1,m f 2 i1 is subjected to:

α 2 1 (t)dt = ∥α 1 ∥ 2 = 1
-Next, the weight function α 2 (t) is computed and the principal component score maximizing i=1,m f 2 i2 is subjected to ∥α 2 ∥ 2 = 1 and to the additional constraint:

α 2 (t)α 1 (t)dt = 0
-Then, the process is repeated for as many iterations.

Functional clustering analysis

The two-stage approaches for functional data clustering consist of a first step, referred to as the filtering step, in which the dimension of data is reduced, and a second step in which classical clustering tools for finite dimensional data are used. The reducing dimension step consists in general in approximating the curves to a finite basis of functions. Because of their superior properties, spline bases are a popular choice. Clustering algorithms, such as the kmeans algorithm, are commonly used to estimate clusters of functional data. Nonparametric approaches for functional data clustering include hierarchical and k-means clustering, which are tools that use standard nonparametric clustering techniques with specific distances or dissimilarities. Several works [START_REF] Strandberg | A comparison of spatiotemporal and functional kriging approaches[END_REF][START_REF] Jacques | Functional data clustering: a survey[END_REF] consider the following measures of proximity between two curves x i and x j :

d l (x i , x j ) = T (x l i (t) -x l j (t)) 2 dt 1/2
, where x l is the l -th derivative of x.

It has been proposed [START_REF] Strandberg | A comparison of spatiotemporal and functional kriging approaches[END_REF][START_REF] Jacques | Functional data clustering: a survey[END_REF][START_REF] Ieva | Multivariate functional clustering for the morphological analysis of electrocardiograph curves[END_REF] to use hierarchical clustering in conjunction with the distance d 0 (the L 2 -metric) or d 1 (the semi-norm in the Hilbert space) or with the semi-metric d 2 . Another proposal is the use of k-means algorithm with d 0 , d 1 and with (d 2 0 +d 2 1 ) 1/2 , as well as d 0 with k-means for Gaussian processes. It was demonstrated, in particular, that cluster centers are linear combinations of functional principal component analysis eigenfunctions.

The same distance d 0 with k-means is considered when defining time-dependent clustering. Following the approach used to estimate the distance d 0 , nonparametric analysis can be applied to raw-data clustering or to two-stage analysis. Indeed, if d 0 is approximated directly using discrete observations of curves, nonparametric analysis is equivalent to raw-data clustering analysis. Similarly, if an approximation of the curves into a finite basis is used to approximate d 0 , nonparametric analysis is equivalent to two-stage methods with the same basis approximation.

Functional linear regression analysis

This section describes two functional linear regression analysis methodologies which are (i) function to scalar linear regression and (ii) function-on-function linear regression.

To describe the function to scalar linear regression model, Let

Y =< θ, X > +ε, (2.21)
where θ is the unknown function of the model, X is a functional covariate belonging to some functional space H which is endowed with an inner product < •, • >, Y is the response variable and ε is the random error term. Usually, H is L 2 ([a, b]) of square integrable functions on some real compact interval [a, b] and

< f, g >= b a f (t)g(t)dt, (2.22)
is the corresponding inner product, where the functions f, g ∈ L 2 ([a, b]). Considering C = [0, 1], so the Equation (2.21) can be written as:

Y = 1 0 θ(t)X(t)dt + ε, (2.23)
where θ is a square integrable function which is defined on C and ε is a random variable such that E(ε) = 0 and E(X(t)ε) = 0, then Equation (2.23) can be rewritten as:

Y = (ΨX) + ε, (2.24)
where Ψ represents the integral.

If the functional linear regression model is considered as a functional input/output regression model where y(t) is response variable and x(t)'s are predictors at each time t, i.e., x(t) → y(t).

The functional linear model with an intercept is of the form:

y(t) = β 0 (t) + β(s, t)x(s)ds + ε. (2.25)
Chapter 3

Mathematical Modeling of COVID-19 Outbreak

This chapter deals with a wide range of mathematical models developed to model the COVID-19 pandemic. Section 3.2 describes the use of matrix algebra to estimate the daily reproduction numbers during the COVID-19 outbreak. The discrete dynamics of the epidemic's growth equation is presented with the estimation of the daily reproduction numbers using a de-convolution technique on a series of new COVID-19 cases, then the discussion of the role of noise on the stability of the epidemic's dynamics using a stochastic framework is proposed. The shapes of the daily reproduction number distribution in the COVID-19 outbreak dynamics are similar to other viral diseases, such as influenza, with two successive reactions from two immune defense barriers, innate cellular immunity first, which is insufficient if symptoms persist, then adaptive immunity, resulting in a transient decrease in contagiousness between the two phases. In Section 3.3, the estimation of maximum reproduction number using continuous formulation of the epidemic's growth equation is presented and applied to different countries in order to show the effectiveness of the method. In Section 3.4 the identification of the first point of inflexion using Bernoulli SI equation and the estimation of transmission rate of the COVID-19 outbreak is presented. The method is applied to real data from Cameroon and its regions, then to the world data. The results derived from this analysis shows its applicability to countries whose epidemic curves did not reach the turning point. In Section 3.5 a discrete epidemic model on the ageing and lifespan loss in COVID-19 outbreak and the role of comorbidities of infected patients using the Usher's model for different age classes and gender (male and female) at sub-population level is presented. Also, the proportion of sensitivity of the model parameters so as to see the most important parameter at different age groups is established. The result is consistent with the observed cases and the semi-quantitative mathematical modeling approach at the population level. The Section 3.6 describes a mathematical age dependent model for two age groups (20-64 years and people above 64) named Susceptible-Infectious-Goneanewsusceptible-Recovered (SIGR) model for COVID-19 outbreak and then some mathematical results by showing the positivity, boundedness, stability, existence and uniqueness of the solution are proved. Numerical simulations of the model with parameters coming from Kuwait, France and Cameroon are presented. The results are centred around the role of different parameters used in the model mainly the effect of vaccination on the epidemic dynamics. Finally, Section 3.7 discusses the perspectives of this chapter and some conclusions based on the models developed.

Introduction

Mathematical modeling of infectious diseases is important to better understand the evolution of any pandemic most especially now that the world is faced with the COVID-19 outbreak which was first discovered in Wuhan, China in December, 2019 and since then we have had different waves of the pandemic as most countries is battling with their fifth wave due to the most recent variant, the omicron variant, which is known for its high contagiousness but not as deadly as previous variant Delta. Recently, in Ireland and some other countries, we are having a rise in hospitalization cases and daily new cases among children, primary school children, their adult contacts and their parents which is one of the reason why the vaccination campaign is now extended to younger ones. COVID-19 pandemics has affected every facet of our lives. The introduction of various mitigation measures has helped to curtail the spread of the disease but this has had adverse effects on the well being of many people due to isolation, psychological effect, lost of jobs and many more impacts on the people. The patterns of cases have shown that the unvaccinated are at far greater risk and there is a lot of debates on declaring that COVID-19 is endemic since the managing of the spread of the disease is getting difficult by the day due to an extended period.

As the virus keeps mutating, the onus lies on us as researchers to keep studying the patterns and the spread of the virus hence, the reason for which we propose different mathematical approaches to better understand the basic reproduction number, the transmission rate from the first point of inflection and the dynamics of the spread of COVID-19 virus at the subpopulation (age group) level.

Estimation of Daily Reproduction Rates in COVID-19 Outbreak

The basic reproduction number

The ability of an infectious disease to spread is quantified by the basic reproduction number R 0 (also called the average reproductive rate), a classical epidemiologic parameter that describes the transmissibility of an infectious disease and is equal to the number of susceptible individuals that an infectious individual can transmit the disease to during his contagiousness period.

Given an SI Equation of the form:

dS dt = -νSI, dI dt = νSI -µI, (3.1)
where S and I are respectively the size of the susceptible and infectious populations. µ is the death rate and ν is the rate of contact. If we assume that S is approximately 1 i.e., everyone in the population is susceptible to the disease, then dI/dt = νI -µI. An individual is expected to infect a number of ν/µ secondary cases which represents the basic reproduction number R 0 at the individual level modeling i.e., R 0 = ν/µ. It has been established that if

R 0 = ν/µ > 1,
the pandemic is an epidemic, and if R 0 = ν/µ < 1, the pandemic is extinct or the disease can be control but a times that is not the case. Following the outbreak of the COVID-19 disease caused by coronavirus SARS CoV-2 which is the third coronavirus outbreak to occur in the past two decades. SARS CoV-2 can be transmitted from person to person by respiratory droplets and through contact and fomites. Therefore, the severity of the disease symptoms, such as cough and sputum (see Figure 3.2), and their viral load, are often the most important factors in the virus's ability to spread, and these factors can change rapidly within only a few days during an individual's period of contagiousness. For contagious diseases like COVID-19, the transmissibility is not a biological constant: it is affected by numerous factors, including endogenous factors, such as the concentration of the virus in aerosols emitted by the patient (variable during his contagiousness period), and exogenous factors, such as geo-climatic, demographic, socio-behavioral and economic factors governing pathogen transmission (variable during the outbreak's history) [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF][START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF][START_REF] Seligmann | Unpredictable, counter-intuitive geoclimatic and demographic correlations of covid-19 spread rates[END_REF] which will be discussed and taking into consideration in the modeling of the COVID-19 disease in some part of this thesis.

In epidemiology, there are essentially two broad ways to calculate R 0 , which correspond to the individual-level modeling and to the population-level modeling. At the individual level, if we suppose the susceptible population size constant (hypothesis valid during the exponential phase of an epidemic), the daily reproduction rates of an individual are typically non-constant over his contagiousness period, and the calculations we present define a new method for estimating R 0 , as the sum of the daily reproduction rates. This new approach allows us to have a clearer view on the respective influence on the transmission rate by endogenous factors (depending on the level of immunologic defenses of an individual) and exogenous factors (depending on environmental conditions). Due to these exogenous factors, R 0 might change seasonally, but these factor variations are not significant if a very short period of time is considered. R 0 depends also on endogenous factors such as the viral load [START_REF] Pan | Viral load of sars-cov-2 in clinical samples[END_REF] of the infectious individuals during their contagiousness period, and the variations in this viral load (see Figure 3.2) must be considered in both theoretical and applied studies on the COVID-19 outbreak, in which the authors estimate a unique reproduction number R 0 linked to the Malthusian growth parameter of the exponential phase of the epidemic, during which R 0 is greater than 1 (Figure 3.1). The corresponding model has been examined in depth, because it is useful and important for various applications, but the distribution of the daily reproduction number R j at day j of an individual's contagiousness period is rarely considered within a stochastic framework. We therefore defined a partial reproduction number for each day of an individual's contagiousness period, and, assuming initially that this number was the same for all individuals, we obtained the evolution equation for the number of new daily cases in a population. Assuming that the distribution of partial reproduction numbers (referred to as daily for simplicity) was subject to fluctuations, we calculated the consequences for their estimation, and we estimated them for a large number of countries, taking a duration of contagiousness of 3 followed by 6 and 7 days. When this distribution is considered, it is possible to calculate its entropy as a parameter quantifying its uniformity and to simulate the dynamics of the infectious disease either using a Markovian model such as that defined in Delbrück's approach [START_REF] Delbrück | Statistical fluctuations in autocatalytic reactions[END_REF] or a classical discrete or ODE SIR deterministic model. In the Markovian case, R 0 can be calculated from the evolutionary entropy defined by L. Demetrius as the Kolmogorov-Sinaï entropy of the corresponding random process [START_REF] Rubio | Stability analysis of the modified levenberg-marquardt algorithm for the artificial neural network training[END_REF], which measures the stability of the invariant measure, dividing the population into the subpopulations S (individuals susceptible to but not yet infected with the disease), I (infectious individuals) and R (individuals who have recovered from the disease and now have immunity to it). In the deterministic case, R 0 corresponds to the Malthusian parameter quantifying its exponential growth, and the stability of the asymptotic steady state depends on the subdominant eigenvalue [START_REF] Rubio | Adapting h-infinity controller for the desired reference tracking of the sphere position in the maglev process[END_REF].

The contagion mechanism from a first infectious case zero

In [START_REF] Böttcher | Connectivity disruption sparks explosive epidemic spreading[END_REF], Eric Goles and his colleagues investigated the spread of an epidemics where an infected patient can recover only if he is not isolated from a central "node" (agent or set of agents), which could represent a public health or medical authority able to prescribe mitigation measures, treatments or vaccines. They used a SIS (for Susceptible-Infected-Susceptible) model, on an underlying network of individuals susceptible to enter in contact or to have interactions implying a physical proximity. Then, the infected individuals contaminate healthy people, and the epidemic spreads. The infected people can only recover when it remains connected to the central node through a path of contacts involving only healthy individuals.

In such an epidemic, clusters of infected people will contaminate their healthy strict interior (with empty intersection with cluster boundary) because it has no contact with the central node no path. For example, in an Apollonian network (i.e., an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles) an infection without reinfection (i.e., with a perfect immunization) starting at the central nodes of such a network will spread until the periphery and stops; on the contrary, an infection starting on the network boundary will reach the central node, or be blocked if there is simultaneously a start in different central nodes (Figure 3.1 Top left). A more realistic network, called the "school network" authorizes long distance links breaking the regular vertices topology of an ideal Apollonian network and seems to be more realistic for modelling interaction networks in which a real epidemic diffuses [START_REF] Chow | Methods of bifurcation theory[END_REF].

Let us suppose that the secondary infected individuals are recruited from the centre of the sphere of influence of an infectious case zero and that the next infected individuals remain on a sphere centred on case 0, by just widening its radius on day 2. Therefore, the susceptible individuals C(j), which each infectious on day j -1 can recruit, are on a part of the sphere of influence of case 0 reached at day j (rectangles on Figure 3.1).

Let us suppose now that the daily reproduction number denoted as a depends on the day j : a j => C(j), where C(j) represents the number of susceptible individuals, which can be met by one contagious individual at day j. If infected individuals (supposed to all be contagious) at day j are denoted by I j , we have:

X j = ∆I j ∆j = I j+1 -I j j + 1 -j = νC(j)I j (3.2)
Let us suppose that the first infectious individual 0 recruits from the centre of its sphere of influence secondary infected individuals remaining in this sphere, and that the susceptible individuals recruited by the I j infectious individuals present at day j are located on a part of the sphere of centered on the first infectious 0 obtained by widening its radius (Figure 3.1). Then, we can consider that the function C(j) increases, then saturates due to the fact that an infectious individual can meet only a limited number of susceptible individuals as the sphere grows. We can propose for C(j) the functional form C(j) = S(j)/(c + S(j)), where S(j) is the number of susceptible individuals at day j. Then, we can write the following equation, taking into account the mortality rate µ:

X j = ∆I j ∆j = νC(j)I j -µI j = νI j S(j) c + S(j) -µI j (3.3)
This discrete version is used much less than the classic continuous version, corresponding to the ODE SIR model, with which we will show a natural link. Indeed, the discrete Equation (3.3) is close to SIR Equation (3.4) , if the value of c is greater than that of S:

dI dt = ν IS (c + S)
-µI (3.4) Figure 3.1: Top left: Spread of an epidemic from a first infectious case 0 (located at its influence sphere center) progressively infecting its neighbours in some regions (rectangles) on successive spheres. Top right: Apollonian interaction graph with 1096 nodes and average degree of connectivity k approximately 5.99 (after [START_REF] Böttcher | Connectivity disruption sparks explosive epidemic spreading[END_REF]). Bottom left: Spread of an epidemic from a first infectious "patient zero" (in red) located on the centre of its influence sphere made of the successive generations of infected individuals, with same value of the reproduction number R 0 = 3 and same inter-generation time interval i between two successive generations of infected. Bottom right: same epidemic, but with a stochastic dynamics defined by a uniform distribution of the reproduction number on an interval centred in R 0 with a standard deviation σ, and a random inter-generation time interval i (after [START_REF] Lacoude | Le début de la fin?[END_REF]).

The biphasic pattern of the virulence curve of coronaviruses

Mostly, the clinical course of patients with seasonal influenza shows a biphasic occurrence of symptoms with two distinct peaks. Patients have a classic influenza disease followed by an improvement period and a recurrence of the symptoms [START_REF] Wölfel | Virological assessment of hospitalized patients with covid-2019[END_REF]. The influenza RNA virus shedding (the time during which a person might be contagious to another person) increases sharply one half to one day after infection, peaks on day 2 and persists for an average total duration of 4.5 days, between 3 and 6 days, which explains why we will choose in the following contagiousness duration these extreme values, i.e., either 3 to 7 days, depending on the positivity of the estimated daily reproduction numbers. It is common to consider this biphasic evolution of influenza clinically: after incubation of one day, there is a high fever (39-40 • C), then a drop in temperature before rising, hence the term "V" fever. The other symptoms, such as coughing, often also have this improvement on the second day of the flu attack: after a first feverish rise (39-39.5 • C), the temperature drops to 38 • C on the second day, then rises before disappearing on the 5th day, the fever being accompanied by respiratory signs (coughing, sneezing, clear rhinorrhea, etc.). By looking at the shape of virulence curves observed in coronavirus patients [START_REF] Lacoude | Le début de la fin?[END_REF][START_REF] Liu | Prolonged virus shedding even after seroconversion in a patient with covid-19[END_REF][START_REF] Ferretti | Quantifying sars-cov-2 transmission suggests epidemic control with digital contact tracing[END_REF][START_REF] Cheng | Contact tracing assessment of covid-19 transmission dynamics in taiwan and risk at different exposure periods before and after symptom onset[END_REF][START_REF] Xi | Temporal dynamics in viral shedding and transmissibility of covid-19[END_REF], we often see this biphasic pattern. Figure 3.2 shows a U-shaped evolution for the viral load in real [START_REF] Chao | Flute, a publicly available stochastic influenza epidemic simulation model[END_REF] and in simulated [START_REF] Itoh | Emergence of h7n9 influenza a virus resistant to neuraminidase inhibitors in nonhuman primates[END_REF] COVID-19 patients, and in real influenza-infected animals for the viral load and the body temperature [START_REF] Demongeot | Evolution of social networks: the example of obesity[END_REF]. [START_REF] Chao | Flute, a publicly available stochastic influenza epidemic simulation model[END_REF] and (c) in real influenza-infected animals (red curve [START_REF] Itoh | Emergence of h7n9 influenza a virus resistant to neuraminidase inhibitors in nonhuman primates[END_REF]), and (d) body temperature in real influenza-infected animals (red curve [START_REF] Itoh | Emergence of h7n9 influenza a virus resistant to neuraminidase inhibitors in nonhuman primates[END_REF]).

Weekly patterns in daily infected cases

Daily new infected cases are highly affected by weekdays, such that case numbers are lowest at the start of the week and increase afterwards. This pattern is observed at the world level, as well as at the level of almost every single country or USA state. Hence, in order to estimate biologically meaningful reproduction numbers, clean of weekly patterns due to administrative constraints, analyses have to be restricted to specific periods shorter than a week, or at rare occasions when patterns escape the administrative constraints. This weekly phenomenon occurs during exponential increase as well as decrease phases of the pandemic and during endemic periods in numbers of daily cases (Figure 3.3). In addition, the daily new infected case record is discontinuous for many countries/regions, which frequently publish, on Monday or Tuesday, a cumulative count for that day and the weekend days. For example, Sweden typically publishes only four numbers over one week, the one on Tuesday cumulating cases for Saturday, Sunday and the two first weekdays. Discontinuity in records further limits the availability of data enabling detailed analyses of daily reproduction numbers and can be considered as extreme weekday effects on new case records due to various administrative constraints.

We calculated Pearson correlation coefficients r between a running window of daily new case numbers of 20 consecutive days and a running window of identical duration with different intervals between the two running windows. These Pearson correlation coefficients r typically peak with a lag of seven days between the two running windows. The mean of these correlations are for each day of the week from Tuesday (data making up for the weekend underestimation) to Monday: 0.571, 0.514 (0.081), 0.383 (0.00008), 0.347 (0.000003), 0.381 (0.000006), 0.468 (0.000444) and 0.558 (0.03916), with, in parentheses, the p-value of the one-tailed paired t-test showing that the correlation observed with running windows starting Tuesday are more than the others (see Table A.1 in Appendix A). This could reflect a biological phenomenon of seven infection days. However, examination of the frequency distributions of lags for r maxima reveals, besides the median lag at 7 days, local maxima for multiples of 7 (14, 21, 28, 35, etc.). About 50 percent of all local maxima in r involve lags that are multiples of seven (seven included). This excludes a biological causation, except if data periodicity comes from an entrainment by the weekly "Zeitgeber" of census, near the duration of the contagiousness interval. We tried to control for weekdays using two methods, and combinations thereof. For the first method, we calculated z-scores for each weekday, considering the mean number of cases for each weekday, and subtracted that mean from the observed number for a day (Figure 3.3). This delta was then divided by the standard deviation of the number of cases for that weekday. The mean and standard variation are calculated across the whole period of study for each weekday.

The second method implies data smoothing using a running window of 5 consecutive days, where the mean number of new cases calculated across the five days is subtracted from the number of new cases observed for the third day. Hence, data for a given day are compared to a mean including two previous, and two later days (Figure 3.4). We constructed two further datasets, where z-scores are applied in the first to data after smoothing from the second method and are applied in the second data after smoothing from the first method (not shown) (Figure 3.5 and Figure 3.6). These four datasets from daily new cases database [119] transformed according to different methods and combinations thereof designed to control for weekday were analysed using the running window method. Despite attempts at controlling for weekday effects, the median lag was always seven days across all four transformed datasets, and local maxima in lag distributions were multiples of seven. After data transformations, about 50 percent of all local maxima were lags that are multiples of seven, including seven. Visual inspection of plots of these transformed data versus time for daily new infected cases from the whole world shows systematic local biases in daily new infected cases (after transformation) on Sundays and Mondays, for all four transformed datasets, with Sundays and/or Mondays as local minima and/or local maxima, according to which method or combination thereof was applied to the data. For each specific day j, the mean number of confirmed daily new cases calculated for days j -1, j -2, j, j + 1 and j + 2 is subtracted from the number for day j. For specific day j, the mean number of confirmed new cases calculated for days j -1, j -2, j, j + 1, j + 2 is subtracted from the number for day j.

Hence, the methods we used failed to neutralize the weekly patterns in daily new cases due to administrative constraints. This issue highly limits the data available for detailed analyses of daily new cases aimed at estimating biologically relevant estimates of reproduction numbers at the level of short temporal scales. By smoothing on five consecutive days of raw data (confirmed world daily new infected cases [119]) and applying the z-transformation, we obtain a better result in Figure 3.7 than in Figure 3.6 in order to neutralize the weekly pattern. The reason is that the smoothing largely eliminates the counting defect during weekends due either to fewer hospital admissions and/or less systematic PCR tests or to a lack of staff at the end of the week to perform the counts.

A deterministic contagion discrete mechanism

In the following, we suppose that the susceptible population size remains constant, which constitutes a hypothesis valid during the exponential phase of epidemic waves. The ODE deterministic approach is linked by a common background consisting of the birth and death process approach used in the kinetics of molecular reactions by Delbrück [START_REF] Delbrück | Statistical fluctuations in autocatalytic reactions[END_REF], then in dynamical systems theory by numerous authors [START_REF] Rubio | Stability analysis of the modified levenberg-marquardt algorithm for the artificial neural network training[END_REF][START_REF] Rubio | Adapting h-infinity controller for the desired reference tracking of the sphere position in the maglev process[END_REF], namely in modeling of the epidemic spread in exponential growth. In the ODE approach, the Malthusian parameter is the dominant eigenvalue. The methodology chosen starts from an attempt to reconstruct an epidemic dynamic only from the observation of the number R ikj of people infected at day j by a given infectious individual i during the k th day of his period of contagiousness of length r. By summing on the number of new infectious individuals X j-k at day j -k where their contagiousness started, the number of new infected people on day j is equal to:

X j = k=1,...,r i=1,...,X j-k R ikj (3.5)
We assume now that R ikj is the same, equal to R k , for all individuals i and day j, and depends only on day k. Then, we have:

X j = k=1,...,r X j-k R k (3.6) X j-1 = k=1,...,r R k X j-1-k , . . . X j-r+1 = k=1,...,r R k X j-r-k
The convolution Equation 3.6 is the basis of our modelling of the epidemic dynamics and the k=1,...,r R k plays the same role as the classical basic reproduction number R 0 [START_REF] Ferretti | Quantifying sars-cov-2 transmission suggests epidemic control with digital contact tracing[END_REF]. Let us choose a simple deterministic rule of infection, and denote by X j the number of new infected cases at day j(j ≥ 1), and R k (k = 1, . . . , r) the daily reproduction number at day k of the contagiousness period of length r for all infectious individuals. Then, we have obtained in Equation (3.6) by supposing that contagiousness behaviour is the same for all the infectious individuals: for example, if r = 3, for the number X 5 of new cases at day 5, the equation

X 5 = R 1 X 4 + R 2 X 3 + R 3 X 2
means that for example new cases at day 4 have contributed to new cases at day 5 with the term R 1 X 4 , R 1 being the reproduction number at first day of contagiousness of new infected individuals at day 4.

In matrix form, we get:

X = M R (3.7)
where X = (X j , . . . , X j-r-1 ) and R = (R 1 , . . . , R r ) are r-dimensional vectors and M is the r -r matrix:

M =     X j-1 X j-2 • • • X j-r X j-k X j-k-1 • • • X j-k-r+1 • • • X j-r X j-r-1 • • • X j-2r+1     (3.8)
It is easy to show that, if X 0 = 1 and r = 5 we obtain:

X 5 = R 5 1 + 4R 3 1 R 2 + 3R 2 1 R 3 + 3R 1 R 2 2 + 2R 2 R 3 + 2R 1 R 4 + R 5 (3.9)
The length r of the contagiousness period can be estimated from the ARIMA series of the stationary random variables Y j 's, equal to the X j 's without their trend, by considering the length of the interval on which the auto-correlation function remains more than a certain threshold, e.g., 0.1 [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF]. For example, by assuming r = 3, if R 1 = a, R 2 = b and R 3 = c, we obtain:

X 0 = 1, X 1 = a, X 2 = a 2 + b, X 3 = a 3 + 2ab + c, X 4 = a 4 + 3a 2 b + b 2 + 2ac, X 5 = a 5 + 4a 3 b + 3ab 2 + 3a 2 c + 2bc, X 6 = a 6 + 5a 4 b + 4a 3 c + 6a 2 b 2 + 6abc + b 3 + c 2 , X 7 = a 7 + 6a 5 b + 5a 4 c + 10a 3 b 2 + 12a 2 bc + 4ab 3 + 3b 2 c + 3ac 2 (3.10)
If R 1 and R 2 , respectively, to a and b, and if a = b = R/2, c = 0, then X 5 is given by:

X 5 = 1 32 R 5 + 1 4 R 4 + 3 8 R 3 (3.11) If R = 2, {X j } i=1,.
..,∞ is exactly the Fibonacci sequence, and more generally, for R > 0, the generalized Fibonacci sequence.

In the general case, where different clusters appear independently at different times, the problem is much more difficult. It can be simplified if it is assumed that these different clusters appear at the same time with the same age of infection. If we suppose that there is more than a unique infected, i.e., if the initial number of infected is I 0 = X 0 > 1 at time 0 and if these I 0 first infected are infectious at same age of infection at time 0, then the Equation (3.5) can be written as:

X j = d=1,...,t R d X j-d , ∀ j > 0 or: X j = d=1,...,r R d X j-d for all t > 0 with X j-d = 0, if j -d < 0
This problem of the initial conditions is crucial. It is relatively simple in the case of a single infected appearing at day 0 (unique cluster) or in the case of multiple infected individuals but with the same age of infection, in which case the infection dynamics proceeds according to Equation (3.5), where X j is the number of new infected individuals at day j and R d is the daily basic reproduction number at the day d of the period of contagiousness of length r (R d and r being supposed to be the same for all individuals and constant during the growth phase of new cases). Another problem is the presence of noise on the registered data. Let us consider X 0 , the size of the first cluster of infectious individuals as a Gaussian random variable with a normal distribution N (µ 0 , σ 0 ). Then, if R d represents the daily basic reproduction number at day d of the contagiousness period of length r and if R d and r, supposed to be not random real numbers are identical for all infectious individuals and constant during the growth epidemic period, we have, for m = 6:

X 1 = R 1 X 0 X 2 = R 1 X 1 + R 2 X 0 = (R 2 1 + R 2 )X 0 X 3 = R 1 X 2 + R 2 X 1 + R 3 X 0 = (R 3 1 + 2R 1 R 2 + R 3 )X 0 X 4 = R 1 X 3 + R 2 X 2 + R 3 X 1 + R 4 X 0 = (R 4 1 + 3R 2 1 R 2 + 2R 1 R 3 + R 2 2 + R 4 )X 0 X 5 = R 1 X 4 + R 2 X 3 + R 3 X 2 + R 4 X 1 + R 5 X 0 = (R 5 1 + 4R 3 1 R 2 + 3R 2 1 R 3 + 3R 1 R 2 2 + 2R 2 R 3 + 2R 1 R 4 + R 5 )X 0 X 6 = R 1 X 5 + R 2 X 4 + R 3 X 3 + R 4 X 2 + R 5 X 1 + R 6 X 0 = (R 6 1 + 5R 4 1 R 2 + 4R 3 1 R 3 + 3R 2 1 R 4 + 6R 2 1 R 2 2 + 6R 1 R 2 R 3 + R 3 2 + R 2 3 + 2R 2 R 4 + 2R 1 R 5 + R 6 )X 0 X 7 = R 1 X 6 + R 2 X 5 + R 3 X 4 + R 4 X 3 + R 5 X 2 + R 6 X 1
and more generally:

X t = d=1,...,r R d X t-d for t ≥ 0 with X d-k = 0, for d < k, then X t = K(t)X 0 and µ t = d=1,...,r R d µ t-d , for t ≥ 0 with µ d-k = 0, for d < k, then µ t = K(t)µ 0
where µ t is the expectation of X t and K(t) the general term of the generalized Fibonacci (or discrete Volterra) series. Because X 0 is Gaussian, all the X t 's are Gaussian with distribution N (K(t)µ 0 , K(t)σ 0 ). If we consider now the vectors R = (R 1 , . . . , R r ) T , X t = (X t , . . . , X t-r+1 ) T and µ t = (µ t , . . . , µ t-r+1 ) T for t > r, then we have:

X t = M t R
where the matrix M t is given by:

M t =     X t-1 X t-2 • • • X t-r X t-k-1 X t-k-2 • • • X t-k-r • • • X t-r X t-r-1 • • • X t-2r+1    
and, for the expectations:

µ t = M t R
where the matrix M t is given by:

M t =     µ t-1 µ t-2 • • • µ t-r µ t-k-1 µ t-k-2 • • • µ t-k-r • • • µ t-r µ t-r-1 • • • µ t-2r+1    
If random variable X t 's are observed, a deconvolution of equations above allows obtaining the vector R. The deconvolution of the first matrix equation leads to random values of the daily basic reproduction numbers R d and of the second leads to deterministic values of the numbers R d , but it is clear that the first matrix equation can only be obtained by simulation or by observing epidemic in different countries supposed to have the same spread characteristics (concerning in particular r and R d parameters), which requires extensive investigation work on reliable data all around the world. As regards the case of several clusters at the start of the epidemy with different ages of infection, a model generalized to multiple cohorts would make it possible to consider an estimation of the R d 's for the various initial clusters, if it would be possible to define a non-degenerate deconvolution operator.

A stochastic contagion discrete mechanism: obtained from the SIR equation

In the following, we suppose that the susceptible population size remains constant, which constitutes a hypothesis valid during the exponential phase of epidemic waves. The Markovian stochastic approach is linked by a common background consisting of the birth and death process approach used in the kinetics of molecular reactions by Delbrück [START_REF] Delbrück | Statistical fluctuations in autocatalytic reactions[END_REF], then in dynamical systems theory by numerous authors [START_REF] Rubio | Stability analysis of the modified levenberg-marquardt algorithm for the artificial neural network training[END_REF][START_REF] Rubio | Adapting h-infinity controller for the desired reference tracking of the sphere position in the maglev process[END_REF], namely in modeling of the epidemic spread in exponential growth. In the Markovian approach, the Malthusian parameter is equivalent to the Kolmogorov-Sinai entropy (called evolutionary entropy in [START_REF] Demetrius | Boltzmann, darwin and directionality theory[END_REF][START_REF] Rhodes | Evolutionary entropy determines invasion success in emergent epidemics[END_REF][START_REF] Demongeot | Complexity and stability in biological systems[END_REF]). Another way to derive the SIR equation is the probabilistic approach, which comes from the microscopic equation of molecular shocks by Delbrück [START_REF] Delbrück | Statistical fluctuations in autocatalytic reactions[END_REF] and corresponds to a classical birth-and-death process: if at least one event (with rates of contact ν, birth f , death µ or recovering ρ) occurs in the interval (t, t + dt), and by supposing that births compensate deaths, leaving constant the total size N of the population, we have:

P ({S(t + dt) = k, I(t + dt) = N -k}) = P (S(t) = k, I(t) = N -k)[1 -[µk + νk(N -k) -f k -ρ(N -k)]dt] +P (S(t) = k -1, I(t) = N -k + 1)[f (k -1) + ρ(N -k + 1)]dt (3.12) -P (S(t) = k + 1, I(t) = N -k -1)[µ(k + 1) + ν(k + 1)(N -k -1)]dt
Hence, we have, if

P k (t) denotes Probability({S(t) = k, I(t) = N -k}): dP k (t) d = [P (S(t + dt) = k, I(t + dt) = N -k) -P (S(t) = k, I(t) = N -k)]/dt = -P (S(t) = k, I(t) = N -k)[µk + νk(N -k)n -f k -ρ(N -k)] +P (S(t) = k -1, I(t) = N -k + 1)[f (k -1) + ρ(N -k + 1)] -P (S(t) = k + 1, I(t) = N -k -1)[µ(k + 1) + ν(k + 1)(N -k -1)],
and we obtain:

dP k (t) dt = -[µk + νk(N -k) -f k -ρ(N -k)]P k (t) + [f (k -1) + ρ(N -k + 1)]P k-1 (t) -[µ(k + 1) + ν(k + 1)(N -k -1)]P k+1 (t)
Then, by multiplying by s k and summing over k, we obtain the characteristic function of the random variable S. If births do not compensate deaths, we have:

P ({S(t + dt) = k, I(t + dt) = j}) = P (S(t) = k, I(t) = j)(1 -[µk + νkj -f k -ρj]dt) +P (S(t) = k -1, I(t) = j + 1)[f (k -1) + ρ(j + 1)]dt -P (S(t) = k + 1, I(t) = j -1)[µ(k + 1) + ν(k + 1)(j -1)]dt
If S and I are supposed to be independent and if the coefficients ν, f, µ and ρ are sufficiently small, S and I are Poisson random variables [START_REF] Garcia | Birth and death processes as projections of higher-dimensional poisson processes[END_REF], whose expectations E(S) and E(I) verify:

dE(S) dt = f E(S) -νE(SI) -µE(S) + ρE(I), or, if f = µ, dE(S) dt ≈ E(I)[-νE(S) + ρ],
leading to the SIR equation for the variables S, I and R considered as deterministic:

dS dt = -νSI + ρR, dI dt = νSI -kI -µI, (3.13
)

dR dt = kI -ρR

A stochastic contagion discrete mechanism: obtained from Boolean networks

If we choose now a stochastic Hopfield-like transition for the calculation of the Boolean weight state O i (t + 1) of an individual i (equal to 0 in case of normal weight and to 1 in case of overweight or obesity) as a function of O i (t), by using a potential function P i :

P i (t) = k∈N (i) W ik O k (t) T k (3.14)
where N (i) is the set of nodes linked to i in the interaction network between the individuals of the studied population, T i is a tolerance parameter (varying between 0 and +∞) quantifying the level of indifference shown by i to the weight state of the nodes belonging to his neighbourhood N (i), and W ik is the interaction coefficient measuring the influence (positive or negative) individual k has on individual i (its value can be equal to 1, 0 or -1, in absence of precise estimation of this influence). Then, the stochastic transition rule with H the Heaviside function

(H(x) = 1, if x ≥ 0; H(x) = 0, if x < 0) is as follows:    O i (t + 1) = H(P i (t)), if T i = 0 P robability({O i (t + 1) = 1}) = e P i (t) 1 + e P i (t) , if T i > 0 (3.15)
In Equation (3.15), the function P i is the analogue of the Hamiltonian function in the Hopfield model [START_REF] Cheng | Contact tracing assessment of covid-19 transmission dynamics in taiwan and risk at different exposure periods before and after symptom onset[END_REF], and the tolerance parameter T i is analogue to the temperature: higher the indifference, nearer 1 /2 the probability to become obese, which corresponds to a quasiabsence of influence on the individual i by his neighbours regardless of their weight status. There are four classical types of centrality in an interaction graph G with n nodes (like the graph of Figure 3.1 Top right). The first is the betweenness centrality, defined for a node k as follows [START_REF] Xi | Temporal dynamics in viral shedding and transmissibility of covid-19[END_REF]:

C B (k) = i̸ =j̸ =k∈G β ij (k) β k (3.16)
where β ij (k) is the total number of shortest paths from node i to node j that pass-through node k, and

β k = i̸ =j∈G β ij (k).
The second type of centrality is the degree centrality, defined from the notions of out-, inor total-degree of a node i, corresponding to the number of arrows of the interaction graph, respectively outing from or entering in the node i, or both. For example, the total-degree centrality is defined by the following formula:

C in-deg i = j=1,...,n;j̸ =i |a ij | n -1 (3.17)
where a ij = sign(W ij ) denotes the general coefficient of the signed incidence matrix A of the graph G.

The third type of centrality is the closeness centrality. The closeness is the inverse of the average farness, which is defined by averaging over all nodes j of the network else than i the distance between i and j ̸ = i:

C clo i = n -1 j=1,...,n;j̸ =i L(i, j) (3.18)
where the distance chosen here is L(i, j), the length of the closest path between i and j.

The last type of classical centrality is the spectral centrality or eigen-centrality, which takes into account the fact that neighbours of a node i can be also highly connected to the rest of the graph, considering that connections to highly connected nodes contribute to the centrality of i more than connections to weakly connected nodes. Hence, the eigenvector centrality of the node i measures more the global influence of i on the whole network and verifies [START_REF] Breban | Theory versus data: How to calculate r0?[END_REF]:

C eigen i = j∈N (i) W ij V j λ (3.19)
where λ is the greatest eigenvalue of the incidence matrix of the graph and V its eigenvector. The four centralities above can be very different (Figure 3.8), but they have each their intrinsic interest: i) betweenness centrality relates to the global connectivity with all nodes of the network, ii) degree centralities corresponds to the local connectivity with only nearest nodes, iii) closeness centrality measures the relative proximity with other nodes for a given distance on the interaction graph, and iv) spectral centrality corresponds to the ability to be connected to possibly a few number of nodes, but having a high connectivity, for example important hub-relays controlling wide sub-networks. Despite the already complementary properties of the classical centralities, we introduce a new notion of centrality, called entropy centrality, taking into account the heterogeneity of the distribution of states of the neighbours of a node i, and not only its connectivity in the graph, and highlighting the symmetric subgraphs:

C entropy i = - k=1,...,s i ν k log 2 ν k (3.20)
where ν k denotes the k th frequency among s i frequencies of the histogram of the observed state values in the neighbourhood N (i) of the node i, set of the nodes out-or in-linked to the node i.

A stochastic contagion discrete mechanism: application to comorbidity risk factor (obesity)

We are interested in modeling the social contagion mechanism through which the obesity can propagate from individual to individual in same or near classes from a high school through the pupil population, pupils changing their weight under the influence of their school friends connected to them through a declared friendship link (possibly not reciprocal).

Obesity in the friendship graph of a Tunisian high school

In Figure 3.8 , we have represented on the left the observed friendship interactions graph in two classes (of level 4 and 5) of a Tunisian high school with 274 pupils having 524 friendship links between them [START_REF] Wu | Estimating clinical severity of covid-19 from the transmission dynamics in wuhan, china[END_REF][START_REF] Demongeot | Discrete dynamics of contagious social diseases: Example of obesity[END_REF][START_REF] Demongeot | From susceptibility to frailty in social networks: The case of obesity[END_REF]. The Tunisian sample contains 18 individuals overweight or obese. We will represent first the friendship network (Figure 3.8a) by putting in blue the links between the nodes of the network, which represent the school children in question.

The links are directed, and they can be unique between two nodes, when a friendship without reciprocity has been detected (for the sake of simplicity, arrows have not been represented, but in a couple of potential friends, the final node has in general an area more important than the first one). To visually represent the connectivity of the network, we have chosen to give the nodes a surface proportional to their in-degree centrality in Figure 3.8a,b, totaldegree centrality in Figure 3.8c, and spectral centrality in Figure 3.8d. In the three last representations, we have leaved the friendship network evolve to its asymptotic state following the Equation (3.15), we have kept the proportionality of the node area to its centrality and put together the nodes of the network having their centrality in the same value interval. As we can see on Figure 3.8b and Figure 3.8c, the total-degree and in-degree centralities give about the same feature, but for the spectral centrality in Figure 3.8d, the neighbors of a node i play contribute more to its area if they are highly connected to the other nodes, because these second level highly connected nodes contribute to the spectral centrality of i more than the weakly connected ones. We see in Figure 3.8 that the spectral centrality (b) gives priority to the nodes of the contagion network (a) connected to hubs on the left bottom part of the contagion network. On the contrary, the total degree centrality (c) favors all the hubs even when they are connected to a relatively small number of nodes and the in-degree centrality (d) gives an importance to all the nodes depending to the number of other nodes expressing their friendship for them.

If the aim of such a study is to diminish the number of obese, the identification of the obese hubs is important: a nutritional reeducation on a part of these obese or overweight hubs can reduce drastically the number of obese or overweight in their neighbourhood. For example, if we diminish their influence by increasing their tolerance parameter T i , we can suppress this influence, then reduce the risk to become obese at their contact. Suppose that we start with an interaction network of 100 individuals including 100 obese or overweight and zero normal weight, with interaction weights w ij chosen at random either 0 or 1. The Figure 3.9 shows that, by increasing the value of the tolerance parameter T i for the 8 most connected nodes, the mean percentage of normal weight individuals increases until the value 1 /2, corresponding to a random choice equilibrated between the normality on one hand, and overweight or obesity on the second hand. That shows that the states of the nodes of the network can be regularized to the normality after a campaign targeted at the most influential obese or overweight individuals of the interaction network.

Obesity in the friendship graph of a French high school

The top image of Figure 3.10 shows the main connected component (89 nodes) of a friendship interaction graph is observed in two classes of a French high school with 104 pupils having 348 friendship links between them [START_REF] Wu | Estimating clinical severity of covid-19 from the transmission dynamics in wuhan, china[END_REF][START_REF] Demongeot | Discrete dynamics of contagious social diseases: Example of obesity[END_REF][START_REF] Demongeot | From susceptibility to frailty in social networks: The case of obesity[END_REF]. The French sample contains 17 overweight or obese individuals. We see in Figure 3.10 that the spectral centrality (middle-left) of the French sample friendship interaction graph gives priority to the big hubs and nodes connected to these hubs. On the contrary, the total-degree centrality (middle-right) favors all hubs, even those connected to a relatively small number of nodes.

The entropy centrality identifies nodes having different states in their neighbourhood and gives them an importance higher than for the nodes having only one type of state in their neighbourhood. Once stabilized after changing the state of the 20 most influential nodes (for example after a dedicated nutritional education) on the left and of the 21 most influential in the right image of the bottom of Figure 3.10, we can observe that the stationary state of the network is fully overweight or obese in the first case of 20 state changes, and fully normal in the second case of 21 state changes, showing a transition of the invariant measure of the contagion network. In the two applications of Boolean dynamics we have presented in this section, we can see some simplifications:

-The small number of states simplifies the dynamics but complicates the examination of realistic solutions to stop the epidemic. We should for example distinguish overweight and obese states in the second case, because the return to a normal weight is much easier for an overweight individual than for an obese one. In the first case, the two-state variable "vaccinated or not" should be added, in order to properly take into account, the effect of the vaccination policy.

-The diffusion through fixed nearest neighbours is often unrealistic, hence it is necessary to build interaction networks with variable links over time, to take into account for example the movement of individuals. This non-constancy of the links can be the subject of a periodic dynamic (linked to school or professional constraints), but it adds a high complexity to the classical Boolean modeling with fixed links over time. Modeling of the spread of epidemics using Boolean dynamics on an ideal Apollonian network or on a more realistic school network was done by Eric Goles and his colleagues [START_REF] Böttcher | Connectivity disruption sparks explosive epidemic spreading[END_REF]. The current context of comorbidity risk factor (obesity) closely monitored by the WHO, a contagious social pandemic shows that this problem is now highly relevant. The particular chapter of Boolean dynamics applied to pandemics is far from over and the combined efforts of several research communities will continue to make decisive contributions (like we did in this thesis) to this new disciplinary field in the future.

Distribution of the daily reproduction numbers along the contagiousness period of an individual: a theoretical approach

If R 0 denotes the basic reproduction number (or average transmission rate) in a given population, we can estimate the distribution V (whose coefficients are denoted V j = R j /R 0 ) of the daily reproduction numbers R j along the contagious period of an individual, by remarking that the number X j of new infectious cases at day j is equal to X j = I j -I j-1 , where I j is the cumulated number of infectious at day j, and verifies the convolution equation (equivalent to Equation (3.6):

X j = k=1,...,r R k X j-k , giving in continuous time : X(t) = r 1 R (s) X (t -s) ds, (3.21)
where r is the duration of the contagion period, estimated by 1/(ρ + µ), ρ being the recovering rate and µ the death rate in SIR Equations:

dS dt = -νSI, (3.22
)

dI dt = νSI -(k + µ)I, (3.23)
where S and I are respectively the size of the susceptible and infectious populations. Also, r and S can be considered as constant during the exponential phases of the pandemic, and we can assume that the distribution V is also constant; then, V which is an entropy can be estimated as the Kolmogorov-Sinaï entropy of the Markovian Delbrück scheme ruling the X j 's and giving new parameters for characterising pandemic dynamics, namely for quantifying its robustness and stability ( [START_REF] Rhodes | Evolutionary entropy determines invasion success in emergent epidemics[END_REF][START_REF] Demongeot | Complexity and stability in biological systems[END_REF]) and by solving the linear system (equivalent to Equation (3.7):

R = M -1 X (3.24)
where M is given by Equation (3.8). Equation (3.24) can be solved numerically, if the pandemic is observed during a time greater than 1/(ρ + µ).

For showing the existence of a unique solution to the deconvolution problem defined to get the daily basic reproduction numbers R d in simple cases, take the example of r = 2, with R 1 = a and R 2 = b. In this case, if X 0 = 1, the matrix M 5 is defined by:

M 5 =   X 4 X 3 X 2 X 3 X 2 X 1 X 2 X 1 X o   =   a 4 + 3a 2 b + b 2 a 3 + 2ab a 2 + b a 3 + 2ab a 2 + b a a 2 + b a 1   Then, det M 5 = a 6 + 3a 4 b -a 4 -2a 3 b + 2a 2 b 2 -a 2 b -2ab 2 is
a polynomial in a and b, which vanishes on a set of Borel measure zero of R 2 . In the particular case a = b, we have det M 5 = a 3 (a -1)(a 2 + 4a + 3) and this set is equal to {0,1}, because the roots of the polynomial a 2 + 4a + 3 are complex conjugates.

The entropy H of V is equal:

H = - k=1,...,r V k log(V k )
If there are negative V j 's like the Kuwait example in Section 3.2.12, it is still possible to define an index of proximity to uniformity of V by considering the entropy of the distribution W defined by:

W j = [(V j -min{V k ≤ 0})/ i=1,...,r W i ].
We will first demonstrate an example of how the matrix M can be repeatedly calculated for consecutive periods of length equal to that of the contagiousness period (supposed to be constant during the outbreak), giving matrix series M 1 , M 2 , . . . Following Equation (3.8), we put the values of X i 's in the two matrices below, with r = 3 for two periods, the first from day 1 to day 3 and the second from day 4 to day 6.

M 1 =   X 4 X 3 X 2 X 3 X 2 X 1 X 2 X 1 X 0   , M 2 =   X 6 X 5 X 4 X 5 X 4 X 3 X 4 X 3 X 2   , . . .
where, after Equation (3.10), M 1 can be calculated from the R j 's as:

M 1 =   R 4 1 + 3R 2 1 R 2 + 2R 1 R 3 + R 2 2 R 3 1 + 2R 1 R 2 + R 3 R 2 1 + R 2 R 3 1 + 2R 1 R 2 + R 3 R 2 1 + R 2 R 1 R 2 1 + R 2 R 1 1   ,
Additionally, from Equation (3.6), if, for instance, j = 8 and r = 3, then we have the expression below, which means that the new cases on the 8th day depend on the new cases detected on the previous days 7, 6 and 5, supposed to be in a period of contagiousness of 3 days:

X 8 = k =1,2,3 R k X 8-k = R 1 X 7 + R 2 X 6 + R 3 X 5 (3.25)
Let us suppose now that the initial R j 's on a contagiousness period of 3 days, are equal to:

  R 1 R 2 R 3   =   2 1 2   , then matrix M defined by M ij = X 7-(i+j)
gives the R j 's from Equation (3.24), hence allows the calculation of

X j = k=1,2,3 R k X j-k .
The inverse of M is denoted by M -1 and verifies: R = M -1 X, where X = (X 6 , X 5 , X 4 ), with X 1 = 1, X 2 = 2, X 3 = 5, X 4 = 14, X 5 = 37, X 6 = 98 and we obtain:

M -1 1 =   37 14 5 14 5 2 5 2 1   -1 =   -1/4 1 -3/4 1 -3 1 -3/4 1 11/4   ,
and a deconvolution gives the resulting R j 's:

  -1/4 1 -3/4 1 -3 1 -3/4 1 11/4     98 37 14   =   2 1 2   =   R 1 R 2 R 3   ,
thanks to the following calculation:

R 1 = -49/2 + 37 -21/2 = 2 R 2 = 98 -111 + 14 = 1 R 3 = -147/2 + 37 + 77 = 2
We obtain for the resulting distribution of daily reproduction numbers the exact replica of the initial distribution. We obtain the same result by replacing M 1 by the matrix M 2 .

Distribution of the daily reproduction numbers: a stochastic approach

Let us consider a stochastic version of the deterministic toy model corresponding to Equation (3.25), by introducing an increasing noise on the R j 's, e.g., by randomly choosing their values following a uniform distribution on the three intervals:

[2 -a, 2 + a], [1 -a/2, 1 + a/2]
and [2-a, 2+a] (for having a U-shape behavior), with increasing values of a, from 0.1 to 1, in order to see when the deconvolution would give negative resulting R j 's, with conservation of the average of their sum R 0 , if the random choice of the values of the R j 's at each generation is repeated, following the stochastic version of Equation (3.6):

X j = k=1,r (R k + ε k )X j-k ,
where r is the contagiousness period duration and ε k is a noise perturbing R k , whose distribution is chosen uniform on the interval [0, 2a] for k = 1, 3, and [0, a] for k = 2. This choice is arbitrary, and the main reason of the randomization is to show that the deconvolution can give negative results for R k 's, as those observed for increasing values of a, from 0.1 to 1, with explicit calculations for three consecutive periods, from day 1 to day 3, from day 4 to day 6, and from day 7 to day 9.

For each random choice of the values of the daily reproduction numbers R j 's, we can calculate a matrix M 1 corresponding to Equation (3.7). Its inversion into the matrix M -1 1 makes it possible to solve the problem of deconvolution of Equation (3.6)-that is to say, to obtain new R j 's as a function of the observed X k 's. We can then calculate a new matrix M 2 from these new R j 's and thus continue during an epidemic the estimation of the daily reproduction numbers R j 's from the successive matrices M 1 , M 2 , . . . , and observed X k 's.

1. For a = 0.1, let us randomly and uniformly choose the initial distribution of the daily reproduction numbers R 1 in the interval [1.9, 2.1], R 2 in [0.95, 1.05] and

R 3 in [1.9, 2.1] as R 1 = 2.1, R 2 = 0.95, R 3 = 2.1.
Then, the transition matrix M 1 is equal to:

M 1 =   41
M -1 1 =   -0.2154195 0.92857143 -0.7953515 0.92857143 -2.95 1.2178571 -0.7953515 1.2178571 2.705584  
From X 6 = 113.491, X 5 = 41.7391, X 4 = 15.351, resulting R j 's are:

R 1 = 2.1, R 2 = 0.95 and R 3 = 2.1.
The next initial R j 's are chosen as: R 1 = 2, R 2 = 0.95, R 3 = 1.9 and we have: The next initial R j 's are: R 1 = 2, R 2 = 1.05, R 3 = 1.9 and we have: From these values of X 9 and X 10 , we obtain the matrices M 3 and M -1 3 :

X 7 = 2X
X 9 = 2X 8 + 1.05X 7 +
M 3 =   2083.

For

a = 1, let us choose the initial R 1 in [1, 3], R 2 in [0.5, 1.5] and R 3 in [1, 3], e.g., R 1 = 1, R 2 = 1.355 and R 3 = 1.1.
Then, the transition matrix M 1 is equal to:

M 1 =   9.101 4.81 2.355 4.81 2.355 1 2.355 1 1  
and its inverse is given by:

M -1 1 =   -1.11983471 2.02892562 0.60828512 2.02892562 -2.93801653 -1.84010331 0.60828512 -1.84010331 1.40759184   New cases are: X 6 = 18.209, X 5 = 9.101, X 4 = 4.81, X 3 = 2.355, X 2 = 1, X 1 = 1,
and by deconvoluting, we obtain the resulting R j 's equal to: R 1 = 1, R 2 = 1.355, R 3 = 1.1, i.e., the exact initial distribution.

Let us now consider new initial R j 's: R 1 = 1, R 2 = 1, R 3 = 1. That gives a new matrix M 2 , with new X 7 and X 8 calculated from the new initial R j 's, by using the former values of X 6 , . . . , X 2 : Table 3.1: Simulation results obtained for extreme noises a = 0.1 and a = 1, showing great variations of deconvoluted distribution of daily reproduction numbers X j 's and a qualitative conservation of their U-shaped distribution along contagiousness period. We calculate X 9 and X 10 using new initial R j 's: R 1 = 3.0, R 2 = 0.5, R 3 = 2.9 : More precise simulation results are given in Table 3.1, which summarizes computations made for random choices of R j 's distributions, for a = 0.1 and a = 1 and until time 20. These simulations show a great sensitivity to noise, but a qualitative conservation of their U-shaped distribution along the contagiousness period of individuals. More precisely, because of the presence of noise on the R j 's, we cannot always obtain positive values from the data for the R j 's by applying the deconvolution, which explains the presence of negative values in empirical examples, as in the theoretical noised examples. A way to solve this problem could be to suppose that noise is stationary during all of the growth period of a wave, then calculate the R j 's for all running time windows of length equal to the contagiousness duration and then obtain the mean of the R j 's corresponding to these windows. As this stationary hypothesis is not widely accepted, we prefer to keep negative values and focus on the shape of the distribution of the R j 's.

X 7 = X 6 + X 5 + X 4 =
a Initial R j 's t X t X t+1 X t+2 Resulting R j 's R o U-shaped 0.1 2
X 9 = 3X

Distribution of the daily reproduction numbers: application to real data from several countries

A case of 3 days for the duration of the contagiousness period Figure 3.11 gives the effective transmission rates R e calculated between 20-25 October 2020 just before the second lockdown in France [126,[START_REF] Scire | Reproductive number of the covid-19 epidemic in switzerland with a focus on the cantons of basel-stadt and basel-landschaft[END_REF]. As the second wave of the epidemic is still in its exponential phase, it is more convenient (i) to consider the distribution of the marginal daily reproduction numbers and (ii) to calculate its entropy and simulate the epidemic dynamics using a Markovian model [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF]. By using the daily new infected cases given in [119], we can calculate, as in Section 3.2.9, the inverse matrix M -1 for the period from 20 to 25 October 2020 (exponential phase of the second wave), by choosing 3 days for the duration of contagiousness period and the following raw data for new infected cases: 20,468 for 20 October, then 26,676, 41,622, 42,032, 45,422 and 52,010 for 25 October. Then, for France between 15 February and 27 October 2020, we obtain the daily reproduction numbers given in Figure 3.11 with a U-shape as observed for influenza viruses. We have:

M -1 =   45
  =   R 1 R 2 R 3  
The effective reproduction number is equal to R 0 ≈ 1.174, a value close to that calculated directly (Figure 3.11), giving V = (0.7, 0.085, 0.215), with a maximal daily reproduction number the first day of the contagiousness period. The entropy H of V is equal to:

H = - k=1,...,r V k log(V k ) = 0.25 + 0.21 + 0.33 = 0.79.
A case of 6 days for the duration of the contagiousness period By using the daily new infected cases given in [119] for Chile, we can calculate M -1 for the period from 1 to 12 November 2020 (endemic phase), by choosing 6 days for the duration of the contagiousness period and the following 7-day moving average data for the new infected cases (Figure 3.12): 1400 for 1 November, then 1370, 1382, 1359, 1362, 1405, 1389, 1385, 1384, 1387, 1394 and 1408 for 12 November. We have:

M -1 =        
        -1 =        
-0.05714222 0.01016059 -0.00901664 0.01474588 0.01016059 -0.01827291 0.0106261 -0.00763363 -0.00901664 0.0106261 -0.00544051 0.02150289 0.01474588 -0.00763363 0.02150289 -0.01796266 0.00640175 0.02139586 -0.01468484 -0.00553414 0.03539322 -0.01613675 -0.00286391 -0.00509801 0.00640175 0.03539322 0.02139586 -0.01613675 -0.01468484 -0.00286391 -0.00553414 -0.00509801 -0.00305831 -0.00452917 -0.00452917 -0.00686198

       
Hence, after deconvolution, we obtain:

R =         -0.36256122 0.22645436 0.01488726 0.33918287 0.28557502 0.50696243        
The effective reproduction number is equal to R 0 ≈ 1.011, a value close to that calculated directly, with a maximal daily reproduction number the last day of the contagiousness period. The quasi-endemic situation in Chile since the end of August, which corresponds to the increase of temperature and drought at this period of the year [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF], gives a cyclicity of the new cases occurrence whose period equals the length of the contagiousness period of about 6 days, analogue to the cyclic phenomenon observed in simulated stochastic data of Section 3.2.10. with a similar U-shaped distribution of the R j 's.

R =           0.466 0.584 1.547 -1.044 0.174 0.297 0.692          
The evolution of the R j 's shows on Figure 3.13 a U-shape on day 4 with a sum of the R j 's equal to 2.72, less than the effective reproduction number R e = 3. 27 [126]. This U-shape is similar to the classical shape observed for the influenza disease [START_REF] Chao | Flute, a publicly available stochastic influenza epidemic simulation model[END_REF].

The biphasic pattern of distribution of the daily reproduction numbers

A case of U-shaped of the evolution of the daily R j 's

If we use the daily new infectious cases given in [119] during the exponential phase of the second wave of COVID-19 outbreak in Kuwait, we can calculate M -1 for the period from December 30 2020 to January 4 2021, by choosing 3 days for the duration of the infectiousness period and the following raw data for the new infected cases are X 1 = 205 the 30th December, X 2 = 286 the 31th, X 3 = 285 the 1st January, X 4 = 205 the 2nd, X 5 = 269 the 3rd and X 6 = 372 the 4th. Hence, giving for the matrix M : Because

M =   269
X =   X 6 X 5 X 4   =   372 269 205   ,
we get R = M -1 X, where:

R 1 = 0.58388424 R 2 = -0.83734838 R 3 = 1.35646161
Then, we can give a graphical representation of the R j 's (Figure 3

W = [(R 1 -R 2 )/(R 1 -2R 2 + R 3 ), 0, (R 3 -R 2 )/(R 1 -2R 2 + R 3 )] = (0.393, 0, 0.607),
which is equal to H = 0.691, the same value than for the first wave.

A case of increased shape of the evolution of the daily R j 's By using the daily new infected cases given in [119] for COVID-19 outbreak in Nigeria, we can calculate M -1 for the period from 5 November to 10 November 2020 (endemic phase), by choosing 3 days for the duration of the contagiousness period and the following raw data for the new infected cases (Figure 3.15): 141 for 5 November, then 149, 133, 161, 164, and 166 for 10 November.

We have: The effective reproduction number is equal to R 0 ≈ 1.129, value close to that calculated directly, with a maximal daily reproduction number the last day of the contagiousness period. The distribution V equals (0.143, 0.342, 0.515) and its entropy H is equal to:

M -1 =   164
H = - k=1,...,r
V k log(V k ) = 0.29 + 0.37 + 0.34 = 1. The effective reproduction number is equal to R 0 ≈ 0.90, value close to that calculated directly, with a maximal daily reproduction number the last day of the contagiousness period. In Appendix A, Table A.2 gives the shape of the R j 's distribution for 194 countries. Table A.2 is built from new COVID-19 cases at the start of the first and second waves for 194 countries; it shows 42 among these 194 countries having a U-shape evolution of their daily R j 's twice, for 12.12 ± 6 expected with 0.95 confidence (p < 10 -12 ), and 189 times, a U-shape evolution for all countries and waves (397), for 99.3 ± 9 expected with 0.95 confidence (p < 10 -24 ). Hence, the U-shape is the most frequent evolution of daily R j 's, which confirms the comparison with the behavior of seasonal influenza. The duration of the contagiousness period, as well as the daily virulence, are not constant over time. Three main factors, which are not constant during a pandemic, can explain this:

-In the virus transmitter, the transition between the mechanisms of innate (the first defense barrier) and adaptive (the second barrier) immunity may explain a transient decrease in the emission of the pathogenic agent during the phase of contagiousness [START_REF] Xi | Temporal dynamics in viral shedding and transmissibility of covid-19[END_REF],

-In the environmental transmission channel, many geophysical factors that vary over time can influence the transmission of the virus (temperature, humidity, altitude, etc.) [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF][START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF][START_REF] Seligmann | Unpredictable, counter-intuitive geoclimatic and demographic correlations of covid-19 spread rates[END_REF],

-In the recipient of the virus, individual or public policies of prevention, protection, eviction or vaccination, which evolve according to the epidemic severity and the awareness of individuals and socio-political forces, can change the sensitivity of the susceptible individuals [START_REF] Seligmann | Unpredictable, counter-intuitive geoclimatic and demographic correlations of covid-19 spread rates[END_REF].

It is therefore very important to seek to estimate the average duration of the period of contagiousness of individuals and the variations, during this phase of contagiousness, of the associated daily reproduction numbers [START_REF] Breban | Theory versus data: How to calculate r0?[END_REF][START_REF] Demetrius | Demographic parameters and natural selection[END_REF][START_REF] Demetrius | Statistical mechanics and population biology[END_REF][START_REF] Demongeot | Sélection naturelle et dérive démographique. Étude empirique de la france de 1860 à 1965[END_REF][START_REF] Demongeot | Biological boundaries and biological age[END_REF][START_REF] Gaudart | Demography and diffusion in epidemics: Malaria and black death spread[END_REF][START_REF] Demongeot | Random modelling of contagious diseases[END_REF]. If the duration of the contagiousness phase is more than 3-5 days, for example ±7 days, the periodicity of seven days observed for the new daily cases could result of an entrainment of the dynamics of new cases driven by the social "Zeitgeber" represented by the counting of new cases, less precise during the weekend (probably underestimated in many countries not working at this time). That questions the deconvolution over 3 and 5 days, giving some negative R j . We have extended our study using a duration r = 3 of contagiousness to r = 7, the result which is the example of USA in Section 3.2.11 show the same existence of identical variations of U-shape type but specify the values of R j 's, more often positive and of more realistic magnitude, while keeping a sum approximately equal to R 0 . Rhodes and Demetrius have pointed out the interest of the distribution of the daily reproduction numbers [START_REF] Demetrius | Boltzmann, darwin and directionality theory[END_REF] with respect to the classical unique R 0 , even time-dependent [START_REF] Rhodes | Evolutionary entropy determines invasion success in emergent epidemics[END_REF]. In particular, they found that this distribution was generally not uniform, which we have confirmed here by showing many cases where we observe the biphasic form of the virulence already observed in respiratory viruses, such as influenza. The entropy of the distribution makes it possible to evaluate the intensity of its corresponding U-shape. This entropy is high if the daily reproduction numbers are uniform, and it is low if the contagiousness is concentrated over one or two days. If some R j are negative, it is still possible to calculate this uniformity index of proximity, by shifting their distribution by a translation equal to the inverse of the negative minimum value. We have neglected in the present study the natural birth and death rates by supposing that they are identical, but we could have taken into account the mortality due to the COVID-19. The discrete dynamics of new cases can be considered as Leslie dynamics governed by the matrix equation:

X j = LX j-1 ,
where X j is the vector of the new cases living at day j and L is the Leslie matrix given by:

L =          R 1 R 2 R 3 . . . . .
0 0 0 . . . b r-1 R r 0 0 . . . . . . 0          and X j-1 =          X j-1 X j-2 X j-3 . . . . . . X j-r         
, where b j = 1 -µ j ≤ 1, ∀i = 1, . . . , r, is the recovering probability between days j and j + 1.

The dynamical stability for L 2 distance to the stationary infection age pyramid

P = lim j X j i=j,j-r+1 X i
is related to |λ -λ ′ |, the modulus of the difference between the dominant and sub-dominant eigenvalues of L, namely λ = e R and λ ′ , where R is the Malthusian growth rate and P is the left eigenvector of L corresponding to λ. The dynamical stability for the distance (or symmetrized divergence) of Kullback-Leibler to P considered as stationary distribution is related to the population entropy H [START_REF] Seligmann | Unpredictable, counter-intuitive geoclimatic and demographic correlations of covid-19 spread rates[END_REF]119,[START_REF] Demongeot | Complexity and stability in biological systems[END_REF][START_REF] Garcia | Birth and death processes as projections of higher-dimensional poisson processes[END_REF]126,[START_REF] Scire | Reproductive number of the covid-19 epidemic in switzerland with a focus on the cantons of basel-stadt and basel-landschaft[END_REF], which is defined if l j = i=1,j-1 b i and p j = l j R j /λ j , as follows:

H = - j=1,...,r p j j=1,...,r jp j log(p j ) (3.26)
The mathematical characterization by the population entropy defined in Equation (3.24) of the stochastic stability of the dynamics described by Equation (3.24) has its origin in the theory of large deviations [START_REF] Ventsel | On small random perturbations of dynamical systems[END_REF][START_REF] Donsker | Asymptotic evaluation of certain markov process expectations for large time, ii[END_REF][START_REF] Freidlin | Random perturbations[END_REF]. This notion of stability pertains to the rate at which the system returns to its steady state after a random exogenous and/or endogenous perturbation and it could be useful to quantify further the variations of the distribution of the daily reproduction numbers observed for many countries [START_REF] Demongeot | Si epidemic model applied to covid-19 data in mainland china[END_REF][START_REF] Gaudart | Factors associated with the spatial heterogeneity of the first wave of covid-19 in france: a nationwide geo-epidemiological study[END_REF][START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF][START_REF] Liu | Understanding unreported cases in the covid-19 epidemic outbreak in wuhan, china, and the importance of major public health interventions[END_REF][START_REF] Griette | Clarifying predictions for covid-19 from testing data: The example of new york state[END_REF]. In summary, the main limitations of the present study are:

-The hypothesis of spatio-temporal stationarity of the daily reproduction numbers is no longer valid in the case of rapid geo-climatic changes, such as sudden temperature rises, which decrease the virulence of SARS CoV-2 [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF], or mutations affecting its transmissibility. The medium of atmospheric, cutaneous or intestinal transmission is of great importance because it depends on numerous environmental conditions, such as humidity.

-The still approximate knowledge of the duration r of the period of contagiousness necessitates a more in-depth study at variable durations, by retaining the value of r, which makes all of the daily reproduction numbers positive.

-The choice of uniform random fluctuations of the daily reproduction numbers is based on arguments of simplicity. A more precise study would undoubtedly lead to a unimodal law varying throughout the contagious period, the average of which following a Ushaped curve, of the type observed in the literature on a few real patients [START_REF] Pan | Viral load of sars-cov-2 in clinical samples[END_REF][START_REF] Chao | Flute, a publicly available stochastic influenza epidemic simulation model[END_REF][START_REF] Itoh | Emergence of h7n9 influenza a virus resistant to neuraminidase inhibitors in nonhuman primates[END_REF] -Another limitation is the comparison of the results of the simulations with the real behaviour of the epidemic, of which we see, in the COVID-19 outbreak, that most of the parameters of transmissibility and contagiousness can change over time. These changes are linked to [START_REF] Demongeot | Si epidemic model applied to covid-19 data in mainland china[END_REF][START_REF] Gaudart | Factors associated with the spatial heterogeneity of the first wave of covid-19 in france: a nationwide geo-epidemiological study[END_REF][START_REF] Griette | Clarifying predictions for covid-19 from testing data: The example of new york state[END_REF][START_REF] Demongeot | Stability, complexity and robustness in population dynamics[END_REF][START_REF] Griette | A robust phenomenological approach to investigate covid-19 data for france[END_REF][START_REF] Demongeot | Biological networks entropies: Examples in neural memory networks, genetic regulation networks and social epidemic networks[END_REF][START_REF] Demongeot | Modeling vaccine efficacy for covid-19 outbreak in new york city[END_REF][START_REF] Griette | What can we learn from covid-19 data by using epidemic models with unidentified infectious cases?[END_REF][START_REF] Magal | A model of vaccination for dengue in the philippines 2016-2018[END_REF][START_REF] Rachdi | Entropy as a robustness marker in genetic regulatory networks[END_REF]: i) the source host of the virus (in whom the intensity of the symptoms varies during the period of contagiousness),

ii) the infectious agent (whose virulence is variable, due for example to the mutations observed),

iii) the future virus host (in whom, the immunological state depends on multiple factors as age, comorbidities, cross immunity, etc.).

The long reaction time to mitigation or vaccination measures decided on the basis of the models makes it very difficult to adapt them to the new propagation conditions and considerably limits the operational character of the models, even if their short-term predictions are often of very good quality.

Among other possible approaches is the use of the Caputo fractional derivative [START_REF] Ogunrinde | Dynamic model of covid-19 and citizens reaction using fractional derivative[END_REF], which is particularly interesting. The Caputo fractional derivative of a function f equals

D q c (f (t)) = T 0 f n (y) (t -y) q+1-n dy
where the integers n and p verify n -1 < q < n.

The use of the fractional derivative in the models makes it possible to obtain simulations of the increase in the cumulative cases of infected individuals, which fits the data better than the classical derivative, particularly when the phenomenon of growth of the infected is damped (at the end of an epidemic wave), i.e., after the very early exponential phase where the classical approach presents a slightly lower error than the fractional one [START_REF] Ogunrinde | Dynamic model of covid-19 and citizens reaction using fractional derivative[END_REF].

Maximal Reproduction Number Estimation in COVID-

19 Outbreak

Maximal reproduction number model

Suppose that the observation time is continuous and that the new cases of infectious X(t) depend on the distribution of the quantities O(s) = R(s)/E(R), where the quantity

E(R) = R R(t)dt : X(t) = R R(s)X(t -s)ds = E(R) R X(t -s)O(s)ds (3.27)
(a) suppose in a first case that the curve O(s) is a Gaussian curve N (m, σ) then:

R(s) = R max exp - (s -m) 2 2σ 2 , (3.28)
where R max is the maximum of the daily reproduction numbers, and to show that the continuous equivalent of Equation (in Section 3.2) can be solved if X(t) is supposed to be exponential: X(t) = 100e B(t-t 0 ) = c e Bt , for all t ≥ t 0 , where c = 100e -Bt 0 , Equation (3.28) can be written as:

ce Bt = R R max exp - (s -m) 2 2σ 2 c e B(t-s) ds,
from which the following equations can be deduce:

1 = R max R exp - (s -m) 2 2σ 2 -Bs ds 1 = R max R exp - (s -m + Bσ 2 ) 2 2σ 2 -Bs + B 2 σ 2 /2 -Bm) ds 1 = R max σ(2π) 1/2 exp(B 2 σ 2 /2 -Bm) R exp - (s -m + Bσ 2 ) 2 2σ 2 /σ(2π) 1/2 ds Hence, 1 = R max σ(2π) 1/2 exp(B 2 σ 2 /2 -Bm) (3.29)
and if σ(4σ ≈ r, the duration of the contagiousness period), m ≈ 2σ and B(the transmission rate), then R max , the value at day m(the middle of the contagiousness period) of the maximal daily reproduction number. (b) if the curve of the daily reproduction numbers R(s) is the sum of two Gaussian-shaped curves, the first corresponding to the activation of the innate immunity and the second to the activation of the adaptive immunity, then:

R(s) = R max 1 exp(-(s -m 1 ) 2 /2σ 2 1 + R max 2 exp(-(s -m 2 ) 2 /2σ 2
in this case, Equation (3.28) can be written as:

c e Bt = R R max 1 exp(-(s -m 1 ) 2 /2σ 2 1 ) + R max 2 exp(-(s -m 2 ) 2 /2σ 2 2 ) c e B(t-s) ds (3.30)
Then,

1 = R max 1 exp(B 2 σ 2 1 /2 -Bm 1 + R max 2 exp(B 2 σ 2 2 /2 -Bm 2 ) (3.31)
formula from which the calculation of the value of a parameter in function of the value of others. If 4σ 1 + 4σ 2 can be estimated by the duration of the contagiousness period and if suppose that σ 1 is close to σ 2 , with m 1 = 2σ 1 and m 2 = 4σ 1 + 2σ 2 , then the only remaining unknown parameters are R max 1 and R max 2 . If these two maximal daily reproduction numbers are supposed to be close, the Equation (3.31) allows the final calculation of a unique R max , which could be compared to the one obtained by the discrete method of deconvolution presented in Section 3.2.

Application of the model

The method derived in Section 3.3 will first be applied to Cameroon's data concerning the COVID-19 outbreak from the Cameroon Situation Reports (weekly regional new cases and deaths [150]) and from the public database Worldometers (weekly national new cases and deaths [119]) and then to six other countries.

-From Equation (3.29), the maximal daily reproduction number R max can be estimated

1 = R max σ(2π) 1/2 exp(B 2 σ 2 /2 -Bm),
Then, from [START_REF] Nkamba | Modelling and prediction of the spread of covid-19 in cameroon and assessing the governmental measures (march-september 2020)[END_REF], σ = 1, m = 2, B = 0.4. Then, R max = 1/(2π) 1/2 exp(0.8 -0.08) ≈ 0.9, which corresponds to the case of a Gaussian shape for the daily reproduction numbers R ′ j s along the contagiousness period of length r ≈ 4σ to a basic reproduction number

R 0 = k=1,...,r R k ≈ 2.1.
-To calculate R max for six other countries using same procedure as above using the transmission rate of the first wave in 2020: Australia = 4.6214 × 10 -8 , Italy = 1.2292 × 10 -8 , Spain = 3.3182 × 10 -7 , United States = 3.1828 × 10 -7 , United Kingdom = 1.1526 × 10 -7 and Canada = 2.0731 × 10 -7 given in [START_REF] Rahman | Modelling the transmission dynamics of covid-19 in six high burden countries[END_REF], with σ = 1 and σ = 2, corresponding to a duration of the contagiousness period respectively 4 and 8 days. It is also possible to calculate R max 2 by using Equation (3.31) as presented in Table 3.3. The values of R max in Table 3.2 is used for R max 1 while still maintaining the values for the transmission rate and σ.

dS dt = β S 0 S(t)I(t) + dS(t), (3.32)
dI dt = β S 0 S(t)I(t) -νI(t),
S(t) = S 0 1 + e B(t+t 0 ) and I(t) = S 0 e B(t+t 0 ) 1 + e B(t+t 0 ) , (3.34)
with

S + I = S 0 . If I(0) = 1, then if t 0 = -log(S 0 -1)/B, S(0) = S 0 /1 + e Bt 0 ≈ S 0 -1.
If S 0 is large, then during the time lapse at which S(t) remains close to S 0 , if µS 0 = f, then Equation (3.33) becomes close to the linear system:

dS dt = (-BS 0 + ρ)I and dI dt = (BS 0 -ν -ρ)I, (3.35)
with the solution:

S(t) = (-BS 0 + ρ)e (BS 0 -ν-ρ)t (BS 0 -ν -ρ) and I(t) = e (BS 0 -ν-ρ)t , (3.36)
where BS 0 represents the basic reproduction number of the epidemic disease, usually denoted R 0 . The present models are in general used to simulate an epidemic wave, as an excursion in the phase plane of an ordinary two-dimensional differential system having some excitable ability, i.e., the existence in his phase portrait of large return trajectories after perturbation of its stable stationary state or of its parameters, these trajectories returning to either an endemic or an eradicated final state. The difference between these two asymptotic behaviors is that in endemic case, the final value of the infected, back to the stationary state is non-zero and, in eradicated case, this value is zero. These behaviors are illustrated by simulations of the Bernoulli model in the possible final states corresponding to epidemic (transients in Figure 3.21), eradication (end state in Figure 3.21 top) and endemic (low level end in Figure 3.21 middle and high level in Figure 3.21 bottom). Suppose that a parameter of the Bernoulli model like the transmission rate B is evolving in time due to various influences as geo-climatic factors (temperature, humidity, elevation, etc.), socio-demographic determinant (density and median age of the target population) and economic variables (GDP, Gini's index, inequality index, etc.), the actual curve of the new infected cases observed during the epidemic outbreak is in fact also driven by the slow dynamics of the seasonally varying or linearly growing parameters. Then, the observed dynamics of new infected cases results from the epidemic dynamics combined with slow evolution of these external factors on variation surfaces mixing both epidemic variables and parameters dynamics. The Figure 3.20 shows such a surface corresponding to the variations of the transmission rate B due to climate changes (transition between winter and summer times) or public health policies (quarantine, vaccination, etc.), and the actual (S, I) trajectory lies on the corresponding surface. The influence of these different factors on the variations of the trajectories (S, I) are difficult to take into account in a differential model of the type Equation (3.33), which is why Section 4.5 will treat the statistical approach providing a better understanding of the correlations between the exogenous determinants of the epidemic and the level of new cases of infected. Graphical representation of the inflection point P I of the curve of I(t) is given in Figure 3.

Mathematical formulation of the identification model

The problem can be formulated in a way that all parameters can be identified from the set Θ = {(S 0 , I 0 , a, β)} ⊂ (0, ∞) 4 if the reported daily infectious cases I ′ (t) is known for all times t ≥ 0. It should be noted that it is not sufficient alone to know the value of I ′ (t) at any time t ≥ 0, because the knowledge about the value of S 0 , the number of susceptible people in the population before the epidemic outbreak is also necessary to allow to obtain a precise information about the values of I 0 , a, β (see [START_REF] Demongeot | Si epidemic model applied to covid-19 data in mainland china[END_REF][START_REF] Ducrot | Identifying the number of unreported cases in sir epidemic models[END_REF]). If the solution of the Bernoulli Equation (3.32) with (d = ν = 0)(fertility and mortality rates are negligible):

I(t) = S 0 e β(t-a)
1 + e β(t-a) and S(t) = S 0 1 + e β(t-a) , where S 0 , β and a represent respectively the susceptible size at endemic start, transmission rate and endemic start time. For any time t, we have:

I(t) + S(t) = S 0 e β(t-a) 1 + e β(t-a) + 1 1 + e β(t-a) = S 0 e β(t-a) = I(t) S 0 -I(t)
If the epidemic wave starts at time 0, where I(0) = 1, then:

1 =
S 0 e -βa 1 + e -βa Then, S(0) = S 0 -1 and a is given by the following equation:

a = log(S 0 -1) β (3.37)

Derivation of the equation of the inflection point

From the Bernoulli's equation, where ν = 0, we have:

I ′ (t) = dI dt = β S 0 SI × S 0 1 + e β(t-a) × S 0 e β(t-a) 1 + e β(t-a) = βS 0 e β(t-a) (1 + e β(t-a) ) 2
The sufficient existence condition for a point of inflection of order 2 for I ′ (t) in the case that I(t) is 3-times continuously differentiable in a certain neighbourhood of a point t i , with I ′′′ (t i ) = 0, and I ′′ (t i ) ̸ = 0. Then, I ′ (t) has an inflection point (P I ) of order 2 at t i , and by differentiating I ′ (t) twice, then:

I ′′ (t) = d 2 I dt 2 = β 2 S 0 e β(t-a) -2e 2β(t-a) (1 + e β(t-a) ) 3 I ′′′ (t) = d 3 I dt 3 = β 3 S 0 e β(t-a) (1 -6e β(t-a) + 2e 2β(t-a) ) (1 + e β(t-a) ) 4
Then, the equation given by β and a from its lowest root

x i = 3 - √ 7 = e β(t i -a) = I(t i ) S 0 -I(t i ) (depending on I(t i ) is: 1 -6x i + 2x 2 i = 1 -6e β(t-a) + 2e 2β(t-a) = 0 (3.38)
If I(0) = 100, then from the same calculations, the values of β and a can be identified from Equation (3.37) and Equation (3.38).

Identification method and its application

If the value of the time t i is calculated from the first P I of an observed wave and if the value of S 0 is known, then it is easy to estimate the maximal reproduction number as I ′′ (t) and identify β and a from Equation (3.37) and Equation (3.38). The steps of identification method are as follows:

• From an interpolation of the curve of the daily reported new cases (see for example red curve in Figure 3.19) of a particular country epidemic wave, calculate the lowest t i of the wave at which I ′′′ (t) = 0,

• From the value of t i , estimate the maximal reproduction number as I ′′′ (t i ),

• From the knowledge of S(0), the number of susceptible people in the population at the start of a particular country epidemic wave, one can get S 0 ,

• From Equation (3.37) and Equation (3.38), β and a can be identified.

The identification method is first applied to Cameroon COVID-19 data and then applied to the world COVID-19 outbreak. To calculate the time t i , of the first inflection point P I of the second wave (which presents a shoulder and no maximum) of the new infectious cases X(t) for the entire world: this time corresponds to a second wave shoulder, in which the maximum of the new cases is not existing for this wave, hence cannot be used for identifying the epidemiologic parameters as in [START_REF] Rahman | Modelling the transmission dynamics of covid-19 in six high burden countries[END_REF][START_REF] Ducrot | Identifying the number of unreported cases in sir epidemic models[END_REF]. For that, the smoothing of the raw data is done for the weekly reported new infectious cases in Figure 3.23 by using a cubic spline and found t i = 4 (weeks). Then, from Equation (3.37) and Equation (3.38): a = log(S 0 -1)/β and 1 -6e β(t i -a) + 2e 2β(t i -a) = 0, with S 0 = 7, 000, 000, 000 and t i = 4. Then a = 10/β and 1-6e 4β-10 +2e 8β-20 = 0, from which β = 2.575 and a = 3.883, which is consistent with the values of the effective reproduction number calculated in [START_REF] Yu | Assessment of basic reproductive number for covid-19 at global level[END_REF].

To conclude this section, methods developed in Section 3.2, Section 3.3 and Section 3.

i + 1, v i + α i = 1 -µ i ≤ 1, ∀ i = 1, • • • , 4
, where µ i is the death rate at age i. and v j . The probabilities P i is on the interval 0

≤ P i ≤ 1∀ i = 1, • • • , 4.
We can also say that (v i + α i ) is the general probability of survival. If the two first only being fertile with non-zero fertility rates f 1 and f 2 , and if a disease like an epidemic outbreak concerning all the age classes occur adding to the natural mortality rate of each of the three first classes a fatality rate, the sum of the natural and disease dependent mortality, denoted 1 -α i = µ i . We present in Figure 3.29 the flow chart illustrating the epidemiology ageing model. The Usher matrix can be written as the following epidemic matrix E * :

E * = (e ij ) =     f 1 + v 1 f 2 0 0 α 1 v 2 0 0 0 α 2 v 3 0 0 0 α 3 v 4    
The characteristic polynomial of the matrix E * is given by:

P (λ) = (f 1 + v 1 -λ)(v 2 -λ)(v 3 -λ)(v 4 -λ) -f 2 α 1 (v 3 -λ)(v 4 -λ) = [λ 2 -λ(f 1 + v 1 + v 2 ) + v 2 (f 1 + v 1 ) -f 2 α 1 ](v 3 -λ)(v 4 -λ)
Then, we can calculate explicitly the values of the spectrum of E * :

v 3 , v 4 , (f 1 + v 1 + v 2 ) ± [(f 1 + v 1 + v 2 ) 2 -4(v 2 (f 1 + v 1 ) -f 2 α 1 )] 1/2 /2
The population growth is stable if λ = 1, increases if λ > 1 and decreases if λ < 1.

The proportion of sensitivity s ij of λ (the dominant eigenvalue of E * ) to a variation of the general element e ij of E * , is given by:

s ij = e ij λ ( V * i V v j k V * k V v k )
where V v is the eigenvector corresponding to the eigenvalue v and V * is the eigenvector corresponding to λ. Then, the total sensitivity equals 1:

i,j s ij = i,j e ij V * i V v j λV * .V v = λ j V * j V v j λV * .V v = 1

Application to COVID-19 outbreak in some countries

We first start by describing the model formulated in terms of normal ageing in Cameroon without considering an outbreak of a viral disease in the population and then apply the model formulated to COVID-19 outbreak in Cameroon, France and Ireland. In the case of Ireland, we extend the age class to five age classes since this country has more data relating to more age groups and most especially the children and teenagers.

Normal ageing in Cameroon

From [START_REF] Tabutin | The demography of sub-saharan africa in the 21st century. transformations since 2000, outlook to 2050[END_REF]156,157,158,[START_REF] Ins | Enquête démographique et de Santé Cameroun[END_REF]160], all the values of parameters like fertility and mortality rates needed to complete the Usher matrix for Cameroon can be found. Table 3.5 gives the list of all the parameters with their signification and Table 3.7 summarizes their values. Concerning the mortality, the death rate for 1,000 in 2019 is equal to 9.059 [158] and life expectancy at birth is 59.292 years [START_REF] Ins | Enquête démographique et de Santé Cameroun[END_REF]. Values of coefficients f 1 , f 2 and f 3 have been calculated in Table 3.7, taking in demographic databases the age-specific fertility rates when available, estimating this rate for the age sub-classes [START_REF] Guttmann | Performance map of a cluster detection test using extended power[END_REF][START_REF] Tariq | Real-time monitoring the transmission potential of covid-19 in singapore, march 2020[END_REF] and [START_REF] Webb | A covid-19 epidemic model predicting the effectiveness of vaccination in the us[END_REF][START_REF] Gavalas | A Lattice Model for COVID-19 Epidemic[END_REF], considering that the last age class (≥ 60) has no fecundity and weighting all these age-specific fertility rates by the proportions of women in each age class. Then, considering the values of normal fecundity, ageing and mortality (neglecting infantile mortality), we can calculate the Usher matrix (see Table 3 The dominant eigenvalue of U is λ = 1.026. It is equal to the exponential of the Malthusian parameter of the Cameroon population growth, the real value given in [START_REF] Tabutin | The demography of sub-saharan africa in the 21st century. transformations since 2000, outlook to 2050[END_REF] being equal to 1.0258 in 2020. We suppose that the fecundity does not change during the epidemy, and we will show the influence of the epidemy during the period of virulence in the host for the infected population. For the sake of simplicity, we suppose that all the infected individuals have the same characteristic of ageing and that the increase of the Malthusian parameter log λ, where λ is the dominant eigenvalue of E, concerns a constant percentage of the whole population (equal to the small proportion 2 × 10 -4 after [START_REF]Rapport de situation COVID-19 n o 84[END_REF]). By neglecting the effect of the viral disease on the fecundity and by taking into account the specific ageing and in worse cases the mortality due to the disease for the fraction of the population affected by COVID-19 [119,[START_REF]Rapport de situation COVID-19 n o 84[END_REF], with 1418 deaths during the three waves after the first one, between the 1st February 2021 and the 31 January 2022, the matrix E * becomes with the same reasoning that for the normal population: The coherence between the orders of magnitude of the calculated and observed death numbers in France during the fifth wave constitutes a semi-quantitative validation of the mathematical modelling approach at the population level.

E * =     0.

Application to COVID-19 outbreak in Ireland

In Ireland, we will consider five age classes (from estimated data in 2020 [96]) and used the same method for calculating and estimating parameters used in Cameroon, which is presented in Table 3.7, while using raw data from Ireland. Figure 3.31 shows a notable difference of incidence rate due to both age and sex.

If we neglect sex influence, the epidemiologic matrix E corresponding to the 5 age classes is:

E * =       0

Role of comorbidities

Let us consider now the distribution of age classes of the subpopulation of infected patients presenting the most frequent comorbidity, i.e., cardiovascular pathologies (cf.

Sensitivity analysis and lifespan loss due to a viral disease

Let consider now as toy example the following epidemiologic matrix of the same form as the previous ones:

E s =    
0.93 0.07 0 0 0.035 v 2 0 0 0 0.03 0.7 0 0 0 0.03 0.7

   
The dominant eigenvalue of E s is equal to 0.9847, if v 2 = 0.94, and 0.9467, if v 2 = 0.8. Using the method proposed in [START_REF] Caswell | A general formula for the sensitivity of population growth rate to changes in life history parameters[END_REF], Figure 3.34(a) shows dominant eigenvalues while varying value of probability v 2 of remaining in second age class [START_REF]World health organization[END_REF][START_REF] Altakarli | China's response to the covid-19 outbreak: A model for epidemic preparedness and management[END_REF][START_REF] Mackolil | Logistic growth and sir modelling of coronavirus disease (covid-19) outbreak in india: Models based on real-time data[END_REF][START_REF] Khanday | Mathematical analysis on the dynamics of covid-19 in india using sir epidemic model[END_REF][START_REF] Yaghoby | A new nonlinear mathematical model for forecasting the waves of the COVID-19 epidemic[END_REF][START_REF] Kaxiras | Multiple epidemic wave model of the covid-19 pandemic: Modeling study[END_REF][START_REF] Meiksin | Using the seir model to constrain the role of contaminated fomites in spreading an epidemic: An application to covid-19 in the uk[END_REF][START_REF] Cámara | An epidemic model for economical impact predicting and spatiotemporal spreading of COVID-19[END_REF][START_REF] Paunovic | Mathematical modeling and statistical analysis in post -covid-19 tourism recovery[END_REF][START_REF] Duan | Simulations of the covid-19 epidemic in nigeria using sir model[END_REF][START_REF] Padmapriya | Fuzzy fractional mathematical model of covid-19 epidemic[END_REF][START_REF] Kozioł | Fractional-order sir epidemic model for transmission prediction of covid-19 disease[END_REF][START_REF] Deforche | An age-structured epidemiological model of the Belgian COVID-19 epidemic[END_REF][START_REF] Balabdaoui | Age-stratified discrete compartment model of the covid-19 epidemic with application to switzerland[END_REF][START_REF] Verrelli | Two-age-structured covid-19 epidemic model: Estimation of virulence parameters to interpret effects of national and regional feedback interventions and vaccination[END_REF][START_REF] Liu | Focus on the mental health of pediatric medical workers in china after the covid-19 epidemic[END_REF][START_REF] Li | Epidemic risk perception, perceived stress, and mental health during covid-19 pandemic: A moderated mediating model[END_REF][START_REF] Rihan | Stochastic sirc epidemic model with time-delay for covid-19[END_REF][START_REF] Kang | Epidemic spreading and global stability of a new sis model with delay on heterogeneous networks[END_REF][START_REF] Dell'anna | Solvable delay model for epidemic spreading: the case of covid-19 in italy[END_REF]: when v 2 is greater than 0.96, the dominant eigenvalue is greater than 1 and the population of observed COVID-19 cases exhibits an exponential growth. The second age class is the only class having a large effect on the population growth, because others have no birth rate. Figure 3.34(b) shows the proportion of sensitivity of λ to various changes in ageing, death and fecundity rates. Figure 3.34(b) confirms that the second age class which contributes the most to population growth is also that which is the most sensitive to its fecundity rate f 2 . In Figure 3.34(b), the b j curve shows also that the probability of remaining in the third and fourth age class does not have effect on λ value while the first age class has the most effect which aligns with our assumption in the equation describing the epidemic ageing model. Biologically, it means that changes of the ageing parameter b in two last age classes do not affect the growth rate of the population. We are also able to deduce from the v j curve that the largest effect on the sensitivity of λ values is due to the first age class while the second age class has an effect reduced by half and others have no effect, which means that changes in survivability of the first age class is of great importance for the growth of the population. Then, we added one more age class in this last example. On the other hand, to refine the coefficients of the epidemiological matrix E * , it is necessary to better understand the specific ageing processes due to the SARS-CoV-2 virus, in particular that which affects anti-apoptosis proteins such as Gaf1. Indeed, these processes affect the mortality specific to COVID-19, hence the coefficient of the matrix E * which quantifies the transition from an age class i to the following age class i + 1. In what follows, we will seek to lay the foundations for future study of these specific ageing processes.

Cells targeted by SARS-CoV-2 and its implications in cell lifespan loss

In SARS-CoV-2, the main target cells are the alveoli cells of the lungs, the pneumocytes. The turn-over of the 300-700 million of human alveoli [START_REF] Ochs | The number of alveoli in the human lung[END_REF] is about 4 months [START_REF] Wright | Macro and micro structure of the lung[END_REF]. The Type I pneumocytes constitute the major part (95%) of the alveolar surface: they are large (approximately 200µm) and thin (less than 0.2µm) cells, so their barrier to drug transport is at least one order of magnitude lower than typical mucosal or epithelial membranes [START_REF] Colombo | Biological in vitro models for absorption by nonoral routes[END_REF].

Then, each type I pneumocyte covers in mean 5, 000µm 2 at the alveolar surface [START_REF] Ochs | The number of alveoli in the human lung[END_REF]. The diameter of an alveolus is between 200 and 500µm [START_REF] Wiebe | Human lung volume, alveolar surface area, and capillary length[END_REF]. If we retain the value of 200µm, their surface is about 280000µm 2 , hence we have about 6 pneumocytes/alveolus. For the whole pulmonary tissue, we have then 18 × 10 8 pneumocytes, from which natural loss each day is about 18 × 10 8 /120 = 1.5 × 10 6 cells. Because the SARS CoV-2 is infecting at most 10 7 cells each day [START_REF] Sender | The total number and mass of sars-cov-2 virions[END_REF], the COVID-19 viral disease causes an accelerated ageing in days of the pulmonary tissue, equal to 1.5 times the duration of the acute phase of virulence in the host.

Another origin of specific ageing comes from the hybridization of mRNA of proteins involved in vital metabolisms. In search of hybridization germs, we have inspected viral RNA sequences from different databases [171, 172, 173], using the classic BLAST software. For example, we have already noticed in a previous work [START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF] that miR 129-5p was a known inhibitor of the biosynthesis of gamma-globin 2, a subunit of human fetal hemoglobin, replaced in adults by beta-globin, also dysregulated in some blood diseases, like the other subunit alpha-globin, by several miRs, including miR 451a [START_REF] Xu | Regulation of gene expression by mir-144/451 during mouse erythropoiesis[END_REF][START_REF] Wang | mir-144/451 in hematopoiesis and beyond[END_REF][START_REF] Lai | Genome-wide analysis of aberrantly expressed lncrnas and mirnas with associated co-expression and cerna networks in β-thalassemia and hereditary persistence of fetal hemoglobin[END_REF]. The search for hybridization germs having the same inhibition potential as that of miRs 129-5p and 451a has led to the identification of two subsequences of RNA-dependent RNA polymerase and S genes of SARS-CoV-2. Figure 3.35 shows these subsequences identified as inhibitors of the biosynthesis of human beta-globin. Figure 3.36 shows, for its part, two hybridizations, by subsequences of the gene of the S protein of the Omicron variant of SARS-CoV-2, parts of human interferon mRNA and anti-aging human Gaf1 protein mRNA by subsequences from S protein gene of Omicron variant of SARS-CoV-2 [START_REF] Rodríguez-López | The gata transcription factor gaf1 represses trnas, inhibits growth, and extends chronological lifespan downstream of fission yeast torc1[END_REF]. The primary targets of many viruses are the cells of the most sensitive tissue developing the viral disease, such as in the case of SARS-CoV-2, the cells presenting the ACE2 (for Angiotensin-Converting Enzyme 2) receptor, or in the case of HIV, the cells most infected by the virus, i.e., immune cells CD4+ T cells and macrophages, as well as cells of microglia. In both cases, the viral disease causes a pathologic ageing, even if the patient survives (the death being often due to an opportunistic superinfection). Cytoplasmic nucleases (e.g., RNases) in the cells targeted by the virus are indeed enzymes capable of cleaving the phosphodiester bonds of viral RNA, and the viral genome fragments thus obtained can subsequently form complexes with mRNAs and/or proteins in the host cell, preventing ribosomal translation of proteins, just as miRs do. When targeted proteins are vital, pathogenicity may be greater than that due to viral replication. RNA viruses reproduce their capsid proteins in host cells and duplicate their genome leaving behind RNA fragments, which can behave like miRs in the host genome, if they bind to Argonaut proteins facilitating hybridization to mRNA and then its hydrolysis [START_REF] De Boer | Quantifying t lymphocyte turnover[END_REF][START_REF] Changeux | A nicotinic hypothesis for covid-19 with preventive and therapeutic implications[END_REF][START_REF] Nishiura | Relative reproduction number of sars-cov-2 omicron (b.1.1.529) compared with delta variant in south africa[END_REF][START_REF] Demongeot | Micro-rnas: viral genome and robustness of gene expression in the host[END_REF][START_REF] Bandiera | Mitomirs delineating the intracellular localization of micrornas at mitochondria[END_REF][START_REF] Demongeot | Mitomirs, chloromirs and modelling of the microrna inhibition[END_REF][START_REF] Pfeffer | Identification of micrornas of the herpesvirus family[END_REF][START_REF] Kincaid | Rna virus microrna that mimics a b-cell oncomir[END_REF][START_REF] Cullen | Viruses and micrornas[END_REF].

In HIV, the main target cells are the T cells of the immune system. HIV can infect up to 2 billion of T cells per day, while no more than 2% (from in mean 500 billion in whole blood of an individual) can be reconstituted per day (by division), that is, about one billion of cells [START_REF] Dykxhoorn | Determinants of specific rna interference-mediated silencing of human β-globin alleles differing by a single nucleotide polymorphism[END_REF]. Then, the HIV virus causes an accelerated aging of the immune system, equal to 2 times the duration of the phase of virulence in the host, which can be chronic in absence of tri-therapy. If we consider the organ level, the high rate of death in COVID-19 patients with cardiac or pulmonary chronic comorbidities (Figure 3.33) indicates that the corresponding organs (heart and lung, respectively) struggle to compensate for the loss of cells destroyed by the SARS-CoV-2 virus, resulting sometimes in a pathologic ageing followed by a failure of these critical organs (critical, because their collapse causes the death of the whole organism of the patient). A source of supplementary pathologic ageing is the inhibition of the biosynthesis of the protein Gaf1, involved in the processes preventing the cell apoptosis, when the viral RNA contains subsequences capable, if it is fragmented by nucleases of the host, to hybridize the mRNA of Gaf1, protein necessary for survive because deeply involved in anti-apoptosis processes [START_REF] Rodríguez-López | The gata transcription factor gaf1 represses trnas, inhibits growth, and extends chronological lifespan downstream of fission yeast torc1[END_REF].

A first example of that is given by fragments of the SARS-CoV-2 virus in Figure 3.36. A second example of the existence of accumulation of small RNA fragments exists in Sclerotinia sclerotiorum infected with the SsHV2-L virus. These virus-derived small RNA fragments measure about 22 nt, the same length as the miRs, suggesting a cleavage by a Dicer-like protein [START_REF] Yekta | Microrna-directed cleavage of <i>hoxb8</i> mrna[END_REF]. Regarding SARS-CoV-2, such an influence on protein translation has already been described [START_REF] Brennecke | Principles of microrna-target recognition[END_REF][START_REF] Li | H5n1 influenza virus-specific mirna-like small rna increases cytokine production and mouse mortality via targeting poly(rc)-binding protein 2[END_REF][START_REF] Marzano | Transcriptional and small rna responses of the white mold fungus sclerotinia sclerotiorum to infection by a virulence-attenuating hypovirus[END_REF][START_REF] Yu | Annotation of mirnas in covid-19 coronavirus[END_REF][START_REF] Mirzaei | The emerging role of micrornas in the severe acute respiratory syndrome coronavirus 2 (sars-cov-2) infection[END_REF][START_REF] Demirci | Computational analysis of microrna-mediated interactions in sars-cov-2 infection[END_REF][START_REF] Abedi | Micrornas and sarscov-2 life cycle, pathogenesis, and mutations: biomarkers or therapeutic agents?[END_REF][START_REF] Gasparello | Tackling the covid-19 "cytokine storm" with microrna mimics directly targeting the 3'utr of pro-inflammatory mrnas[END_REF][START_REF] Chauhan | Covid-19: fighting the invisible enemy with micrornas[END_REF][START_REF] Zhang | Cellular nanosponges inhibit sars-cov-2 infectivity[END_REF][START_REF] Nersisyan | Potential role of cellular mirnas in coronavirus-host interplay[END_REF][START_REF] Satyam | mirnas in sars-cov 2: A spoke in the wheel of pathogenesis[END_REF][START_REF] Khan | Epigenetic regulator mirna pattern differences among sars-cov, sars-cov-2, and sars-cov-2 world-wide isolates delineated the mystery behind the epic pathogenicity and distinct clinical characteristics of pandemic covid-19[END_REF], causing observed effects on the concentration of certain proteins, such as a dramatic decrease in hemoglobin as in other blood diseases [START_REF] Xu | Regulation of gene expression by mir-144/451 during mouse erythropoiesis[END_REF][START_REF] Wang | mir-144/451 in hematopoiesis and beyond[END_REF][START_REF] Lai | Genome-wide analysis of aberrantly expressed lncrnas and mirnas with associated co-expression and cerna networks in β-thalassemia and hereditary persistence of fetal hemoglobin[END_REF]. If we assume that these short RNA subsequences from the genes of the SARS-CoV-2 virus can bind to Argonaut proteins and hybridize the mRNA of key human proteins involved in important metabolisms such as oxygen metabolism, it follows that mutations and/or deletions observed in the SARS-CoV-2 genome (such as those which appeared in the United Kingdom, South Africa, France or spontaneously in vitro [START_REF] Nishiura | Relative reproduction number of sars-cov-2 omicron (b.1.1.529) compared with delta variant in south africa[END_REF][START_REF] Khan | Epigenetic regulator mirna pattern differences among sars-cov, sars-cov-2, and sars-cov-2 world-wide isolates delineated the mystery behind the epic pathogenicity and distinct clinical characteristics of pandemic covid-19[END_REF]) reinforce the possible existence of these RNA fragments, capable of hybridizing, for example, the mRNA of hemoglobin subunits (Figure 3.35 and Figure 3.36), such as beta-globin [START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF], impacting oxygen transport in infected patients and of Gaf1 protecting against apoptosis [START_REF] Rodríguez-López | The gata transcription factor gaf1 represses trnas, inhibits growth, and extends chronological lifespan downstream of fission yeast torc1[END_REF]. This mechanism can be marginal but has to be considered in future studies on ageing due to viral infections, which could include a part dedicated to prevention and therapy [START_REF] Bozgeyik | Therapeutic potential of mirnas targeting sars-cov-2 host cell receptor ace2[END_REF] involving circular RNAs, which serve as "sponges" or "decoys" to small RNA fragments, to prevent them from hybridizing certain proteins vital to the body [START_REF] Bernardo | Generation of microrna-34 sponges and tough decoys for the heart: Developments and challenges[END_REF].

Cell ageing due to the virulence: a discrete approach

To model the virus impact on the cell lifetime, causing cell death or disrupting the cell cycle, (when it can be documented), the Hahn model can be used to quantify the loss of function of an organ targeted by the virus. If a certain proportion of cells are destroyed, the virus causes critical organ failure, resulting in the patient's death. Taking the heart as an example, the inhibition of Gaf1 by SARS-CoV-2 (as already proposed in [START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF]) could activate apoptosis by shortening the cell cycle [START_REF] Rodríguez-López | The gata transcription factor gaf1 represses trnas, inhibits growth, and extends chronological lifespan downstream of fission yeast torc1[END_REF]. Then, using the Hahn model, we could quantify the viral influence on the cell replacement rate in the organ, allowing us to refine the specific mortality rate due to the virus by organ already weakened due to comorbidities.

Cell population growth has been already modeled by Hahn [START_REF] Hahn | A formalism describing the kinetics of some mammalian cell populations[END_REF], using a discrete dynamic ruled by the matrix equation:

u(t) = Au(t -1),
with Hahn matrix defined by:

A = (a ij ) =          α 1 0 0 • • • 2Qγ n-1 2Qβ n β 1 α 2 0 • • • 0 2Qγ n γ 1 β 1 α 3 • • • 0 0 . . . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • β n-1 α n         
where u i (t) represents the size of the cell population in state i of the cell cycle at time t, Q is the mitotic abortive coefficient (0 < Q ≤ 1), α i ( resp. β i and γ i ) the probability to remain in state i (respectively to go to state (i + 1) and (i + 2)) between times t and t + 1, and µ i mortality rate with:

α i + β i + γ i = 1 -µ i ≤ 1, for all i = 1, . . . , n.
As with the Usher model, the L 2 dynamical stability modulus of the invariant measure w of A is equal to e -|λ-λ , | , where the dominant and sub-dominant eigenvalues of the Hahn matrix if matrix coefficients do not depend on i equal:

λ = α + (2Q) 1/n β + (2Q) 2/n γ and λ , = α + (2Qϕ) 1/n β + (2Qϕ) 2/n γ
where ϕ is the second largest absolute value of the n th root of 1. The Kullback-Leibler stability modulus is defined as the cell evolutionary entropy H defined by [START_REF] Demongeot | Stability, complexity and robustness in population dynamics[END_REF][START_REF] Demongeot | Dynamical confinement in neural networks and cell cycle[END_REF]:

H = -α/λ log(α/λ) -(2Q) 1/n β/λ log((2Q) 1/n β/λ) -(2Q) 2/n γ/λ log((2Q) 2/n γ/λ)
In certain cases, cell therapies could cause a rejuvenation of the tissue damaged by the virus, and the value of α(a) could in this case increase and change the value of H.

Cell ageing due to the virulence: a continuous approach

Several works have introduced continuous models with the demographic variable age in order to differentiate the reactions to the virulence of the different age groups of a population, in particular because of an immune response which gradually decreases with age [START_REF] Chu | Hopf bifurcation in a size-structured population dynamic model with random growth[END_REF][START_REF] Tang | De-escalation by reversing the escalation with a stronger synergistic package of contact tracing, quarantine, isolation and personal protection: Feasibility of preventing a covid-19 rebound in ontario, canada, as a case study[END_REF][START_REF] Griette | Unreported cases for age dependent covid-19 outbreak in japan[END_REF].

We consider also as an ultimate perspective the building of a global continuous model integrating both the cell cycle of the organ cells and the age of the patient. For that purpose, we recall that the continuous equivalent of Usher model derives from the classical von Foerster equation [START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF], where u(a,t,s) is the cell concentration at age a, time t and space s:

∂u ∂a + ∂u ∂t = -µ(a)u
By adding a second order term ∂ 2 u ∂a 2 taking into account the existence of the parameter γ in the Hahn matrix, the dynamical behavior of this continuous equation is the same as for the discrete Hahn model. The main interest of the continuous formulation is the possibility to add a diffusion term [START_REF] Dilao | Diffusion in the mckendrick-von foerster equation[END_REF], if cells are moving to repair an organ damaged by the virus:

∂u ∂a + ∂u ∂t -(a) ∂ 2 u ∂a 2 -σ(a)∆u = -µ(a)u ⇔ ∂u ∂a + ∂u ∂t -□u-= -µ(a)u,
where □u = ∂ 2 u ∂a 2 -σ(a)∆u can be considered as a Dalembertian operator taking into account accelerated ageing and diffusion in space. Then, we obtain the most general continuous operator including pathological cell ageing and cell motion represented by what J. Besson and J.P. Caubet [START_REF] Caubet | Relativistic brownian motion[END_REF][START_REF] Porte | Passion des Formes. A René Thom[END_REF] called the charge of Sinbad the Porter (the Dalembertian symbol □) and the sail of Sinbad the Sailor (the Laplacian symbol ∆).

Age Dependent Epidemic Modeling of COVID-19 Outbreak

The purpose of this section is to improve age dependent modeling which was presented in the previous section by using a deterministic approach with a system of ordinary differential equations and we have been able to also show in the previous section the influence of the median age of a country on the incidence of COVID-19 which has been highlighted, in particular through the appearance of many asymptomatic cases in the age groups below 50.For example, on the site of Johns Hopkins University dedicated to COVID- 19 [214], data clearly shows this influence on the case fatality rate corresponding to the cumulative deaths recorded 5 months after the beginning of the outbreak (12 May 2020) vs. the median age of many countries in 2017 (Figure 3.37). This first observation has been confirmed by studies in France (Figure 3.40), and we will confirm in this section that the age of the patients suffering from COVID-19 is a good predictor of severity. The data used in this section is from public epidemiologic and demographic databases [157, 215] and the reference methods are both from classical demographic (such as Leslie) and epidemiologic (such as Ross and Kermack-McKendrick) models. In Figure 3.38 and Figure 3.39, we present visualizations for pandemic dynamics in different age groups and sexes for Kuwait and Cameroon, respectively, to support the motivation for this section that pandemic evolution and severity are related to age classes, and thus modeling is important as various researchers and health experts are investigating SAR-CoV-2 mutations. The countries under consideration have a higher proportion of young people, while the elderly have a lower proportion. Only 2% of the total population in Kuwait is over 65 and vulnerable to the pandemic, whereas 20.8% of the population in France is over 65 and vulnerable to the pandemic, and 2.7% of the population in Cameroon is over 65 and vulnerable to the pandemic. This severity among different age groups increased COVID-19 incidence at the start of the pandemic in terms of hospitalization cases, intensive care unit (ICU) cases, prompting many countries to quickly implement intervention measures such as increasing the number of beds and ventilators in their various hospitals and setting up new isolation centers in order to mitigate its spread and save lives, particularly the elderly, who were disproportionately among those who were highly susceptible.

Related Work

The heterogeneity in the age of the populations studied here (Kuwait, France, and Cameroon) leads to considering a model in age groups with specific susceptibilities for each age class, which makes the prediction problem of the new infectious growth more difficult. Basically, there are three age groups of interest in the COVID-19 outbreak, which tween demographic parameters such as age pyramid, fertility and mortality, and epidemiological parameters such as the basic reproduction number R0 and vaccination rate.

In subsequent sections, we present (i) in Section 2, the materials and methods, (ii) in Section 3, the formulation of a new age-dependent dynamical model for the spread of COVID-19, (ii) in Section 4, we provide some mathematical analyses of the model, (iii) in Section 5, we carry out numerical simulations for some COVID-19 parameters from Kuwait, France, and Cameroon; lastly (iv), in Sections 6 and 7, we provide a discussion and some conclusions.

Material and Methods

Material

The data material is from public epidemiologic and demographic databases [START_REF] Khanday | Mathematical analysis on the dynamics of covid-19 in india using sir epidemic model[END_REF][START_REF] Yaghoby | A new nonlinear mathematical model for forecasting the waves of the COVID-19 epidemic[END_REF] and the reference methods are both from classical demographic (such as Leslie) and epidemiologic (such as Ross and Kermack-McKendrick) models. In Figures 3 and4, we present visualizations for pandemic dynamics in different age groups and sexes for Kuwait and Cameroon, respectively, to support the motivation for this article that pandemic evolution and severity are related to age classes, and thus modeling is important as various researchers and health experts are investigating SAR-CoV-2 mutations. The countries under consideration have a higher proportion of young people, while the elderly have a lower proportion. Only 2% of the total population in Kuwait is over 65 and vulnerable to the pandemic, whereas 20.8% of the population in France is over 65 and vulnerable to the pandemic, and 2.7% of the population in Cameroon is over 65 and vulnerable to the pandemic.

Leslie Model

The first population growth model was proposed by Leslie in 1945 using the age pyramid vector p t p t , ,…, , where p t represents the size of the age class j at time t, with j ranging from the birth age 1 to the maximal death age m, whose discrete dynamics This severity among different age groups increased COVID-19 incidence at the start of the pandemic in terms of hospitalization cases, intensive care unit (ICU) cases, prompting many countries to quickly implement intervention measures such as increasing the number of beds and ventilators in their various hospitals and setting up new isolation centers in order to mitigate its spread and save lives, particularly the elderly, who were disproportionately among those who were highly susceptible.

Related Work

The heterogeneity in the age of the populations studied here (Kuwait, France, and Cameroon) leads to considering a model in age groups with specific susceptibilities for each age class, which makes the prediction problem of the new infectious growth more difficult. Basically, there are three age groups of interest in the COVID-19 outbreak, which are, respectively, 0-19 years, 20-64 years, and >64 years, but here we only consider two We propose a model which is an improvement of the Ross and Kermack-McKendrick model by trying to compensate some deficiencies in this model. We will first introduce two age classes to account for adults and the elderly in the population and then take account of vaccination before applying our model to some countries chosen as examples.

New SIGR model formulation

The The description above can be illustrated by the following set of non-linear differential Equation (3.39), while the graphical representation of the model is given in Figure 3.41:

dS 1 dt = β 11 (S 1 + G 1 + R 1 ) + β 21 (S 2 + G 2 + R 2 ) -a S 1 + θ G 1 + θ R 1 +φ 1 (I 1 + I 2 ))S 1 dS 2 dt = a S 1 S 1 -µ 2 S S 2 -θ G 2 + θ R 2 +φ 2 (I 1 + I 2 ))S2 + β 22 (G 2 + R 2 ) dI 1 dt = φ 1 (I 1 + I 2 ) S 1 -(a I 1 + η 1 )I 1 dI 2 dt = φ 2 (I 1 + I 2 ) S 2 + a I 1 I 1 -η 2 I 2 -µ 2 I I 2 dG 1 dt = η 1 I 1 + θ G 1 S 1 -a G 1 + γ 1 + ϵ 1 )G 1 dG 2 dt = η 2 I 2 + a G 1 G 1 + θ G 2 S 2 -γ 2 + µ 2 G + ϵ 2 + β 22 )G 2 dR 1 dt = γ 1 G 1 + θ R 1 S 1 -a R 1 R 1 dR 2 dt = γ 2 G 2 + θ R 2 S 2 + a R 1 R 1 -(µ 2 R + β 22 )R 2 (3.39) with S 1 (t) ≥ 0, S 2 (t) ≥ 0, I 1 (t) ≥ 0, I 2 (t) ≥ 0, G 1 (t) ≥ 0, G 2 (t) ≥ 0, R 1 (t) ≥ 0, R 2 (t) ≥ 0.

Positivity and boundedness of the solution

Lemma 1. Let the initial conditions be given as follows:

{S 1 (0) , S 2 (0) , I 1 (0) , I 2 (0) , G 1 (0) , G 2 (0) , R 1 (0) , R 2 (0) ≥ 0}
, then solutions of the system of equations (3.39) are positive for all t > 0.

Proof. From the first equation in the model Equation (3.39), we obtain:

dS 1 /dt = β 11 (S 1 + G 1 + R 1 ) + β 21 (S 2 + G 2 + R 2 ) -a S 1 + θ G 1 + θ R 1 +φ 1 (I 1 + I 2 ))S 1 , Hence, dS 1 /dt ≥ β 11 -a S 1 -θ G 1 -θ R 1 )S 1
. By using the separating variable method and then, integrating, we obtain:

dS 1 S 1 ≥ β 11 -a S 1 -θ G 1 -θ R 1 dt ln S 1 ≥ β 11 -a S 1 -θ G 1 -θ R 1 t
+ k Finally, by writing S 1 (0) = e k , we have:

S 1 (t) ≥ S 1 (0) e (β 11 -a S 1 -θ G 1 -θ R 1 )t ≥ 0.
By applying the same process to other equations in (1), we have:

S 2 (t) ≥ S 2 (0) e (a S 1 -θ G 2 -θ R 2 -µ 2 S )t ≥ 0 I 1 (t) ≥ I 1 (0) e -(a I 1 +η 1 )t ≥ 0 I 2 (t) ≥ I 2 (0) e -(η 2 I 2 + µ 2 I )t ≥ 0 G 1 (t) ≥ G 1 (0) e -(a G 1 +γ 1 +ϵ 1 )t ≥ 0 G 2 (t) ≥ G 2 (0) e -(γ 2 + µ 2 G +ϵ 2 )t ≥ 0 R 1 (t) ≥ R 1 (0) e -a R 1 t ≥ 0 R 2 (t) ≥ R 2 (0) e -µ 2 R t ≥ 0
Then, the solutions of the system of equations (3.39)

{S 1 (t) , S 2 (t) , I 1 (t) , I 2 (t) , G 1 (t) , G 2 (t) , R 1 (t) , R 2 (t)}
are positive for all t > 0. □ Also, Let denote by S the total size of all individuals:

S(t) = S 1 (t) + S 2 (t) + I 1 (t) + I 2 (t) + G 1 (t) + G 2 (t) + R 1 (t) + R 2 (t)
Then, by adding all the model Equation (3.39), we have:

dS dt = β 11 (S 1 + G 1 + R 1 ) + β 21 (S 2 + G 2 + R 2 ) -(µ 2 S S 2 + µ 2 I I 2 + µ 2 G G 2 + ϵG 2 + µ 2 R R 2 Let us denote µ = inf {µ S 2 , µ I 2 , µ G 2 + ϵ, µ R 2 }
. By neglecting the fecundity rate of the young class and if β = β 21 ≤ µ, we have:

dS dt ≤ β(S 2 + I 2 + G 2 + R 2 ) --µ(S 2 + I 2 + G 2 + R 2 ) ≤ 0,
Then, we can conclude that the total size S is bounded, which implies the boundedness of the partial sizes

S 1 (t) , S 2 (t) , I 1 (t) , I 2 (t) , G 1 (t) , G 2 (t) , R 1 (t) , R 2 (t).

Disease-Free (Eradication) equilibrium and stability of the endemic state

Setting the right hand side of the equations in the model Equation (3.39) to zero, i.e.,

dS 1 dt = dS 2 dt = dI 1 dt = dI 2 dt = dG 1 dt = dG 2 dt = dR 1 dt = dR 2 dt = 0
and supposing that all infectious class sizes are equal to zero which means that there is no disease (eradication) in the studied population, thus the disease-free equilibrium state is:

(S 1 * , S 2 * , I 1 * , I 2 * G 1 * , G 2 * , R 1 * , R 2 * ) = β 21 S 2 a S 1 + θ G 1 + θ R 1 -β 11 , a S 1 S 1 µ 2 S + θ G 2 + θ R 2
, 0, 0, 0, 0, 0, 0

To show the stability of the endemic state, let us fix a set of values for the model parameters,

where k is a scale parameter:

φ 1 = φ 2 = 4k 2 100 , µ 2 S = ϵ 1 = β 22 = 0, µ 2 I = 499k 100 , µ 2 G = 49k 100 , µ 2 R = 2k 5 , β 11 = β 21 = k 5 , θ G 1 = θ R 1 = γ 1 = γ 2 = 0.1k, θ G 2 = θ R 2 = k, a S 1 = a I 1 = a G 1 = a R 1 = 98k 96 , ϵ 2 = 4k 5 , η 1 = η 2 = 0.2k
The two stationary points are labeled with * (resp. **) for the eradication (resp. endemic) state:

(S 1 * , S 2 * , I 1 * , I 2 * G 1 * , G 2 * , R 1 * , R 2 * ) = β 21 S 2 a S 1 + θ G 1 + θ R 1 -β 11 , a S 1 S 1 µ 2 S + θ G 2 + θ R 2
, 0, 0, 0, 0, 0, 0 and [START_REF] Kremer | Authors' response: Estimating the generation interval for covid-19 based on symptom onset data[END_REF][START_REF] Kremer | Authors' response: Estimating the generation interval for covid-19 based on symptom onset data[END_REF][START_REF] Kremer | Authors' response: Estimating the generation interval for covid-19 based on symptom onset data[END_REF][START_REF] Kremer | Authors' response: Estimating the generation interval for covid-19 based on symptom onset data[END_REF][START_REF]World health organization[END_REF][START_REF] Bianconi | Epidemic spreading[END_REF][START_REF] Chan | Surveillance and epidemiology of infectious diseases using spatial and temporal lustering methods[END_REF][START_REF] Chan | Surveillance and epidemiology of infectious diseases using spatial and temporal lustering methods[END_REF] , if k = 1.

(S 1 * * , S 2 * * , I 1 * * , I * * 2 , G 1 * * , G 2 * * , R 1 * * , R 2 * * ) =
Let us show that the endemic state is locally stable. With the chosen parameter values and k = 1, from the model Equation (3.39), we have:

dS 1 dt = 0.2G 1 -1.02S 1 + 0.2R 1 + 0.2S 2 + 0.2G 2 + 0.2R 2 -0.04I 1 S 1 -0.04I 2 S 1 dS 2 dt = 1.02S 1 -2S 2 -0.04I 1 S 2 -0.04I 2 S 2 dI 1 dt = 0.04I 1 S 1 + 0.04I 2 S 1 -1.04I 1 dI 2 dt = 0.04I 1 S 2 + 0.04I 2 S 2 + 1.02I 1 -5.19I 2 dG 1 dt = 0.2I 1 + 0.1S 1 -1.12G 1 (3.40) dG 2 dt = 0.2I 2 + 1.02G 1 + S 2 -1.39G 2 dR 1 dt = 0.1G 1 + 0.1S 1 -1.02R 1 dR 2 dt = 0.1G 2 + S 2 + 1.02R 1 -0.4R 2
By calculating the Jacobian matrix M of the system of Equation (3.40) at the endemic state, where I is the identity matrix and finding the roots of its characteristic polynomial P M , we have for the second stationary point the expression as follows:

M -λI =             -1.02 -λ 0.2 -0.4 -0.4 0.2 0.2 0.2 0.2 1.02 -2 -λ -0.4 -0.4 0 0 0 0.8 0 -1.04 -λ 0 0 0 0 0 0.8 1.02 -5.19 -λ 0 0 0 0.1 0 0.2 0 -1.12 -λ 0 0 0 1 0 0.2 1.02 -1.39 -λ 0 0.1 0 0 0 0.1 0 -1.02 -λ 0 1 0 0 0 0.1 1.02 -0.4 -λ            
The roots of the characteristic polynomial of M, P M , satisfy: P M (λ) = det (M -λI) = (λ + 0.706923) (λ + 5.13103) λ 2 + 0.973528λ + 0.34052 λ 2 + 2.0371λ + 1.03861 λ 2 + 3.69142λ + 3.53083 = 0

The real parts of the eigenvalues of the matrix M are all negative, equal to:

-0706923, -5.13103, -0.486764, -1.01855, -1.84571 Hence, with the set of chosen parameter values, the stability of the endemic state is proved.

Existence and unicity of the solution

We want to establish existence and unicity of the solution for the model Equation (3.39)

= (S 1 , S 2 , I 1 , I 2 , G 1 , G 2 , R 1 , R 2 ) , with Z (0) = Z 0 , Z (t, ϵ) = H (t, ϵ) Z (t) .
Let us define the Volterra integral equation of the second kind which is given as:

S (t) = S (0) + t 0 H (t, ϵ) S (ϵ) dϵ
where H (t, ϵ) is a kernel and S (t) is the function to be solved. The Volterra integral equation formulation of the model Equation (3.39) is the following:

S 1 (t) = S 1 (0) + t 0 H 1 (t, ϵ) S 1 (ϵ) dϵ S 2 (t) = S 2 (0) + t 0 H 2 (t, ϵ) S 2 (ϵ) dϵ I 1 (t) = I 1 (0) + t 0 H 3 (t, ϵ) I 1 (ϵ) dϵ I 2 (t) = I 2 (0) + t 0 H 4 (t, ϵ) I 2 (ϵ) dϵ G 1 (t) = G 1 (0) + t 0 H 5 (t, ϵ) G 1 (ϵ) dϵ (3.41) G 2 (t) = G 2 (0) + t 0 H 6 (t, ϵ) G 2 (ϵ) dϵ R 1 (t) = R 1 (0) + t 0 H 7 (t, ϵ) R 1 (ϵ) dϵ R 2 (t) = R 2 (0) + t 0 H 8 (t, ϵ) R 2 (ϵ) dϵ
Let S 1 be the solution for S 1 and kernels H i , i = 1, 2, . . . , 8 satisfy Lipschitz conditions:

sup 0<t≤1 ∥S 1 ∥ ≤ c 1 , sup 0<t≤1 ∥S 2 ∥ ≤ c 2 , sup 0<t≤1 ∥I 1 ∥ ≤ c 3 , sup 0<t≤1 ∥I 2 ∥c 4 , sup 0<t≤1 ∥G 1 ∥ ≤ c 5 , sup 0<t≤1 ∥G 2 ∥ ≤ c 6 , sup 0<t≤1 ∥R 1 ∥ ≤ c 7 , sup 0<t≤1 ∥R 2 ∥ ≤ c 8
and c i > 0, for i = 1, 2, . . . , 8. Then, following inequalities hold, using triangle inequality and properties of the H ′ i s norm:

∥H 1 (S 1 ) -H 1 (S 1 ) ∥ ≤ ∥ β 11 -a S 1 + θ G 1 + θ R 1 + φ 1 (I 1 + I 2 ) S 1 -β 11 -(a S 1 + θ G 1 + θ R 1 + φ 1 (I 1 + I 2 )) S 1 ∥ ≤ ∥β 11 -(a S 1 + θ G 1 + θ R 1 + φ 1 (I 1 + I 2 ))∥∥S 1 -S 1 ∥ ≤ ∥β 11 -(a S 1 + θ G 1 + θ R 1 + φ 1 c 3 + φ 1 c 4 )∥∥S 1 -S 1 ∥ = ∂ 1 ∥S 1 -S 1 ∥ where ξ = β 11 (G 1 + R 1 ) + β 21 (S 2 + G 2 + R 2 ), ∂ 1 = β 11 -a S 1 -θ G 1 -θ R 1 -φ 1 c 3 -φ 1 c 4 .
Therefore, H 1 satisfies the Lipschitz conditions. We can show in the same way that other functions H i , i = 2, . . . , 8 in the model Equation (3.39) satisfy the Lipschitz conditions as follows:

∥H 2 (S 2 ) -H 2 (S 2 ) ≤ ∥ ∂ 2 ∥S 2 -S 2 ∥ ∥H 3 (I 1 ) -H 3 (L 1 ) ≤ ∥∂ 3 ∥I 1 -L 1 ∥ ∥H 4 (I 2 ) -H 4 (L 2 ) ≤ ∥ ∂ 4 ∥I 2 -L 2 ∥ ∥H 5 (G 1 ) -H 5 (G 1 ) ≤ ∥ ∂ 5 ∥G 1 -G 1 ∥ ∥H 6 (G 2 ) -H 6 (G 2 ) ≤ ∥ ∂ 6 ∥G 2 -G 2 ∥ ∥H 7 (R 1 ) -H 7 (ℜ 1 ) ≤ ∥ ∂ 7 ∥R 1 -ℜ 1 ∥ ∥H 8 (R 2 ) -H 8 (ℜ 2 ) ≤ ∥ ∂ 8 ∥R 2 -ℜ 2 ∥
Let us now consider the following Neumann series:

S 1m (t) = S 1 (0) + t 0 H 1 (t, ϵ) S 1m-1 (ϵ) dϵ S 2m (t) = S 2 (0) + t 0 H 2 (t, ϵ) S 2m-1 (ϵ) dϵ I 1m (t) = I 1 (0) + t 0 H 3 (t, ϵ) I 1m-1 (ϵ) dϵ I 2m (t) = I 2 (0) + t 0 H 4 (t, ϵ) I 2m-1 (ϵ) dϵ G 1m (t) = G 1 (0) + t 0 H 5 (t, ϵ) G 1m-1 (ϵ) dϵ G 2m (t) = G 2 (0) + t 0 H 6 (t, ϵ) G 2m-1 (ϵ) dϵ R 1m (t) = R 1 (0) + t 0 H 7 (t, ϵ) R 1m-1 (ϵ) dϵ R 2m (t) = R 2 (0) + t 0 H 8 (t, ϵ) R 2m-1 (ϵ) dϵ
These Neumann series are convergent due to the Lipschtizian character of H ′ s, then:

∥S 1m+1 -S 1m ∥ ≤ t 0 ∥H 1 (t, ϵ) S 1m (ϵ) -H 1 (t, ϵ) , S 1m-1 (ϵ) dϵ∥ ≤ t 0 ∥S 1m (t) -S 1m-1 (t) ∥dϵ ≤ ∂ 1 ∥S 1m (t) -S 1m-1 (t) ∥ ∞
Other equations are given as follows:

∥S 2m+1 -S 2m ∥ ≤ ∂ 2 ∥S 2m (t) -S 2m-1 (t) ∥ ∞ ∥I 1m+1 -I 1m ∥ ≤ ∂ 3 ∥I 1m (t) -I 1m-1 (t) ∥ ∞ ∥I 2m+1 -I 2m ∥ ≤ ∂ 4 ∥I 2m (t) -I 2m-1 (t) ∥ ∞ ∥G 1m+1 -G 1m ∥ ≤ ∂ 5 ∥G 1m (t) -G 1m-1 (t) ∥ ∞ ∥G 2m+1 -G 2m ∥ ≤ ∂ 6 ∥G 2m (t) -G 2m-1 (t) ∥ ∞ ∥R 1m+1 -R 1m ∥ ≤ ∂ 7 ∥R 1m (t) -R 1m-1 (t) ∥ ∞ ∥R 2m+1 -R 2m ∥ ≤ ∂ 8 ∥R 2m (t) -R 2m-1 (t) ∥ ∞
The above inequalities prove the existence of the function H. We now show the uniqueness of the solution by assuming that the kernels H i , i = 1, 2, . . . , 8 are separable, i.e., H 1 (t, ϵ) = ∅ (t) ς (ϵ). By denoting:

ζ (t) = t 0 S 1 (ϵ) ς (ϵ) dϵ 110 then, because S 1 (t) = S 1 (0) + t 0 H 1 (t, ϵ) S 1 (ϵ) dϵ = S 1 (0) + t 0 ∅ (t) ς (ϵ) S 1 (ϵ) dϵ, we have: ζ ′ (t) = ∅ (t) ς (t) ζ (t) + S 1 (0) ς (t) If ς (0) = 0, the solution ζ (t) follows ζ (t) = e t 0 ∅(ϵ)ς(ϵ)dϵ
and from the definition of ζ (t), the solution of S 1 (t) is given as:

S 1 (t) = S 1 (0) + ∅ (t) t 0 S 1 (ϵ) ς (ϵ) dϵ = S 1 (0) + ∅ (t) ζ (t) = S 1 (0) + ∅ (t) e t 0 ∅(ϵ)ς(ϵ)dϵ
Hence, by the unicity of the solution ζ (t), there exists just only one continuous solution for S 1 (t). Following the same approach, we can obtain a unique solution for the remaining equations of the system in Equation (3.41).

Basic reproduction number

It is possible to apply the idea of a next-generation matrix by linearizing the model Equation (3.39) near the endemic state, for the infectious part of the system in Equation (3.39), and then obtain:

dI 1 dt = φ 1 (I 1 + I 2 ) S 1 * * -(a I 1 + η 1 )I 1 , dI 2 dt = φ 2 (I 1 + I 2 ) S 2 * * + a I 1 I 1 -η 2 I 2 µ 2 I I 2 , (3.42
)

dG 1 dt = η 1 I 1 + θ G 1 S * * 1 -a G 1 + γ 1 + ϵ 1 G 1 , dG 2 dt = η 2 I 2 + a G 1 G 1 + θ G 2 S 2 * * -γ 2 + µ 2 G + ϵ 2 + β 22 G 2
By summing Equation (3.42) and by denoting I as the size of all infectious, we have:

I(t) = I 1 (t) + I 2 (t) + G 1 (t) + G 2 (t),
and

dI dt = φ 1 (I 1 + I 2 ) S 1 * * + φ 2 (I 1 + I 2 ) S 2 * * -µ 2 I I 2 + θ G 1 S 1 * * -(γ 1 + ϵ 1 )G 1 -(γ 2 + ϵ 2 + β 22 )G 2
The matrix J of the linearized system near a state (S 1 , S 2 ) is:

J =     φ 1 S 1 -a I 1 -η 1 φ 1 S 1 0 0 φ 2 S 2 + a I 1 φ 2 S 2 -η 2 -µ 2 I 0 0 η 1 0 -a G 1 -γ 1 -ϵ 1 0 0 η 2 a G 1 -γ 2 -µ 2 G -ϵ 2 -β 22    
The corresponding characteristic polynomial P J (λ) is equal to:

(φ 1 S 1 -a I 1 -η 1 -λ)(φ 2 S 2 -η 2 -µ 2 I -λ) -a G 1 -γ 1 -ϵ 1 -λ -µ G 2 -γ 2 -ϵ 2 -β 22 -λ -φ 1 tS 1 φ 2 S 2 + a I 1 -a G 1 -γ 1 -ϵ 1 -λ -µ G 2 -γ 2 -ϵ 2 -β 22 -λ = P J (λ)
The positive eigenvalues of J are roots of the following polynomial:

λ 2 -φ 1 S 1 -a I 1 -η 1 + φ 2 S 2 -η 2 -µ 2 I λ-φ 1 S 1 φ 2 S 2 + a I 1 +(φ 1 S 1 -a I 1 -η 1 )(φ 2 S 2 -η 2 -µ 2 I )
These roots are equal to: B ± (B 2 -C) 1/2 , where B and C are equal to:

B = φ 1 S 1 -a I 1 -η 1 + φ 2 S 2 -η 2 -µ 2 I 2 C = -φ 1 S 1 φ 2 S 2 + a I 1 + (φ 1 S 1 -a I 1 -η 1 )(φ 2 S 2 -η 2 -µ 2 I )
Then, we have:

B + (B 2 -C) 1/2 = φ 1 S 1 -a I 1 -η 1 + φ 2 S 2 -η 2 -µ 2 I 2 + (φ 1 S 1 -a I 1 -η 1 + φ 2 S 2 -η 2 -µ 2 I ) 2 4 + φ 1 S 1 φ 2 S 2 + a I 1 -φ 1 S 1 -a I 1 -η 1 φ 2 S 2 -η 2 -µ 2 I 1/2
Hence, near unstable endemic state the positive dominant eigenvalue Λ is equal to:

Λ = φ 1 S 1 -a I 1 -η 1 + φ 2 S 2 -η 2 -µ I 2 2 + (φ 1 S 1 -a I 1 -η 1 -φ 2 S 2 + η 2 + µ I 2 ) 2 4 + φ 1 S 1 φ 2 S 2 + a I 1 1/2
Therefore, the basic reproduction number R 0 equal to Λ near the endemic stationary state depends mainly on the infection rates φ 1 and φ 2 , when the sizes S 1 and S 2 are sufficiently important. If after a change of parameter values, the endemic state becomes unstable and R 0 is becoming more than 1, then an epidemic wave starts.

Numerical simulation of the model: examples of the COVID-19 outbreak in some countries

The numerical simulation of the model developed is done using data coming from Kuwait, France and Cameroon. First, we provide some explanations for the data used for the simulations, including how we assumed some of the parameters, calculated others, and selected some from the literature cited in this article. In order to determine the susceptible classes sizes, we used Wikipedia data on the three studied countries to obtain their total population size. We then calculated the ratio between the young and elderly from the data presented in We present the visualization results for the simulated values in Figure 6.

COVID-19 Outbreak in France

We have chosen the following set of parameter values corresponding to the COVID-19 outbreak at the start of the fifth wave in December 2021 in France: We present the visualization results for the simulated values in Figure 7.

COVID-19 Outbreak in Cameroon

We choose the following set of parameter values corresponding to the COVID-19 outbreak at the start of the second wave in January 2021 in Cameroon: We present the visualization results for the simulated values in Figure 8. We present the visualization results for the simulated values in Figure 8.

rical simulation of the variables I1 and I2, R1 and R2, G1, G2 for Came ts for the three considered countries present some similarities at we will discuss in the following. ach case, the exponential growth of the infectious I1 + I2 and 2 populations sizes correspond roughly to the data given in [5 We present the visualization results for the simulated values in Figure 3.43. The results for the three considered countries present some similarities but also some differences that we will discuss in the following. First, in each case, the exponential growth of the infectious I 1 + I 2 and completely recovering R 1 + R 2 populations sizes correspond roughly to the data given in [119]. The model simulations give more, i.e., allows to see the part brought by each age class to the global growth. Second, for Kuwait, Figure 3.42 shows a faster growth of infectious (I) in the young class (≤65 years) than in the older class (>65 years), which is also the case for France (Figure 3.43) and for Cameroon (Figure 3.43). On the other hand, this phenomenon is reversed in the three countries, with regard to the growth of populations immunized in a transient manner (G) and in a lasting manner (R). The phenomenon is more marked for Kuwait than for France, France itself having a more marked difference than for Cameroon. This is partly explained by the better vaccination rate of Kuwait than that of France and Cameroon in the young class, the effectiveness of the vaccination (which decreases in the older class) having been assumed to be equal for the three countries. Taking age into account in modeling the COVID-19 pandemic makes it possible to simulate the differential dynamic behavior of the growth of infectious and immune populations, young and old, in order, for example, to adjust the vaccine policy according to the age.

Conclusion

Concerning contagious diseases, public health physicians are constantly faced with three challenges. The first concerns the estimation of the basic reproduction number R 0 . The systematic use of R 0 simplifies the decision-making process by policymakers, advised by public health authorities, but it is too much of a caricature to account for the biology behind the viral spread. We have observed in the COVID-19 outbreak that it was non-constant during an epidemic wave due to exogenous and endogenous factors influencing both the duration of the contagiousness period and the daily transmission rate during this phase [START_REF] Kawasuji | Transmissibility of covid-19 depends on the viral load around onset in adult and symptomatic patients[END_REF][START_REF] Kim | Viral kinetics of sars-cov-2 in asymptomatic carriers and presymptomatic patients[END_REF][START_REF] Murphy | Evaluation of influenza a/hong kong/123/77 (h1n1)ts-1a2 and cold-adapted recombinant viruses in seronegative adult volunteers[END_REF]. Then, the second challenge concerns the estimation of the mean duration of the infectious period for infected patients. As for the transmission rate, realistic assumptions made it possible to obtain an upper limit to this duration [START_REF] Demongeot | Si epidemic model applied to covid-19 data in mainland china[END_REF], mainly due to the lack of viral load data in large patient cohorts (see Figure 3.2 from [START_REF] Chao | Flute, a publicly available stochastic influenza epidemic simulation model[END_REF][START_REF] Itoh | Emergence of h7n9 influenza a virus resistant to neuraminidase inhibitors in nonhuman primates[END_REF][START_REF] Demongeot | Evolution of social networks: the example of obesity[END_REF]), in order to better guide the individual quarantine measures decided by the authorities in charge of public health. This upper bound also makes it possible to obtain a lower bound for the percentage of unreported infected patients, which gives an idea of the quality of the census of cases of infected patients, which is the second challenge facing specialists of contagious diseases. The third challenge is the estimation of the daily reproduction number over the contagiousness period, which was one of the issue we tackle in this chapter. Eventually, our approach using marginal daily reproduction numbers involving a certain level of noise in the dynamics of new daily infected cases defines a stochastic framework which describes phenomenologically the exponential phase as our results show for countries such as France, Russia, Sweden, etc. This stochastic modeling allows a better understanding of the role of the contagiousness period length and of the heterogeneity (e.g., the U-shape) of its daily reproduction number distribution in the COVID-19 outbreak dynamics. On the medical level, the important message about the U-shape is that COVID-19 is similar to other viral diseases, such as influenza, with two successive reactions from the two immune defense barriers, innate cellular immunity first, which is not sufficient if symptoms persist, then adaptive immunity (cellular and humoral), which results in a transient decrease in contagiousness between the two phases. The medical recommendations are, in this case, never to take a transient improvement for a permanent disappearance of the symptoms. One could indeed, for a public health use, be satisfied after estimating the sum of the R j 's, that is to say, R 0 or the effective R e . For an individual health use, it is important to know the existence of a minimum of the R j 's, which generally corresponds to a temporary clinical improvement, after the partial success of the innate immune defenses. This makes it possible to prevent the patient from continuing to respect absolute isolation and therapeutic measures, even if a transient improvement occurs; otherwise, they risk, as in the flu, a bacterial pulmonary super-infection (a frequent cause of death in the case of COVID-19). On the theoretical level, the interest of the proposed methods is its generic character: it can be applied to all contagious diseases, within the very general framework of Equation (3.6), which makes no assumption about the spatial heterogeneity or the longitudinal constancy of the daily reproduction numbers. The deconvolution of Equation (3.6) poses a new theoretical problem when it is offered in this context. The current context of the pandemic closely monitored by the WHO, a contagious social pandemic and a contagious infectious pandemic, COVID-19, shows that this problem is now highly relevant. To be able to be both retro-predictive on past pandemics (like the great plague of 1348 in Europe) or anticipatory on present and future pandemics, it is necessary to be able, from observed data (which are daily in the case of the COVID-19 [119] and annual in the case of obesity [214]), to estimate the parameters of the model, then simulate it and interpret the scenarios of the simulations within the framework of concrete public health measures (change in eating behaviours, preventive preparation of the system immunity, implementation of mitigation measures, vaccination, preventive education etc.). This part is the most difficult and is only effective within the framework of multidisciplinary teams, with very good mathematicians of dynamical systems and very good nutritionists and infectiologists. Solid knowledge in the genetics of infectious agents and in the monitoring of environmental conditions is necessary because:

-the dynamic phenomena of propagation present isolated or periodic waves of the new cases, with bifurcations (in the deterministic case) or phase transitions (in the stochastic case) often difficult to study theoretically and to simulate numerically [START_REF] Demongeot | Boundary conditions and phase transitions in neural networks. theoretical results[END_REF][START_REF] Demongeot | Boundary conditions and phase transitions in neural networks. simulation results[END_REF][START_REF] Demongeot | Phase transitions in stochastic non-linear threshold boolean automata networks on z 2 : The boundary impact[END_REF] -the changes in the infectious genius of the agents at the origin of pandemics need a new knowledge about mechanisms often partly unknown, because linked to uneasily predictable and often hypothetical [START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF][START_REF] Marchi | The role of micrornas in modulating sars-cov-2 infection in human cells: a systematic review[END_REF] modifications in their genome (such as point mutations, deletions and insertions of more or less long sequences, re-combinations with other genomes, passages in the host nuclear genome, etc.). Furthermore, we have been able to use a continuous approach in estimating the maximal reproduction number in COVID-19 outbreak and by extension estimating the transmission rate, a parameter which helps to know how a disease is spread by identifying the point of inflexion during the exponential phase of the pandemic wave, we have then deduced the value of the basic reproduction rate at this point of inflexion while the method was validated by linking the correlation of these epidemiology parameters with some socio-demo-economic parameters. Finally, we used a matrix population growth modeling on countries where age data were available for this study and also propose some panorama on cells targeted by SARS-CoV-2 and cell lifespan loss and also on how the method can be extended to ageing in cell populations using discrete and continuous approach. The model developed in this study differs from many previously constructed agent-based and dynamical system models of population dynamics, particularly when considering viral infection. Its greatest utility is that it can be used to highlight specific characteristics of the life span caused by viral infection, which have a significant impact on population growth. As a result, the findings of this study may be useful in developing management programs to reduce the potential for COVID-19 epidemics, particularly when comorbidities and ageing are considered. We used data from Cameroon to investigate the role of comorbidities in age-dependent COVID-19 modeling, while data from France were used to assess the proportion of parameter sensitivity at the sub-population level. Data from Ireland were used to examine the evolution of the pandemic in young people because this country has seen a sharp increase in cases among children and teenagers as at the time of this study. Age dependent modeling is important to better understand the dynamics of viral diseases, and we have been able to present another perspective to this research direction using a discrete approach, matrix algebra and an ODE compartment model. It is a known fact that due to a new variant of the disease, there is an increase in infection rate in different age groups. Our work was able to provide a link between the dominant eigenvalue which represents the exponential growth parameter and the COVID-19 affected sub-population and also the differential dynamic behavior of the growth of infectious and immune populations between young and old.

Chapter 4 Statistical Modeling of COVID-19 Outbreak

In epidemiology, statistical modeling helps to understand the mechanisms that influence the spread of infectious diseases, and it suggests prevention and control strategies. It can be used to comprehend how a virus spreads across a large region or country hence, this chapter is devoted to large scale analysis and application of different statistical tools such as machine learning and deep learning to model the COVID-19 pandemics so as to predict and forecast the evolution of the disease at the population level. COVID-19 cases prediction models are currently divided into three categories: theoretical models, single artificial intelligence models, and decomposition integration models. The majority of statistical methods are used in the case study of COVID-19 to make prediction of the pandemic evolution at different phases. We present in this chapter some useful quantitative statistical tools for analysing epidemiological data. Section 4.1 examines the application of ARIMA, regression models, exponential smoothing model and prophet model on several time series. The slope of the regression line is calculated using an exponential model, while the initial auto-correlation slope of new cases were forecasted using an ARIMA model for the two waves observed at the time of this study. The best statistical model to describe the daily count of new cases and deaths due to COVID-19 infections in Kuwait is presented by testing the model's performance, accuracy, and validation, as a result, the best model to estimate the number of new cases as well as a model for COVID-19 infection-related mortality were utilized. The findings indicate that the COVID-19 outbreak in Kuwait is divided into two stages i.e., first wave and transition to second wave. In Section 4.2, the application of deep learning to COVID-19 pandemic time series data has been done for the United States (USA), United Kingdom (UK), India, Russia, Brazil, France, and Turkey, which are countries where the pandemic is more prevalent globally. Multi-layer Perceptron (MLP), Long Short-term Memory (LSTM), Gated Recurrent Unit (GRU), Convolution Neural Network (CNN), and Deep Neural Network (DNN) methods were used on time series data from the beginning of the pandemic until September 3, 2021 to predict daily new cases and daily deaths at different waves of the pandemic in countries considered, with MLP forecasted until December 2021 due to the model performance. The performance of the methods was accessed using Root Mean Square Error (RMSE) and relative Root Mean Square Error (rRMSE). The results have depicted the impact of COVID-19 management across the countries studied. In Section 4.3 spectral analysis method is used to convert time (days) for COVID-19 pandemic time series data from United States (USA), United Kingdom (UK), India, Russia, Brazil, France, and Turkey to analyze the frequency peaks and periodicity of the time series data. It was discovered that the pandemic's peaks detection was central in the analysis and short-term forecasting. In Section 4.4 the modelling of the COVID-19 pandemic against some socio-economic parameters is presented by examining the time series of infection spread throughout the first and second waves in all countries (OECD, developing and developed countries namely African countries), as well as their socio-economic statistics. The proof of a mathematical and statistical relationship between Theil and Gini indices is presented, and then multiple machine learning and deep learning approaches investigate the association between epidemiological data and socioeconomic variables. Also the relationship between epidemiologic and socio-economic variables such as CHE/GDP data is studied thanks to ordinary least square multivariate modelling and classifying countries is using principal component analysis, K-means, and hierarchical clustering. The study demonstrates that there are correlations between growth parameters directly linked to the occurrence of new cases of COVID-19 and socio-economic variables, specifically current health expenditure as a percentage of gross domestic product (CHE/GDP) is anti-correlated with basic reproduction number, demonstrating the effectiveness of public health mitigation measures, even if they involve significant medico-economic costs. The result reveals a disparity between developed and developing countries, despite the fact that rapid implementation of isolation and vaccination measures allowed them to anticipate and mitigate the effects of the second wave. Section 4.5 deals with the techniques of functional data analysis applied to model daily hospitalized, deceased, Intensive Care Unit (ICU) cases, and return home patient numbers along the COVID-19 outbreak, considered as functional data across different departments in France, while response variables are numbers of vaccinations, deaths, infected, recovered, and tests numbers. The first functional principal component well characterized the forms seen for the data connected to the first three principal components, according to the functional principal component analysis. This discovery confirms the significance of the first component in explaining and qualitatively predicting the observed data. The influence of vaccination is visible because the shapes are attenuated after vaccination and do not resemble the shapes observed in Section 3.2 for seasonal influenza. Finally, this chapter is concluded in Section 4.6 with some perspectives and conclusions.

Data Analysis and Forecasting of COVID-19 Pandemic Based on Daily Observation Dynamics

This section deals with the application of several statistics tools for modelling epidemic data such as ARIMA model, Exponential smoothing model, Holt's method, Prophet forecasting model and machine learning models. Among machine learning models, log-linear, polynomial and support vector regressions were used in order to compare these models. The main objective and motivation is to present the best statistical model to describe the daily count of new cases and deaths due to COVID-19 outbreak. The best model is therefore used to predict the number of new cases and to model the mortality due to the COVID-19 pandemic.

Another objective is to decompose the COVID-19 data into two main waves and model each wave separately using different tools along with the classical time series model that depends on the basic reproduction number R o , followed by many different additive models to predict the time series such as Exponential regression, log-linear and Polynomial with different degrees.

In addition, several smoothing methods, Prophet and Support vector methods were applied to COVID-19 data. A comparison is made between these different techniques for the best fit. These different tools have been applied to some countries data and the models are supported by validity and accuracy tests. The sensitivity analysis for the parameters used in the ARIMA model and exponential models are presented.

ARIMA modeling for first wave and second wave

This section deals with the application of ARIMA model to calculate the initial autocorrelation slope and also to forecast new cases for both waves for some countries. The ARIMA model shows more than 95% confidence interval as it can be seen in Mali shows a quasi-endemic behavior with a weakly varying autocorrelation function and Luxembourg a frank epidemic wave with a classic shape. For the second wave in fall 2020, Mali presents an attenuated epidemic shape (due probably to specific geoclimatic conditions in western Africa [START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF]) and a country from central Europe, Slovenia, shows at this period an endemic behavior with an oscillatory occurrence of new cases. Figure 4.3 proposes a forecasting based on ARIMA decomposition for the first and second waves in Mali with a better approximation for the epidemic second wave than for the quasi-endemic first wave. It is the same for Luxembourg with an inversion of the phases order, an epidemic wave followed by an endemic state well predicted. On the contrary, for Slovenia, the endemic state with oscillations is badly predicted.

ARIMA Model Forecast for First and Second Wave

The forecast using the ARIMA method shows a good retrospective adjustment to past data, but a weak predictive power of the future trend of new cases, in particular for the prediction of the entry into the endemic phase after an epidemic wave (Figure 9). The comparison during the first wave between two countries (Figure 4.1), one developed (Luxembourg) and one developing (Mali) shows a difference in length of contagiousness period (linked to the value of the opposite to the slope at origin of the autocorrelation function) and shape of the growth curve, indicating a lower virulence of the SARS Cov-2 in Mali, possibly due to the influence of the temperature [START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF]. This tendency is reversed during the second wave between Mali and Slovenia (Figure 4.2). The forecast using the ARIMA method shows a good retrospective adjustment to past data, but a weak predictive power of the future trend of new cases, in particular for the prediction of the entry into the endemic phase after an epidemic wave (Figure 4.3). The initial negative autocorrelation slope of the epidemic spread averaged on six days is also calculated and presented for Kuwait while other countries is summarized in Appendix B, Table B.1. As expected, the 100 first days of the first wave start show a positive slope (0.0687) and the 100 first days of the second wave transition show a negative slope (-0.0094). The results is compared using parameters (p,d,q) and the best result gave lower RMSE and p-value. We trained 60% of the data and forecast 100 days while the rest was tested to show the validation and accuracy of the model. It was observed that while the RMSE of the tested data was high, by checking the residual using Ljung-Box test, the Q * is large with p-value < 0.05 which shows that the autocorrelation did not come from white noise. The visualisation result is in

Clustering of Countries from Epidemic and Economic Variables

Hierarchical Clustering

The hierarchical clustering allows developed and developing countries to be grouped into 5 separate clusters (Figure 10) and Principal Component Analysis into 3 separate clusters (Figure 11), one being a singleton corresponding to Spain (Figure 12).

Clustering of Countries from Epidemic and Economic Variables

Hierarchical Clustering

The hierarchical clustering allows developed and developing countries to be grouped into 5 separate clusters (Figure 10) and Principal Component Analysis into 3 separate clusters (Figure 11), one being a singleton corresponding to Spain (Figure 12).

Regression analysis for first wave and second wave

In this Section, regression models are applied on the daily new cases for both waves observed in Kuwait and it is also applied to some countries epidemiologic variables (maximum reproduction number, exponential slope calculated in this section and the initial autocorrelation slope calculated in the previous section) in order to establish the relationship among them. Looking at the pattern of COVID-19 data of Kuwait, one realises that it consists of two main waves. Each wave lasts approximately three months. Based on available data, 100 days both in the first wave and second wave was considered. First wave new cases were considered from 25/02/2020 to 03/06/2020 while second wave new cases were considered from 15/10/2020 to 22/01/2021. For the first wave, daily deaths data is from 04/04/2020 to 12/07/2020, while for the second wave is from 15/10/2020 to 22/01/2021. The slopes from the log-linear regression analysis is calculated using exponential model. The result of exponential slope for Kuwait is presented in this section while other countries is summarized in Appendix B, Table B.1. For first wave, the exponential model: y = 0.8e 0.07x , based on the following results:

• Log-linear regression: slope = 0.0686852051631, intercept=0.800565672854, r = 0.939152595486, p-value = 2.079801359e -45 , standard error = 0.002591135543915, R-squared = 0.882008, RMSE = 0.711400960815.

• Using polynomial regression of order 4 we have: R-squared = 0.787421, RMSE = 0.586403581239, p-value< 2.2e -16 , standard error = 0.6069, F-statistics = 256.7.

• Using Support vector regression we get: R-squared = 0.98382, RMSE = 0.263424533991, slope = 0.0128466, intercept= 3.697591, p-value< 4.33e -15 .

For second wave transition, the exponential model is y = 6.5e -0.009x , based on the following results:

• Log-linear Regression: slope = -0.0093674219195, intercept = 6.5198283885, r = -0.57106504118, p-value = 5.50386624132e-10, standard error = 0.00136023603, R-squared = 0.326115, RMSE = 0.388700386316,

• Polynomial of order four Regression: R-squared = 0.0927367, RMSE = 0.1818719990, p-value< 2.2e -16 , standard error = 0.186, F-statistics = 137.2,

• Support Vector Regression: R-squared = 0.9680774158, RMSE = 0.0846001608, slope = -0.2570435, intercept= -1.370541, p-value = 0.2775. All coefficients for the first wave and transition to second wave for both Log-linear and polynomial regression are significant with p-value less than 0.001. The residuals of the log-linear regression were examined and it was discovered that the median for both the cases were close to zero with median = 0.06031 for the first wave and median = 0.07629 for transition to the second wave. The normality of the residual was tested using Jarque-Bera test and with high p-value we fail to reject the null hypothesis that the skewness and kurtosis of the residuals are significantly equal to zero. Also, the median of the residual for both first wave and transition to second wave of polynomial regression of order four is close to zero with median = 0.0271 and median = 0.00669 respectively. For support vector regression, we tested the normality of the residual using the Jarque-Bera test and it was discovered that it's skewness and kurtosis are significantly equal to zero since the p-value is large. We now apply linear regression model by using some historic data (100 days infectivity period in this case) on some countries epidemiologic variables and consider a linear relationship between them while polynomial regression models use a similar approach but the dependent variable is modeled as a degree n (6 ≥ n ≥ 2) polynomial in x. It is presented as follows:

• Linear Regression - More precisely, on Figure 4.7(a)-(b) first and second waves of the COVID-19 pandemic are compared using linear and parabolic or cubic regression, showing a significant positive (resp. negative) correlation between the opposite of the initial autocorrelation slope and exponential regression slope of the first (resp. second) wave for developed (resp. all) countries. This opposition between the two waves could result from the application of a more severe lockdown in developed countries during the second wave. On Figure 4.7(c), the opposite of the initial autocorrelation slope decreases significantly if the start of the first wave in a country is late with respect to the start of the COVID-19 outbreak in China due probably to the progressive implementation of mitigation measures in that country taking into account the experience of the countries starting first wave before.

-For The curves show a different behavior between the two waves (a) and (b), probably due to an increase in the contagion parameter, the basic reproduction number R 0 (linked to the Malthusian parameter of the exponential growth phase), despite a shortening of the duration of contagiousness (linked to the slope at the origin of the auto-correlation function, which is all the stronger as the distance from the start of the epidemic increases, no doubt because of the mitigation measures, which decrease the duration of the contagiousness period).

The meaning of the abbreviations used in Figure 1 is the following: LinregressResult slope = slope of the linear regression, intercept = ordinate at origin of the regression curve, r value = correlation coefficient, p value = p value of the nullity test of correlation coefficient, stderr = standard error of the regression, RMSE = root of mean square error. • Quartic Regression

Sextic Regression

We studied the correlation between the value of the opposite of the slope at the origin of the autocorrelation function of the first wave and the economic and health index CHE/GDP, by studying a polynomial regression of degree 6 (Figure 3). It shows an anticorrelation in the linear regression and a local maximum for countries with an average CHE/GDP ratio of around 7. Countries with a high CHE/GDP ratio (such as France and the United States) have a low value of l in opposite to this slope. The explanation for this phenomenon may come from the correlation reported in the introduction between the CHE/GDP and Gini indices, the poor classes having a longer duration of contagiousness due to a less important state of immunological defense and perhaps less compliance with mitigation measures.

Performance, accuracy and validation of the regression models

In order to know the performance of the regression models we trained 80% of the data from Kuwait and test 20% percent of the data and also did cross validation to be sure of the accuracy. The predicted and the observed values are very close to the result presented for all the regression models. For Log linear model we present the cross validation result in Figure 4.9(a) whose average mean square errors for the 5 portion folds are 0.5027791 for first wave on the left and 0.1533665 for second wave transition on the right. We observed high correlations between the tested and the predicted values (R-squared = 0.9278587 for first wave cases and R-squared = 0.5312499 for second wave transition cases) for both cases. For polynomial regression of order four, we present the performance of the test model as follows: for first wave cases, multiple R-squared= 0.9437, p-value= 3.448e -09 , relative standard error is 0.5152 and the residual median is 0.03408, value close to zero which shows that the model performs very well; for second wave transition cases, multiple R-squared= 0.9039, p-value= 1.829e -07 , relative standard error is 0.194 and the residual median is 0.00386, value close to zero which shows that the model performs optimally.

Exponential smoothing method

Simple exponential model (SES or ANN) is a good forecasting tool and in this case we used α = 0.2 as one of the parameters while other parameters varies depending on the case in consideration. The degree of freedom (df) is 8 while the model degree of freedom is 2 with 10 lags. Also, 60% of the Kuwait data was trained with 100 days forecasting while 40% was tested. It was observed that while training and testing the model, the best exponential model is when the trend is removed from the test set as it can be seen in Figure 4.11 even though the test RMSE is quite large but the p-value is low. Other exponential models like diff.SES (removing trend from SES), simple exponential smoothing with multiplicative errors (MNN) and simple exponential smoothing with additive errors (MAN) did not give better performance when their RMSE was compared which were extremely larger than the simple exponential model without trend. The median of the residual is zero for all cases except first wave cases whose median is -55, which shows that residual is normally distributed. The parameters and result for each case is as follows: for first wave cases the parameters used are l = 0. It is also important to present the visualisation of different types of exponential smoothing models (see Figure 4.12(a)-(d)) to give significance for recent observations and produce accurate forecasts of 100 days while Holt's model (AAN) as shown in Figure 4. 13(a)-(d) gives the trend and level of a time series and is computationally more efficient than double moving average. Holt's-Winters' model considers randomness using efficient smoothing process and is computationally efficient too. Holt's linear method with additive errors did not give better performance when we compared with the optimal Holt's model because the RMSE was extremely larger.

Performance, accuracy and validation of the exponential models

We give an explicit analysis of the performance and accuracy of the simple exponential without trend and Holt's model using Kuwait data. To conclude this Section, it is very important to give a critical look at the RMSE comparison in Figure 4.15a-d because it guides the choice of parameters to be used in the model. In Figure 4.15a we observed that for simple exponential model without trend on the left, if the α value is increased beyond 0.2, the RMSE value is on the increasing trend while for Holt's model on the right, it is best to use β of about 0.05 and after then the RMSE is on the increasing trend. In Figure 4.15b we notice that for the simple exponential model without trend on the left the choice for α must be below 0.2 and after that the RMSE continues to increase while for Holt's model on the right, β must be below 0.1 else the RMSE value becomes large. The Figure 4.15c is a bit tricky because for the choice of α makes the values of RMSE to be stationary till the point where α is 0.6 where the increasing trend begins for simple exponential model without trend on the left while for Holt's model on the right there was turning point when the value of β is 0.15 but the value gives minimum RMSE when β = 0.04 and β = 0.38, which means that the choice of β for this case is very critical and must be precise. In

Sensitivity analysis

Here, we present the sensitivity analysis of the parameters used in the modelling of our study. The support vector model performance is sensitive to the choice of the cost function, γ and ϵ parameters for the data set and that is why we used the idea of cross validation and optimal model to choose the best parameters that best fit the model. From Table 4.1, we present the sensitivity parameters for the cases we considered. We discovered that for ARIMA models, all the cases are more sensitive to the choice of the order of the auto regression and moving average while it is least sensitive to the trend difference.

For exponential models, the data is very sensitive to the choice of σ and α but least sensitive to the number of lags and choices of β. The optimal algorithm was able to choose the best parameter that fit the model.

Prophet and neural prophet forecasting

Prophet is a procedure for forecasting time series data based on an additive model where non-linear trends fit yearly, weekly, and daily data seasonal and holiday effects while neural prophet is a auto-regressive deep learning tool use for forecasting by using classical neural networks components. We observed that the death data from Kuwait has a better performance than the new case data. We present the visualisation and 100 days forecasting of the results in Figure 4.16(a)-(d) and the trend plots in Figure 4.17 We have also presented in Figure 4.17(a)-(d) a loss plot for our neural network model which helps to know the performance of the model. It was observed that the loss plot shows a good fit and convergence of the model. The plot of training loss decreases to a point of stability and also the plot of validation loss decreases to a point of stability and has a small gap (generalisation gap) with the training loss. In Figure 4.17e-f we provide the visualisation of the neural forecast of 100 days for the first wave cases and second wave transition cases which aligns with the trend of results we have presented in other forecasting results in the previous Section, and also from the observed results from worldometer, we noted a decrease in daily cases from end of April to May 2021 and this also aligns with the result presented in Table 4.2.

The two phases of the COVID-19 outbreak in Kuwait present differences: (a) the first wave start shows an increase of the daily new cases (with a peak of 1000 daily new cases) during about 60 days with a slope of the exponential regression equal to 0.07, followed by an endemic phase with a high mean number of daily new cases (about 600) and with a delay of about 15 days an increase of deaths and (b) the second wave transition shows a decrease of the new cases during about 50 days with a slope of the exponential regression equal to -0.009, followed by an increase during about 20 days.

Approach to COVID-19 Time Series Data Using Deep Learning Method

This section aims to use a data-driven approach to retro-predict and forecast the daily new cases and deaths of the COVID-19 outbreak from the start of the pandemic until September 3, 2021 at various stages of the waves in USA, India, Brazil, France, Turkey, UK, and Russia, which are the top seven countries in the world where the pandemic is the most prevalent. For prediction, various deep learning methods was used, while for short term forecasting, ELM, MLP and spectral analysis were used, and comparison of their performance was done using root mean square error (RMSE) and relative root mean square error (rRMSE).

Time series and evaluation metrics

Here, the time series visualization (see Figure 4.19) for Turkey, the United Kingdom, the United States, Russia, India, France, and Brazil for daily new cases, 3-day moving average, and 7-day moving average from [119] is presented. This time series data from [119] will be used throughout the analysis. When analyzing a model, it is critical to measure its performance in order to draw the best conclusion and interpretation for the time series data. Two errors were used to estimate the prediction and forecasting precision for the models, where for i = 1, 2, . . . , n, Y i 's are the observed values, n is the number of data points and y i 's are the predicted values given below: -Root Mean Square Error (RMSE), given as:

RMSE = 1 n n i=1 (Y i -y i ) 2 ,
-relative Root Mean Square Error (rRMSE), given as:

rRMSE = 1 n n i=1 (Y i -y i ) 2 max(Y i ) -min(Y i ) .

Parameters for the modeling in LSTM, GRU, CNN, and DNN methods

To evaluate the model, 80% of the data set was trained and 20% was tested. The MinMaxScaler feature in Python was used to normalize the data. The errors are shown in Table 4.4 and Table 4.5. Figure 4.20 shows the visualization of the results for daily new cases and Figure 4.21 for daily deaths. Other specific details of the parameter used for each model is presented as follows:

• LSTM: The best hyperparameter tuning for the LSTM, as determined by a manual search, is: Batch size of 32, epochs of 100, drop out = 0.2, and units = 50. Adam optimizer was used, and mean square error measures the effectiveness of the loss. The Adam optimizer was found to slightly outperform other learning algorithms.

• GRU: the best hyperparameter tuning for the model, as determined by a manual search, is: Batch size of 30, epochs of 100, drop out = 0.2, and units = 50. Adam optimizer was used and tanh activation function. Mean square error measures the effectiveness of the loss. The Adam optimizer was found to slightly outperform other learning algorithms.

• CNN: An input layer with three neurons was found, followed by convolution layers with 128 units, 64 units, and 16 units, a max-pooling layer with 64 units, and a kernel size of 3. The best hyperparameter tuning for the model, as determined by a manual search, is: Batch size of 32, epochs of 100, verbose = 0.2 but 0 for checkpoint, and validation split = 0.2. Adam is the optimizer in use. While the mean square error was used to assess the effectiveness of the loss, the relu activation function was used for all layers expecting the output with linear function. Adam optimizer was found to slightly outperform other learning algorithms.

• DNN: The DNN has two hidden layers with 32 and 8 neurons, respectively, and it was discovered that if the model goes beyond 32 neurons, the model tends to give much higher error with poorer prediction. The best hyperparameter tuning for the model, as determined by a manual search, is: Batch size of 32, epochs of 100, verbose = 0.2 but 0 for checkpoint, period = 1, and validation split = 0.2. The optimizer used is Adam, and mean square error measures the effectiveness of the loss. The relu activation function was used for all layers expecting the output with linear function. Adam optimizer was found to slightly outperform other learning algorithms.

Parameters for the modeling in ANN method

Here, a visualization of the network weights and the bias between the data is presented. As shown in Figure 4.22 and Figure 4.23, the weights are good with low bias. Also, some of the data were predicted (the test data) and the daily new cases prediction scores in Figure 4.22 for (a) France is 1.0, (b) India is 0.99, (c) USA is 1.00, (d) Brazil is 0.99, (e) Russia is 0.99, (f) Turkey is 0.10, and (g) UK is 0.99. In Figure 4.23, shows the daily deaths prediction scores for (a) France is 0.93, (b) India is 0.85, (c) USA is 0.99, (d) Brazil is 0.88, (e) Russia is 0.15, (f) Turkey is 0.99, and (g) UK is 0.51. It was discovered that the daily deaths errors were lower than those of the daily new cases.

Parameters for the modeling in MLP and ELM methods

Here, the description of the best parameter used for each case based on performance and forecasting of daily new cases and deaths. For Turkey daily deaths and Russia daily new cases and deaths, 12 lags, five hidden layers, and 20 repetitions were used, with series modelled in difference and forecast combined using the median operator, as shown in Figure 4.24(d), For daily new cases, 24 lags for Turkey, India, and Brazil was used, as well as for death cases for the United Kingdom while keeping 12 lags and testing the rest for inclusion. Other parameters, such as those shown in Figure 4.24(c), Figure 4.24(f), 25(e), were kept. In addition, for daily new cases in France, United States, United Kingdom, and Brazil, 24 lags was used while retaining all lags, as shown in Figure 4.24(a), Figure 4.24(b), Figure 4.24(g) and Figure 4.25(f). For India daily death cases, four hidden layers was used, 20 repetitions, and 12 lags while keeping the same parameters for forecasting and series modelling as shown in Figure 4.24(e). Figure 4. 25(a) shows the use of 24 lags for daily deaths in the United States, with no testing for inclusion. Finally, for France daily deaths, the trend with no differencing was removed while also using regressors and 12 lags. The errors are shown in Table 4.4 and Table 4.5.

According to Table 4.4 and Table 4.5, DNN has the least error for the daily new cases in the United States, and by looking at Figure 4.20(a), one can say that the estimation is good and it reflects the situation of the pandemic in the country as new cases fluctuate, just like daily deaths, which are best predicted by GRU with the least error. For the UK, MLP performed better than other models with the least error, as shown in Figure 4.24(b), where daily new cases are decreasing from November 2021 as observed from real data, while CNN best predicts daily deaths, as shown in Figure 4.21(b). For Russia, the daily new cases and daily deaths are best predicted by MLP, which has the least error, and from Figure 4.24(d) and Figure 4.25(d), we can see that the forecast curve is increasing, which corresponds to the current situation in the country, and with the festive season approaching in December 2021, it is expected that cases will increase if not controlled and also as new variants emerges because the virus keeps mutating. For France, CNN best predicts daily deaths, indicating that, despite a decreasing trend, cases will fluctuate as shown in It is also observed that all the methods are sensitive to the parameters being used: for instance, for DNN if we go beyond 32 neurons, the model tends to give much higher error with poorer prediction. Also, for MLP if we use too many lags and neurons for some of the data, the prediction will be poorer, which was explained earlier in Section 4.2.2. The training time used in the modelling varies across models and that is one factor that enhances their performance: it is observed that LSTM training time was the lowest, hence, it does not perform better in any of the data set. Eventually, it was noticed also that the model performance depends highly on the data size.

Approach to COVID-19 Time Series Data Using Spectral Analysis Method

In this section, spectral analysis is used to measure the different frequencies in COVID-19 data set presented in Section 4.2.1 during the contagiousness period by estimating the time series periodicity and analyzing their peaks. Spectral analysis allows to remove the weekly and seasonal components discovered on the epidemiological data of incidence and mortality. Firstly, the stationarity and normality is checked before applying spectral analysis to the set of time series data. The Box-Lung test, the Jarque-Bera normality test, and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test was used and it was found that the p-value ≤ 0.01 for all of these tests and for all of the time series considered in this article, leading to the conclusion that the data set is stationary and its residual is normal. So, the estimation of the spectral density and smoothed the periodogram was done and also the plotting of the spectral density and spectrum and then performed harmonic regression because the goal of spectral analysis is to decompose a time series into periodic components. This was considered by performing a regression, in which the time series was regressed on a set of sine and cosine waves and then attempted to regress the time series on harmonic waves, including daily harmonics as well as other harmonics. For the entire data set, a p-value ≤ 0.05 is obtained. The daily variation were accurately described and the median of the residual was negative and close to zero, confirming the normality. The scaling of both x and y using a continuous scale for x and log scale for y is done. Log-scaling has some theoretical advantages. The periodogram values should be roughly normally distributed in the log scale, and log scaling can be useful because it spreads out the low frequencies while squashing the high frequencies. Due to the noisy nature of the periodogram, it was smoothed by using 9 moving averages for the kernel functions of the periodogram. To estimate the spectral density, tapering and multi-taper was used, which is a method that allows to test for peaks using the F-test. It was discovered that p-value ≤ 0.001 for the entire data set, indicating that there are peaks in the model. The time bandwidth parameter (NW) used is 16 and the number of tapers is 31. Figure 4.27 and Figure 4.26 show the results of the spectral analysis as well as a forecasting using the analysis, methods and theoretical study given in [START_REF] Golyandina | Basic singular spectrum analysis and forecasting with r[END_REF].

Modeling of COVID-19 Pandemic vis-à-vis Some Epidemiology, Socio-economic, Geo-climatic and Demographic Factors

The use of machine learning methods to analyze data has been helpful over the years to get a proper view on how a model behaves. In this Section, some supervised and unsupervised machine learning methods were used and also tried to use two deep learning methods. To jointly interpret the socio-economic, geo-climatic, demographic and epidemiological data, these main classes of the descriptive statistics was chosen, which allows to compare them. Supervised learning is used in its regression function (prediction of a quantitative variable from annotated examples) and unsupervised learning (in which the data is not labeled) in its classification function. As for deep learning, it makes it possible to create a model from large-scale unlabeled data. The supervised machine learning methods used are first univariate polynomial regression, linear regression, lasso regression, and ridge regression. Also, some of these methods was used to make prediction by training the model and testing some percentage of the values. Lasso regression helps to know the best variables to be used in the modeling. After the univariate regressions, multivariate least square methods was introduced, allowing to test much more complex relations between variables.

After the supervised learning methods, unsupervised learning approaches was applied to cluster variables across countries and the methods proposed to validate the results were K-means clustering, Hierarchy clustering, and Principal Component Analysis (PCA). Correlation calculations was performed among parameters used in the modeling step. An optimization method called Ordinary Least Square (OLS) for the epidemiology, socio-economic, geo-climatic and demographic determinants of COVID-19. Eventually, the deep learning methods used were Neural Network (NN) and Multi-Layer Perceptron (MLP) regressor, which is a class of feedforward Artificial Neural Network (ANN).

Socio-economic, geo-climatic and demographic variables

Socio-economic, geo-climatic and demographic variables constitute a strong determinant of the spread of the pandemic. These variables were extracted from [96,160,214,[START_REF]Life expectancy at birth in africa as of 2019, by country[END_REF]227,228,229,230, 231], while some were calculated like the socio-economic fracture index. Some of the observed variables used are immigration rate (IR), average life expectancy (LE), Tuberculosis incidence (TB), temperature, percentage of gross domestic product devoted to health expenditure (CHE/GDP), percentage of 10% lowest (LI) and 10% highest incomes (HI), government response stringency index (SI), sustainable development goal (SDG) index, human development index (HDI), environmental performance index (EPI), consumer confidence index (CCI), stringency index (SI), Theil index (TI), and Gini index (GI). Other variables used are in Table 4.6, precisely for Cameroon regions. The data was collated based on the available countries and most recent years in public databases [96,160,214,[START_REF]Life expectancy at birth in africa as of 2019, by country[END_REF]227,228,229,230,231]. The calculated socio-economic variables are as follows:

-Social fracture (SF) index is the ratio between the 10% highest income and the 10% lowest income. In brief, it is expressed by the equation below:

SF = 10%HI 10%LI
-Demo-economic (DI) index is the ratio between the percentage of GDP devoted to health expenditure and social fracture index. It is expressed by the equation below:

DI = CHE/GDP SF
The precise value of all variables are in Table 4.8, Table 4.9, Table 4 Lastly, the statistical approach is in two folds: (1) to visualize the weights of the neural network and the bias between Theil and Gini index, and as it can be seen in Figure 4.29H, the weights are good with low bias and (2) to use linear and polynomial regression model of order 2 in order to see the relationship between Theil index and Gini index. For the linear regression as shown in Figure 4.29A, the intercept is 31.03, p-value is 0.0181, R-squared is 0.4881, residual standard error is 3.116, and all coefficients are significant with pvalue < 0.05 for both the train and test data for linear and polynomial regression. The median of the residual plot in Figure 4.29B and Figure 4.29F are 0.2111 and 0.2566, respectively, for both linear and polynomial regression, which are low values. The normality of the residual was tested using Jarque-Bera and Durbin-Watson tests, which gave a high p-value, and the null hypothesis was rejected, that is, the skewness and kurtosis of the residuals are statistically equal to zero. In order to know the performance of the linear regression model, 80% of the data was trained and 20% was tested and also did cross-validation to be sure of the accuracy. The predicted and the observed values are very close to the results presented for the regression models used. For the linear model, the cross-validation result is in Figure 4.29E whose average mean square error for the five portion folds is 11.72794. The correlation between the tested and the predicted values has high accuracy (R-squared = 0.97). The test set p-value is 0.02 with a residual standard error of 3.528. For polynomial regression of order 2, the train set has the following results: R-squared = 0.6, p-value = 0.002, and residual standard error = 2.935. The test set has the following results: R-squared = 0.99, p-value = 0.008, and residual standard error = 0.5639.

Correlations between epidemiologic, socio-economic and demographic parameters

In this section, two sets of correlation is done: (i) correlation between Theil and Gini indices with all epidemiologic, demographic, and socio-economic variables for Latino-American countries and (ii) correlation between epidemiologic, demographic, and socio-economic variables for Cameroon regions.

In ). The parameters list is given in Table 4.6 and their values are extracted from Table 3.4 and from [126,227]. The correlations between the parameters of Table 3.2 considered as variables on the 10 regions of Cameroon can be analyzed thanks to Figure 4.30 and Figure 4.31, which show that the epidemiologic parameters (from parameter Dose 1 to parameter Recovered) are significantly anti-correlated (pink color) with many socio-economic and demographic parameters, namely the three first ones (Subnational Human Development Index, International Wealth Index and Gross National Income per capita) and to those linked to the house holding, showing the influence of a good socio-economic state on the resistance against the virus. By exploiting the data corresponding to the parameters of Table 4.6, it is possible to analyze the correlations existing between three families of parameters observed in the different regions of Cameroon during the second wave of the COVID-19 outbreak. In Figure 4.30, it can be observed that the socio-economic factors are highly positively correlated with each other, but negatively correlated with demographic parameters and not correlated with epidemiologic parameters, except those corresponding to vaccination and death which are anticorrelated with the signs of wealth. This can be explained by a higher socio-cultural level, which pushes individuals to monitor their health more and to get vaccinated. In Figure 4.31, it can be deduce that a greater number of demographic parameters are anti-correlated with socio-economic factors. These are in particular infant mortality, mean age, fertility and poverty indicators, which have positive or zero correlation coefficients between them and very negative with wealth indicators, such as GNI or possession of material goods or land (television and house), because in general a rise in the standard of living causes fertility and infant mortality decrease.

Multivariate analysis, clustering and prediction for Latino-American countries

The multivariate analysis using ordinary least square method, clustering analysis using PCA, K-means and Hierarchy clustering and the prediction of the Gini index using MLP regressor which is a deep learning method, linear, lasso and ridge regression for Latino-American countries is presented. The multivariate analysis using ordinary least square method, clustering analysis using PCA, K-means and Hierarchy clustering and the prediction of the percentage GDP devoted to health expenditure using MLP regressor which is a deep learning method, linear, lasso and ridge regression for developed and developing countries is presented. • Figure 4.39E-H correspond to the ordinary multivariate least square method with R-squared = 0.76. Figure 4.39E shows some developing countries as outliers, while Belgium is the only developed country, which does not fit the data. Cross-validation method helps to choose the best parameter α for the modeling as shown in Figure 4.40C.

For ridge regression, α = 0.012 with a mean square error = 2.32 and α = 0.029 for the lasso regression with a mean square error = 2.21. For Also, sextic regression (polynomial regression of order six) was used to studied the correlation between the value of the opposite of the slope at the origin of the autocorrelation function of the first wave and the economic and health index CHE/GDP, by studying a polynomial regression of degree 6 (Figure 4.42) whose results is as follows:

-Linear regression: slope = 0.01117, intercept = 0.0664, r -value = 0.47219, p-value = 0.0097, standard error = 0.004, RMSE = 0.0387, -Polynomial regression (order 6): RMSE = 0.04399, R-squared = 0.4.

It shows an anti-correlation in the linear regression and a local maximum for countries with an average CHE/GDP ratio of around 7. Countries with a high CHE/GDP ratio (such as France and the United States) have a low value in opposite to this slope.

The explanation for this phenomenon may come from the correlation reported in the introduction between the CHE/GDP and Gini indices, the poor classes having a longer duration of contagiousness due to a less important state of immunological defense and perhaps less compliance with mitigation measures. • The correlation between the first wave exponential regression slope and the CHE/GDP index for developed and developing countries is significantly positive (R = 0.57) in Figure 4.44 with the result as follows: slope = 0.026632, intercept = -0.1052912, r -value = 0.5661, p-value = 7.60 ×10 -5 , standard error = 0.00605655, R-squared = 0.320470, RMSE = 0.095836.

• In Figure 4.43, the regression analysis result is as follow:

Multivariate analysis, clustering and prediction for African countries

The multivariate analysis using ordinary least square method, clustering analysis using PCA, K-means and Hierarchy clustering and the prediction of the temperature (which is a variable seen to influence the spread of the pandemic in this continent due to low cases and deaths observed) using MLP regressor which is a deep learning method, linear, lasso and ridge regression for African countries is presented.

• Figure 4.47A-D correspond to the ordinary multivariate least square method with R 2 = 0.60.

• Figure 4.47A shows Botswana and Tanzania as outliers not fitting the data. Crossvalidation method helps to choose the best parameter α for the modeling as shown in Figure 4.45C.

• For ridge regression, α = 1.005 with a mean square error of 19.13,

• For lasso regression, α = 6.018 with a mean square error = 16.93. For CHE/GDP index for developed and developing countries is significantly positive (R = 0.57) on Figure 4. Figure 5 shows the same but for developed countries only with a still higher correlation (R = 0.65). It is evident from these results that the lasso regression best predicts temperature with the highest test score, and predicted values of temperature for lasso and ridge regression are close. All the regression methods give about the same result with the maximum accuracy for the ridge regression. In Figure 4.46E and Figure 4.46F, the first cluster has 40 countries, the second cluster has 13 countries, while the last cluster has only 1 country, which is Botswana.

Only two cluster dendrograms with many countries are presented. In

Multivariate analysis, clustering and prediction for OECD countries

The multivariate analysis using ordinary least square method, clustering analysis using PCA, K-means and Hierarchy clustering and the prediction of the percentage GDP devoted to health expenditure using MLP regressor which is a deep learning method, Linear, Lasso and Ridge regression for OECD countries is presented.

• Figure 4.48 corresponds to the ordinary multivariate least square method with R 2 = 0.90. • Figure 4.48A shows Iceland, United States, Austria, and Belgium as outliers not fitting the data.

• The partial regression plots in Figure 4.48C shows that the best correlation observed is between CHE/GDP and the demo-economic index DI as observed before in [START_REF] Demongeot | Data fusion for analysis of persistence in pervasive actimetry of elderly people at home, and the notion of biological age[END_REF].

• Cross-validation method was used to choose the best parameter α for the modeling as shown in Figure 4.49D.

• For ridge regression, α = 0.005 with a mean square error of 1.905, and for Lasso regression, α = 0.027 with a mean square error = 1.657.

• For It is evident from these results that the lasso regression best predicts percentage of GDP devoted to health expenditure with the highest test score and predicted values are very close.

All the regression methods give about the same result with the maximum accuracy for the ridge regression. Only the two highest cluster dendrograms is presented. In Figure 4.50C, the Gini index and social fracture index have the highest positive correlation of 0.45 and 0.46, respectively, in PC 1 while the percentage of GDP devoted to health expenditure and demo-economic index have the highest positive correlation in PC 2, whose values equal to 0.65 and 0.41, respectively. The two main clusters correspond both to developed countries, but in the first, countries are more continental, and in the second, countries are more maritime, which could be explained by their difference in consumer confidence index (CCI), which is less important in maritime countries than in continental ones.

ARIMA Model for First and Second Wave

The ARIMA model shows more than 95% confidence interval as it can be seen in Figures 7a to 7d with p value for Mali for first wave is p = 0.01 and for second wave it is p = 6.3𝑒 -10 while for first wave for Slovenia p = 0.01 and for second wave in Luxembourg p = 0.01.

First wave ARIMA Model

The comparison during the first wave between two countries (Figure 7), one developed (Luxembourg) and one developing (Mali) shows a difference in length of contagiousness period (linked to the value of the opposite to the slope at origin of the autocorrelation function) and shape of the growth curve, indicating a lower virulence of the SARS Cov-2 in Mali, possibly due to the influence of the temperature [START_REF] Desjardins | Rapid surveillance of covid-19 in the united states using a prospective space-time scan statistic: Detecting and evaluating emerging clusters[END_REF]. This tendency is reversed during the second wave between Mali and Slovenia (Figure 8). The clustering and multivariate analysis using ordinary least square method for all countries from epidemic and economic variables is described in

Discussion

Discussion

There are a lot of differences between the first and second wave results concerning 3. The analysis of parallel coordinates for cluster centroids also shows the importance of the deterministic R0 in the discrimination of clusters (Figure 12); 4. The analysis of the residuals shows a good explanatory power of the first three principal components (60% of the total variance in Figure 11c, confirmed by the projections on the two first principal planes of Figures 11d and11e), and a weak correlation of the principal components with these residuals (Figures 13a and13b). The multivariate analysis using ordinary least square method, clustering analysis using PCA, K-means and Hierarchy clustering and the prediction of the percentage GDP devoted to health expenditure using MLP regressor which is a deep learning method, linear, lasso and ridge regression for all countries is presented.

Parallel coordinates plot for the centroids

To conclude this section, a new approach to the modeling of epidemiology, socio-economic, geo-climatic, and demographic determinants of COVID-19 outbreak during the exponential phase has been developed. Some of these determinants have shown high correlation with epidemiologic parameters as it can be seen in the heatmap diagrams in For developed and developing countries, the lasso regression reduced the correlation between the social fracture index and the 10% highest income, while for OECD countries, the correlation between the Gini index and social fracture index was reduced to zero. Some of the variables were not used in the optimization method-OLS due to multicollinearity observed on results summary. For the two sets of countries, consumer confidence index, opposite of the initial autocorrelation slope averaged on 6 days for the first and second wave, 10% lowest income, and 10% highest income were not used in the modeling. The R-squared for OLS results for developed, developing, and OECD countries are 0.76 and 0.90, respectively, which shows a high significance rate ( What can be deduce from all these observations is that the socio-economic determinants are key to the modeling of infectious diseases like COVID-199 various countries ( [START_REF] Gaudart | Factors associated with the spatial heterogeneity of the first wave of covid-19 in france: a nationwide geo-epidemiological study[END_REF][START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF][START_REF] Griette | A robust phenomenological approach to investigate covid-19 data for france[END_REF][START_REF] Soubeyrand | Towards unified and real-time analyses of outbreaks at country-level during pandemics[END_REF]). Similarly, there are a lot of differences between the first and second wave results concerning the exponential regression slope and the auto-correlation initial slope: while some countries have higher figures for the first wave, others have lower figures for the second wave and vice versa. This was also evident for the regression plot where some countries have negative correlation values for the first wave of some growth parameters with the CHE/GDP and positive for the second wave, and vice versa for other countries. These phenomena prove that the way the pandemic spread in the second wave is different from what was experienced in the first wave. In the principal component analysis, we discovered that first wave deterministic R 0 and CHE/GDP health had high weights in first and second principal components (PC1 and PC2), which are dominant components in the PC analysis. More precisely, on Figure 4.42, for the first wave the opposite of the initial auto-correlation slope is positively (resp. negatively) correlated with the CHE/GDP (resp. maximum R 0 ) for developed countries, which could correspond to the efficiency of the mitigation measures decided in these countries. This is confirmed on Figure 4.44, where the first wave exponential regression slope is positively correlated with the CHE/GDP in a mix of developed and developing countries. The Figure 4.43a shows the same type of effect of public health policies in developed countries for the first wave, where CHE/GDP increases with the first wave exponential regression slope, but this result is inverted on Figure 4.43b for the second wave perhaps due to a rationalization of the care activity between the first two waves. For the first wave of all countries, Figure 4.52 shows the same positive correlation as Figure 4.43a between the exponential regression slope and CHE/GDP. Clustering of all countries is then studied on (c) corresponding to more "developing" (in red with some notable exceptions such as the Czech Republic and Germany) and (c) more "developed" (in green and partially in orange) countries parts of the hierarchy tree, with a small "exotic" cluster for Tanzania and Mauritius. Figure 4.54(a)-(e) shows the results of the principal component analysis (PCA), with (a) the three principal components declined on the initial variables calculated for all countries (first and second waves maximum R 0 's denoted first wR 0 and second wR 0 , deterministic R 0 's denoted first wR det 0 and second wR det 0 , Arima slopes denoted first wArima, second wArima slopes, and the current health expenditure as gross domestic product percentage denoted CHE/GDP), (b) the projection of the points corresponding to countries of the PCA's plot on the first PC plane, (c) the explained variance plot and (d,e) the correlation circles for the first three principal components with projection of the initial variables as vectors (having 195 components corresponding to the 195 countries of the table in appendix on the corresponding principal planes. In Figure 4.54(a), the main initial variable in the linear combination of the first (resp. the second) principal component is the first wave deterministic R det 0 (resp. the CHE/GDP) and these two initial variables R det 0 and CHE/GDP are anti-correlated as we have already noticed when commenting before on the Figure 4.42 (a country devoting a large share of its GDP to health expenditure reduces the occurrence of new cases). The main purpose of this section is to revisit the data on COVID-19 from public databases using methods that are still little used, such as Functional Data Analysis (FDA), in which there is a great deal of theoretical work [START_REF] Ferraty | Curves discrimination: a nonparametric functional approach[END_REF][START_REF] Ferraty | Nonparametric models for functional data, with application in regression, time series prediction and curve discrimination[END_REF][START_REF] Ferraty | Functional nonparametric statistics: a double infinite dimensional framework[END_REF][START_REF] Demongeot | Functional data: local linear estimation of the conditional density and its application[END_REF][START_REF] Rachdi | Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data[END_REF][START_REF] Demongeot | On the local linear modelization of the conditional distribution for functional data[END_REF][START_REF] Demongeot | Relative-error prediction in nonparametric functional statistics: Theory and practice[END_REF][START_REF] Demongeot | Local linear regression modelization when all variables are curves[END_REF][START_REF] Altendji | Functional data analysis: estimation of the relative error in functional regression under random lefttruncation model[END_REF][START_REF] Henien | Test d'hétéroscédasticité quand les covariables sont fonctionnelles[END_REF], but practical applications are still rare. In particular, the generalization at the FDA of classical finite-dimensional methods, such as estimation, regression and principal component analysis, shows that it is possible to process epidemic data obtained from a large sample (approximately one million data) concerning the incidence, mortality and exogenous or endogenous factors associated with the COVID-19 epidemic. This sampling concerns about ten variables (sometimes missing, in particular those concerning risk factors) and includes longitudinal (about 600 days all over the world but in this case about 469 days for French departments) and cross-sectional data (about 200 countries, some comprising up to a hundred documented regions while for France which is the main focus of this section there are 101 departments), for the monitoring and prediction of a pandemic whose origin and end are still uncertain, but which, by its magnitude and its dramatic consequences (around 5 million deaths) justifies such a descriptive statistical investment. The descriptive study of many factors associated with the epidemic (namely the socio-economic and geoclimatic ones) makes it possible to understand closely linked mechanisms, those exogenous or endogenous to the viral pathogeny of propagation and endogenous ones of pathogenicity [START_REF] Demongeot | Temperature decreases spread parameters of the new covid-19 case dynamics[END_REF][START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF][START_REF] Seligmann | Unpredictable, counter-intuitive geoclimatic and demographic correlations of covid-19 spread rates[END_REF][START_REF] Demongeot | Sars-cov-2 and mirna-like inhibition power[END_REF][START_REF] Soubeyrand | Towards unified and real-time analyses of outbreaks at country-level during pandemics[END_REF]. The seminal paper by Bernoulli [START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF] on the SI model is the origin of all future discussions about epidemic modelling, in particular the first by d'Alembert [START_REF]Opuscules mathématiques[END_REF] in his Opuscules mathématiques and Lambert until its last recent refinements. It contains the explicit solution of the SI model given in Equation (3.36) and permits to consider the empiric S(t) and I(t) curves as functions. It can be manipulated (after smoothing) as elements in a functional space. The goal here is to apply to these elements the FDA descriptive techniques like Canonical Correlation Analysis (CCA), K-means clustering and Functional Principal Component Analysis (FPCA) in order to compare these functions among different departments in France before vaccination started and after vaccination has started and to finally make some predictions about the evolution of the disease in France. The functional variables are numbers of ICU cases, daily deceased (DD), daily return home (DRH) and hospitalization which are given as

X 1 , X 2 , X 3

Time series and curve fitting

The time series analysis of daily new cases in France, daily hospitalization in three French departments out of 101 in France, and curve fitting for two French departments is presented. The RMSE for other curve fittings is shown in

Data smoothing

The first step in analyzing functional data is to smooth the curves. In this section, different smoothing techniques which shall be illustrated and basic explanation of the techniques deployed for smoothing the functional data will be presented. The mean of the data set which corresponds to elastic mean and the cross-sectional mean is plotted (see Figure 4.57), which corresponds to the karcher-mean under the L2 distance. The karcher-mean has an important role in the warping framework and is used to align the functions in order to improve the matching of features (peaks) across functions for a given set of warping functions

∂ 1 , ∂ 2 , • • • , ∂ n ∈ ∆ as ∂ n = argmin ∂∈L2=S∆ n i=1 dist(∂, ∂ i ) 2 ,
where S is a quotient space, ∆ is the differential geometry, ∂ ∈ ∆, ∂ i is the set of warping functions [START_REF] Srivastava | Functional data and elastic registration[END_REF].

It was observed that the elastic mean better captures the geometry of the curves compared to the standard L2 mean for some of the functional data set considered, since it is not affected by the deformations of the curves. This phenomenon can be seen in In Figure 4.59, the correlation coefficient between all the departments in France based on the functional data in consideration is plotted, in order to see how well the data is well correlated between the departments and it was observed that there is a high correlation between various departments with except in few cases where low correlation is observed as we can see in the contour plots presented in Figure 4.

59(a)-(h).

The further smooth the data, spline interpolation of order 3 and smoothing the interpolation using the smoothness parameter equal to 1.5 in the cubic spline smoothing was used. Also the monotone technique and a piece-wise cubic Hermite interpolating polynomial (PCHIP) was used. The visualization of the results is in Figure 4.60.

From graphs Figure 4.603(a)-(f) one can deduce that cubic spline smoothing curves exhibit oscillations and oscillations are important to know the low and high data thresholds in case the consecutive data points experience a significant change in slope. It was also observed that PCHIP is smooth and non-oscillatory despite some sharp increase as the U-shape of the curve deepens. Figure 4.61 shows the Kernel smoothing for the smoothed curves of the functional data set for three different smoothing methods to see how cross validation score varies over a range of different parameters used in smoothing methods. The essence of this smoothing is to estimate the smoothing parameter h that better represents functional data. It has been selected by Generalised Cross-Validation criteria (GCV). The results show a comparable behavior of these scores by varying the smoothing parameter h.

Functional principal component analysis (FPCA)

Functional principal component analysis (FPCA) is a dimension reduction with high correlation in functional data analysis which completes the statistical tools used in the modelling of epidemiology data. In this section, the 4 PCs values plot is presented throughout the days considered and the principal component scores plot for all the different departments providing functional data being before vaccination started and during vaccination. In Figure 4.62(a) it was observed that PC 1 peaked in the early days of the pandemic between February and March 2020 and then there was a decline after about 50 days becoming stationary till day 150 possibly due to mitigation measures promulgated during this period. The same phenomenon has been observed for PC 2. In Figure 4.62(a), PC 4 shows a sinusoidal shape, peaked at day 100 which is around June 2020 with least values at day 30 and day 180 which are respectively in March and August 2020. Figure 4.62(b) shows the same sinusoidal shape for PC 4 and same shape for PC 3 but with a drift in the observation with a difference between the dynamics of hospitalization cases before and after vaccination has started in France. PC 1 in Figure 4.62(b) shows a decline across the infective period which may be due to the aggressive vaccination campaign in the country. In Figure 4.62(c) it was observed that from day 50 (around April 2020) till day 150 (around July 2020), the PC 1 value which is the major PC is stable throughout this period of various confinement measures in France and all PCs tend to show increasing behavior after the confinement measures have been relaxed and in In Table 4.12, the PCs variance proportion is presented and it was observed that PC 1 is the most important principal component. This description shows that there is a difference between the vaccination period in France and the period when measures like lockdown, social distancing etc. were only used to control the spread of the virus despite the fact that it has been proven medically that people can be vaccinated and still be infected. The diagrams in Figure 4.63a-h show the same shift toward positive PC 1 values. These shifts demonstrate the effect of various mitigation measures differences in the departments and based on population and migration in this area, particularly the five departments (Mayotte, Guadeloupe, La Réunion, Guiana and Martinique) outside France's metropole where the rules in France are not strictly enforced. It also demonstrates that PC 1 is the most important PC, from which the majority of the analysis information can be obtained.

Canonical correlation analysis (CCA)

Canonical correlation is a statistical analysis method that is used to simultaneously correlate several metric dependent variables and several metric independent variables measured on or observed with similar experimental units. It helps to find linear combinations within a data set with the goal of maximizing the correlation between these linear combinations. The variables used in this section are presented in Table 4.13. X are the variables listed in the first row of Table 4.13 i.e., total number of hospitalizations, daily return home, deceased and ICU cases for all departments before and after vaccination has started in France and Y variables are the response variables described earlier as presented in the first column of Table 4.13. The visualization of the results can be found in Figure 4.64 and Figure 4.66 and also the correlation scores is presented in tabular form (see Table 4.13).

The helio plot in Figure 4.64 depicts the relationships between the different departments in France, as well as the epidemiology variables and control measures (vaccination). 4.13 there is high correlations between the variables considered. The Figure 4.65 presents the redundancy between the canonical variates. The canonical correlation is tested and the null hypothesis is when the canonical correlation is equal to zero.

Clustering method

The clustering of functional data is one method that statisticians are always interested in and in this section the K-means and fuzzy K-means techniques whose algorithm is already in Python was used. These methods will enable us to visualize how various departments are clustered based on our functional data and to give it the best interpretation based on their geographical location. The basic function used for the K-means clustering is a B-spline and results of the clusters are presented. The result in the cluster form and also on the map of France with indication of the membership to the 3 (i.e., K=3) clusters (0, 1 or 2) to get a clearer view of the result are presented. Even if the value of K is increased, it won't improve the results as K is optimal. Only the result for two cases (daily hospitalized and daily deceased) for the period before vaccination begins in France and two cases (daily return and ICU cases) for the period when vaccination has started in France is presented. In Figure 4.67(e) and Figure 4.67(g) the same pattern for the data points in the clustering were observed which means that the same way that there are more hospitalization cases before vaccination, also there are more people returning home during vaccination which affirm the fact that the vaccination campaign in France has helped to mitigate the spread of the disease. Figure 4.67(f) and Figure 4.67(h) is a bit tricky because of the pattern they followed but it is not surprising that there is more deceased in most departments before vaccination period and less ICU cases during the vaccination period. These patterns of having more cluster points attaining the highest peak in PC 1 is distinct in the results presented.

Function-on-function linear regression modeling

The modeling of the functional data using function-on-function linear regression of the form of Equation (2.25) i.e., y AllfdData (t) = β 0 (t) +

Conclusion

Modern statistics applied to large-scale data analysis is computationally intensive, because many routine statistical tasks, such as distance matrix calculations, hierarchical clustering, principal component analysis, and so on, are difficult to calculate. Data analysis is widely used to gain new insights into viral disease evolution and to develop better mitigation methods and approaches. The current study demonstrated the ability of machine learning models and deep learning to accurately identify major components of viral disease such as COVID-19 and several determinants. Through the feature selection process, some of the regression models predicted some epidemiology variables and socio-economic variables, as dominant determinants in statistical modeling of the COVID-19 outbreak. The role of social inequalities in the current COVID-19 pandemic is becoming increasingly clear, both in terms of disease incidence and severity. Although the exact magnitude of these health disparities is difficult to quantify, the populations most at risk or vulnerable are frequently underrepresented in epidemiological studies and data. Many countries are in the grip of a severe and destructive recession. A significant number of firms and businesses have been downsized or closed. We have proposed a set of different regression methods in order to find the best ones in the Kuwait context, both for daily new cases and deaths, without a priori about the degree of nonlinearity and the stochastic structure of noise behind the data. Surprisingly, we discovered that often the best regression method was the support vector one and that the stochasticity of the data at the start of the two waves was the same. This result confirmed our choice of comparing several regression methods (exponential regression being the most commonly, and often the only one, chosen) and showed that once the trend due to epidemic dynamics and its seasonality has been removed, the random factors explaining the variations compared to the deterministic model of Bernoulli-Ross-McKendrick (namely the uncertainties related to the counting of new cases and deaths) were expressed through a similar noise for the first and the second wave. More, these two phases show a certain homogeneity in their stochastic structure at their beginning, because they have the same level for the variation coefficient (around 0.5), both for daily new cases number and for death number, and for the auto-correlation initial slope (-0.031 for the first phase and -0.038 for the second).

We have shown in this chapter that there exist correlations between the growth parameters directly linked to the occurrence of new cases of COVID-19 and socio-economic variables, in particular the ratio between the current health expenditures and gross domestic product percentage (CHE/GDP) anti-correlated with the basic reproduction time R 0 , which shows the effectiveness of public health mitigation measures, even if they involve significant medicoeconomic costs. Larger perspectives are offered by combining this study with others on geoclimatic and demographic severity factors of the COVID-19 outbreak [START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF][START_REF] Tang | Estimation of the transmission risk of the 2019-ncov and its implication for public health interventions[END_REF] with the present socio-economic determinants, in order to obtain the most comprehensive and accurate picture of non-biological exogenous influences on the expanding COVID-19 pandemic.

In general, we have also been able to access the interplay between socio-demo-economic variables and epidemiologic variables for explaining how inequalities in these socio-demoeconomic variables across countries, i.e., in terms of developed countries and developing countries responded to the COVID-19 outbreak. Also, from the results presented we can see that, there is a nexus between economic decisions taken by these countries and COVID-19 pandemic effects on the general economic activities in those countries.

Chapter 5

General Conclusion and Future Work

Scientific Contribution of the Research

The contributions made by this thesis are as follows:

■ We have been able to provide the equation of the discrete dynamics of the epidemic's growth and obtain an estimation of the daily reproduction rates by using a deconvolution technique on a series of new COVID-19 cases from many countries. We established that the daily reproduction rates has four unique shapes but the U-shape is the most frequent evolution of daily reproduction rates, which confirms the comparison with the behavior of seasonal influenza.

■ We have proved that socio-demo-economic determinants play a crucial role and that there is a relationship between these determinants during the exponential phase of the COVID-19 pandemic modeling.

■ We have employed functional data analysis method to analyze the shapes of COVID-19 outbreak incidence rate curves in different departments in France and statistically group them into distinct clusters according to their shapes. The result reveals that pandemic curves often differ substantially across departments of same country, and we show that it is possible to analyze epidemic data obtained from a large sample concerning the incidence, mortality and exogenous or endogenous factors associated with the COVID-19 epidemic.

■ We used the spectral analysis method to convert time (days) to frequency in order to analyze the peaks of frequency and periodicity of the time series data. Some of the peaks observed in the time series data correspond with the proven pattern of weekly peaks that is unique to the COVID-19 time series data and is due to the weekly incidence reporting.

■ We proposed a new mathematical age-dependent SIGR (Susceptible-Infectious-Goneanewsusceptible-Recovered) model for the COVID-19 outbreak. We were able to simulate the differential dynamic behavior of the growth of infectious and immune populations, young and old in different countries, in order, for policy makers, to be able to have an idea of when to adjust the vaccine policy according to the age groups.

■ We have been able to use different deep learning methods to retro-predict COVID-19 time series data from different countries. The results are closed and coincide often with the observed values of the time series based on our methods.

■ We have been able to propose a continuous formulation of a unique maximum reproduction number estimate with an assumption that the epidemic curve is in form of the Gaussian curve and then compare the model with the discrete forms and the basic reproduction numbers observed during the contagiousness period considered. We also estimated the transmission rate from identification of the first inflection point of a wave on the curve of daily new infectious cases using the Bernoulli S-I (Susceptible-Infected) equation. We applied this new method to the real data from Cameroon COVID-19 outbreak both at national and regional levels and to the second wave COVID-19 outbreak for the world data which is a period the phenomena we are considering were observed.

■ We developed an epidemic ageing model in human populations in order to estimate the lifespan loss due to the disease and ageing. This model was applied to COVID-19 outbreak data from different countries as well as the role of comorbidities and sensitivity of the model to various parameters was established.

■ We applied the ideas in the Boolean dynamic context to comorbidity risk factor (obesity) by establishing a modeling epidemic propagation in a Boolean framework (Hop field-type propagation equation, notion of centrality to interpret interaction graphs, existence of stationary states), we presented an application of the model chosen within the framework of the observation of real data and we show the contribution of the model in their interpretation in order to increase the efficiency of the model currently used, for the understanding of the mechanisms and the prediction of the dynamic behaviours of the pandemics present and future.

General Conclusion and Some Perspectives

We have provided in this thesis the intertwine between mathematical and statistical models for the modeling of infectious disease. Concerning the contagious diseases, public health physicians and policy-makers are constantly faced with four challenges. The first concerns the estimation of the basic reproduction number R 0 . The systematic use of R 0 simplifies the decision-making process by policy-makers, advised by public health authorities, but it is too caricature to account for the biology behind the viral spread. We have observed that R 0 was not constant during an epidemic wave due to exogenous and endogenous factors influencing both the duration of the contagiousness period and the transmission rate during this phase. Then, the first challenge concerns the estimation of the mean duration of the contagiousness period for infected patients. As for the transmission rate, realistic assumptions made it possible to obtain an upper limit to this duration, in order to better guide the individual quarantine or lockdown measures decided by the authorities in charge of public health. This upper bound also makes it possible to obtain a lower bound for the percentage of unreported infected patients, which gives an idea of the quality of the census of cases of infected patients, which is the second challenge facing specialists of contagious diseases. The third challenge is the estimation of the daily reproduction numbers over the contagiousness period and the fourth and final interesting challenge is the extension of the methods developed in the present paper to contagious noninfectious diseases (i.e., those without causal infectious agents), such as social contagious diseases, the best example being that of the pandemic linked to obesity, for which many concepts and modelling methods presented here remain available. Also, the severity of public health mitigation strategies implemented in different cities or regions across countries demonstrates that demographic fluctuations are important factors for the spread and prevalence of COVID-19 due to its stochastic transmission nature. Environmental factors have an impact on these demographic events. We discovered that demographic factors, regardless of reopening or closing strategies, can change the time and scale of infection. The public health mitigation, in conjunction with our mathematical and statistical models of the COVID-19 pandemic, is used to build a detailed picture of the distribution of immunity to SARS-CoV-2 and determine the severity of consecutive waves of infection given the specific public health mitigation programs. Test, trace and isolate approaches will better control the disease spread but observations from different countries showed how limited capacities for this approach can imply a fast growth in undetected incidence cases. Adaptions of non-pharmaceutical interventions were associated with changes and fluctuation in the effective reproduction number and also in the daily reproduction rates our modeling approach has calculated in this thesis. The perspective generated by studying epidemic ageing model in human populations support the continued use of matrix population models for understanding the epidemiology of viral infections across the life span, by noticing that each time the virus mutates, it is necessary to re-estimate the coefficients of the matrix model again.

Finally, we have presented a large scale data analysis and also assessed the efficiency of many statistical models. We have adopted a retrospective modelling approach which has shown to us that the results are coherent with observations based on different parameters or counts in the COVID-19 pandemic having in mind that the prevention protocols varied during the different phases of the pandemic throughout the epidemic period. Interestingly, the results of the prediction models show these models are more or less adapted for COVID-19 data set with different fluctuations. Through the fitting and prediction of the data set, we reveal that some of the models have more advantages in the prediction of the epidemiological data coming from COVID-19 outbreak.

Recommendations

While previous research by our research network indicated that geo-climatic factors such as temperature change is likely to extend the geographical spread of the disease and has had the greatest influence on the spread of the epidemic, we were able to demonstrate that socioeconomic factors such as GDP dedicated to health expenditures and demographic factors such as age also had the greatest influence on the spread of the epidemic. We observe a correlation of about 42 percent when GDP dedicated to health expenditures is correlated with epidemiologic variable, and of course, using ordinary least square method by adding more socioeconomic variable to the modeling, a strong correlation is observed, proving that GDP dedicated to health expenditures takes a chunk of this correlation. In this thesis, we used different modeling strategies to demonstrate the impact of vaccination on older and younger age groups, as well as to demonstrate a faster growth of infectious in the younger sub-population than the older sub-population since younger population with active social activities are more likely to be exposed to COVID-19. Our modeling approach was able to confirm what was observed in real life, indicating that age has an impact on epidemic spread. The role of vaccination has been globally positive, most notably in the reduction of mortality and hospitalization cases, which we studied using a functional data analysis approach to study the dynamics of diseases in French departments before and after vaccination. As a result, it is critical to encourage people to get vaccinated in order to avoid severe illness and hospitalization. One key reason we predicted the test response variable using functional linear model is because multiple testing campaigns helps to improve COVID-19 surveillance day by day, particularly in the case of "false" peaks where one observe only shoulders detected by our inflection point estimation method, reason we recommend that researchers and modelers should not only estimate parameters or be interested in the dynamics of the disease at the exponential phase alone but also at the shoulders before the different waves of the pandemic which we have investigated in this thesis since many countries don't attain the different peaks detected during the spread of the disease.

We also observed that socio-economic demographic variables show an anti-correlation for vaccination (Dose 1 and Dose 2) and number of tests. This demonstrates that individuals with a high socio-cultural level are more likely to monitor their health by getting tested and vaccinated when vaccines were introduced because these socioeconomic factors we considered affects the quality of life. We investigated all of these in order to localize mitigation measures and how to allocate intervention resources for future epidemics using sub-population modeling approach, and if government policies can aid in the increase in GDP dedicated to health expenditures, allowing countries to strengthen their health infrastructures, it will aid in proper preparedness for future pandemics and preventing future pandemic outbreaks because the COVID-19 epidemic has wreaked havoc on the healthcare system as well as the socioeconomic standing of many countries.

These modeling approaches we have presented will be useful for predicting, preventing, and controlling the spread of COVID-19, and thus future epidemics.

Future Work

There are several possible extensions of the two approaches in this thesis. On the mathematical modeling side, in the future, we will compare the results generated from the estimation of daily reproduction rates by varying contagiousness duration between 3 and 12 days in order to obtain possibly more realistic values for the R j 's, and hence, have perhaps a double explanation for the 7 days periodicity, both sociological and biological. One interesting challenge for infectious disease modeling is the improvement and extension of the decovolution method developed here to the contagious non-infectious diseases (i.e., without causal infectious agent), such as social contagious diseases, the best example being that of the pandemic linked to obesity, for which many concepts and modelling methods remain available and our future research will propose new avenues of research in this field.

In the first case (discrete), the two-state variable "vaccinated or not" should be added, in order to properly take into account the effect of the vaccination policy. We intend to extend the epidemic ageing model in human population to more age classes (6, 7, etc.) in more countries in which data are available in order to better understand the evolution of the disease in subpopulations; now that we are observing a sharp increase in the number of children infected and those who interact with them, such as parents and teachers in some countries; then, we will consider the role of comorbidities in the subpopulations of more The most important pitfall in simulations of such a model is the estimation (by observation, calculation, or assumption) of its parameters, already difficult with two age groups. A random choice of the values of the parameters in plausible intervals followed by a study of the sensitivity to the parameters of the model, could make it possible to partly overcome the constraint of parameter estimation in a more precise future epidemiological-demographic model, in particular with regard to age groups. Also we can introduce an accelerated ageing due to infection, as well as an influence of exogenous determinants such as geo-climatic, socio-economic, and health-related factors, which weight differently on the different age groups, therefore, change the growth dynamics specific to each of the sub-populations studied in this thesis. It is possible to estimate cell ageing due to the virulence which supplementary cell lifespan loss which could cause pathological events like infectious diseases, by using discrete and continuous formulation of Hahn's model.

From endotracheal cytometry we count the alveolar cells in order to have information about their loss and also from the protein profile we will get information on proteins involved in apoptosis like Gaf2 and then this information will be fielded into the mathematical model (Hahn's model) in order to estimate the lifespan loss. Also, we can introduce the influence of altitude in the modeling of the cell lifespan loss because it has been established that the beneficial effects of altitude on lung function could be due to changes in the expression level of ACE2, which is not only the primary route of entry for the COVID-19 virus into cells, but also appears to play a key role in acute lung injury and other organ damages.

On the statistical modeling side, to benchmark the statistical studies in this thesis, mixed effect models, generalized additive models and spatial data analysis will be used to further predict the evolution and dynamics of the spread of COVID-19 pandemic. As demonstrated by the COVID-19 pandemic, spatial epidemiology is an important field of study. For instance, Hidden Markov models are popular time series models. It can be defined over discrete or continuous time. In the mathematical modeling in this thesis, we have been able to develop an epidemic ageing model for human populations which we can now exploit using a statistical approach by studying the State-dependent distributions and state estimations, developing transition probability matrix, marginal distributions and temporal dependence, estimating the initial state distribution and assessing model adequacy using forecast (pseudo-)residuals and posterior predictive checks. Also, spatial statistics and disease mapping are well-developed in literature, however, there are notable computational bottlenecks when it comes to the application of spatially-fine grained epidemiological models. We will investigate the application of recently proposed deep learning approaches to efficiently encode spatiotemporal data before standard (non-spatial) epidemiological modeling approaches, such as SEIR, renewal process, and ODE-based models. Spatial epidemiology is designed to find patterns, and in this case, it will help us detect peaks at different waves of the pandemic while taking demographic factors into account. The goal is to investigate the evolution and dynamics of disease spread over time along the epidemic curve in various regions, countries, and continents. Also, we can investigate the relationship between comorbidities like obesity, diabetes, cardiovascular diseases or asthma and epidemic-socio-demo-economic determinants, using several machine learning and deep learning methods and then predict some of these determi-nants since we have already shown in this thesis how they were associated with COVID-19 prevalence. In addition to improving the section of the thesis dedicated to spectral analysis methods in which time (days) was converted to frequency, it is possible to examine the periodicity and frequency peaks of time series data by converting time series data (pandemic) from a time domain to a frequency domain and applying a Dense Convolutional Neural Network (DenseCNN) or CNN and producing the scalogram generated from Discrete Fourier transform (DFT) transformation of the pandemic data and training the DenseCNN for a time series by performing a regression or classification target. Also, another future work could be to try to model the change-points between different phases of a pandemic, namely the transition between an endemic phase and the future epidemic wave. The estimation and forecast of this transition time is a difficult challenge rarely faced by the modelers (using in general change-points estimation in linear models or algorithms for detection of abrupt changes in signals and systems) and could constitute if it provides accurate and reliable methods a huge progress in epidemic studies.

Finally, the population distribution of viral loads observed under random or symptom-based surveillance reverse transcription quantitative polymerase chain reaction (RT-qPCR) tests, which provide semiquantitative results in the form of cycle threshold (Ct) values that change during an epidemic and are an important predictor of pandemic trajectory, should be investigated. We intend to estimate the epidemic's trajectory using both statistical modeling (in this case, the branching process) and mathematical modeling (in this case, the exponential model and the SEIR model).

The publications in preparation are as follows:

• K. Oshinubi, J.P. Olumuyiwa, M. Rachdi and J. Demongeot. The modeling of COVID-19 data sets using generalized addictive models and mixed models (in preparation).

• K. Oshinubi, M. Table A.2: Calculation of the daily R j 's and shape of their distribution for 194 countries and for the two first waves.

Country Name

First wave Second wave R 0 R j 's U-shape R 0 R j 's U-shape AFGHANISTAN 0.65 0.17; 0.09; 0.

Figure 1 . 1 :

 11 Figure 1.1: Schematic diagram of a simple SEIR model.

Figure 2 . 1 :

 21 Figure 2.1: Structure of Bernoulli's epidemiological model for a population divided into susceptible and immunized. At age x, the probability of surviving the infection is z(x), transmission rate is β(x) and natural death rate is µ(x).

Figure 2 .

 2 1 shows the structure of Bernoulli's epidemiological model. The differential equation is then satisfied by p(x): dp dx = -(β(x) + µ(x))p, with the initial condition p(0) = 1.

Figure 2 . 2 :

 22 Figure 2.2: Structure of D'Alembert's epidemiological model where the two deaths ratesµ(x) (natural death rate) and λ(x) (death rate due to the infectious disease) are assumed to act independently.

  u * (j) = S * j P * , v * (j) = I * j P * , w * (j) = R * j P * , where P * = i=1,...,m

f 1 f 2 f 3

 3 . . . . . . b 1 0 0 . . . . . . 0 b 2 0 . . . . . . . . . . . . . . . . . . . . .

Figure 2 .

 2 3 shows the life cycle representation of the Usher projection matrix model.

Figure 2 . 3 :

 23 Figure 2.3: Life cycle representation of theUsher projection matrix model, where p i is the probability for an individual to stay in class i, q i is the probability to move up from class i to i + 1, m i is the probability of dying and f is the average fertility rate supposed to be constant (after[START_REF] Mortier | Population dynamics of species-rich ecosystems: the mixture of matrix population models approach[END_REF]).

Figure 2 . 4 :

 24 Figure 2.4: Block diagram of the statistical methodology.

Figure 2 . 5 :

 25 Figure 2.5: Data pre-processing of all data-set used in the study.

Figure 2 . 6 :

 26 Figure 2.6: Flowchart of supervised machine learning technique.

Figure 2 . 7 :

 27 Figure 2.7: Flowchart for k-fold cross-validation strategy to evaluate supervised machine learning technique.

a 1 =

 1 (a 11 , a 21 , . . . , a p1 ) T and b 1 = (b 11 , b 21 , . . . , b q1 ) T , in order to maximize the correlation β = corr(Xa, Y b), while U = Xa and V = Y b. Linear combinations of X and Y components respectively, constitute the first pair of canonical predictors. Then, the second pair of canonical response can be found in the same way subject to the constraint that they are uncorrelated with the first pair of variables. By repeating this procedure, the r = min{p, q} pairs of the canonical response can be found and we will finally get the matrix A = [a 1 , a 2 , . . . , a r ] and the matrix B = [b 1 , b 2 , . . . , b r ] to transfer X and Y to canonical responses U and V following the below expression:

Figure 2 . 8 :

 28 Figure 2.8: Flowchart of clustering technique.

Figure 2 . 9 :

 29 Figure 2.9: LSTM and GRU architectures (after ([96]). The calculation unit uses cell state (C) and hidden state (H) variables for LSTM and only H for GRU, with a control variable (X t ) for both.

Figure 2 .

 2 Figure 2.10: (a) A CNN sequence illustration (after [96]). (b) Deep neural network with two or more hidden layers.

Figure 3 . 2 :

 32 Figure A1

Figure 3 . 3 :

 33 Figure 3.3: Confirmed world daily new cases (from [119]) as a function of days since 26 February until 23 August 2020 + indicates Sundays, x indicates Mondays.

Figure 3 . 4 :

 34 Figure 3.4: Z-transformed scores of confirmed world daily new cases [119], from Figure 3.3, as a function of days since 26 February 2020 until 23 August 2020 + indicates Sundays, x indicates Mondays. Z-transformations are specific to each weekday.

Figure 3 . 5 :

 35 Figure 3.5: Smoothed confirmed world daily new cases [119], from (Figure 3.3, as a function of days since 26 February 2020 until 23 August 2020 + indicates Sundays, x indicates Mondays.For each specific day j, the mean number of confirmed daily new cases calculated for days j -1, j -2, j, j + 1 and j + 2 is subtracted from the number for day j.

Figure 3 . 6 :

 36 Figure 3.6: Smoothed confirmed world daily new cases [119] applied to z-scores from (Figure 3.4, as a function of days since 26 February 2020 until 23 August 2020 + indicates Sundays, x indicates Mondays. Z-transformations are specific to each weekday.For specific day j, the mean number of confirmed new cases calculated for days j -1, j -2, j, j + 1, j + 2 is subtracted from the number for day j.

Figure 3 . 7 :

 37 Figure 3.7: Z-transformed scores of smoothed confirmed world daily new cases [119] smoothed data from Figure 3.5, as a function of days since 26 February 2020 until 23 August 2020. + indicates Sundays, x indicates Mondays. Z-transformations are specific to each weekday.

Figure 3 .

 3 Figure 3.8: a): Friendship empirical network representing a Tunisian sample, with 274 pupils among which 18 overweight or obese (in red) observed from classes of a Tunisian high school, with 524 links between them (only the links having a weight over a fixed threshold have been represented for the sake of visibility). The vertex size is proportional to the in-degree centrality of the corresponding node in the friendship interaction graph. b): Representation of the Tunisian friendship interaction graph at its stationary state, iterated from the empirical network as initial state to the asymptotic state under the stochastic transition rule given in Equation (3.15), with node size proportional to its in-degree centrality. c): Representation of the total-degree centrality of the nodes of the graph b). d): Representation of the spectral centrality of the nodes of the graph b).

Figure 3 . 9 :

 39 Figure 3.9: Percentages of obese or overweight (red) and normal weight (green) individuals evolving from an initial population with 100 obese or overweight with increasing values of T i for the 8 individuals with the most neighbours: (a) T i = 0.1, (b) T i = 0.5, (c) T i = 1, (d) T i = 2, (e) T i = 3, (f) T i = 4, (g) T i = 5, (h) T i = 10 and (i) T i = 100, value for which the percentage of normal is 1 /2.

Figure 3 . 10 :

 310 Figure 3.10: Top. Friendship empirical network representing the largest connected component of a sample of 104 pupils from which 17 overweight or obese (in red) from a French high school, with 348 links between them (only the links having a weight over a fixed threshold have been represented for the sake of visibility). Node sizes are proportional to their in-degree centrality. Middle: Friendship interaction graph iterated from the empirical network as initial state to the asymptotic state under the stochastic transition rule given in Equation (3.15), with node size proportional to spectral centrality (left) and to total-degree centrality (right). Bottom: Friendship interaction graph iterated from the empirical graph with entropy centrality once stabilized by changing the state of the 20 most entropy-central nodes (left) and of the 21 most entropy-central nodes (right): stationary state is fully overweight or obese in first case and fully normal in the second case, showing a "phase transition" of the asymptotic invariant measure.

  resulting R j 's equal: R 1 = 2.486, R 2 = -2.33 and R 3 = 7.38769.

  R j 's equal: R 1 = 3.66898, R 2 = -33.857 and R 3 = 61.32.

ChileFigure 3 . 11 :

 311 Figure 3.11: Top: estimation of the effective reproduction number R e 's for 20 October and the 25 October 2020 (in green, with their 95% confidence interval) [126, 127]. Bottom left: daily new cases in France between 15 February and 27 October [119]. Bottom right: Ushape of the evolution of the daily R j 's along the 3-day contagiousness period of an individual.

  can deduce the daily R j 's, i.e., the vector (R 1 , R 2 , R 3 ) :

Figure 3 Figure 4 ChileFigure 3 . 12 :

 34312 Figure 3

Figure 3 . 13 :

 313 Figure 3.13: Values of the daily reproduction numbers R j along the period of contagiousness of length 7 days.

Figure 3 .

 3 Figure 3.14: U-shape of the evolution of the daily R j 's along the infectious 3-day period of an individual, daily new cases in Kuwait between December 30 2020 and January 4 2021 in worldometer and estimation of the average transmission rate R o for December 30th 2020 and January 4th 2021 with its 95% confidence interval (in green).

Figure 6 FigureFigure 3 . 15 :

 6315 Figure 6

Figure 3 . 16 :

 316 Figure 3.16: Decreasing evolution of the daily R j 's along the infectious 3-day period of an individual, calculated for the period from June 19 to June 24 2020 during the first wave of COVID-19 outbreak in Cameroon.

Figure 3 .

 3 Figure 3.17: Inverted U-shaped evolution of the daily R j 's along the infectious 3-day period of an individual, calculated for the period from October 20 to October 25 2020 during the first exponential phase of COVID-19 outbreak in Morocco.

  where d is the susceptible demographic balance, ν the specific mortality rate due to the disease, β/S 0 the disease transmission rate, S(t) the number of susceptible individuals, I(t) the number of infectious individuals at time t ≥ 0, and the initial conditions of the model are: S(0) = S 0 > 0 and I(0) = 1. By simulating Equation(3.32), the results obtained by neglecting the mortality due to the COVID-19 outbreak and to the demography of the susceptible population are given on Figure 3.19. The Figure 3.18 shows slight differences at the beginning of the epidemic wave between Model 1, corresponding to Equation (3.32) in which the fertility is neglected and the mortality (d = ν = 0) and Model 2, corresponding to Equation (3.32) by fixing the demographic balance of susceptibles d to 0.01 and the specific mortality due to the COVID-19 outbreak ν to 0.01, which are realistic values for Cameroon. Let's assume we want to simulate Bernoulli equation if it is evolving in time due to various influences as geoclimatic, socio-economic and demographic factors but Equation (3.32) must be rewritten as follows: dS dt = BSI -µS + f + σI, (3.33) dI dt = BSI -νI + f + ρI, with S(0) = S 0 , I(0) = I 0 and where S(t) (resp. I(t)) is the size of susceptible (resp. infected) at time t, B the transmission rate, µ (resp. ν) the natural (resp. viral) death rate, f the fecundity, σ (resp. ρ) the post-infected non immunized (resp. immunized) rate. If f = µ = ρ = σ = ν = 0, the solution of Equation (3.33) is given by:

  19, for the first half part of the second wave of the COVID-19 weekly new reported cases between week 5 2021 and week 12 2021 (data from [150]). The red curve represents the cubic spline approximation of the raw data with the following statistics: p-value = 0.03, Multiple Rsquared (MR) = 0.90, Relative Mean Square Error (RMSE) = 0.247. During the same period, it is also possible to smooth the death data (from [119]) with a cubic spline with the following statistics: p-value = 0.12, Multiple R-squared (MR) = 0.99, Relative Mean Square Error (RMSE) = 4.380 (see Figure 3.22).

Figure 3 .

 3 Figure 3.18: Top: simulation of the evolution of the number of new infected I in Model 1 (Equation (3.32) with d = ν = 0) and Model 2 (Equation (3.32) with d = ν = 0.01) Bottom: simulation of Models 1 and 2 with same initial conditions for infected (I(0) = 1) and susceptible individuals (S(0) = 400).

Figure 3 .

 3 Figure 3.19: COVID-19 weekly new cases in Cameroon showing the inflection point P I (red arrow) during the first half part of the second wave between week 5 of 2021 and week 12 of 2021. (after [150]).

Figure 3 . 20 :

 320 Figure 3.20: Surface of evolution of 3 covariables, S, I and B, the transmission rate.

Figure 3 . 21 :

 321 Figure 3.21: Dynamics of the variables S and I in eradication (top) and endemic behavior (bottom) (after https://elsenaju.eu/ Calculator/ODE-System-2-2.htmRWD-1). (a) Eradication B = 0.6, f = 4, µ = 2, ρ = 0, σ = 0, ν = 4, S(0) = 40, I(0) = 1; (b) Endemic behavior B = 0.5, f = 6, µ = 0.5, ρ = 0, σ = 0, ν = 4, S(0) = 40, I(0) = 1.

Figure 3 . 22 :

 322 Figure 3.22: Spline graph for COVID-19 deaths in Cameroon during the second wave between week 5 of 2021 and week 14 of 2021.

Figure 3 . 23 :

 323 Figure 3.23: Spline graph for COVID-19 deaths in Cameroon during the second wave between week 5 of 2021 and week 14 of 2021.

Figure 3 . 24 :

 324 Figure 3.24: Evolution of the daily new cases numbers in Cameroon along the second wave between weeks 1 and 10 of 2021.

Figure 3 . 25 :

 325 Figure 3.25: Evolution of cured cases numbers in Cameroon along second wave between weeks 1 and 10 of 2021.

Figure 3 . 26 :

 326 Figure 3.26: Spline graph for COVID-19 deaths in Cameroon during the second wave between week 5 of 2021 and week 14 of 2021.

  3.2 is applied to Cameroon regions data from[150] in order to obtain the value of the main epidemiologic parameters, using new cases during the beginning of the second wave in 2021 to calculate the discrete R 0 and also the total number of new cases as S 0 in order to calculate the transmission rate β when t i = 4 and finally R max with σ = 1 and σ = 2, corresponding to a duration of the contagiousness period respectively 4 and 8 days. The summary of the result is presented in Table3.4. The evolution of the daily new cases and cured cases for the regions in Cameroon can be seen in Figure3.27 and Figure3.28. It was observed that there is a big difference between the urban regions, essentially the regions of Yaoundé and Douala (i.e., respectively the Center and Littoral regions) whose urban density is high and the other mainly rural regions.

Figure 3 . 27 :

 327 Figure 3.27: Evolution of the daily new cases numbers in the 10 regions of Cameroon along the second wave between weeks 1 and 10 of 2021.

Figure 3 . 28 :

 328 Figure 3.28: Evolution of the cured cases numbers in the 10 regions of Cameroon along the second wave between weeks 1 and 10 of 2021.

Figure 3 . 29 :

 329 Figure 3.29: Flow chart illustrating the epidemiology ageing model.

Figure 3 . 30 :

 330 Figure 3.30: Histogram of weekly COVID-19 incidence rates / age class in France during 10 weeks from the 17th January 2022. The global daily incidence rate on the day 17th January 2022 was equal to 3,264 / 100,000 (taking into account the age class sizes) ([162]).

Figure 3 . 31 :

 331 Figure 3.31: Cumulative age and sex-specific incidence rates of confirmed COVID-19 cases per 100,000 population notified in Ireland between week 51 in 2021 and week 8 in 2022 (after [164]).

  dominant eigenvalue of E * is equal to 1.007, which shows that during the month between 19th December 2021 and 18th January 2022, the loss of population due to COVID-19 deaths has been equal to (1.012 -1.007) × 458342/12 = 191, the real observed death numbers being equal to 6035 -5835 = 200, which once again confirms the realistic nature of the model.

Figure 3 . 32 :

 332 Figure 3.32: Ireland weekly age-specific incidence rates of confirmed COVID-19 cases per 100,000 population among children aged 0 -18 years by notification (left) and from Epiet data (right) from week 51 in 2021 to week 8 in 2022 (after [163]).

Figure 3 .

 3 [START_REF] Balabdaoui | Age-stratified discrete compartment model of the covid-19 epidemic with application to switzerland[END_REF] and where males are most numerous than females[119].For each comorbidity, we can estimate its effect on the Malthusian parameter by taking into account the distribution of the cumulated COVID-19 cases on the age classes. For example, the cardiovascular comorbidity causes the majority of new cases between 40 and 59 years for both sexes, and we have for men (M):The difference between the values of the exponential growth parameters is equal to that between the dominant eigenvalues: λ M = 1.0234 and λ W = 1.0173, indicating that among patients suffering from cardiovascular pathologies, men are more affected by the COVID-19 than women, as confirmed by the statistics on 485 cumulated COVID-19 cardiovascular deaths observed among the 22,421 cumulated new cases on 9th September 2020 in Cameroon after 6 months of pandemic[START_REF]Rapport de situation COVID-19 n o 84[END_REF], from which 278 observed are men and 207 observed are women, that is a sex ratio M / W observed of 1.343 and a differential growth rate ratio calculated equal to (1.026 -1.0226) / (1.026 -1.02352) = 1.369.

Figure 3 . 33 :

 333 Figure 3.33: On the left, Table of cumulated COVID-19 cases by pathologies and sex on the 9th November 2020. On the right, pyramids of ages by sex for the major comorbidity, the cardiovascular diseases.

Figure 3 .

 3 Figure 3.34: (a) Dominant eigenvalue plotted against the varying values of probability v 2 of remaining in the same age class (20-39), during COVID-19 outbreak in France. (b) Sensitivity curves for France. (c) Histogram of changes in dominant eigenvalue λ of E * for the variations in different percentages of the different E * coefficients in Ireland. (d) Pie chart to compare the sensitivities for the different categories of E s coefficients in Ireland.

Figure 3 .

 3 Figure3.34(c) affirms that "stay in same state" coefficient of the first two age classes (v 1 = A1S and v 2 = A2S) in Ireland is crucial to the population growth which is also the same with the results we observed in France. Figure3.22d confirms that λ is sensitive to v i = AiS(i = 1, 2) because corresponding sensitivities are the highest. The last example of Ireland has shown that it was difficult to overlook the class of young people under 18, who shows an incidence reaching a peak of 3% during the fifth wave (Figure3.32) and a cumulative rate of approximately 13000 new cases per 100,000 in 10 weeks of this wave (Figure3.31). Then, we added one more age class in this last example. On the other hand, to refine the coefficients of the epidemiological matrix E * , it is necessary to better understand the specific ageing processes due to the SARS-CoV-2 virus, in particular that which affects anti-apoptosis proteins such as Gaf1. Indeed, these processes affect the mortality specific to COVID-19, hence the coefficient of the matrix E * which quantifies the

Figure 3 . 35 :

 335 Figure 3.35: Complete fetal human hemoglobin beta (HBB) subunit mRNA sequence potentially targeted by SARS-CoV-2 gene fragments, those from RNA-dependent RNA polymerase (also targeted by the β A siRNA in violet[START_REF] Dykxhoorn | Determinants of specific rna interference-mediated silencing of human β-globin alleles differing by a single nucleotide polymorphism[END_REF][START_REF] Dykxhoorn | The silent revolution: Rna interference as basic biology, research tool, and therapeutic[END_REF]) and from protein S (in blue), and by the human miR hsa miR 451a (in red). Also shown (in red) are fragments containing deletions of protein S in its N-terminal domain and mutations N501Y, E484K and D614G (base mutated in green). The probability of a hybridization by chance of length 8 in red (resp. 10), for 624 nucleotides, is equal to 0.04 (resp. 0.005), hybridizations TG and GT counting for 1 /2.

Figure 3 . 36 :

 336 Figure 3.36: At the top, partial sequence of the mRNA of the S protein of the Omicron variant of the SARS-CoV-262 virus with indication (in red) of the subsequences of the human interferon gene and the human Gaf1 gene anti-aging that they hybridize. In the bottom, the corresponding hybridization (in red) of part of the anti-aging human Gaf1 gene [177].

Figure 1 .

 1 Figure 1. Dependence of the case fatality rate (from cumulative deaths on the 20 May 2020) vs. median age of several countries in 2017 (from [1]). The area of a country circle is proportional to the number of cumulated deaths due to COVID-19 on the 20 May 2020, e.g., for the USA: 99,643 (in red).

Figure 2 .

 2 Figure 2. Top left: COVID-19 percentage of death in France by age class [23]. Bottom left: Influence of age (curves with color coding) on COVID-19 hospitalizations in France in the extreme age classes [23]. Bottom right: age classes pyramid in 2020 in France (total population size: 65,273,512) [24].

Figure 3 . 37 :

 337 Figure 3.37: Dependence of the case fatality rate (from cumulative deaths on the 20 May 2020) vs. median age of several countries in 2017 (from [214]). The area of a country circle is proportional to the number of cumulated deaths due to COVID-19 on the 20 May 2020, e.g., for the USA: 99,643 (in red).

Figure 3 .

 3 Figure 3. Curves of COVID-19 weekly ICU admissions in Kuwait by age classes (two on the left and four on the right) and gender from 27 February 2020 to 9 March 2021.

Figure 3 . 38 :

 338 Figure 3.38: Curves of COVID-19 weekly ICU admissions in Kuwait by age classes (two on the left and four on the right) and gender from 27 February 2020 to 9 March 2021. Healthcare 2022, 10, x FOR PEER REVIEW 4 of 18

Figure 4 .

 4 Figure 4. Left: Distribution of cumulated confirmed cases of COVID-19 by age group and gender in Cameroon as of 23 June 2021 [25]. Right: Distribution of deaths due to COVID-19 infection by age group and gender in Cameroon as of 23 June 2021 (after [34]).

Figure 3 . 39 :

 339 Figure 3.39: Left: Distribution of cumulated confirmed cases of COVID-19 by age group and gender in Cameroon as of 23 June 2021 [216]. Right: Distribution of deaths due to COVID-19 infection by age group and gender in Cameroon as of 23 June 2021 (after [150]).

Figure 1 .

 1 Figure1. Dependence of the case fatality rate (from cumulative deaths on the 20 May 2020) vs. median age of several countries in 2017 (from[START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF]). The area of a country circle is proportional to the number of cumulated deaths due to COVID-19 on the 20 May 2020, e.g., for the USA: 99,643 (in red).

Figure 2 .

 2 Figure 2. Top left: COVID-19 percentage of death in France by age class [23]. Bottom left: Influence of age (curves with color coding) on COVID-19 hospitalizations in France in the extreme age classes [23]. Bottom right: age classes pyramid in 2020 in France (total population size: 65,273,512) [24].

Figure 3 . 40 :

 340 Figure 3.40: Top left: COVID-19 percentage of death in France by age class [215]. Bottom left: Influence of age (curves with color coding) on COVID-19 hospitalizations in France in the extreme age classes [215]. Bottom right: age classes pyramid in 2020 in France (total population size: 65,273,512) [157].

Figure 3 . 38 ,

 338 Figure 3.40 and Figure 3.39 and used this ratio to determine the sizes value for the two susceptible classes at the exponential phase considered (for Kuwait 28 December 2020, France 30 October 2021, and Cameroon 19 September 2021), because the Ross and Kermack-McKendrick model is only suited for the exponential growth phase of an epidemic wave. Because the progression rates to the classes reversed recovery and fully recovered in Figure 3.41 are not zero, we assumed that they should have some populations at the start of the wave, even if they were small. The transmission rate for Cameroon was chosen at the 5.1. COVID-19 Outbreak in Kuwait We have chosen the following set of parameter values corresponding to the COVID-19 outbreak at the start of the fourth wave in December 2021 in Kuwait: φ = 1.7, φ = 0.9, μ = 0.003, ϵ = 0.28, β = 0, μ = 0.0025, μ = 0.002, μ = 0.0021 , β = 2.1 , β = 2.3 , θ = 0.765, θ = 0.678, γ = 0.62, γ = 0.74, θ =0.45 , θ = 0.33, a = 0.7, a = 0.54, a = 0.65, a = 0.8, ϵ = 0.19 , η = 0.3 and η = 0.38. with initial values for S1 = 4413099, S2 = 51422, I1 = 65, I2 = 194, G1 = 20, G2 = 17, R1 = 30 and R2 = 73.

Figure 6 .

 6 Figure 6. Numerical simulation of the variables I1 and I2, R1 and R2, G1 and G2 for Kuwait.

φ = 1 . 2 ,

 12 φ = 0.9, μ = 0.009, ϵ = 0.28, β = 0, μ = 0.0025, μ = 0.002, μ = 0.0021 , β = 1.9, β = 2.2, θ = 0.735, θ = 0.678, γ = 0.62, γ = 0.74, θ =0.45 , θ = 0.33, a = 0.7, a = 0.54, a = 0.65, a = 0.8, ϵ = 0.19, η = 0.3 and η = 0.38. with initial values for S1 = 53372880, S2 = 14017120, I1 = 3912, I2 = 1757, G1 = 200, G2 = 170, R1 = 300 and R2 = 730.

Figure 3 . 42 :

 342 Figure 3.42: Numerical simulation of the variables I 1 and I 2 , R 1 and R 2 , G 1 and G 2 for Kuwait.

Figure 7 .

 7 Figure 7. Numerical simulation of the variables I1 and I2, R1 and R2, G1 and G2 for France.

φ = 1 . 3 ,

 13 φ = 0.9, μ = 0.009, ϵ = 0.28, β = 0, μ = 0.0025, μ = 0.002, μ = 0.0021 , β = 4.5, β = 4.1, θ = 0.024, θ = 0.678, γ = 0.62, γ = 0.74, θ =0.45, θ = 0.33, a = 0.7, a = 0.54, a = 0.65, a = 0.8, ϵ = 0.19, η = 0.3 and η = 0.38. with initial values for S1 = 25828973, S2 = 721027, I1 = 151, I2 = 21, G1 = 75, G2 = 33, R1 = 56 and R2 = 64.

Figure 3 . 43 :

 343 Figure 3.43: Numerical simulation of the variables I 1 and I 2 , R 1 and R 2 , G 1 and G 2 for France.

Figure 3 . 43 :

 343 Figure 3.43: Numerical simulation of the variables I 1 and I 2 , R 1 and R 2 , G 1 , G 2 for Cameroon.

Figure 4 . 32 Figure 7 .

 4327 Figure 7. (a) First wave moving average and standard deviation of new cases (left) and (b) autocorrelation curve for Mali (right). (c) First wave moving average and standard deviation of new cases (left) and (d) autocorrelation curve for Luxembourg (right).

Figure 4 . 1 :

 41 Figure 4.1: (a) First wave moving average and standard deviation of new cases (left) plotted against time in days on x-axis and (b) autocorrelation curve for Mali (right). (c) First wave moving average and standard deviation of new cases (left) plotted against time in days on x-axis and (d) autocorrelation curve for Luxembourg (right).

Figure 4 .

 4 Figure 4.1 compares two countries, one from Sahelian Africa, Mali and one from western Europe, Luxembourg during the first wave of COVID-19 outbreak during the spring 2020:Mali shows a quasi-endemic behavior with a weakly varying autocorrelation function and Luxembourg a frank epidemic wave with a classic shape. For the second wave in fall 2020, Mali presents an attenuated epidemic shape (due probably to specific geoclimatic conditions in western Africa[START_REF] Seligmann | Inverted covariate effects for first versus mutated second wave covid-19: High temperature spread biased for young[END_REF]) and a country from central Europe, Slovenia, shows at this period

Healthcare 2021, 9 , x 12 of 32 3. 3 . 2 .Figure 8 .

 932328 Figure 8. (a) Second wave moving average and standard deviation of new cases (left) and (b) autocorrelation curve for Mali (right). (c) Second wave moving average and standard deviation of new cases (left) and (d) autocorrelation curve for Slovenia (right).

Figure 4 . 2 :

 42 Figure 4.2: (a) Second wave moving average and standard deviation of new cases (left) plotted against time in days on x-axis and (b) autocorrelation curve for Mali (right). (c) Second wave moving average and standard deviation of new cases (left) plotted against time in days on x-axis and (d) autocorrelation curve for Slovenia (right).

Figure 4 . 4 : 32 Figure 9 .

 44329 Figure 9. (a) First and (b) second wave forecast for Mali. (c) First wave forecast for Luxembourg. (d) Second wave forecast for Slovenia.

Healthcare 2021, 9 , x 13 of 32 Figure 9 .

 9329 Figure 9. (a) First and (b) second wave forecast for Mali. (c) First wave forecast for Luxembourg. (d) Second wave forecast for Slovenia.

Figure 4 . 3 :

 43 Figure 4.3: (a) First and (b) second wave forecast for Mali. (c) First wave forecast for Luxembourg. (d) Second wave forecast for Slovenia. All plotted against time in days on x-axis.

Figure 4 . 4 :

 44 Figure 4.4: (a) ARIMA forecast for first wave new cases in Kuwait. (b) ARIMA forecast for second wave transition cases in Kuwait. (c) ARIMA forecast for first wave deaths in Kuwait and (d) ARIMA forecast for second wave transition deaths in Kuwait.

Figure 4 . 5 :

 45 Figure 4.5: (a) Log-linear and polynomial regression for first wave cases in Kuwait. (b) Support vector regression for first wave cases in Kuwait. (c) Log-linear and polynomial regression for second wave transition cases in Kuwait and (d) Support vector regression for second wave transition cases in Kuwait. The origin of time corresponds to the corresponding wave start.

Figure 4 .

 4 Figure 4.5 show different regressions for the first and transition to second waves of COVID-19 in Kuwait.All coefficients for the first wave and transition to second wave for both Log-linear and polynomial regression are significant with p-value less than 0.001. The residuals of the log-linear regression were examined and it was discovered that the median for both the cases were close to zero with median = 0.06031 for the first wave and median = 0.07629 for transition to the second wave. The normality of the residual was tested using Jarque-Bera test and with high p-value we fail to reject the null hypothesis that the skewness and kurtosis of the residuals are significantly equal to zero. Also, the median of the residual for both first wave and transition to second wave of polynomial regression of order four is close to zero with median = 0.0271 and median = 0.00669 respectively. For support vector regression, we tested the normality of the residual using the Jarque-Bera test and it was discovered that it's skewness and kurtosis are significantly equal to zero since the p-value is large. We now apply linear regression model by using some historic data (100 days infectivity period in this case) on some countries epidemiologic variables and consider a linear relationship

Figure 4 . 6 :

 46 Figure 4.6: Regression plots for developed countries of (a) first and (b) second wave maximum R 0 of the new cases curve.

Figure 4 .-

 4 6(a): slope = 0.03612, intercept = 0.0062, r-value = 0.3299, p-value = 0.0352, standard error = 0.0165, R-squared = 0.109, RMSE = 0.116. Figure 4.6(b): slope = 0.05223, intercept = -0.0421, r-value = 0.434366, p-value = 0.0051, standard error = 0.01757, R-squared = 0.18867, RMSE = 0.01689. -Figure 4.6(a) and Figure 4.6(b) show a positive correlation between the slope of the logarithmic regression curve of the new cases of COVID-19 as a function of time (a sign of rapid growth of the epidemic if it is high). The early implementation of mitigation measures in developed countries helps in reducing the exponential growth of new cases in the second wave. This trend is confirmed in the study of the correlation between the slope of the logarithmic regression and the maximum R 0 (Figure 4.6(a) and Figure 4.6(b)), which increases during the second wave in developed countries (the correlation coefficient rising from 0.33 to 0.44), showing a growth of the new cases more brutal, but shorter, undoubtedly due to the establishment of a faster and more effective lock-down. • Parabolic and Cubic Regression -Figure 4.7 show the classical linear and polynomial regressions (parabolic for Graphs (a) and (b) and cubic for Graph (c)) between the opposite of the slope at the origin of the autocorrelation function of the ARIMA model and successively the slope of the logarithmic regression line of the new daily cases of COVID-19 of the first wave (a), then that of the second wave (b), and finally the number of days since the start of the outbreak (c).

Figure 4 .

 4 7(a): • Linear regression: slope = -0.193, intercept = 0.102, r -value = -0.394, p-value = 1.026 × 10 -7 , standard error = 0.03467, RMSE = 0.0385, • Polynomial regression (order 2): p-value = 0.00145, standard error = 0.54339, Rsquared = 0.19, , RMSE = 0.046. -For Figure 4.7(b):• Linear regression: slope = 1.867, intercept = 0.089, r -value = 0.487, RMSE = 0.063,• Polynomial regression (order 2): R-squared = 0.37, RMSE = 0.094.-For Figure4.7(c):• Linear regression: slope = 0.000295, intercept = 0.0765, r -value = 0.195469, p-value = 0.01415, standard error = 0.000119, R-squared = 0.038, RMSE = 0.04,• Polynomial regression (order 3): R-squared = 0.1, RMSE = 0.0414825.

Figure 4 .- 1 .

 41 Figure 4.8 is obtained by using a polynomial of degree 4, a fit showing a minimum for the value of the maximum R 0 equal to 3.5, which is considered as the observed value for the maximal effective reproduction number at start of the first wave in many developed countries (France, Germany, Switzerland, UK, USA, etc.)[126], which corresponds to the fact that the opposite of the initial autocorrelation slope (indicating that the length of the contagiousness is short when the absolute value of the slope is high) decreases (the contagiousness duration increases) when the maximum R 0 increases, which seems logical. The result is as follows:-Linear regression: slope = 0.01034, intercept = 0.1019, r -value = -0.3578, p-value = 0.02163, standard error = 0.00433, RMSE = 0.0303, -Polynomial regression (order 4): RMSE = 0.0349, R-squared = 0.33. On Figure4.8, the opposite of the initial autocorrelation slope is significantly negatively correlated with the maximum R 0 observed at the inflection point of the new cases curve, confirming that long contagiousness periods give high exponential increases of the new cases.

Figure 1

 1 Figure 1 aims to show that classical linear and polynomial regressions (pa Graphs (a) and (b) and cubic for Graph (c)) between the opposite of the slope at of the autocorrelation function of the ARIMA model and successively the slope

Figure 4 . 7 :

 47 Figure 4.7: Linear (in red) and parabolic or cubic (in green) regression plots of the opposite of the initial autocorrelation slope vs. (a) first wave exponential regression slope for all countries, (b) second wave exponential regression slope for developed countries and (c) days from the start of the first wave observed for all countries.

Figure 2 .

 2 Figure 2. Linear (in red) and quartic (in green) regression plots of the opposite of the initial autocorrelation slope of the first wave vs first wave maximum R0 for developed countries. LinregressResult slope = 0.01034, intercept = 0.1019, r value = -0.3578, p value = 0.02163, stderr = 0.00433, RMSE for linear regression = 0.0303, RMSE for polynomial regression = 0.0349, R-squared for order four polynomial regression = 0.33.

Figure 4 . 8 :

 48 Figure 4.8: Linear (in red) and quartic (in green) regression plots of the opposite of the initial autocorrelation slope of the first wave vs first wave maximum R 0 for developed countries.

Figure 4 . 9 :

 49 Figure 4.9: (a) Cross validation result for Log linear model for new cases in Kuwait. (b) Comparison of the tuned support vector model and support vector regression model with the test prediction for first wave new cases in Kuwait. (c) Comparison of the tuned support vector model and support vector regression model with the test prediction for second wave transition new cases in Kuwait.

Figure 4 .

 4 Figure 4.10: (a) Optimum support vector model mean square error visualisation for first wave new cases in Kuwait and (b) Optimum support vector model mean square error visualisation for second wave transition new cases in Kuwait.

  48 and σ = 39, RMSE for training = 38.3, RMSE for test = 219.9, Q * = 78 and p-value= 1e -13 ; for second wave transition cases the parameters used are l = 0 and σ = 92.3, RMSE for training = 90.7, RMSE for test = 76.2, Q * = 30 and p-value= 2e -04 ; for first wave deaths the parameters used are l = 0.0106 and σ = 2.73, RMSE for training = 2.69, RMSE for test = 2.07, Q * = 29 and p-value= 4e -04 and for second wave transition deaths the parameters used are l = 0.4891 and σ = 2.49, Root MSE for training = 2.45, RMSE for test = 1.05, Q * = 14 and p-value= 0.09. It can be concluded that the simple exponential smoothing model without trend works better for first wave deaths.

Figure 4 .

 4 Figure 4.11: (a) Simple exponential smoothing for first wave cases in Kuwait. (b) Simple exponential smoothing for second wave transition cases in Kuwait. (c) Simple exponential smoothing for first wave deaths in Kuwait and (d) Simple exponential smoothing for second wave transition deaths in Kuwait.

Figure 4 . 12 :

 412 Figure 4.12: Comparison between different exponential smoothing methods for (a) first wave deaths, (b) first wave cases, (c)second wave transition deaths and (d) second wave transition cases in Kuwait.

Figure 4 . 13 :

 413 Figure 4.13: Holt method for (a) first wave deaths, (b) first wave cases, (c)second wave transition deaths and (d) second wave transition cases in Kuwait.

Figure 4 .

 4 15(a)-(d) gives the comparison between optimal RMSE's of simple exponential model without trend and Holt's model. We trained 60% of the model while others were tested and 10 lags were used for the modelling.

Figure 4 .

 4 14(a)-(d) show visualisation of comparison between the holts model and the optimal Holt's model

Figure 4 . 14 :

 414 Figure 4.14: Comparison of Holt's model and optimal Holt's model forecast for (a) first wave cases, (b) second wave transition cases, (c) first wave deaths and (d) second wave transition deaths in Kuwait.

Figure 4 .

 4 15d, the simple exponential model on the left shows that there was a turning point at α = 0.5 with the highest RMSE and the least RMSE is for values α = 0 and α = 1.0. For Holt's model the minimum RMSE values is for β = 0 and β = 0.1, and any other values increases the RMSE value.

Figure 4 . 15 :

 415 Figure 4.15: Comparison of simple exponential model without trend optimal RMSE and Holt's model optimal RMSE for (a) first wave cases, (b) second wave transition cases, (c) first wave deaths and (d) second wave transition deaths in Kuwait.

2 )

 2 with drift = 74.5 diff.SES = 2.7 diff.SES = 2.5

  (a)-(d). Also we present in Table4.3 and Table4.2 the predicted values for Kuwait daily new cases and deaths for October, 2020 along with predicted range. It is observed that some of the numerical values generated using this model are close to the observed values of the COVID-19 pandemic in Kuwait. The mean square error is given as follows: first wave cases = 79247.71, second wave transition cases = 40288.81. The MSE for first wave deaths = 9.04 and second wave transition deaths = 3.93.

Figure 4 . 16 :

 416 Figure 4.16: Prophet forecast for (a) first wave deaths, (b) second wave transition deaths, (c) first wave cases and (d) second wave transition cases in Kuwait.

Figure 4 . 17 :

 417 Figure 4.17: Trend plots for (a) first wave deaths, (b) second wave transition deaths, (c) first wave cases and (d) second wave transition cases in Kuwait.

Figure 4 .

 4 Figure 4.18: (a) Loss plot for second wave transition deaths in Kuwait, (b) Loss plot for second wave transition cases in Kuwait, (c) Loss plot for first wave deaths in Kuwait, (d) Loss plot for first wave new cases in Kuwait, (e) Neural forecast for first wave cases in Kuwait and (f) Neural forecast for second wave transition cases in Kuwait.

Figure 4 . 19 :

 419 Figure 4.19: Times series visualization of daily new cases for (a) USA, (b) UK, (c) Turkey, (d) Russia, (e) India, (f) Brazil and (g) France.

Figure 4 .

 4 Figure 4.20: LSTM, GRU, CNN, and DNN results of daily new cases for (a) USA, (b) UK, (c) Turkey, (d) Russia, (e) India, (f) Brazil and (g) France.

Figure 4 .

 4 Figure 4.21: LSTM, GRU, CNN, and DNN results of daily deaths for (a) USA, (b) UK, (c) Turkey, (d) Russia, (e) India, (f) Brazil and (g) France.

Figure 4 .

 4 25(c) andFigure 4.25(d).

Figure 4 .

 4 25(b) and Figure 4.

Figure 4 .

 4 21(g), and MLP best predicts daily new cases, indicating a slight increase in September 2021 as shown in Figure Figure4.24(g). We discovered that MLP achieves the best prediction for daily new cases and daily deaths in India, Brazil, and Turkey, as shown in Figure 4.24c, Figure 4.24(e), Figure 4.24(f), Figure 4.25(c), Figure 4.25(e), and Figure 4.25(f). Despite a slight increase in September cases, the pandemic dynamics from October in India, Brazil, and Turkey shows a decreasing but fluctuating trend.

Figure 4 . 22 :

 422 Figure 4.22: Neural network for daily new cases in (a) France, (b) India, (c) USA, (d) Brazil (e) Russia, (f) Turkey, and (g) UK. In figures, x represents daily new cases and t represents days.

Figure 4 . 23 : 151 Figure 4 . 24 :

 423151424 Figure 4.23: Neural network for daily deaths in (a) France, (b) India, (c) USA, (d) Brazil (e) Russia, (f) Turkey, and (g) UK. In figures, x represents daily deaths while t represents days.

Figure 4 . 25 :

 425 Figure 4.25: MLP and ELM results of daily deaths for (a) USA, (b) UK, (c) Turkey, (d) Russia, (e) India, (f) Brazil, (g) France.

Figure 4 .

 4 [START_REF] Cámara | An epidemic model for economical impact predicting and spatiotemporal spreading of COVID-19[END_REF] and Figure4.26 show a clear peak of frequency at value 0.145, which is approximately a 7 days period showing a weekly pattern for countries considered, both for daily new cases and daily deaths, except for Turkey daily deaths as seen in Figure4.26(f), where peaks are not really visible with a slight peak at 0.14 and another at 0.08. India daily deaths, Russia daily new cases, and the United Kingdom daily new cases have only one clear peak (see Figure4.27(e), Figure4.27(g) and Figure4.26(b)), whereas others have smaller peaks at 3.6 days and 3.4 days, which represent approximately mid-week, which is an expected phenomenon due to accumulated cases from the beginning of the week, and as also seen in some countries collating their infection records after three days or more (e.g., one week in Cameroon). Furthermore, the forecasting pattern of the spectral analysis results corresponds to that of the MLP forecasting results presented in Section 4.2.4, with the exception of Russia daily new cases and daily deaths, which have different results for both methods. Forecasting based on spectral analysis also confirms fluctuations in MLP forecasted values presented in Section 4.2.4.

Figure 4 . 26 :Figure 4 . 27 :

 426427 Figure 4.26: Spectral analysis and forecasting of daily deaths for (a) France, (b) India, (c) USA, (d) Brazil (e) Russia, (f) Turkey and (g) UK.

Figure 4 . 28 :

 428 Figure 4.28: Lorenz curve showing the cumulated part of income vs. cumulated part of a population having this cumulated income. The curve in red represents left right triangleshaped Lorenz curve.

Figure 4 .

 4 Figure 4.29: (A) Linear regression line (in green) with the confidence interval (in gray). (B) Residual plot for the linear regression. (C) Partial regression plot. (D) Fit plot. (E) On left-hand side is the cross-validation plot for the linear regression and on right-hand side is the polynomial regression plot. (F) Residual plots for polynomial regression. (G) Heat map for the correlations between all variables. (H) Neural network visualization.

Figure 4 .

 4 29G, Theil and Gini indices are highly positively correlated with coefficient 0.7. By observing the epidemiologic variables (numbers of weekly reported new infectious cases, cured and deceased patients) during the 19 first weeks of 2021 (cf. in Figure3.19, data from[119]

Figure 4 . 30 :

 430 Figure 4.30: Correlations between various epidemiologic, socio-economic and demographic parameters observed in the different regions of Cameroon during the second wave. The false color scale indicates the intensity of the correlation between -1 and 1.

Figure 4 . 31 :

 431 Figure 4.31: Correlations between various epidemiologic, socio-economic and demographic parameters observed in the different regions of Cameroon during the second wave. The false color scale indicates the intensity of the correlation between -1 and 1.

Figure 4 . 32 :

 432 Figure 4.32: Correlations of the cured cases numbers between the 10 regions of Cameroon.

Figure 4 . 33 :

 433 Figure 4.33: Correlations of the deceased cases numbers between the 10 regions of Cameroon.

Figure 4 . 34 :

 434 Figure 4.34: Correlations of the new cases numbers between the 10 regions of Cameroon.

Figure 4 .

 4 37 corresponds to the ordinary multivariate least square methods with R-squared = 0.674.

Figure 4 .

 4 37A shows Paraguay as outlier not fitting data, Figure4.37B normalizes all countries and does not point any country in the plot. Using cross-validation method, the best parameter α for the modeling is shown in Figure4.35C. For ridge regression, α=0.142 with a mean square error of 1.36 and α=0.368 for lasso regression with a mean square error = 5.10. For Figure 4.35E, training score = 1.000 and test score = 0.641; for Figure 4.35F, training score = 0.992 and test score = 0.497; for Figure 4.36A, training score = 0.99 and test score = 0.406; and for Figure 4.36B, training score = 0.984 and test score = -0.077. It is evident from these results that linear regression best predicts Gini index with the highest test score, and predicted values are very close to each other as presented in Table 4.7. Also, the same pattern of prediction is observed in Figure 4.35E-F and Figure 4.36A-B showing that all methods used in this section have the same predictive behavior.

Figure 4 .

 4 38C, the first two clusters have 14 countries and the third has three countries, which are Uruguay and El Salvador on the same hierarchy while Argentina is on another hierarchy.Only the first cluster in the dendrogram is presented. In Figure4.38F, the Gini index has the highest positive correlation of 0.44 with the principal component PC 1 and Theil index has only the value 0.34 with PC 1. The main variable causing the separation into three classes is the Gini index.

Figure 4 .

 4 Figure 4.35: (A) Residual plot for lasso regression. (B) Residual plot for ridge regression. (C) Lasso regression cross-validation error. (D) Prediction error for the lasso regression. (E) Linear regression prediction plot. (F) Lasso regression and (G) ridge regression prediction plots. (H) MLP regression prediction plot.

Figure 4 .

 4 Figure 4.36: (A) Ridge regression prediction plots. and (B) MLP regression prediction plot.

Figure 4 .

 4 Figure 4.37: (A) Influence plot. (B) Leverage vs. Normalized residuals squared plot. (C) Partial regression plot. (D) Component-Component plus residual plot.

Figure 4 .

 4 Figure 4.38: (A) Scree plot. (B) Plot for projection of points for PC1 and PC2. (C) Hierarchy clustering dendrogram. (D) Parallel coordinates plot for the clusters. (E,F) PC's visualization. (G) Box plot for the clusters. (H) Parallel coordinates plot for the centroids. 173

Figure 4 . 39 :

 439 Figure 4.39: Linear regression plots for (A) the first wave slope vs. the demo-economic index for the developed and developing countries, (B) second wave slope vs. immigration rate for developed countries, (C) the opposite of the initial auto-correlation slope for first wave vs. social fracture index, (D) opposite of initial autocorrelation slope for second wave vs. social fracture index. (E) Influence plot. (F) Leverage vs. normalized residuals squared plot. (G) Partial regression plot. (H) Component-Component plus residual plot.

Figure 4 .

 4 Figure 4.40: (A) Residual plot for lasso regression. (B) Residual plot for ridge regression. (C) Lasso regression cross-validation error. (D) Prediction error for lasso regression curve (R = 0.8). (E) Linear regression prediction plot. (F) Lasso regression prediction plot. (G) Ridge regression prediction plot. (H) MLP regression prediction plot.

Figure 4 .

 4 Figure 4.41: (A) Heatmap of the parameter's correlations. (B) Scree plot. (C,D) PCs visualization. (E,F) Hierarchy clustering dendrogram. (G) Boxplot of the clusters. (H) Parallel coordinates plot for the clusters. (I) Parallel coordinates plot for the centroids. (J) Projection points for PC1 and PC2. 177

Figure 4 .----

 4 39C and Figure 4.39D, the type of regression used is polynomial or order 6 while Figure 4.39A and Figure 4.39B is linear regression. • The Figure 4.39 present regression analyses with the following parameters: • Figure 4.39A: Linear regression: slope = 0.11463663009107196, intercept = -0.0037118697103040027, r-value = 0.4387157758684147, p-value = 0.0032517683682962654, standard error = 0.03667137141150123, R-squared = 0.192472 and RMSE = 0.1044724946057671. • Figure 4.39B: Linear regression: slope = -0.002547609589041096, intercept = 0.07888755616438356, r-value = -0.3272886357381478, p-value = 0.03672803730354382, standard error = 0.0011777868896598461, R-squared = 0.107118 and RMSE = 0.03065537183298402. • Figure 4.39C: -Linear regression: slope = 0.0017309145398433248, intercept = 0.06695128460299407, r-value = 0.263675660748951, p-value = 0.08754941979369255, standard error = 0.0009889311191763849, R-squared = 0.069525 and RMSE = 0.0354860744891158, Polynomial regression (order 6): R-squared = 0.3 and RMSE = 0.04060485094256808. • Figure 4.39D: Linear regression: slope = 0.0015999465132904799, intercept = 0.0810899892250729, r-value = 0.2861266574746827, p-value = 0.13239511534872409, standard error = 0.001031140187045727, R-squared = 0.081868 and RMSE = 0.05492215494302141, Polynomial regression (order 6): RMSE = 0.07286590609946085 and R-squared = 0.35.

Figure 4 .

 4 40E, training score = 0.983 and test score = 0.607; for Figure 4.40F, training score = 0.170 and test score = 0.021; for Figure 4.40G, training score = 0.854 and test score = 0.115; and for Figure 4.40H, training score = 0.980 and test score = -2.386.It is evident from the results that linear regression best predicts GDP percentage devoted to health expenditure with the highest test score, and all predicted values are very close. In Figure4.41E and Figure4.41F, the first cluster has 15 countries, the second cluster has 27 countries, while the last cluster has 2 countries, which are Tanzania and Mauritius. Only the first two cluster dendrograms are presented. With PC1, Gini index GI has the highest positive correlation of 0.52 with PC1 and demo-economic index DI has the second highest negative correlation of -0.53, while with PC2, first wave maximum R 0 has the highest positive correlation of 0.70 (Figure4.41C). The first cluster contains a majority of developing countries, and the second cluster contains a majority of developed countries, the main variable causing the separation into two classes being the Gini index in PC1.

6 of 32 Figure 3 .

 6323 Figure 3. Linear (in red) and sextic (in green) regression plots of first wave opposite of initial autocorrelation slope vs CHE/GDP. LinregressResult slope = 0.01117, intercept = 0.0664, r value = 0.47219, p value = 0.0097, stderr = 0.004, RMSE for linear regression = 0.0387, RMSE for polynomial regression = 0.04399, R-squared for order six polynomial regression = 0.4.

Figure 4 .

 4 Figure 4.42: Linear (in red) and sextic (in green) regression plots of first wave opposite of initial autocorrelation slope vs. CHE/GDP.

  (a) slope = 0.0320468, intercept = -0.16158, r -value=0.6481, p-value = 4.62 13 ×10 -6 , standard error = 0.00603, R-squared = 0.42, RMSE = 0.09359760581, and (b) slope = -0.0010489, intercept = 0.01994, r -value = 0.1340845, p-value = 0.4094462, standard error = 0.001258, R-squared = 0.018, RMSE for linear regression = 0.018583749. • Figure 4.43(a) and Figure 4.44 show a positive correlation while Figure 4.43(b) show a negative correlation between the slope of the logarithmic regression curve of the new cases of COVID-19 as a function of time (a sign of rapid growth of the epidemic if it is high) and the economic index CHE/GDP. This is true when we observe the developed and developing countries (Figure 4.44) or the developed countries alone for which the positive correlation is higher, the correlation coefficient increased from 0.57 to 0.65 (Figure 4.43(a)), but this trend is reversed for the second wave (Figure 4.43(b)), where the correlation coefficient equal -0.57, this being possibly due the early implementation of mitigation measures in developed countries, reducing the exponential growth of new cases in the second wave.

Figure 4 .Figure 4 . 43 :

 4443 Figure 4.43: Regression plots for developed countries of (a) first and (b) second wave exponential regression slope versus CHE/GDP. 182

Figure 4 .

 4 Figure 4. Regression plot of first wave exponential regression slope vs CHE/GDP for developed and developing countries. LinregressResult slope = 0.026632, intercept = -0.1052912, r value = 0.5661, p value = 7.60 × 10 -5 , stderr = 0.00605655, Rsquared = 0.320470, RMSE for linear regression = 0.095836.

Figure 4 . 44 :

 444 Figure 4.44: Regression plot of first wave exponential regression slope vs CHE/GDP for developed and developing countries.

Figure 4 .

 4 46C, average life expectancy has the highest positive correlation of 0.46 in PC 1 while first wave deterministic R o (first wave D) has the highest positive correlation in PC 2, equal to 0.47. The two socio-economic variables explaining the most clustering are the average life expectancy (LE) and the stringency index (SI).

Figure 4 .

 4 Figure 4.45: (A) Residual plot for lasso regression. (B) Residual plot for ridge regression. (C) Lasso regression cross-validation error. (D) Prediction error for lasso regression curve. (E) Linear regression prediction plot. (F) Lasso regression prediction plot. (G) Ridge regression prediction plot. (H) MLP regression prediction plot.

Figure 4 .

 4 Figure 4.46: (A) Heatmap of the parameter's correlations. (B) Scree plot. (C,D) PCs visualization. (E,F) Hierarchy clustering dendrogram. (G) Parallel coordinates plot for the centroids. (H) Projection points for PC1 and PC2. (I) Boxplot of the clusters. (J) Parallel coordinates plot for the clusters.

Figure 4 .

 4 Figure 4.47: (A) Influence plot. (B) Leverage vs. normalized residuals squared plot. (C) Partial regression plot of the temperature vs. maximal R 0 of the first wave. (D) Fit plot. The first wave maximal R 0 is negatively correlated with the mean temperature of the country ([101, 102]).

Figure 4 .

 4 49E, training score = 0.993 and test score = 0.535. • For Figure 4.49F, training score = 0.898 and test score = 0.629. • For Figure 4.49G, training score = 0.983 and test score = 0.259. • For Figure 4.49H, training score = -0.072 and test score = -0.196.

Figure 4 .

 4 Figure 4.48: (A) Influence plot. (B) Leverage vs. Normalized residuals squared plot. (C) Partial regression plot. (D) Component-Component plus residual plot.

Figure 4 .

 4 Figure 4.49: (A) Residual plot for lasso regression. (B) Residual plot for ridge regression. (C) Prediction error for lasso regression curve. (D) Lasso regression cross-validation error. (E) Linear regression prediction plot. (F) Lasso regression prediction plot. (G) Ridge regression prediction plot. (H) MLP regression prediction plot. 189

Figure 4 .

 4 Figure 4.50: (A) Heatmap of the parameter's correlations. (B) Scree plot. (C,D) PCs visualization. (E,F) Hierarchy clustering dendrogram. (G) Parallel coordinates plot for the centroids. (H) Projection points for PC1 and PC2. (I) Boxplot of the clusters. (J) Parallel coordinates plot for the clusters. 190

Figure 4 . 32 aFigure 6 .

 4326 Figure 4.52 show a positive correlation between the slope of the logarithmic regression curve of the new cases of COVID-19 and the economic index CHE/GDP and the correlation coefficient for the first wave remains for all countries close to that for developed countries. The result for the linear regression is as follows: slope = 0.01214439, intercept = -0.0159087, r -value = 0.3655, p-value = 2.71 ×10 -6 , standard error = 0.00249223, R-squared = 0.13359, RMSE = 0.0819603345. Healthcare 2021, 9, x 14 of 32

Figure 4 . 52 :

 452 Figure 4.52: Regression plot of first wave exponential regression slope vs CHE/GDP for all countries.

Figure 4 . 2 . 4 .

 424 51 to Figure 4.54 and shows several features: 1. The hierarchical clustering (Figure 4.51 and Figure 4.54(b)) shows a trend common to developed countries (shown in green), with the notable exception of Germany and Czechia, The principal component analysis shows the importance of the CHE/GDP index in the first principal component (Figure 4.54(a),(c),(d)) and of the deterministic R 0 (R det 0 ) of the exponential phase of the first wave in the second principal component and of the second wave in third principal component (Figure 4.54(e)); 3. The analysis of parallel coordinates for cluster centroids also shows the importance of the deterministic R 0 in the discrimination of clusters (Figure 4.55); The analysis of the residuals shows a good explanatory power of the first three principal components (60% of the total variance in Figure 4.54(c), confirmed by the projections on the two first principal planes of Figure 4.54(d)-(e)), and a weak correlation of the principal components with these residuals (Figure 4.53(a)-(b)).

Figure 13 .

 13 Figure 13. (a) Leverage vs normalized squared residuals plot. (b) Residuals regression plots for initial variables.

Figure 13 .

 13 Figure 13. (a) Leverage vs normalized squared residuals plot. (b) Residuals regression plots for initial variables.

Figure 4 .

 4 Figure 4.53: (a) Leverage vs normalized squared residuals plot. (b) Residuals regression plots for initial variables. 196

Figure 10 .Figure 11 .

 1011 Figure 10. (a) Boxplots of the clusters corresponding to the hierarchical clustering. Visualizations of (b) more "developing"(in red with some notable exceptions such as the Czech Republic and Germany) and (c) more "developed" (in green and partially in orange) countries parts of the hierarchy tree.

Figure 4 .Figure 12 .

 412 Figure 4.54: (a) Principal components (PC) plot from the principal component analysis (PCA) on the initial variables: first and second waves maximum R 0 , 1st wave R 0 and 2nd wave R 0 , deterministic R 0 , 1st wR 0 det and 2nd wR 0 det , 1st wave Arima slope, 2nd wave Arima slope, and CHE/GDP. (b) Projection of the points corresponding to 204 countries of the PCA's plot on the first PC plane with more developed countries in green and more developing in orange. (c) Explained variance plot. (d,e) Correlation circles for the two first PC planes. 198

3. 5 . 2 .

 52 Ordinary Least Square Method. The Multivariate Case. The clustering of the countries from epidemic and economic variables is described in Figures 10 to 13 and shows several features: 1. The hierarchical clustering (Figures 10 and 11b) shows a trend common to developed countries (shown in green), with the notable exception of Germany and Czechia, The principal component analysis shows the importance of the CHE/GDP index in the first principal component (Figure 11a, c and d) and of the deterministic R0 (R0det) of the exponential phase of the first wave in the second principal component and of the second wave in third principal component (Figure 11e);

Figure 4 . 55 :

 455 Figure 4.55: Parallel coordinates for cluster centroids.

Figure 4 .

 4 29G, Figure 4.41A, Figure 4.46A, and Figure 4.50A, explaining the role of each variable thanks to these correlations.

Figure 4 .

 4 41E, Figure 4.41F and Figure 4.48). The principal component analysis shows high correlation for the numbers of new cases used in this analysis. The social fracture index has high correlation in PC1 for both cases, while in PC2, percentage of GDP devoted to health expenditure was dominant for OECD countries, and maximum R o for the first wave was dominant for both developed and developing countries (see Figure 4.41C and Figure 4.50C).

Figure 4 . 51 ,

 451 Figure 4.55 and Figure 4.54. Figure 4.51a shows the box-plot of the seven initial variables used in hierarchical clustering: the first and second wave opposite of the initial auto-correlation slope (respectively ARIMAF and ARIMAS), exponential regression slope and maximum R 0 (respectively FirstwaveD, SecondD, FirstwaveR, SecondR), and CHE/GDP. The box-plots contain five clusters represented in Figure 4.51(b)-

Figure 4 .

 4 54(b) gives the projection of 204 countries on the first PC plane and distinguishes two main clusters of 118 and 85 countries, respectively, plus a singleton repre-senting Botswana, with more developed countries in green and more developing countries in orange.

Figure 4 .

 4 54(c) shows that 60% of the variance is explained by the three first PCs, and Figure4.54(d)-(e) presents the correlation circles with projection of the initial variables as vectors on the corresponding two principal planes (PC1, PC2) and (PC2, PC3), showing such as in Figure4.54(a) the preeminence of the opposite vectors, the first wave deterministic R 0 and the CHE/GDP.

Figure 4 .

 4 [START_REF] Pizzuti | Network-based prediction of covid-19 epidemic spreading in italy[END_REF] shows also for the first K-means cluster the importance of the first wave deterministic R 0 . Finally, Figure4.53(a)-(b) corresponds to the ordinary multivariate least square method.

Figure 4 .

 4 53(a) shows the eccentric position of developed countries such as Belgium and USA and developing countries such as Equatorial Guinea and Suriname as outliers not fitting the data bulk, and Figure4.53(b) the concentration of the initial variable CHE/GDP with the first and second waves deterministic R det 0 , in agreement with the fact that they are the most dominant initial variables in PCA and k-means clustering.

4. 5

 5 Functional Data Analysis: Application to Daily Observation of COVID-19 Prevalence

, X 4

 4 while response variables are given as Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 are numbers of recovery, deaths, infected, vaccination, vaccination per 1000 population and number of tests. All the data are from [119, 244, 245]. This approach is complementary of the works estimating the parameters of the functions S and I (namely R 0 and B) presented in Section 3.3.

Figure 4 .

 4 Figure 4.56: (a) Time series modelling of daily new cases between 01/05/2021-15/07/2021 in France. (b) GRU deep learning forecasting method for daily new cases between 25/02/2020-03/09/2021 in France. (c) Daily hospitalization cases in three French departments: Nord, Paris and Essonne. (d) Fit curve for hospitalization cases in Paris and Seine-Maritime.

Figure 4 .

 4 57(a)-(c) and Figure 4.57(e)-(g), but Figure 4.57(d) and Figure 4.57(h) show a bad shape for elastic mean.The B-spline smoothing technique was used in smoothing the functional data where the number of basis function n = 7 which was discovered that it best suit for the modelling of the data as number of elements. The result of this smoothing technique can be seen in Figure4.58.

Figure 4 .

 4 Figure 4.57: (a,c,e,g) -hospitalized, ICU, daily return home and deceased cases before start of vaccination, (b,d,f,h) -hospitalized, ICU, daily return home and deceased cases.

Figure 4 . 58 : 205 Figure 4 . 59 :

 458205459 Figure 4.58: Smoothed curves for the shape of COVID-19 epidemic in all departments in France: (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) daily deceased, (d) daily deceased when vaccination has started, (e) daily return home, (f) daily return home when vaccination has started, (g) ICU cases and (h) ICU cases when vaccination has started.

Figure 4 .

 4 Figure 4.60: (a) Hospitalized cases interpolation smoothing, (b) hospitalized monotone smoothing, (c) ICU cases interpolation smoothing, (d) ICU cases monotone smoothing, (e) daily deceased interpolation smoothing and (f) daily deceased monotone smoothing.

Figure 4 . 61 :

 461 Figure 4.61: Kernel smoothing method for (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination has started, (e) daily return home and (f) daily return home when vaccination has started.

Figure 4 .

 4 62(d), PC 1 has strictly positive values while PCs 2-4 show negative values between February to June 2021. In Figure4.62(e) PC 1 peaked with a positive value at the beginning of the pandemic in France which validates the percentage of recovery in France (about 71% as at 14/11/2021) while PC 1 in Figure4.62(f) shows a positive decline across the days considered, with a disparity between the period of vaccination and without vaccination. On the y-axis of Figure4.62(g) and Figure4.62(h), it was observed that this is the only result with low values for the PCs because the deaths due to COVID-19 in France remain at a low level (about 2% as at 14/11/2021), while all PCs show almost the same pattern as what was observed previously for the other variables.

Figure 4 . 62 :

 462 Figure 4.62: Functional PCs for different functional data before the start of vaccination (19/03/2020-29/10/2020) and when vaccination has started (27/12/2020 to 30/06/2021): (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination has started, (e) daily return home, (f) daily return home when vaccination has started, (g) daily deceased and (h) daily deceased when vaccination has started. 211

Figure 4 .

 4 Figure 4.63: FPCA scores for different functional data before the start of vaccination (19/03/2020-29/10/2020) and when vaccination has started (27/12/2020-30/06/2021). (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination has started, (e) daily return home, (f) daily return home when vaccination has started, (g) daily deceased and (h) daily deceased when vaccination has started. Note that the numbering of points on the diagram are codes for each French department. 212

Figure 4 .

 4 64(a), Figure4.64(c) and Figure4.64(e) showed a negative correlation between the epidemiology variables and the hospitalization, ICU, and daily return home cases across departments, whereas Figure4.64(g) showed a positive correlation between the epidemiology variables and the deceased cases across departments.

Figure 4 .

 4 64(b), Figure 4.64(d) and Figure 4.64(f) confirm the effect of vaccination on the number of hospitalizations, ICU, and daily return home during vaccination, demonstrating a positive correlation and the effect of this control measure, whereas Figure 4.64(h) shows a negative correlation, indicating a negative relationship between the deceased and the vaccination introduced. The results presented in Figure 4.66 show the linear relations in the scatter plot as most of the variables show 95% significance level and from Table

Figure 4 .

 4 66(a) shows hospitalized cases with p-value < 0.05 for all canonical variate, Figure4.66(b) shows hospitalized when vaccination has started with p-value < 0.05 except the last canonical variate with p-value = 0.88, Figure4.66(c) shows ICU cases with p-value < 0.05 for all canonical variate, Figure4.66(d) shows ICU cases when vaccination has started with p-value < 0.05 except the last two Canonical variate with p-value = 0.68 and p-value = 0.87 respectively, Figure4.66(e) shows daily return home with p-value < 0.05 for all canonical variate, Figure4.66(f) shows daily return home when vaccination has started with p-value < 0.05 except the last two canonical variate with p-value = 0.14 and p-value = 0.34 respectively, Figure4.66(g) shows daily deceased with p-value < 0.05 except the last canonical variate with p-value = 0.08 and Figure4.66(h) shows daily deceased when vaccination has started with p-value < 0.05 except the last two canonical variate with p-value = 0.08 and p-value = 0.46 respectively.

Figure 4 . 64 :

 464 Figure 4.64: Helio plot for the correlation of French departments for (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination started, (e) daily return home, (f) daily return home when vaccination started, (g) daily deceased and (h) daily deceased when vaccination started.

Figure 4 . 65 :

 465 Figure 4.65: Canonical variate redundancy plot for (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination has started, (e) daily return home, (f) daily return home when vaccination has started, (g) daily deceased and (h) daily deceased when vaccination has started.

Figure 4 . 66 :

 466 Figure 4.66: Canonical correlation visualization of (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination has started, (e) daily return home, (f) daily return home when vaccination has started, (g) daily deceased and (h) daily deceased when vaccination has started.

Figure 4 .

 4 67(a)-(d) presents the clusters (0, 1 or 2) that each French department belongs to. The result clustered French departments outside metropole or the French hexagon to the same clusters which of course are not binded with mitigation measures and rules used in departments within France[96]. Also, departments close to Paris are in the same cluster which is the same with departments having the same trend of the pandemic prevalence as presented in Figure4.67(a)-(d).

Figure 4 . 67 :

 467 Figure 4.67: Clustering of all departments in France using K-means: (a) hospitalized, (b) daily deceased, (c) daily return home when vaccination has started and (d) ICU cases when vaccination has started, and fuzzy K-means, (e) clusters for hospitalized, (f) clusters for daily deceased, (g) clusters for daily return home when vaccination has started and (h) clusters for ICU when vaccination has started. 218

101 1 β

 1011 (s, t)x AllfdData (s)ds + ε by using the functional data curves of 101 days (predictors) to predict another set of curves of 101 days (response) while also estimating the slope β(s, t), whose results in considered cases are presented in 3D diagrams of Figure4.69.

Figure 4 .

 4 69(a) shows hospitalized cases with the slope on the interval -2.799063 ≤ β(s, t) ≤ 1.980147, Figure 4.69(b) shows hospitalized when vaccination has started with the slope on the interval -1.501887 ≤ β(s, t) ≤ 1.076421, Figure 4.69(c) shows ICU cases with the slope on the interval -1.0733846 ≤ β(s, t) ≤ 0.841100, Figure 4.69(d) shows ICU cases when vaccination has started with the slope on the interval -0.5646148 ≤ β(s, t) ≤ 0.3661280, Figure 4.69(e) shows daily return home with the slope on the interval -0.6755000 ≤ β(s, t) ≤ 0.7030529, Figure 4.69(f) shows daily return home when vaccination has started with the slope on the interval -0.4333295 ≤ β(s, t) ≤ 0.4300995, Figure 4.69(g) shows daily deceased with the slope on the interval -0.3277864 ≤ β(s, t) ≤ 0.4002531 and Figure4.69(h) shows daily deceased when vaccination has started with the slope on the interval -0.3284866 ≤ β(s, t) ≤ 0.3641679. The observation from all the figures in this section is that the 3D surfaces yield results whose shapes look roughly similar to the slope curve, functional predictors curve and functional response curve.

Figure 4 .

 4 Figure 4.68: (a) Functional linear regression model prediction for number of deaths in France as response variable before vaccination begins and (b) Functional linear regression model prediction for number of tests in France as response variable before vaccination begins.

Figure 4 .

 4 Figure 4.69: The 3D visualization of function-on-function regression for (a) hospitalized cases, (b) hospitalized when vaccination has started, (c) ICU cases, (d) ICU cases when vaccination has started, (e) daily return home, (f) daily return home when vaccination has started, (g) daily deceased and (h) daily deceased when vaccination has started.

  Note: PCN is Pearson correlation coefficient of the 140 mobile windows starting at day N, PCD is Pearson correlation coefficient of the 20 mobile windows starting at week day D, PCEW is Pearson correlation coefficient of the 140 mobile windows starting at week day E of the week W, and MCC is Mean of the correlation coefficients between the first window and the 20 mobile windows starting at week day E and number of such correlation coefficients significantly > 0.

  Time series data set can be hourly, daily, monthly or yearly because one of the variables has to do with time and it consists of four components which are trend, cyclical, seasonal and irregular and it depends on the component of interest you want to investigate in the modeling. It can be represented as follows:{p t } t=1,...,Twhere p t is a random variable which represents a repeated observations.The auto-regressive integrated moving average model, abbreviated as ARIMA, is a wellknown and widely used statistical technique for time series forecasting that uses the form of

	Auto-regressive integrated moving average

the general equation presented in Equation (2.1) to forecast future values of a time series based on previous values. ARIMA can model non-seasonal time series with non-white noise patterns. Box and Jenkins introduced the ARIMA model in 1970

[START_REF] Geurts | Time series analysis: Forecasting and control[END_REF]

. ARIMA model has proven to be capable of producing accurate short-term forecasts. This model is based on the idea that a variable's future value of a time series is determined by its past values and errors. The model equation is as follows:

Table 2 .

 2 1: Exponential smoothing models.

	Trend

The Prophet forecasting modeling is an improved version of ARIMA model and exponential smoothing model. It uses three main model components which are trend, seasonality and holidays. It is described by the model equation below:

Table 2 .

 2 2: Model parameters for support vector machine.

	Hyperparameter	Nature/Value
	Type	Eps-regression
	Kernel	Radial
	Cost	1
	Gamma	1
	Epsilon	0.1
	p-value	< 2.2e -16

Table 2 .

 2

	Hyperparameter	Nature
	Number of clusters	Three
	Affinity	Euclidean
	Linkage	Ward
	A Dendrogram was used to depict the hierarchical clustering method. A dendrogram rep-
	resents the results of cluster analysis; it reveals all of the steps taken in the hierarchical
	algorithm, including the distance at which clusters merged [90]. Dendrograms provide a con-
	venient method for exploring various data-combination possibilities. The number of clusters,

3: Hierarchical clustering algorithm parameters. affinity, and linkage are the parameters used in the training of a hierarchical algorithm, as shown in Table 2.3. All of these parameters are optimized to improve model optimization.

Table 2 .

 2 4: Artificial neural network models parameters.

	Hyperparameter	Definition	Values
	Batch size and max-pooling layers Batch size is the number	30, 32, 64, 128, 256, . . .
		of sub-samples given to
		the neural networks after
		which parameters is
		updated
	Number of epoch	It is the number of	100, . . .
		iterations or times in
		training a neural
		network
	Number of hidden layers and units It is the layers between	32, 50, 64, 128, 256, . . .
		input layers and output
		layers
	Dropout	It is used to regularize	20 -50%
		artificial neural network
		models to avoid
		over-fitting
	Optimizer	It is used to change the	Adam, stochastic
		attributes of the neural	gradient and gradient
		networks such as weights	descent
		and learning rate so as
		to reduce losses
	Activation function	It decides whether the	Sigmoid, Rectified
		neurons should be	Linear Unit (ReLU) and
		activated or not	tanh
	Filter size/Kernel in CNN layers	It helps to pad the input	3 × 3 or 5 × 5
		volume with zeros in
		such a way that the
		CNN layers does not
		alter the dimension of
		the input
	Pooling size	It reduces the dimension	2 × 2
		of the feature map
	Verbose	It is a setting that allows	0, 0.2, 1, 2, . . .
		you to see wordy
		information in neural
		network modeling while
		it is training.
	Validation split	It's a float value for	Between 0 and 1
		validation data which is
		not used for the training,
		but to evaluate the loss
		and the accuracy

Figure 2.11: Flowchart of deep structured learning modeling.

  R 1 = 2.0279, R 2 = 7.6158 and R 3 = -16.426.

					295.8 113.491 41.7391	
		M 2 =		113.491 41.7391 15.351	
					41.7391 15.351	5.36
			-0.07779371 0.20964295 0.00524305	
	M -1 2 =		0.20964295 -1.0123552 1.26721348	
			0.00524305 1.26721348 -3.48354228
	Then, the resulting R j 's equal:

6 + 0.95X 5 + 1.9X 4 = 226.982 + 39.652 + 29.17 = 295.8 X 8 = 2X 7 + 0.95X 6 + 1.9X 5 = 591.6 + 107.816 + 79.304 = 778.72 Then, we obtain the matrices M 2 and M -1 2 :

  [START_REF] Ladoy | Size and duration of covid-19 clusters go along with a high sars-cov-2 viral load: A spatio-temporal investigation in vaud state, switzerland[END_REF].209 + 9.101 + 4.81 = 32.12 X 8 = X 7 + X 6 + X 5 = 32.12 + 18.209 + 9.101 = 59.43

	Hence, we obtain:
					32.12 18.209 9.101	
			M 2 =		18.209 9.101 4.81	
					9.101	4.81	2.36
	and		-0.35061537 0.1839519	0.97925345	
	M -1 2 =		0.1839519 -1.47916605 2.31025157	
			0.97925345	2.31025157 -8.0783421
	and the resulting R

j 's equal: R 1 = 2.90, R 2 = 5.4888, R 3 = -14.696.

  .14). The average transmission rate R o equals about 1.1, value close to that calculated directly, with a maximal daily reproduction rate the first day of the infectiousness period. Because of the negativity of R 2 , we have to calculate the entropy H of the distribution

Table 3 .

 3 2: Result for R max for different countries.

	Countries	σ = 1	σ = 2
	United Kingdom	0.39978	0.20031
	Australia	0.40018	0.20071
	Italy	0.39927	0.19980
	Spain	0.40136	0.20190
	Canada	0.40018	0.20098
	United States	0.40126	0.20180

Table 3 .

 3 3: Result for R max 2 while using R max in Table3.2 as R max 1 for different countries.

	Countries	σ = 1	σ = 2
	United Kingdom	0.79842	0.59769
	Australia	0.79742	0.59608
	Italy	0.79970	0.59974
	Spain	0.79440	0.59129
	Canada	0.79673	0.59498
	United States	0.79466	0.59169
	3.4 Estimation of Transmission Rate From Identification
	of the First Inflection Point of New Infectious Cases
	Waves in COVID-19 Outbreak

Following the same approach as in

[START_REF] Ducrot | Identifying the number of unreported cases in sir epidemic models[END_REF]

, the inflection point during the outbreak of COVID-19 pandemic can be derived using Bernoulli equation and to analyze this Susceptible-Infected (SI) compartment model for calculating the time of the inflection point of the curve of new infectious cases. First, let's recall that the SI Bernoulli differential equation model of an outbreak epidemic is given as:

Table 3 . 4

 34 Model for the Lifespan Loss Due to COVID-19 Outbreak3.5.1 Formulation of epidemic ageing model in human populationsLet us consider a population with four age classes(i = 1, • • • , 4) that is (0, 19),[START_REF]World health organization[END_REF][START_REF] Dell'anna | Solvable delay model for epidemic spreading: the case of covid-19 in italy[END_REF],[START_REF] Bianconi | Epidemic spreading[END_REF][START_REF] Prodanov | Analytical parameter estimation of the sir epidemic model. applications to the covid-19 pandemic[END_REF], and (≥ 60) where f 1 , f 2 , f 3 , f 4 are the fertility rates in the age classes, v 1 , v 2 , v 3 , v 4 are the probability to remain in state i, α 1 , α 2 , α 3 , α 4 are the probability to go from state i to state

	Adamaoua	1.30175	1.50	2.60143	0.89396
	Center	1.04135	1.75	2.85711	0.47851
	Est	2.18015	1.50	2.60143	0.89396
	Extreme-Nord	1.74880	1.50	2.60143	0.89396
	Littoral	0.91404	1.75	2.85711	0.47851
	Nord	1.31392	1.50	2.60143	0.89396
	Nord-Ouest	3.64468	1.50	2.60143	0.89396
	Ouest	3.38611	1.50	2.60143	0.89396
	Sud	1.19674	1.50	2.60143	0.89396
	Sud-Ouest	0.88279	1.50	2.60143	0.89396

: Values of epidemiologic parameters for different regions of Cameroon. Cameroon regions Discrete R 0 Transmission rate β R max (σ = 1) R max (σ = 2)

Table 3 .

 3 5: List of variables and parameters considered in the paper. Mortality rate in age i, with v i + α i = 1 -µ i v i (resp. α i ) Probability to remain in age i (resp. go to (i + 1)) between t and t + 1 λ (resp. λ , and v ) Dominant eigenvalue (resp. subdominant and current eigenvalue) of U S, s ij Sensitivity matrix S and its general coefficient s ij P i Probabilities on the interval 0 ≤ P i ≤ 1 ∀ i = 1, . . . , m. u(a,t,s) Cell concentration at age a, time t and space s v i (resp. β i ) Probability to remain in age i (resp. go to (i + 2)) between t and t + 1 Q Mitotic abortive coefficient (0 < Q ≤ 1) ∆ (resp. □) Laplacian (resp. Dalembertian) partial derivative operator Epidemiologic dynamics E * , e ij Epidemiologic matrix E * , its general coefficient e ij E M (resp. E W ) Epidemiologic matrix relative to men (resp. women) w, H Invariant measure of E

	Population dynamics
	u i (t)	Size of the age class i at time t
	U, A	Usher matrix, Hahn matrix
	f i	Fertility rate
	µ i

* , entropy of the invariant measure w of E *

Application to COVID-19 outbreak in Cameroon

Consider Cameroon, which has a portion of its population affected by the COVID-19 outbreak. The data coming from [119,

[START_REF] Tabutin | The demography of sub-saharan africa in the 21st century. transformations since 2000, outlook to 2050[END_REF] 156, 157, 158,[START_REF] Ins | Enquête démographique et de Santé Cameroun[END_REF] 160,[START_REF]Rapport de situation COVID-19 n o 84[END_REF]

allow to calculate the epidemic matrix in the cases of normal ageing and supplementary ageing due to the COVID-19 outbreak.

  menarche and menopause have a 95%-Confidence Interval (CI) in the literature (see Table3.7) and[158,[START_REF] Ins | Enquête démographique et de Santé Cameroun[END_REF]). If we take the left and right limits of these intervals and if we calculate their impact on demographic data, we find for the fertility rate a 95%-CI equal to [0.013, 0.017] at age (0, 19) and [0.014, 0.016] at age[START_REF] Bianconi | Epidemic spreading[END_REF][START_REF] Gavalas | A Lattice Model for COVID-19 Epidemic[END_REF]. Consequently, the dominant eigenvalue of the E * matrix belongs to the interval [1.02476, 1.02736]. The real value given in[START_REF] Tabutin | The demography of sub-saharan africa in the 21st century. transformations since 2000, outlook to 2050[END_REF] is equal to 1.0258 in 2020, and hence, it belongs to this interval. We used the same method for calculating and estimating parameters used in Cameroon, which is presented in Table3.7 for France, while using raw data from France. From [119, 162], we know that France fertility rate in 2020 is equal to 1.85 children / fertile woman and that women of the second age class[START_REF]World health organization[END_REF][START_REF] Altakarli | China's response to the covid-19 outbreak: A model for epidemic preparedness and management[END_REF][START_REF] Mackolil | Logistic growth and sir modelling of coronavirus disease (covid-19) outbreak in india: Models based on real-time data[END_REF][START_REF] Khanday | Mathematical analysis on the dynamics of covid-19 in india using sir epidemic model[END_REF][START_REF] Yaghoby | A new nonlinear mathematical model for forecasting the waves of the COVID-19 epidemic[END_REF][START_REF] Kaxiras | Multiple epidemic wave model of the covid-19 pandemic: Modeling study[END_REF][START_REF] Meiksin | Using the seir model to constrain the role of contaminated fomites in spreading an epidemic: An application to covid-19 in the uk[END_REF][START_REF] Cámara | An epidemic model for economical impact predicting and spatiotemporal spreading of COVID-19[END_REF][START_REF] Paunovic | Mathematical modeling and statistical analysis in post -covid-19 tourism recovery[END_REF][START_REF] Duan | Simulations of the covid-19 epidemic in nigeria using sir model[END_REF][START_REF] Padmapriya | Fuzzy fractional mathematical model of covid-19 epidemic[END_REF][START_REF] Kozioł | Fractional-order sir epidemic model for transmission prediction of covid-19 disease[END_REF][START_REF] Deforche | An age-structured epidemiological model of the Belgian COVID-19 epidemic[END_REF][START_REF] Balabdaoui | Age-stratified discrete compartment model of the covid-19 epidemic with application to switzerland[END_REF][START_REF] Verrelli | Two-age-structured covid-19 epidemic model: Estimation of virulence parameters to interpret effects of national and regional feedback interventions and vaccination[END_REF][START_REF] Liu | Focus on the mental health of pediatric medical workers in china after the covid-19 epidemic[END_REF][START_REF] Li | Epidemic risk perception, perceived stress, and mental health during covid-19 pandemic: A moderated mediating model[END_REF][START_REF] Rihan | Stochastic sirc epidemic model with time-delay for covid-19[END_REF][START_REF] Kang | Epidemic spreading and global stability of a new sis model with delay on heterogeneous networks[END_REF][START_REF] Dell'anna | Solvable delay model for epidemic spreading: the case of covid-19 in italy[END_REF] represents 85% of the whole fertile women during this period of 20 years. Because the second age class size is twice its woman size, the coefficient f 2 of U is equal to ((1.85 / 0.85) / 20) / 2 = 0.108. The value of the coefficient f 1 (resp. f 3 ) is obtained in the same way, by considering that only 6% (resp. 10%) of the woman population of the first (resp. third) age class are able to get children. Using data from France [162], population affected by COVID-19 in acute infectious phase is about 4% of the total population at the middle of the fifth wave and mean value of daily cases between 18th and 23rd January 2022 equals 402,984[119], according to the calculation: Then, the dominant eigenvalue of E * in French COVID-19 population (whose size was 6,100,000 individuals during the month between 19th December 2021 and 18th January 2022) is equal to 1.00107 vs 1.0021 in the general population, and the monthly loss of population due to COVID-19 deaths has been equal to (1.0021 -1.00107) × 6, 000, 000 = 6, 283, the real observed death number being equal to127,638 -121,493 = 6,145.

	Application to COVID-19 outbreak in France
	 0.01 + 0.95 0.06 0.01 0	
	  	0.05 0	0.9 0.045 0.87 0 0 0	  
		0	0	0.035 0.5
		015 + 0.95 0.095 0.015 0	
		0.05 0	0.94 0.044 0.8 0	0 0	  
		0	0	0.035 0.5

The dominant eigenvalue of E is equal to λ = 1.0236, slightly less than the observed value 1.0258, and represents the exponential of the Malthusian parameter for the subpopulation affected by the COVID-19, largely less important than the rest of the Cameroon population, which constitutes the large majority of the whole population

(99.98 % after [119]

). The data used for Cameroon are all coming from public databases (either demographic or epidemiological) and results from counting published without standard errors. Only median ages of (mean daily cases) × (acute phase duration) (population size) = 403, 000 × 7/65, 518, 000 = 0.043 Hence, from Figure

3

.30 and from the fecundity rate (1.85) and mortality rate (9.37 / 1000) given in [119, 162], we can calculate with same method as for Cameroon the epidemic matrix for the COVID-19 affected population:

E * =

Table 3 .

 3 6: Distribution of Irish population into 5 age classes in 2020.

	Age class	Men	Women
	0 -14	560.338	534.570
	15 -24	316.239	308.872
	25 -54	1098.058	1085.794
	55 -64	278.836	278.498
	≥ 65	331.772	383.592

Table 3 .

 3 6 shows a slight difference due to sex. The fertility rate is equal to 1.808 in 2021 [119].

On Figure

3

.32, the epidemiologic data from

[163, 164]

concerning three young age classes show an increase of notified incidence rate in week 2 of 2022 due to a change in test policy, but a decrease in Epiet data incidence rate at the same time. This can be explained by the fact that the notification requires an administrative validation, which is undoubtedly the cause of the discrepancy observed between the reported incidence curves (on the left in Figure

3

.

[START_REF] Deforche | An age-structured epidemiological model of the Belgian COVID-19 epidemic[END_REF]

Table 3 .

 3 7: List of parameter values used for Cameroon. 2W )/(P 1W + P 2W ) F i P iW / i=3,6 P iW 7W + P 8W + P 9W )

	Parameter	Nature/Source	Value/Year
	Fertility rate of class i	[156, 160]	Births ‰ women (2020)
	F 1 at age 13 -14 years	Estimated	[85‰, 95‰]
	F 2 at age 15 -19 years	Raw data	105.8‰
	F 3 at age 20 -24 years	Raw data	211.22‰
	F 4 at age 25 -29 years	Raw data	210.03‰
	F 5 at age 30 -34 years	Raw data	187.84‰
	F 6 at age 35 -39 years	Raw data	138.59‰
	F 7 at age 40 -44 years	Raw data	50.54‰
	F 8 at age 45 -49 years	Raw data	16.57‰
	F 9 at age 50 -52 years	Estimated	[8‰, 12‰]
	Age pyramid(P) M,W	[157]	P iM , P iW (2019)
	(0, 14)P 1M , P 1W	Raw data	21,2%, 20.9%
	(15, 19)P 2M , P 2W	Raw data	5.4%, 5.3%
	(20, 24)P 3M , P 3W	Raw data	4.6%, 4.5%
	(25, 29)P 4M , P 4W	Raw data	4%, 4%
	(30, 34)P 5M , P 5W	Raw data	3.5% 3.5%
	(35, 39)P 6M , P 6W	Raw data	2.9%, 2.9%
	(40, 44)P 7M , P 7W	Raw data	2.3%, 2.3%
	(45, 49)P 8M , P 8W	Raw data	1.8%, 1.8%
	(50, 59)P 9M , P 9W	Raw data	2.4%, 2.5%
	(13, 14)P (50, 52)P	′ 1M , P 1W ′ ′ 9M , P ′ 9W	Estimated Estimated	3.1% [3.5%, 2.5%] 0.4% [0.4%, 0.6%]
	Median age (menarche)	Raw data (95% -CI) [158] 13.03 [12.47, 13.83] (2016)
	Fertility F 1 at age (0,19) =	Calculated	31.7%
	(F 1 P
	Women % W 1 (0,19)	Calculated	26.2 / 52.8
	f 1 = F 1 W 1	Calculated	[0.013, 0.017]
	Fertility F 2 at age (20,39) = i=3,6 Calculated	19.13%
	Women % W 2 (20,39)	Calculated	14.9 / 29.8
	f 2 = F 2 W 2	Calculated	0.095
	Median age (menopause)	Raw data (95% -CI) [159] 48 [44, 52] (2005)
	Fertility F 3 at age (40,52) =	Calculated	29.9%
	(F 7 P 7W + F 8 P 8W ) +
	F 9 P
	Women ratio W 3 (40,59)	Calculated	6.6 / 13.1
	f 1 = F 3 W 3	Calculated	[0.013, 0.017]
	Natural death rate / 1000	Raw data [160]	9.059 (2019)
	Life expectancy (years)	Raw data [160]	59.292 (2019)
	Men / Women ratio	Raw data [119]	59 / 41 (2021)
	COVID-19 deaths	Raw data [161]	1418 (2021)
	Death rate (20,39) M, W	Raw data	2%,7%
	Death rate (40,59) M, W	Raw data	23%,12%
	Death rate (>60) M, W	Raw data	34%,22%

′ 1W + F 2 P ′ 9W /(P

Table 3 .

 3 heterogeneity in the age of the populations studied here (Kuwait, France, and Cameroon) leads to considering a model in age groups with specific susceptibilities for each age class, which makes the prediction problem of the new infectious growth more difficult. Basically, there are three age groups of interest in the COVID-19 outbreak, which are, respectively, 0-19 years, 20-64 years, and >64 years, but here we only consider two (20-64 years and >64 years) age groups because the group 0-19 years is widely seen as being less infectious by SARS-CoV-2 since this age group had a low infection rate throughout the period considered in this study, especially the countries data used for simulation. This heterogeneity in individual age and the reaction to biological and environmental changes that have been observed in COVID-19 dynamics in terms of different reactions to vaccination by age group, severity of infection per age group, hospitalization, and intensive care unit as age pyramid, fertility and mortality, and epidemiological parameters such as the basic reproduction number R 0 and vaccination rate. 8: List of the parameters considered in the SIGR (Susceptible-Infectious-Goneanewsusceptible-Recovered) model.Survival rate from age 1 to age 2 for compartment K of age class 1 µ K McKendrick models including age class and vaccination state for COVID-19 in a given population. Neglecting differences between kids and young adults, we only retain two age classes: adult and elderly, 1 and 2. We assume that all infectious are symptomatic and we consider birth and natural death rates β and µ, as well as specific fatality rate ϵ due to the disease. Age groups (i = 1, 2) concern individuals susceptible, infectious I i , gone anew susceptible G i , and a fully recovered and resistant R i . We denote for each age group (i = 1, 2) the transmission rates φ i , fertility, and loss of resistance rates β ij (supposed to be equal inside an age class, for the sake of simplicity), natural death , specific death rate due to the disease ϵ i , relapsed rate η i and recovery rate γ i (cf. Table3.8).

	Epidemiologic Parameters
	S i	Susceptible individuals of age class i
	I i	Infectious individuals of age class i
	G i	Gone anew susceptible individuals of age class i
	R i	Fully recovered and resistant individuals of age class i
	R 0	Basic reproduction number
	ϕ i	Transmission rate of age class i
	η i	Relapsed rate of age class i
	γ i	Recovering rate of age class i
	a K 1

(ICU) records show different patterns, which is why it is important to improve mathematical models for COVID-19 pandemic prediction to account for different proportions of ages in the population, which is a major factor in epidemic history

[START_REF] Bacaër | A Short History of Mathematical Population Dynamics[END_REF]

. Here, the originality is to propose an improvement of the classical Ross and Kermack-McKendrick model and use it for giving a theoretical and numerical framework for interpreting the relationship between demographic parameters such 2 Natural death rate of compartment K of age class 2 Demographic Parameters β i1 Birth & loss of resistance of recovered from compartments of age class i β 22 Loss of resistance of recovered rate from compartments of age class 2 θ K i Vaccination rate from compartment K of age class i ϵ i Specific fatality rate ϵ due to the disease of age class i

 22  11 Infectious Reversible recovering Definitive recovering Figure 3.41: Age-dependent scheme for COVID-19 outbreak modeling. We propose a Susceptible-Infectious-Goneanewsusceptible-Recovered (SIGR) model as an improvement of the Ross and Kermack-

  . Let us denote the second member of Equation (3.39) by H = (H 1 , H 2 , H 3 , H 4 , H 5 , H 6 , H 7 , H 8 ) and the state vector by Z

Table 4 .

 4 1: The sensitive parameters for the analysis showing their Root MSE results (Note: FWC means first wave new cases, SWC means transition to second wave new cases, FWD means first wave deaths and SWD means transition to second wave deaths).

Table 4 .

 4 2: Kuwait number of daily deaths and daily cases predicted for April and May, 2021 (Note: PRD means predicted range for daily deaths, PVD means predicted values for daily deaths, PRC predicted range for daily cases and PVC means predicted value for daily cases).

	No	Date	PRD	PVD	PRC	PVC
	1	17-04-21	1 ∼ 8	5	344 ∼ 1223
	2	18-04-21	1 ∼ 8	4	467 ∼ 1303
	3	19-04-21	1 ∼ 8	3	267 ∼ 1164
	4	22-04-21	0 ∼ 7	4	380 ∼ 1276
	5	23-04-21	0 ∼ 7	3	237 ∼ 1172
	6	24-04-21	1 ∼ 9	5	345 ∼ 1241
	7	25-04-21	0 ∼ 8	4	445 ∼ 1303
	8	26-04-21	1 ∼ 8	4	268 ∼ 1131
	9	27-04-21	0 ∼ 7	3	336 ∼ 1233
	10	28-04-21	0 ∼ 7	3	397 ∼ 1280
	11	29-04-21	0 ∼ 8	4	395 ∼ 1271
	12	30-04-21	0 ∼ 7	3	281 ∼ 1166
	13	01-05-21	1∼ 8	5	340 ∼ 1216
	14	02-05-21	0∼ 8	4	439 ∼ 1311
	15	03-05-21	0 ∼ 8	4	282 ∼ 1151
	16	04-05-21	0 ∼ 7	3	369 ∼ 1228
	17	05-05-21	0 ∼ 7	3	455 ∼ 1313
	18	06-05-21	0 ∼ 8	4	387 ∼ 1274
	19	07-05-21	0 ∼ 7	3	280 ∼ 1178
	20	08-05-21	1 ∼ 8	5	357 ∼ 1215
	21	09-05-21	0∼ 8	4	478 ∼ 1342
	22	10-05-21	1 ∼ 8	4	291 ∼ 1155
	23	11-05-21	0 ∼ 7	3	371 ∼ 1230
	24	12-05-21	0 ∼ 6	3	457 ∼ 1345
	25	13-05-21	0 ∼ 8	4	447 ∼ 1302
	26	14-05-21	0 ∼ 7	3	330∼ 1211
	27	15-05-21	1 ∼ 9	5	383 ∼ 1259
	28	16-05-21	0 ∼ 8	4	448 ∼ 1369
	29	17-05-21	1 ∼ 8	4	308 ∼ 1193
	30	18-05-21	0 ∼ 7	3	381 ∼ 1245
	31	19-05-21	0 ∼ 7	3	450 ∼ 1333
	32	20-05-21	0∼ 7	4	441 ∼ 1291
	33	21-05-21	0 ∼ 7	3	299 ∼ 1184
	34	22-05-21	1 ∼ 9	5	332 ∼ 1284
	35	23-05-21	0 ∼ 7	4	467 ∼ 1354
	36	24-05-21	1 ∼ 8	4	279 ∼ 1184
	37	25-05-21	0 ∼ 7	3	362 ∼ 1262
	38	26-05-21	0 ∼ 7	3	494 ∼ 1332
	39	27-05-21	0 ∼ 7	8	406 ∼ 1299
	40	28-05-21	0 ∼ 7	3	340 ∼ 1213
	41	29-05-21	1 ∼ 9	5	367 ∼ 1241
	42	30-05-21	0 ∼ 7	4	510 ∼ 1375
	43	31-05-21	1 ∼ 8	4	328 ∼ 1207

Table 4 .

 4 3: Kuwait number of daily deaths predicted for October, 2020 (Note: PR means predicted range, OV means Observed value from worldometer and PV means predicted value).

	Date	PR	OV	PV	In range?
	11/10/20	0 ∼ 5	3	2	YES
	12/10/20	0 ∼ 6	6	3	YES
	15/10/20	0 ∼ 6	8	3	NO
	16/10/20	0 ∼ 6	6	3	YES
	17/10/20	1 ∼ 6	4	3	YES
	18/10/20	0 ∼ 5	7	2	NO
	19/10/20	0 ∼ 6	9	3	NO
	20/10/20	0 ∼ 5	4	3	YES
	21/10/20	1 ∼ 6	7	3	NO
	22/10/20	0 ∼ 6	9	3	NO
	23/10/20	0 ∼ 6	10	3	NO
	24/10/20	1 ∼ 5	4	3	YES
	25/10/20	0 ∼ 5	2	2	YES
	26/10/20	0 ∼ 6	3	3	YES
	27/10/20	0 ∼ 5	7	3	NO
	28/10/20	0 ∼ 6	7	3	NO
	29/10/20	0 ∼ 6	4	3	YES
	30/10/20	0 ∼ 6	6	3	YES
	31/10/20	0 ∼ 6	6	3	YES

Table 4 .

 4 4: Comparison of the prediction performances of daily new cases for deep learning results.

	Country	RMSE	RMSE	RMSE	RMSE	RMSE	RMSE	Mean
	name	LSTM	GRU	CNN	DNN	MLP	ELM	daily
								cases
	USA	15484.45 10401.13 10308.60 9630.55	5031.00 11149.23 73122.00
	India	28990.34 11526.90 76877.87 114120.26 7503.58	7956.88 59147.00
	Brazil	16355.71 11223.93 15638.27 214359.20 5324.19 11700.93 37607.00
	Turkey	2327.30	1577.54	2151.18	3936.00	578.38	1434.25 11595.00
	France	6708.81	4734.50	4358.47	4062.04	1864.17	7163.29 12231.00
	Russia	1728.23	1676.55	1165.26	1115.78	640.81	643.54	12523.00
	UK	5022.61	4381.44	4205.14	5613.93	1330.47	2689.11 12394.00
		rRMSE	rRMSE	rRMSE	rRMSE	rRMSE	rRMSE
		LSTM	GRU	CNN	DNN	MLP	ELM
	USA	0.08	0.06	0.05	0.05	0.11	0.11
	India	0.11	0.04	0.28	0.42	0.02	0.22
	Brazil	0.16	0.11	0.15	0.14	0.11	0.14
	Turkey	0.10	0.07	0.09	0.17	0.03	0.07
	France	0.22	0.16	0.14	0.13	0.10	0.22
	Russia	0.10	0.09	0.07	0.06	0.03	0.23
	UK	0.11	0.08	0.08	0.11	0.07	0.10

Table 4 .

 4 5: Comparison of the prediction performances of daily deaths for deep learning results.

	Country name RMSE	RMSE	RMSE	RMSE	RMSE	RMSE	Mean
		LSTM	GRU	CNN	DNN	MLP	ELM	daily
								deaths
	USA	225.66	105.15	249.55	194.62	130.77	288.84	1202.00
	India	387.64	223.58	1670.94	301.49	95.26	182.50	790.40
	Brazil	739.86	276.26	579.29	397.66	110.27	369.81	1046.00
	Turkey	16.49	14.54	15.96	15.61	8.54	9.98	103.30
	France	35.08	37.80	28.72	31.20	79.30	122.88	206.10
	Russia	97.66	73.74	149.56	122.06	32.66	38.03	333.20
	UK	29.43	37.82	27.32	69.16	35.77	124.91	238.90
		rRMSE	rRMSE	rRMSE	rRMSE	rRMSE	rRMSE
		LSTM	GRU	CNN	DNN	MLP	ELM
	USA	0.12	0.06	0.13	0.10	0.24	0.13
	India	0.08	0.05	0.35	0.06	0.01	0.10
	Brazil	0.29	0.11	0.23	0.50	0.06	0.08
	Turkey	0.06	0.06	0.06	0.06	0.04	0.06
	France	0.17	0.19	0.14	0.15	0.15	0.24
	Russia	0.19	0.15	0.30	0.24	0.11	0.18
	UK	0.14	0.18	0.13	0.33	0.17	0.24

  .10, Table A.2 in the Appendix A, Table B.1 and Table B.2 in the Appendix B.

Table 4 .

 4 6: List of the parameters considered for the different regions of Cameroon.

	Socio-economic parameters
	HDI	Subnational Human Development Index
	IWI	International Wealth Index
	GNI	Gross National Income per capita
	PoorHighest	Percentage poor households (IWI value under 70)
	PoorLowest	Percentage poor households (IWI value under 35)
	MeanEducation	Mean years education population aged 20+
	Household-TV	Percentage households with a TV
	Household-C	Percentage households with a computer
	Household-W	Percentage households with piped water
	Household-E	Percentage households with electricity
	Household-P	Percentage households with a phone
	Household-I	Percentage households with internet access
	HouseholdSize	Average household size
	Patrilocality	Patrilocality index (positive values patrilocal)
	Dependency	Dependency ratio
	Demographic parameters
	FertilityRate	Total fertility rate
	InfantMortality	Infant mortality rate
	Population	Total area population in millions
	PopulationU	Urban population size
	Epidemiologic parameters
	Dose1	Number of vaccinated people that have received first dose
	Dose2	Number of vaccinated people having received second dose
	Pfizer	Number of vaccinated people with Pfizer vaccine
	Sinopharm	Number of vaccinated people with Sinopharm vaccine
	Test	Number of tested people
	PositiveCases	Number of people that tested positive
	ASYM	Patients that are asymptomatic and mild symptoms
	Hmoderate	Patients with moderate symptoms
	Hsevere	Patients with severe symptoms
	Deaths	Number of deaths recorded
	Recovered	Number of recovery recorded
	R 1	Daily reproduction number of the first day of infection
	R max 1	R maximum first wave
	R max 2	R maximum second wave
	BETA	Transmission rate

Table 4 .

 4 7: Comparison of different regression models for the prediction.

	S/N	Country Name	Gini	Linear	Lasso	Ridge	MLP
			Index	Regres-	Regres-	Regres-	Regres-
				sion	sion	sion	sor
	1	PARAGUAY	46.2	46.5	46.3	46.5	46.0
	2	PANAMA	49.2	50.7	51.2	51.3	51.5
	3	BRAZIL	53.9	58.4	58.9	59.6	61.7
	4	BOLIVIA	42.2	42.2	43.2	42.7	42.0
	5	HONDURAS	52.1	56.6	57.6	-	59.6
	6	DOMINICAN	43.7	43.4	43.5	43.4	43.8
	7	CHILE	46.0	46.9	-	47.0	48.3
	8	MEXICO	45.4	-	45.8	45.7	-
	9	COLUMBIA	50.4	51.3	51.4	51.7	52.8

Table 4 .

 4 8: Latino-American countries data.

	Country Name	Gini	10% LI	10% HI	SF	Demographic Theil
		index			index	index	index
	Argentina	41.4	1.8	29.9	16.61	0.58	0.312583
	Bolivia	42.2	1.5	30.4	20.27	0.31	0.474293
	Brazil	53.9	1.0	42.5	42.50	0.22	0.528996
	Chile	46.0	1.8	37.1	20.61	0.44	0.529734
	Colombia	50.4	1.4	39.7	28.36	0.27	0.569899
	Costa Rica	48.0	1.5	36.3	24.20	0.31	0.429539
	Dominican	43.7	2.3	35.2	15.30	0.37	0.429447
	Ecuador	45.4	1.6	34.4	21.50	0.38	0.404237
	El Salvador	38.6	2.4	29.4	12.25	0.58	0.327834
	Guatemala	48.3	1.7	38.0	22.35	0.26	0.509287
	Honduras	52.1	0.9	39.1	43.44	0.16	0.476235
	Mexico	45.4	2.0	36.4	18.20	0.30	0.511219
	Nicaragua	46.2	2.0	37.2	18.6	0.46	0.479821
	Panama	49.2	1.2	37.1	30.92	0.24	0.491384
	Paraguay	46.2	1.7	35.9	21.12	0.31	0.618109
	Peru	42.8	1.8	32.1	17.83	0.29	0.355739
	Uruguay	39.7	2.3	29.7	12.91	0.71	0.307564
	4.4.6 Multivariate analysis, clustering and prediction for developed
	and developing countries

Table 4 .

 4 [START_REF] Ganyani | Estimating the generation interval for coronavirus disease (covid-19) based on symptom onset data, march 2020[END_REF]: Developed and developing countries data.

	Country Name 10%	10%	SF	DI	CCI	GI Country Name Immigration
		LI	HI					rate
	ALBANIA	3.1	24.8 8.00 0.66	-	33.2 AUSTRIA	7.4
	AUSTRIA	3.0	23.0 7.67 1.35 99.15 29.7 BELARUS	1.9
	BELGIUM	3.3	21.9 8.11 1.27 100.75 27.4 BELGIUM	4.2
	BHUTAN	2.7	27.9 10.33 0.30	-	37.4 BRAZIL	0.1
	BULGARIA	1.9	31.9 16.79 0.44	-	40.4 BULGARIA	-0.3
	CANADA	2.7	25.3 9.37 1.15	-	33.3 CANADA	6.6
	CHILE	2.3	36.3 15.78 0.58 96.55 44.4 CHINA	-0.2
	CROATIA	2.7	22.8 8.44 0.81	-	30.4 CYPRUS	4.2
	CYPRUS	3.4	25.5 7.50 0.90	-	31.4 CZECH	2.1
	CZECH	4.2	21.5 5.12 1.49 98.97 24.9 DENMARK	2.6
	DENMARK	3.7	24.0 6.49 1.55 100.50 28.7 ESTONIA	0.3
	DJIBOUTI	1.9	32.3 17.00 0.14	-	41.6 FINLAND	2.5
	EGYPT	3.8	26.9 7.08 0.70	-	31.5 FRANCE	0.6
	ESTONIA	3.0	22.5 7.50 0.89 97.59 30.4 GERMANY	6.6
	FINLAND	3.8	22.6 5.95 1.52 100.27 27.4 GREECE	-1.5
	FRANCE	3.2	25.8 8.06 1.40 98.12 31.6 HUNGARY	1.3
	GABON	2.2	27.7 12.59 0.22	-	38.0 IRELAND	4.9
	GREECE	2.4	25.9 10.79 0.72 98.11 34.4 ISRAEL	1.2
	HUNGARY	3.0	23.9 7.97 0.84 98.88 30.6 ITALY	2.5
	IRAN	2.3	31.3 13.61 0.64	-	40.8 KOSOVO	-
	ITALY	1.9	26.7 14.05 0.62 100.25 35.9 LATVIA	-7.6
	KAZAKHSTAN	4.3	23.0 5.35 0.55	-	27.5 LITHUANIA	-11.6
	KOSOVO	3.8	24.6 6.47	-	-	29.0 LUXEMBOURG	16.3
	LATVIA	2.3	26.9 11.70 0.53 95.05 35.6 MALTA	2.1
	LESOTHO	1.7	32.9 19.35 0.48	-	44.9 MEXICO	-0.5
	LITHUANIA	2.1	28.4 13.52 0.49 100.98 37.3 NETHERLANDS 0.9
	LUXEMBOURG	2.4	25.8 10.75 0.49 98.52 34.9 MACEDONIA	-0.5
	MALTA	3.4	23.3 6.85 1.31	-	29.2 NORWAY	5.3
	MAURITIUS	2.9	29.9 10.31 0.57	-	36.8 POLAND	-0.8
	MYANMAR	3.8	25.5 6.71 0.71	-	30.7 PORTUGAL	-0.6
	N.LANDS	3.5	23.3 6.66 1.50 99.93 28.5 ROMANIA	-3.8
	NORTH.M	1.7	23.8 14.00 0.47	-	34.2 RUSSIA	1.7
	NORWAY	3.3	21.6 6.55 1.53	-	27.0 SERBIA	0.5
	POLAND	3.2	23.5 7.34 0.86 98.70 29.7 SLOVAK	0.3
	PORTUGAL	2.7	26.7 9.89 0.95 97.41 33.8 SLOVENIA	1.0
	ROMANIA	1.6	24.9 15.56 0.36	-	36.0 SPAIN	0.9
	SERBIA	1.4	25.6 18.29 0.47	-	36.2 SWEDEN	4.0
	SAO TOME	1.3	49.1 37.77 0.17	-	56.3 SWITZERLAND	6.1
	SLOVENIA	4.1	20.4 4.98 1.67 96.34 24.2 TURKEY	3.5
	SPAIN	2.1	25.4 12.09 0.74 96.61 34.7 UKRAINE	0.2
	SWEDEN	3.0	22.3 7.43 1.47 101.89 28.8 UK	3.9
	SWITZERLAND 3.1	25.5 8.23 1.44 97.47 32.7 USA	2.9
	TANZANIA	2.9	33.1 11.41 0.32	-	40.5 -	-
	ZIMBABWE	2.5	34.8 13.92 0.34	-	44.3 -	-

Table 4 .

 4 10: OECD countries data.

	S/N	Country Name	10%	10%	SF	CCI	Gini	Demographic
			LI	HI	index		index	Index
	1	AUSTRALIA	2.8	26.1	9.32	100.86	32.5	1.00
	2	AUSTRIA	3.3	22.5	6.82	99.15	28.0	4.52
	3	BELGIUM	3.6	20.6	5.72	100.75	25.8	1.80
	4	BULGARIA	1.9	31.9	16.79	-	40.8	0.44
	5	CANADA	2.6	24.2	9.31	-	30.3	1.15
	6	CHILE	1.8	37.1	20.61	96.55	46.0	0.44
	7	COSTA RICA	1.5	36.3	24.20	98.94	47.8	0.31
	8	CZECH	4.0	22.2	5.55	98.97	24.9	1.38
	9	DENMARK	4.0	21.2	5.30	100.50	26.4	1.90
		ESTONIA	2.3	26.3	11.43	97.59	30.5	0.59
		FINLAND	4.0	21.2	5.30	100.27	26.9	1.71
		FRANCE	3.5	24.2	6.91	98.12	30.1	1.63
		GERMANY	3.5	23.5	6.71	98.97	28.9	1.70
		GREECE	2.2	25.4	11.55	98.11	30.6	0.67
		HUNGARY	3.1	22.5	7.26	98.88	28.9	0.92
		ICELAND	4.1	20.6	5.02	-	25.0	1.69
		IRELAND	3.1	24.4	7.87	100.48	29.5	0.88
		ISRAEL	2.0	26.3	13.15	99.15	34.8	0.57
		ITALY	2.1	24.4	11.62	100.25	33.4	0.75
		KOREA	2.2	22.0	10.00	99.43	34.5	0.76
		LATVIA	2.4	26.1	10.88	95.05	35.1	0.57
		LITHUANIA	2.1	28.4	13.52	100.98	36.1	0.49
		LUXEMBOURG	3.5	22.1	6.31	98.52	31.8	0.84
		MEXICO	1.7	36.4	21.41	99.42	45.8	0.25
		NETHERLANDS	3.3	22.7	6.88	99.93	28.5	1.45
		NORWAY	3.4	20.6	6.06	-	26.2	1.66
		POLAND	3.1	23.4	7.55	98.70	28.1	0.84
		PORTUGAL	2.4	26.2	10.92	97.41	31.7	0.86
		ROMANIA	1.6	24.9	15.56	-	35.0	0.36
		RUSSIA	2.9	29.1	10.03	96.12	33.1	0.53
		SLOVAK	3.3	21.6	6.55	97.71	23.6	1.02
		SLOVENIA	3.6	20.4	5.67	96.34	24.9	1.46
		SPAIN	2.0	24.7	12.35	96.61	33.0	0.73
		SWEDEN	3.5	22.6	6.46	101.89	27.5	1.69
		SWITZERLAND	3.4	24.1	7.09	97.47	29.9	1.68
		UK	2.7	28.6	10.59	98.71	36.6	0.94
		USA	1.6	29.2	18.25	98.68	39.0	0.93

Table 4

 4 time series by accepting the alternative hypothesis with a p-value less than the 5% (p-value = 0.02) critical value, so it is 95 percent certain that this is a stationary series. Also, Figure 4.56(c) shows the plotting of three French departments (Nord, Paris and Essonne) with more prevalent COVID-19 hospitalization cases and Figure 4.56(d) shows the fitted curve of two of the French departments (Paris and Seine-Maritime) while all departments have RMSE in the interval 0.51 ≤ RM SE ≤ 17.38 with Essonne department having the highest RMSE and Lozère department having the lowest RMSE. Other RMSE values are in Table 4.11. A deep learning forecasting result using GRU for France data between the beginning of the pandemic in France till September 3 2021 by training 80% of the data and testing 20% is presented Figure 4.56(b). The predicted cases curve values decline over the whole-time.

.11 and it was computed using the formula presented in Section 4.2.1. Figure 4.56(a) gives the time series of recent daily cases of COVID-19 in France which shows stationarity with rolling values (window = 12) appearing to be varying slightly. Also, Augmented Dickey-Fuller Test was used to test for stationarity of the

Table 4 .

 4 11: RMSE confidence interval for all French departments for the fitness curve of the four functional data.

		RMSE before vaccination	RMSE after vaccination has
		started	started
	Hospitalized	0.51 ≤ RM SE ≤ 17.38	1.00 ≤ RM SE ≤ 18.00
	ICU	0.05 ≤ RM SE ≤ 2.60	0.35 ≤ RM SE ≤ 5.20
	Daily return home	0.25 ≤ RM SE ≤ 12.49	1.10 ≤ RM SE ≤ 17.50
	Daily deceased	0.04 ≤ RM SE ≤ 4.52	0.32 ≤ RM SE ≤ 4.10

Table 4 .

 4 12: PCA variance proportion for 4 PCs before and after vaccination started in France.For functional principal component scores and clusters, the focus will be on the departments where the pandemic is most prevalent, as well as on PCs 1 and 2, while ignoring the other PCs. Recall that French departments have code numbers, for example, Nord is code number 59, and these codes will be used this to described the departments as well as in the visualization of results shown in Figures6 and 7. For a complete list of all French department code numbers see[96]. In Figure 4.63a, the Paris department (code number 75) and Nord department (code number 59) have a positive score in PC 1 and negative score in PC 2 while the Essonne department is positive in both PCs. In Figure 4.63b, the Paris department and Essonne department (code number 91) are negative in both PCs while the Nord department is positive in PC 2 with the highest score and negative in PC 1.Figure 4.63c, Nord and Essonne departments are negative in PC 2 but positive in PC 1 while the Paris department is positive in both PCs. The Paris department and Essonne department are negative in both PCs in Figure 4.63d while the Nord department is positive in PC 2 and negative in PC 1. In Figure 4.63e, Paris and Nord departments have positive scores in both PCs while the Essonne department is negative in PC 2 and positive in PC 1. Nord department has the highest positive score in PC 1 for Figure 4.63f and negative for PC 1, Paris department is positive in PC 2 and negative in PC while Essonne department is negative in both PCs. The Paris department has the highest positive score in PC 1 and negative in PC 2 in Figure 4.63g, Nord department is positive in both PCs while Essonne department is negative in PC 2 but positive in PC 1. Finally, in Figure 4.63h while Nord department is positive and highest in PC 2, Paris department is the lowest with negative score in PC 2. Both departments are negative in PC 1. Essonne department is positive in PC 2, but negative in PC 1.

			Before				After
		PC1	PC2	PC3	PC4	PC1	PC2	PC3	PC4
	Hospitalized	0.945	0.039	0.008	0.005	0.938	0.041	0.012	0.004
	ICU	0.960	0.028	0.009	0.001	0.962	0.023	0.008	0.004
	Daily return home 0.925	0.045	0.015	0.007	0.953	0.030	0.009	0.004
	Daily deaths	0.965	0.017	0.013	0.003	0.914	0.055	0.016	0.010

Table 4 .

 4 [START_REF] Guttmann | Performance map of a cluster detection test using extended power[END_REF]: Canonical correlation scores for the variables before and after vaccination started in France.

				Before			After
		Hospitalized ICU DRH Daily	Hospitalized ICU DRH Daily
				deaths	deaths
	Deaths	0.996	0.926 0.859 -	0.989	0.689 0.745 -
	Recovered	0.970	0.973 0.941 0.961 0.838	0.865 0.846 0.797
	Test	0.950	0.937 0.911 0.816 0.685	0.750 0.776 0.736
	Vaccination	-	-	-	-	0.924	0.942 0.939 0.936
	Infected	-	0.998 0.992 0.987 -	0.980 0.971 0.970
	Vaccination/1000 -	-	-	-	0.901	0.885 0.917 0.841

  4.5.6 Function to scalar linear regression modelingThis section describes the used function to scalar linear regression model to predict two of our response variables which is of the form:The data considered in this section are data before vaccination started in France and 80% was trained and 20% was tested. By applying the model presented Equation(2.23), in this case, the functional data is X (hospitalization) which is a curve whose prediction is linked to a scalar Y (number of deaths and tests between 19/03/20 -13/09/20 which is the time interval representing[START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF][START_REF] Rodríguez-López | The gata transcription factor gaf1 represses trnas, inhibits growth, and extends chronological lifespan downstream of fission yeast torc1[END_REF]) response variable. The time interval for in which the response was predicted is from 14/09/20 -29/10/20 which represent[START_REF] De Boer | Quantifying t lymphocyte turnover[END_REF][START_REF] Marchi | The role of micrornas in modulating sars-cov-2 infection in human cells: a systematic review[END_REF] (the test data). The visualization of the results is presented in Figure4.68 and the tabular form of the numerical results can be found in TableB.3 in the Appendix B. The prediction affirms the fact that the relaxation in the mitigation measures during the studied period increases the number of deaths and tests in France, the predicted results of deaths being systematically higher than the observed values as seen in Figure4.68a and TableB.3 in the Appendix B.

	177
	Y Deaths =	θ(t)X fdDataHospitilization (t)dt + ε
	1
	177
	Y Tests =	θ(t)X fdDataHospitilization (t)dt + ε
	1

  countries, and build a global continuous model. It is possible to use a matrix population model that considers Intensive Care Unit (ICU) and hospitalization cases due to COVID-19 and then see the effect of therapies at different life history points. Another future work for age dependent modeling should, for example, take into account more age groups, at least four: children (age ≤ 12 years), adolescents (12 years < age ≤ 18 years), young adults (18 years < age ≤ 65 years), and older adults (>65 years).

  Rachdi and J. Demongeot. Spatial Epidemiology Modeling of COVID-19 Pandemic (in preparation).

			PCN PCN	D D	PCD PCD	E W PCEW E W PCEW	MCC MCC
	106 153	93736	0.506118 4 0.506118 3 13 0.674633 0.605019 2 0.605019 5 20 -0.04435
	107 154	86990	0.607988 5 0.607988 3 14 0.59476 0.575404 3 0.575404 6 1 0.789508 0.557804 14/20
	108 155	78677	0.702738 6 0.702738 3 15 0.575404 0.592655 4 0.592655 6 2 0.276242
	109 156	71674	0.771323 7 0.771323 3 16 0.488039 0.667777 5 0.667777 6 3 0.311795
	110 157	84952	0.778017 1 0.778017 3 17 0.402127 0.706037 6 0.706037 6 4 0.307003
	111 158	88726	0.71472 2 0.71472 3 18 0.009761 0.692452 7 0.692452 6 5 0.527497
	112 159	96296	0.698072 3 0.698072 3 19 -0.11507 0.653118 1 0.653118 6 6 0.499389
	113 160	99237	0.692351 4 0.692351 3 20 -0.22695 0.494937 2 0.494937 6 7 0.699566
	114 161	95176	0.726056 5 0.726056 4 1 0.774313 0.38065 12/20 0.488039 3 0.488039 6 8 0.702738
	115 162	81646	0.75754 6 0.75754 4 2 0.311278 0.59711 4 0.59711 6 9 0.75754
	116 163	90580	0.783995 7 0.783995 4 3 -0.17193 0.683387 5 0.683387 6 10 0.666455
	117 164	95791	0.719714 1 0.719714 4 4 -0.05802 0.7602 6 0.7602 6 11 0.746575
	118 165	103269	0.571106 2 0.571106 4 5 0.320394 0.783989 7 0.783989 6 12 0.795896
	119 166	108305	0.527833 3 0.527833 4 6 0.102804 0.656816 1 0.656816 6 13 0.777163
	120 167	107714	0.562433 4 0.562433 4 7 0.470877 0.477868 2 0.477868 6 14 0.736841
	121 168	100380	0.610005 5 0.610005 4 8 0.506118 0.402127 3 0.402127 6 15 0.706037
	122 169	97159	0.666455 6 0.666455 4 9 0.692351 0.462685 4 0.462685 6 16 0.7602
	123 170	91816	0.686601 7 0.686601 4 10 0.562433 0.537366 5 0.537366 6 17 0.600851
	124 171	93546	0.625795 1 0.625795 4 11 0.602733 0.600851 6 0.600851 6 18 0.195316
	125 172	107449	0.56017 2 0.56017 4 12 0.667945 0.574631 7 0.574631 6 19 0.282929
	126 173	117149	0.574066 3 0.574066 4 13 0.685452 0.368153 1 0.368153 6 20 0.016546
	127 174	126792	0.602733 4 0.602733 4 14 0.594878 0.071667 2 0.071667 7 1 0.767306 0.571253 15/20
	128 175	125224	0.634611 5 0.634611 4 15 0.592655 0.009761 3 0.009761 7 2 0.230716
	129 176	109948	0.746575 6 0.746575 4 16 0.59711 0.043801 4 0.043801 7 3 0.39616
	130 177	105959	0.788147 7 0.788147 4 17 0.462685 0.114956 5 0.114956 7 4 0.298721
	131 178	116523	0.66546 1 0.66546 4 18 0.043801 0.195316 6 0.195316 7 5 0.526067
	132 179	121422	0.609405 2 0.609405 4 19 0.000826 0.208246 7 0.208246 7 6 0.548743
	133 180	131498	0.621132 3 0.621132 4 20 -0.1457 0.004649 1 0.004649 7 7 0.715578
	134 181	131555	0.667945 4 0.667945 5 1 0.763159 0.467941 13/20 -0.18503 2 -0.18503 7 8 0.771323
	135 182	129557	0.726055 5 0.726055 5 2 0.266571 -0.11507 3 -0.11507 7 9 0.783995
	136 183	114896	0.795896 6 0.795896 5 3 -0.00811 0.000826 4 0.000826 7 10 0.686601
	137 184	108855	0.811018 7 0.811018 5 4 0.116351 0.161079 5 0.161079 7 11 0.788147
	138 185	122083	0.780229 1 0.780229 5 5 0.440548 0.282929 6 0.282929 7 12 0.811018
	139 186	136934	0.718404 2 0.718404 5 6 0.320819 0.258542 7 0.258542 7 13 0.784775
	140 187	139155	0.674633 3 0.674633 5 7 0.632728 0.000278 1 0.000278 7 14 0.759236
	141 188	143260	0.685452 4 0.685452 5 8 0.607988 -0.23757 2 -0.23757 7 15 0.692452
	142 189	135048	0.733599 5 0.733599 5 9 0.726056 -0.22695 3 -0.22695 7 16 0.783989
	143 190	124422	0.777163 6 0.777163 5 10 0.610005 -0.1457 4 -0.1457 7 17 0.574631
	144 191	126372	0.784775 7 0.784775 5 11 0.634611 -0.04435 5 -0.04435 7 18 0.208246
	145 192	144280	0.72027 1 0.72027 5 12 0.726055 0.016546 6 0.016546 7 19 0.258542
	146 193	146482	0.602925 2 0.602925 5 13 0.733599 0.038824 7 0.038824 7 20 0.038824
	147 194	141526	0.59476 3 0.59476 5 14 0.668216
	148 195	182683	0.594878 4 0.594878 5 15 0.667777
	149 196	158002	0.668216 5 0.668216 5 16 0.683387
	150 197	131120	0.736841 6 0.736841 5 17 0.537366
	151 198	140363	0.759236 7 0.759236 5 18 0.114956
	152 199	164750	0.707716 1 0.707716 5 19 0.161079

Table B .

 B 1: Parameters for first and second waves for 195 countries.

	Country Name S/N Country Name SDG First wave R 0 R j 's First wave R o Exp. slope A-slope R o Exp. slope Second wave U-shape R 0 R j 's Second wave A-slope First wave Second wave CHE/GDP U-shape CHE/GDP R o Exp. slope A-slope R o Exp. slope A-slope First wave Second wave CHE/GDP R o Exp. slope A-slope R o Exp. slope A-slope First wave Second wave CHE/GDP R o Exp. slope A-slope R o Exp. slope index A-slope EPI TBI HDI SI LE Temperature
	39 1.25 3.93; -6.21; 3.53 5.46 10.31; -39.32; 34.47 YES YES YES 1.36 1.00; 0.79; -0.43 DECR 0.63 0.33; 1.42; -1.12 INV 1.92 0.00; 1.25; 0.67 INV 0.96 0.48; 0.50; -0.02 INV 0.73 0.57; -1.28; 1.44 YES 4.43 17.99; -36.99; 23.43 YES 2.79 -1.02; 3.47; 0.34 YES 1.17 -1.78; -0.05; 3.00 INCR 1.16 1.23; -1.32; 1.25 YES 0.57 -0.13; -0.98; 1.68 YES 1.10 -0.74; 0.28; 1.56 INCR BANGLADESH 1.04 2.37; -2.97; 1.64 ALGERIA ARUBA ANDORRA ANGOLA ANTIGUA ALBANIA ARGENTINA ARMENIA AUSTRALIA AUSTRIA AZERBAIJAN BAHAMAS BAHRAIN YES BARBADOS 1.86 0.86; -0.64; 1.64 YES BELARUS 1.57 -2.37; -4.58; 8.52 YES BELGIUM 0.43 11.66; -15.63; 4.41 YES BELIZE 0.99 0.80; 0.42; -0.23 DECR BENIN 0.85 0.81; 0.47; -0.43 DECR BHUTAN 15.00 14.00; 15.00; -14.00 INV BOLIVIA 2.17 8.47; -1.17; -5.13 DECR BOSNIA 0.09 -1.06; -1.05; 2.20 INCR BOTSWANA 28.47 0.22; 0.00; 28.25 YES BRAZIL 0.77 0.31; 1.08; -0.62 INV BRUNEI 1.08 0.10; -0.15; 1.13 YES BULGARIA 5.06 14.73; -66.02; 56.35 YES BURKINA FASO 1.08 0.72; -0.34; 0.70 YES BURUNDI 1.33 1.33; -0.67; 0.67 YES CABO VERDE 0.82 -0.08; -0.26; 1.16 YES CAMBODIA 0.34 0.08; 0.25; 0.01 INV CAMEROON 2.17 2.36; 1.25; -1.44 DECR CANADA 1.10 -0.55; -0.72; 2.37 YES CAR 1.66 -0.07; 0.64; 1.09 INCR CHAD 1.19 0.77; -1.15; 1.57 YES CHILE 1.00 0.72; 0.17; 0.11 DECR CHINA 1.10 0.90; -0.49; 0.69 YES COLUMBIA 1.00 1.75; -0.86; 0.11 YES COMOROS 3.75 0.00; -2.75; 6.5 YES CONGO DEM. 0.03 -0.37; -0.39; 0.79 YES CONGO REP. 0.92 0.92; 0.92; -0.92 DECR COSTA RICA 0.50 -2.79; -3.84; 7.13 YES COTE D'VOIRE 1.18 -0.49; -0.63; 2.30 YES CROTIA 0.75 0.53; 0.79; -0.57 INV CUBA 0.48 -37.25; 16.17; 21.56 INCR CURACAO 0.50 3.00; -1.00; -1.50 DECR CYPRUS 0.69 0.27; 2.49; -2.07 INV CZECH 0.16 -0.16; 3.88; -3.56 INV DENMARK 0.80 -0.11; 0.41; 0.50 INCR DJIBOUTI 0.17 1.23; 0.24; -1.30 DECR DOMINICAN 1.02 1.05; -0.31; 0.28 YES DOMINICA 7.75 2.00; -4.00; 9.75 YES ECUADOR 1.46 -0.47; 1.06; 0.87 INV EGYPT 0.84 0.30; 0.37; 0.17 INV EL SALVADOR 1.70 -0.20; 0.59; 1.31 INCR EQUATORIAL G. 0.38 0.85; -0.20; -0.27 DECR ERITREA 1.18 1.44; -0.05; -0.21 DECR ESTONIA 0.87 1.96; 0.82; -1.91 DECR ESWATINI 0.94 1.41; -1.42; 0.95 YES ETHIOPIA 0.80 -0.56; -1.45; 2.81 YES FIJI 2.00 0.00; 1.00; 1.00 INCR FINLAND 1.14 0.91; -0.42; 0.65 YES FRANCE 1.17 0.82; 0.10; 0.25 YES GABON 0.97 0.20; 0.47; 0.30 INV GAMBIA 0.83 -0.25; 0.43; 0.65 INCR GEORGIA 1.23 0.16; 0.43; 0.64 INCR CAMBODIA 5.55 -0.0129 -0.129 3.12 0.0010 0.04 -1.38; -0.36; 1.78 INCR 0.91 1.28; -1.06; 0.69 YES 1.10 1.54; -1.60; 1.16 YES 0.12 4.34; -1.63; -2.59 DECR 1.70 9.22; -1.58; -5.94 DECR 2.13 -0.40; 1.33; 1.20 INV 0.66 1.98; -0.56; -0.76 DECR 0.36 1.27; 0.75; -1.66 DECR 0.86 1.41; -0.97; 0.42 YES 1.50 -0.88; 0.68; 1.70 INCR 2.08 0.62; -3.55; 5.01 YES 0.37 10.36; -6.45; -3.54 YES 1.22 0.22; -0.86; 1.86 YES 1.14 1.98; -2.69; 1.85 YES 0.99 0.86; -0.69; 0.82 YES 1.14 0.22; -0.81; 1.73 YES 1.07 -0.33; 0.24; 1.16 INCR 2.23 1.17; -2.39; 3.45 YES 0.51 1.77; -0.21; -1.05 DECR 0.85 1.17; 0.22; -0.54 DECR 1.08 0.80; 0.57; -0.29 DECR 1.61 0.96; -0.30; 0.95 YES 1.56 -0.57; -0.51; 2.64 INCR 28.43 0.22; -0.05; 28.26 YES 0.46 1.21; 0.16; -0.91 DECR 1.00 1.00; -1.00; 1.00 YES 0.75 1.34; -0.98; 0.39 YES 0.94 0.31; 0.24; 0.39 YES 2.18 0.53; 1.80; -0.15 INV 0.19 0.56; 1.37; -1.74 INV 0.27 0.06; 0.15; 0.06 INV 2.48 0.50; -0.25; 2.23 YES 0.44 2.36; -0.44; -1.48 DECR 0.33 0.44; -0.22; 0.11 YES 0.77 1.19; 0.25; -0.67 DECR 1.64 0.37; -4.45; 5.72 YES 0.87 1.16; 0.60; -0.89 DECR 1.47 -1.14; 3.08; -0.47 INV 1.65 -0.58; 1.24; 0.99 INV 0.88 0.66; 0.74; -0.52 INV 0.39 -0.12; 0.19; 0.32 INCR 1.26 1.21; -0.85; 0.90 YES 2.09 4.32; -7.09; 4.86 YES 0.57 0.68; -0.64; 0.53 YES 0.78 0.34; -0.73; 1.17 YES 4.19 1.93; -4.01; 6.27 YES 0.45 -0.42; 1.76; -0.89 INV 0.88 1.88; -1.41; 0.41 YES 0.64 -0.03; 4.65; -3.98 INV 0.36 0.64; 0.41; -0.69 DECR 1.57 0.32; -0.06; 1.31 YES 0.67 -0.36; 0.72; 0.31 INV 1.14 0.73; -0.14; 0.55 YES 0.51 11.99; -3.76; -7.72 DECR 0.66 -0.76; -14.49; 15.91 YES 1.48 0.81; -0.66; 1.33 YES 0.80 1.02; 0.20; -0.42 DECR 3.04 -0.70; -1.80; 5.54 YES 0.71 -0.02; 1.52; -0.79 INV 1.24 0.34; 0.13; 0.77 YES 0.50 0.75; -0.50; 0.25 YES 2.41 0.56; -2.38; 4.23 YES 2.17 0.88; -0.86; 2.15 YES 0.19 -0.51; 0.00; 0.70 INCR 0.37 -0.38; 0.00; 0.75 INCR 0.79 1.52; -0.49; -0.24 -0.158 6.03 YES Appendix B Appendix Country Name First wave Second wave CHE/GDP R o Exp. slope A-slope R o Exp. slope A-slope AFGHANISTAN 1.78 0.1070 -0.025 0.79 0.0017 -0.097 9.40 ALGERIA 2.19 0.1594 -0.040 0.86 0.0316 -0.100 6.22 ARUBA ----0.0172 -0.112 -ANDORRA --0.0313 -0.121 --0.0067 -0.155 6.71 ANGOLA 2.06 0.0100 -0.095 1.13 -0.0135 -0.057 2.55 ANTIGUA 4.23 --3.30 0.0051 -0.177 5.23 ALBANIA 1.61 0.0091 -0.138 0.99 0.0058 -0.085 5.26 ARGENTINA 2.06 0.1485 -0.060 1.19 0.0427 -0.240 9.62 ARMENIA 1.51 0.0809 -0.050 0.80 0.0570 -0.090 10.03 AUSTRALIA 2.45 0.1832 -0.054 1.11 0.0037 -0.136 9.28 AUSTRIA 2.93 0.2825 -0.053 1.05 0.0034 -0.053 10.33 AZERBAIJAN 2.11 0.1422 -0.071 0.63 0.0676 -0.130 3.51 BAHAMAS 6.33 --1.48 -0.0250 -0.077 6.25 BAHRAIN 1.81 0.1884 -0.079 1.24 0.0012 -0.053 4.13 BANGLADESH 3.67 0.0799 -0.033 0.92 -0.0086 -0.046 2.34 BARBADOS 4.63 --1.99 0.0378 -0.109 6.56 BELARUS 3.15 0.0043 -0.060 1.02 0.0159 -0.026 5.64 BELGIUM 8.28 0.1963 -0.047 0.88 -0.0182 -0.063 10.32 BELIZE 3.74 --1.34 -0.0004 -0.140 5.69 BENIN 2.16 0.0226 -0.133 1.55 0.0020 -0.125 2.49 BHUTAN 2.10 0.0021 -0.118 2.49 0.0126 -0.099 3.06 BOLIVIA 1.46 0.0647 -0.045 1.45 0.0152 -0.087 6.30 BOSNIA 1.70 0.0088 -0.110 0.97 -0.0118 -0.106 8.90 BOTSWANA 3.76 --1.43 0.0030 -0.186 5.85 BRAZIL 3.10 0.0389 -0.048 0.92 0.0092 -0.188 9.51 BRUNEI 5.00 -0.0165 -0.120 3.66 --2.41 BULGARIA 1.97 0.0178 -0.087 0.78 0.0049 -0.110 7.35 BURKINA FASO 2.44 -0.0227 -0.123 1.18 0.0360 -0.058 5.63 BURUNDI 2.80 --1.69 0.0226 -0.063 CABO VERDE 1.54 0.0247 -0.091 1.71 -0.0064 -0.110 5.36 GUYANA 2.49 0.0005 -0.152 1.54 -0.0021 -0.163 5.94 MOROCCO 2.05 0.1161 -0.114 0.84 -0.0159 -0.065 5.31 ST VINCENT 5.86 --2.17 0.0407 -0.080 4.47 7.74 CAMEROON 2.56 0.0338 -0.123 1.64 0.0085 -0.207 -CANADA 2.95 0.2432 -0.029 1.05 0.0153 -0.047 10.79 CAR 2.45 -0.0130 -0.096 4.99 --10.99 CHAD 2.43 -0.0108 -0.114 1.44 0.0222 -0.050 4.10 CHILE 2.42 0.1906 -0.034 1.16 0.0586 -0.090 9.14 CHINA 2.05 -0.0602 -0.088 1.07 0.0137 -0.068 5.35 COLUMBIA 1.86 0.0384 -0.040 0.99 0.0061 -0.126 7.64 COMOROS 1.93 -0.0094 -0.153 1.58 0.0397 -0.076 4.59 CONGO DEM. 1.48 0.0384 -0.052 1.10 0.0252 -0.089 3.30 CONGO REP. 2.39 0.0294 -0.152 1.43 0.0064 -0.118 2.14 COSTA RICA 1.51 0.0142 -0.110 1.08 -0.0022 -0.209 7.56 COTE D'VOIRE 1.47 0.0309 -0.080 1.35 0.0253 -0.078 4.19 CROTIA 3.95 -0.0042 -0.069 0.72 -0.0115 -0.106 6.83 CUBA 2.23 0.0706 -0.063 1.30 0.0517 -0.040 11.19 CURACAO -----0.0060 -0.074 -CYPRUS 2.21 -0.0056 -0.131 1.30 0.0273 -0.089 6.77 CZECH 2.40 0.2570 -0.067 1.22 0.0474 -0.197 7.65 DENMARK 1.60 -0.0024 -0.087 0.90 0.0092 -0.048 10.07 DJIBOUTI 2.73 0.0144 -0.094 1.47 -0.0045 -0.169 2.32 DOMINICAN 2.09 0.0309 -0.088 1.10 0.0151 -0.081 5.73 DOMINICA ------6.59 ECUADOR 2.22 0.0157 -0.140 1.18 -0.0045 -0.175 8.14 EGYPT 1.69 0.0527 -0.042 1.33 0.0243 -0.023 4.95 EL SALVADOR 1.58 0.0783 -0.052 1.29 0.0535 -0.113 7.11 EQUATORIAL G. 10.0 0.0454 -0.190 2.41 0.0142 -0.177 3.00 ERITREA 2.57 0.0083 -0.216 0.74 0.0222 -0.146 4.09 ESTONIA 2.10 -0.0254 -0.116 1.03 0.0279 -0.099 6.69 ESWATINI 2.08 0.0317 -0.071 1.34 0.0412 -0.034 6.54 ETHIOPIA 2.42 0.1259 -0.054 1.11 -0.0041 -0.136 3.30 FIJI -----3.42 FINLAND 1.66 -0.0030 -0.093 1.04 -0.0010 -0.119 9.04 FRANCE 2.68 0.2898 -0.110 1.00 -0.0096 -0.081 11.26 GABON 1.83 0.0404 -0.077 1.44 0.0187 -0.143 2.75 GAMBIA 3.21 -0.0026 -0.094 2.29 0.0145 -0.099 3.09 GEORGIA 2.19 0.2536 -0.136 0.76 0.0293 -0.057 7.11 GERMANY 2.84 0.2624 -0.050 0.98 0.0050 -0.195 11.43 GHANA 1.85 0.0463 -0.099 1.09 0.0118 -0.117 3.45 GREECE 1.72 -0.0189 -0.091 1.05 -0.0111 -0.069 7.72 GRENADA 5.78 --1.08 0.0106 -0.167 4.46 GUADELOUPE --0.0131 -0.130 --0.0084 -0.137 -GUATEMALA 1.67 0.0880 -0.044 1.08 0.1109 -0.197 5.71 GUIANA FRENCH -0.0391 -0.102 -0.0238 -0.124 -GUINEA 1.50 0.0097 -0.111 1.36 -0.0108 -0.126 3.93 GUINEA BISSAU 3.56 0.0230 -0.145 4.66 --7.00 HAITI 2.32 0.0565 -0.047 1.66 0.0217 -0.082 7.69 HONDURAS 1.96 0.0532 -0.086 1.59 0.0016 -0.141 7.05 HONG KONG --0.0003 -0.060 -0.0285 -0.041 -HUNGARY 2.25 0.0018 -0.093 0.77 -0.0081 -0.088 6.70 ICELAND 2.89 -0.0261 -0.056 1.86 -0.0174 -0.079 8.47 INDIA 2.43 0.0331 -0.050 0.91 -0.0151 -0.048 3.54 INDONESIA 2.04 0.0391 -0.071 1.07 0.0127 -0.051 2.87 IRAN 3.61 0.2641 -0.063 1.00 0.0438 -0.140 8.66 IRAQ 1.81 0.1184 -0.084 0.96 0.0410 -0.150 -IRELAND 2.63 -0.0021 -0.058 1.45 0.0188 -0.057 6.93 ISRAEL 2.86 -0.0047 -0.049 1.33 0.0339 -0.037 7.52 ITALY 2.99 0.2475 -0.040 1.06 -0.0057 -0.072 8.67 JAMAICA 2.43 -0.0031 -0.089 1.22 0.0034 -0.174 6.06 JAPAN 1.91 0.0872 -0.055 1.21 0.0260 -0.052 10.95 JORDAN 2.16 -0.0006 -0.155 0.93 -0.0138 -0.053 7.79 KAZAKHSTAN 2.85 0.0856 -0.064 1.05 0.0933 -0.210 2.92 KENYA 1.57 0.0413 -0.067 1.26 -0.0237 -0.310 5.17 KOREA REP. 6.06 0.1664 -0.076 0.90 0.0585 -0.090 7.56 KOSOVO 1.90 --0.82 ---KUWAIT 2.25 0.0687 -0.031 1.27 -0.0094 -0.038 5.00 KYRGYZSTAN 2.27 0.0671 -0.091 0.86 0.0271 -0.200 6.53 LAO PDR ------2.25 LATVIA 2.32 -0.0179 -0.087 1.10 0.0224 -0.136 6.19 LEBANON 1.91 0.2286 -0.112 1.27 0.0757 -0.180 8.35 LESOTHO 1.99 0.0053 -0.206 1.36 0.0398 -0.087 9.28 LIBERIA 1.76 0.0151 -0.114 3.08 0.0046 -0.159 6.74 LIBYA 3.12 0.0493 -0.047 1.09 -0.0059 -0.099 -LITHUANIA 1.63 0.0394 -0.096 0.98 0.0554 -0.230 6.57 LUXEMBOURG 1.99 -0.0401 -0.061 0.83 -0.0174 -0.105 5.29 MACAO --0.0019 -0.190 ----MADAGASCAR 2.48 0.0377 -0.057 1.54 0.0060 -0.211 4.79 MALAWI 3.55 0.0478 -0.088 1.66 0.0583 -0.087 9.33 MALAYSIA 2.86 0.1042 -0.101 1.15 0.0794 -0.260 3.76 MALDIVES 1.96 0.0031 -0.154 1.41 0.0007 -0.116 9.41 MALI 1.61 0.0158 -0.100 0.97 0.0148 -0.115 -MALTA 4.46 0.0712 -0.114 1.29 0.0536 -0.330 8.96 MAURITANIA 1.66 -0.0033 -0.055 0.82 0.0362 -0.037 4.58 MAURITIUS 5.40 -0.0209 -0.120 9.32 -0.0032 -0.143 5.83 MAYOTTE -0.0129 -0.103 -0.0065 -0.154 -MEXICO 2.03 0.1759 -0.100 0.98 0.0117 -0.109 5.37 MOLDOVA 2.03 0.0324 -0.086 0.83 -0.0037 -0.127 6.60 MONACO 5.48 -0.0044 -0.147 1.66 0.0134 -0.136 1.60 MONGOLIA 3.12 0.0116 -0.204 1.98 0.0195 -0.127 3.79 MONTENEGRO 8.16 -0.0114 -0.171 1.07 0.0040 -0.085 8.42 MOZAMBIQUE 2.14 0.0260 -0.109 1.59 0.0152 -0.068 8.17 MYANMAR 2.70 -0.0028 -0.113 0.83 -0.0137 -0.050 4.79 NAMIBIA 2.10 0.0315 -0.049 1.03 0.0315 -0.039 7.95 NEPAL 2.28 0.2070 -0.035 0.91 -0.0264 -0.065 5.84 NETHERLANDS 2.40 0.2485 -0.043 0.92 0.0002 -0.074 9.97 NEW ZEALAND 5.63 -0.0426 -0.087 1.89 0.0140 -0.099 9.21 NICARAGUA 5.76 --1.39 --8.56 NIGER 2.58 -0.0231 -0.083 0.96 0.0390 -0.048 7.33 NIGERIA 1.91 0.0502 -0.046 1.06 0.0333 -0.047 3.89 MACEDONIA 1.84 0.0858 -0.092 0.87 0.0528 -0.230 6.58 NORWAY 2.40 0.2716 -0.055 1.14 0.0052 -0.145 10.05 OMAN 1.73 0.0972 -0.092 1.13 0.0936 -0.130 4.13 PAKISTAN 1.90 0.1301 -0.060 1.02 0.0113 -0.047 3.20 PALESTINE --0.0053 -0.202 -0.0063 -0.050 -PANAMA 2.08 0.1443 -0.063 1.13 0.1195 -0.070 7.27 PAPAU NEW G. 1.95 -0.0081 -0.115 2.45 --2.37 PARAGUAY 2.22 0.0196 -0.147 0.97 0.0032 -0.168 6.65 PERU 2.35 0.0915 -0.010 1.26 -0.0077 -0.111 5.24 PHILIPPINES 2.29 0.1627 -0.082 0.91 0.1772 -0.174 4.40 POLAND 2.17 0.1562 -0.079 0.99 0.0094 -0.072 6.33 POLYNESIA ------0.075 -PORTUGAL 2.92 0.0301 -0.140 1.15 0.0431 -0.190 9.41 QATAR 2.61 0.0694 -0.070 1.16 -0.0019 -0.094 2.49 ROMANIA 2.26 0.0218 -0.056 0.91 -0.0072 -0.121 5.56 RUSSIA 2.41 0.0775 -0.020 1.00 0.0046 -0.037 5.32 RWANDA 2.03 0.0615 -0.146 1.26 0.0382 -0.064 7.54 SAO TOME 3.09 -0.0218 -0.153 3.33 0.0162 -0.127 6.27 SAN MARINO 5.88 -0.0157 -0.137 1.14 -0.0028 -0.154 SUDAN 1.97 0.0193 -0.094 0.24 0.0407 -0.039 4.51 SURINAME 1.41 0.0214 -0.061 1.27 0.0379 -0.042 DRC 41.6 30.41 3.20 0.1 -61.0 24.00 7.97 SWEDEN 2.10 0.2572 -0.106 1.05 0.0123 -0.162 Congo 48.7 42.39 3.73 -47.22 61.0 24.55 10.90 SWITZERLAND 2.86 0.2388 -0.044 0.95 -0.0082 -0.041 Ivory Coast 55.6 45.25 1.37 21 25.93 61.0 26.35 11.88 SYRIA 2.80 0.0311 -0.030 1.06 0.0086 -0.041 Djibouti 49.7 40.04 -40 37.96 65.0 28.00 -TAIWAN 3.42 0.0036 -0.084 1.84 0.0053 -0.123 Egypt 63.8 61.21 -3.6 54.63 74.0 22.10 -TAJIKISTAN 1.68 0.0418 -0.066 0.54 -0.0016 -0.131 Equatorial 43.4 60.40 1.81 --66.0 24.55 7.24 TANZANIA 5.00 0.1205 -0.125 18.4 --Eritrea 43.3 39.34 0.86 0.1 86.11 66.0 25.50 3.63 THAILAND 3.42 -0.0201 -0.055 1.63 0.0496 -0.100 Eswatini 52.4 -3.63 5.0 68.52 59.0 21.40 3.79 TIMOR LESTE ------Ethiopia 53.2 44.78 1.40 8.4 50.93 68.0 22.20 4.33 TOGO 2.09 0.0093 -0.084 1.41 0.0083 -0.112 Gabon 59.4 45.05 5.21 20 50.93 69.0 25.05 6.17 TRINIDAD 5.35 -0.0025 -0.139 1.30 -0.0102 -0.113 Gambia 51.9 42.42 1.58 4.3 44.44 66.0 27.50 6.93 TUNISIA 2.64 -0.0122 -0.084 1.15 0.0053 -0.117 Ghana 61.2 49.66 1.44 2.5 50.93 68.0 27.20 7.29 TURKEY 4.32 0.0120 -0.040 0.81 0.0078 -0.030 Guinea 49.4 46.62 1.76 3.1 52.78 63.0 25.70 4.12 UAE 2.33 0.0484 -0.080 1.22 0.0085 -0.055 Guinea Bissau 45.5 44.67 3.61 1.3 -63.0 26.75 4.23 UGANDA 2.18 --0.88 0.0047 -0.154 Kenya 56.6 47.25 2.67 0.9 50.93 69.0 24.75 6.53 UKRAINE 2.16 0.0325 -0.130 0.89 -0.0032 -0.093 Lesotho 50.9 33.78 6.54 6.8 65.74 53.0 11.85 7.72 UK 2.89 0.2223 -0.037 1.25 0.0106 -0.035 Liberia 48.0 41.62 3.08 -40.74 65.0 25.30 10.00 USA 3.85 0.2882 -0.030 0.99 0.0121 -0.060 Libya -49.79 -34 76.85 77.0 21.80 16.89 URUGUAY 2.76 -0.0228 -0.086 1.15 0.0389 -0.039 Madagascar 45.6 33.73 2.33 4.9 37.04 67.0 22.65 9.20 UZBEKISTAN 1.82 0.1231 -0.088 0.71 0.0238 -0.170 Malawi 52.3 49.21 1.46 1.9 48.15 63.0 21.90 5.29 VENEZUELA 2.57 0.0389 -0.073 0.94 0.0002 -0.134 Mali 51.7 43.71 0.52 2.5 48.15 62.0 28.25 3.56 VIETNAM 3.59 -0.0166 -0.158 1.94 -0.0040 -0.158 Mauritania 51.3 39.24 0.89 14 29.63 65.0 27.65 5.92 WEST GAZA 3.73 --0.87 --Mauritius 66.2 56.63 0.12 3.0 93.52 76.0 22.40 -YEMEN 1.57 0.0049 -0.164 2.84 0.0006 -0.150 Morocco 64.4 63.47 -19 76.85 73.0 18.10 -ZAMBIA 2.80 0.0265 -0.134 1.73 0.0372 -0.046 Mozambique 51.4 46.37 3.61 5.1 55.56 56.0 23.80 4.93 ZIMBABWE 1.98 0.0367 -0.087 1.40 0.0438 -0.045 Namibia 57.1 58.46 4.86 1.4 42.59 65.0 20.95 4.73 Niger 50.3 35.74 0.84 7.3 34.26 59.0 27.15 7.14 SAUDI ARABIA 2.31 0.0607 -0.060 0.90 -0.0138 -0.029 6.36 SENEGAL 2.02 0.0351 -0.047 1.24 0.0387 -0.047 Nigeria 47.1 54.76 2.19 0.1 45.37 60.0 26.80 Table B.2: Africa countries data. Rwanda 57.9 43.68 0.57 1.8 71.30 65.0 18.85 3.98 SERBIA 2.13 0.0042 -0.053 0.79 0.0123 -0.038 8.54 SEYCHELLES 2.68 --1.94 0.0313 -0.134 5.11 SIERRA LEONE 1.50 0.0143 -0.107 1.52 0.0291 -0.063 S/N Country Name SDG EPI TBI HDI SI LE Sao Tome 61.8 54.01 1.14 --66.0 23.75 Temperature Senegal 57.0 49.52 1.17 12 13.89 63.0 27.85 index Seychelles -66.02 0.16 -65.74 76.0 27.15 16.06 SINGAPORE 2.06 0.0551 -0.030 1.52 0.0641 -0.080 1 Algeria 65.8 57.18 -21 72.22 78.0 18.79 Sierra Leone 49.7 42.54 2.95 6.6 48.15 60.0 23.82 4.46 SLOVAK 1.74 -0.0286 -0.123 0.92 0.0028 -0.193 2 Angola 49.3 37.44 3.51 -58.33 61.0 21.55 Somalia 40.1 -2.58 -47.22 54.0 27.05 6.69 SLOVENIA 1.78 -0.0345 -0.079 1.08 -0.0004 -0.263 3 Benin 51.5 38.17 0.55 3.2 43.52 61.0 27.55 South Africa 60.4 44.73 6.15 1.0 48.15 65.0 18.75 8.30 SOLOMON ISL. ------4 Botswana 61.6 51.70 -20 62.04 65.0 21.50 South Sudan 29.2 -2.27 -72.22 ---4.47 SOMALIA 1.95 -0.0085 -0.091 2.55 --5 Bukina Faso 53.5 42.83 0.47 0.6 22.22 63.0 28.29 Sudan 47.4 51.49 0.67 5.4 24.07 67.0 26.90 -SOUTH AFRICA 2.54 0.257 -0.110 1.15 0.0303 -0.039 6 Burundi 50.3 27.43 1.07 -20.37 67.0 20.80 Tanzania 55.9 50.83 2.37 16 8.33 64.0 22.35 8.25 SOUTH SUDAN 2.99 0.0007 -0.152 1.59 0.0095 -0.133 7 Cabo Verde 64.1 56.94 0.46 19 45.37 73.0 23.30 Togo 52.7 41.78 0.37 1.1 67.59 67.0 27.15 6.40 SPAIN 3.85 0.3350 -0.035 1.16 0.0029 -0.080 8 Cameroon 51.6 40.81 1.79 1.4 35.19 62.0 24.60 Tunisia 66.1 62.35 -5.4 62.04 76.0 16.30 8.98 SRI LANKA 4.14 0.0144 -0.159 1.04 0.1347 -0.160 9 CAR 36.7 36.42 5.40 2.9 18.52 54.0 24.90 Uganda 54.9 44.28 2.00 0.5 53.70 68.0 22.80 3.76 ST KITTS NEVIS ------10 Chad 38.7 45.34 1.42 -22.22 58.0 26.55 Zambia 53.0 50.97 3.33 20 36.11 54.0 21.40 5.31 ST LUCIA 1.34 --2.86 0.0157 -0.082 4.40 11 Comoros 47.6 44.24 0.35 4.2 -66.0 25.55 Zimbabwe 54.8 43.41 1.99 16 53.70 62.0 21.00

Table B .

 B 3: Comparison between the predicted and actual values.

	Day	Deaths		Tests
		Actual/%gain	Predicted	Actual	Predicted
	1	49/43%	86	269886
	2	46/66%	136	251301
	3	50/42%	86	248354
	4	153/-32%	116	248910
	5	26/68%	82	96177
	6	11/72%	39	30345	75319
	7	53/45%	97	247760
	8	78/42%	136	222942
	9	43/65%	124	206626
	10	52/70%	171	207651
	11	150/14%	175	214336
	12	39/69%	126	86361
	13	27/36%	42	23804	78612
	14	81/-12%	72	223293
	15	85/51%	172	199948
	16	63/62%	168	191917
	17	63/74%	246	196259
	18	130/11%	146	210495
	19	49/71%	167	90639
	20	32/56%	59	25699
	21	69/62%	181	240612
	22	66/64%	185	217585
	23	80/57%	188	214258
	24	76/61%	194	231306
	25	109/50%	219	259073	64680
	26	54/74%	209	114369
	27	46/41%	78	32368	45122
	28	95/52%	198	299121
	29	108/63%	292	276013
	30	104/37%	165	279376
	31	88/68%	275	301465
	32	178/44%	320	322468
	33	89/63%	238	140298
	34	85/50%	169	40313
	35	146/67%	447	355160
	36	262/30%	376	321373
	37	163/60%	410	330328
	38	162/51%	329	357368
	39	298/38%	484	388884
	40	137/33%	206	165764
	41	116/64%	318	47485
	42	257/40%	430	430644
	43	523/-63%	320	387569
	44	244/55%	548	379590

• K. Oshinubi, S. Flaxman, M. Rachdi and J. Demongeot. Estimating Viral Load Distribution using a Time-varying Branching Process and Epidemic Transmission Models (in preparation).

Acknowledgements

Without the support of my sponsors, Petroluem Technology Development Fund (PTDF) and Campus France

start of the second wave in January 2021 from [119,[START_REF] Nkamba | Modelling and prediction of the spread of covid-19 in cameroon and assessing the governmental measures (march-september 2020)[END_REF], for France at the start of the fifth wave in December 2021 [START_REF]Life expectancy at birth in africa as of 2019, by country[END_REF], and for Kuwait at the start of the fourth wave in December 2021 [214]. The values of specific death rates due to disease and natural death rates for Kuwait and France were taken from [160,214,[START_REF]Life expectancy at birth in africa as of 2019, by country[END_REF]. For Cameroon, the natural death rate was taken from [218], while the specific death rate due to disease was calculated from the cumulated deaths number due to the disease in two years divided by the cumulated infectious number in these two years [218]. The vaccination rate was chosen from [126], while the loss of resistance was chosen from [214]. Other parameters were assumed.

Numerical simulation based on COVID-19 outbreak in Kuwait

We have chosen the following set of parameter values corresponding to the COVID-19 outbreak at the start of the fourth wave in December 2021 in Kuwait: We present the visualization results for the simulated values in Figure 3.42.

Numerical simulation based on COVID-19 outbreak in France

We have chosen the following set of parameter values corresponding to the COVID-19 outbreak at the start of the fifth wave in December 2021 in France:

We present the visualization results for the simulated values in Figure 3.43.

Numerical simulation based on COVID-19 outbreak in Cameroon

We choose the following set of parameter values corresponding to the COVID-19 outbreak at the start of the second wave in January 2021 in Cameroon:

The relationship between Theil index and Gini index: mathematical and statistical approach

Firstly, the relationship between Theil index and Gini index is proved mathematically. The Gini index is defined as follows [96]:

where x k (resp. y k ) denotes the kth cumulative part of the population (respectively income).

On choosing the population increments, d k = x k -x k-1 are equal to 1/n, and if E(∆) represents the expectation of the increment, ∆ k = y k -y k-1 for the distribution d k . Then, the Theil index applied to the percentage y k of the total income relative to a percentage x k of the total population ([96]) is defined by the following equation:

If the first increment of y, ∆ 1 = y 1 ≤ 1, is close to 1 [which corresponds to a square-shaped Lorenz curve, i.e., close to a left right triangle-shaped income vs. population curve (in red on Figure 4.28), or to a high Gini index close to 1], then: -log(∆ 1 ) ∼ 1 -∆ 1 and then:

the equality being available only if the Lorenz curve presents a perfect left right triangle shape.