Keywords: Identiability, SIR models, Epidemic, SIR models with nonlinear incident rate, Parameter Estimation Identiabilité, Les modèles de SIR, Epidémie, Les modèles SIR avec un taux d

Cette thèse est consacrée à l'étude de l'identiabilité et de l'estimation des paramètres d'une classe de modèles SIR et se divise en deux parties.

Dans la première partie (chapitre 1) nous tentons à estimer les paramètres de modèles SIR simples à partir des données réelles. Dans le chapitre 1, nous construisons un schéma qui permet de retrouver tous les paramètres des modèles SIR sous l'hypothèse que nous avons la connaissance des données des cas déclarés chaque semaine lors d'une épidémie. Ce schéma a été utilisé pendant des épidémies qui ont eu lieu en Amérique, en France et en Inde.

Dans la deuxième partie (chapitre 2, chapitre 3) nous étudions l'identiabilité d'une classe de modèles SIR avec un taux d'incidence non linéaire. Dans le chapitre 2, nous fournissons quelques préliminaires d'algèbre diérentielle, et une méthode pour trouver les paramètres identiables d'un système dynamique. Dans le chapitre 3, nous appliquons la méthode du chapitre 2 pour certains modèles SIR à taux d'incidence non linéaire.

Acknowledgement

Firstly, I would like to express my sincere gratitude to my supervisors Professor Pierre Magal and Professor Arnaud Ducrot for their continuous support of my doctoral study and related research. Their patience, motivations, and immense knowledge inspired me to take a long journey into the research. Their suggestions also allowed me to improve the manuscript of this thesis. I could not have imagined having better supervisors and mentors for my doctoral study.

Besides my supervisors, I would like to thank Prof. Andreas Hartmann, and Prof. David Lannes not only for their insightful comments and encouragement but also for the hard question which encouraged me to widen my research from various perspectives.

My sincere thanks also go to Dr. Xiaoming Fu, Dr. Minh Phan Tran Duc, Dr. That Tu Ton, Dr. Duc Trung Hoang, Dr. Lara Abi Rizk who helped me a lot when I got some problems concerning the study. I also want to thank my doctoral student friends Tuan Thuong Dang, Abhinandan, Trung Hieu Vu as well as the other doctoral students, who give me a lot of valuable advice in the researches. Without their precious support, it would not be possible to conduct this research.

I also thank my fellow labmates Felipe Negreira, and Rolando Iii Perez for the stimulating discussions and all the fun we have had in the last four years. 2 Some preliminaries of dierential algebra and application to system identiability

Résume

Tout au long de l'histoire de l'humanité, on a été témoin de tant d'épidémies qui ont causé beaucoup de décès. Dans une petite maison datant de 5000 ans découverte dans un village préhistorique en Chine [77], se trouve les os des 97 corps étant entassés et ensuite brûlés et comprenant de tout âge : des adultes, des adolescents et des enfants. C'est très probalement une trace d'une épidemie, constatent les anthropologues. Autres exemples : la peste bubonique (connue au monde comme "Black Deaths"), entre 1346 et 1350, qui bouleversait l'Occident et retirait environ un tier de la population; la pandémie grippale (également appelée "Grippe espagnole" [START_REF] Brauer | Mathematical epidemiology: Past, present, and future[END_REF]), pour la période 1918-1919, qui a causé plus de 50 millions de décès dans le monde entier; et plus récemment, la pandémie de coronavirus, le Covid-19 (SARS-CoV-2) d'origine Wuhan, en Chine (identié pour la première fois en décembre 2019), répands rapidemment dans le monde entier et a causé 1 160 416 décès au total compté jusqu'au 25 octobre 2020 [76].

Pour comprendre la mécanisme d'une épidemie et pour y faire face, l'épidémiologie mathématique a été née et devenue un domaine de recherche très actif, et typiquement, dans la situation sanitaire récente dans le monde. L'épidémiologie mathématique, par son nom, joue le rôle essentiel dans la mise en place des modèles mathématiques qui permettent de clarier et expliquer les mécanismes de transmission et de propagation des maladies. Depuis les travaux de Daniel Bernoulli en 1760, en mettant en ouvre un modèle mathématique sous forme d'équations diérentielles ordinaires [START_REF] Bacaër | A short history of population dynamics[END_REF], jusqu'à près d'un siècle et demi, en 1906 Hamer proposa un modèle à temps discret [START_REF] Hethcote | The mathematics of infectious diseases[END_REF], qui peut être considéré comme l'idée pionnière des modèles compartimentaux, où il supposa que le nombre de nouveaux cas par unité de temps d'une épidémie dépend du nombre d'individus sensibles et du nombre d'individus infectés. Aujourd'hui, la plupart des modèles compartimentaux développés pour les systèmes déterministes ainsi que les systèmes stochastiques se basent au modèle SIR très connue de Kermack et McKendrick introduit en 1927 [START_REF] Kermack | Contributions to the mathematical theory of epidemics, part I[END_REF], [START_REF] Kermack | Contributions to the mathematical theory of epidemics, part II[END_REF]. Dans ce modèle, une population est divisée en trois compartiments (ceux qui donnent le nom SIR du modèle) : les individus sensibles, abrégés par le symbole S, les individus infectés, abrégés par le symbole I, les individus retirés, abrégés par le symbole R. Ici, S(t), I(t) et R(t) indiquent respectivement le nombre de personnes sensibles, infectées et retirées, au moment t > 0. Le paramètre τ > 0 correspond au taux de transmission de la maladie et le paramètre ν > 0 correspond au taux d'élimination des individus infectés. Tous les paramètres et le conditions initiales considérées comme inconnues.

En réalité, pour étudier une épidémie, nous ne disposons que des données fournies par un tableau contenant le nombre d'individus infectés déclarés. Pour comprendre les causes d'une épidémie et estimer sa taille, nous devons trouver un modèle approprié qui correspond aux données dans un certain sens. Cela se pose naturellement l'un des problèmes les plus importants de l'épidémiologie mathématique : l'ajustement des données au modèle.

Supposons que nous disposions des données sur les cas signalés chaque semaine lors d'une épidémie, comme le montre le tableau suivant Résume Semaine (t) Les cas signalés

1 N 1 2 N 2 ... ... k N k
Table 1: Données sur les cas signalés chaque semaine pour une épidémie de la semaine 1 à la semaine k.

An d'ajuster les données, nous déterminons d'abord les paramètres τ, ν, S 0 , I 0 , R 0 de (0.0.1) pour que nous puissions obtenir un modèle SIR approprié. Certains travaux tentent d'estimer ces paramètres par la méthode des moindres carrés, la méthode des vraisemblances,...tels que [START_REF] King | Introduction to inference: parameter estimation, ICTP Workshop on Mathematical Models of Climate Variability[END_REF], [START_REF] Tchavdar | Inverse problem for coecient identication in SIR epidemic models[END_REF], [START_REF] Capaldi | Parameter Estimation And Uncertainty Quantication For An Epidemic Model[END_REF], [START_REF] Hadeler | Parameter identication in epidemic models[END_REF], [START_REF] Capistran | Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus[END_REF], [START_REF] Hadeler | Parameter estimation in epidemic models: simplied formulas[END_REF]. L'inconvénient de ces méthodes est qu'elles ne peuvent pas fournir une vue générale des paramètres de (0.0.1). Cela signie que l'on ne peut pas trouver tous les paramètres de (0.0.1) et qu'elles ne mentionnent pas l'identiabilité des paramètres de (0.0.1). En 2018, Pierre Magal et Glenn Webb [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] ont fourni une nouvelle approche pour estimer les paramètres de (0.0.1) en utilisant le point tournant. Dans leur travail, ils considèrent les cas non signalés en ajoutant le nouveau paramètre ν 1 , le taux retiré des personnes infectées en raison des données sur les cas signalés. Le paramètre ν est considéré comme le taux retiré de personnes infectées de tous cas, y compris les cas déclarés et les cas non déclarés, puis un autre nouveau paramètre apparaît ν 2 = ν -ν 1 , le taux retiré de personnes infectées en raison des cas non déclarés. La deuxième colonne du tableau de données ci-dessus contient maintenant les valeurs de la fonction ν 1 I(t), et en construisant la fonction CR(t) = t 0 ν 1 I(s)ds, le nombre cumulé de cas signalés au moment t, Pierre Magal et et Glenn Webb ont fourni un algorithme pour calculer les paramètres numériquement. Cependant, le calcul numérique dans leurs travaux nécessite de connaître les valeurs de S 0 et I 0 , et une fois de plus, l'identiabilité des paramètres est négligée.

L'identiabilité des paramètres des systèmes dynamiques est un domaine actif étudié par de nombreux mathématiciens, par exemple [START_REF] Evans | Identiability of uncontrolled nonlinear rational systems[END_REF], [START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Ljung | On global identiability for arbitrary model parametrizations[END_REF], [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Denis-Vidal | Some eective approaches to check the identiability of non-linear systems[END_REF], [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Denis-Vidal | System identiability (symbolic computation) and parameter estimation (numerical computation[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF]. An de comprendre le rôle important de l'identiabilité d'un système dynamique pour l'estimation des paramètres, nous présentons ici la dénition pour (0.0.1). Denition 0.0.1. Considérons le système (0.0.1) avec la fonction de sortie CR(t) = t 0 ν 1 I(s)ds. Les paramètres τ ,ν, ν 1 , S 0 , I 0 , R 0 sont supposés être positifs et nous désignons p = (τ, ν, ν 1 , S 0 , I 0 , R 0 ), p = (τ , ν, ν 1 , S 0 , I 0 , R 0 ). Si nous avons p = p de la condition CR(t, p) = CR(t, p) pour chaque t ≥ 0, le (0.0.1) est appelé pour être identiable à partir de la fonction de sortie CR(t).

À partir des données sur les cas déclarés chaque semaine, on considère que la fonction de sortie CR(t) est déterminée. Si le système (0.0.1) est identiable, on peut alors armer que les paramètres peuvent être déterminés de manière unique à partir de la fonction de sortie CR(t). Malheureusement, d'après le résultat que nous présentons au Chapitre 1, (0.0.1) n'est pas identiable . De plus, nous avons la relation ν = ν, τ ν 1 = τ ν1 , τ S 0 = τ S 0 , τ I 0 = τ I 0 . (0.0.2) si CR(t, p) = CR(t, p) pour chaque t ≥ 0. Cela signie qu'à partir des données des cas déclarés chaque semaine, il y a tant de vecteurs de paramètres p donnant la même fonction de sortie CR(t). Cette analyse nous montre l'importance de prendre en compte l'identiabilité avant d'essayer d'estimer les paramètres à partir des données. Dans le Chapitre 1, du point de vue de [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] et en considérant l'identiabilité de (0.0. [START_REF] Capasso | A generalization of the Kermack-McKendrick Deterministic Epidemic Model[END_REF] ont généralisé le modèle (0.0.1) comme suit

     S (t) = -S(t)g(I(t)), I (t) = S(t)g(I(t)) -νI(t), R (t) = νI(t) (0.0.3)
avec la même condition initiale de (0.0.1). La fonction g : [0, +∞) → [0, +∞) est supposé remplir les conditions suivantes i. g(0) = 0.

ii. Il existe c > 0 tel que g(x) ≤ c pour tous les x ∈ [0, +∞).

iii. La dérivée de g existe et est bornée sur tout intervalle compact de [0, +∞), avec

g (0) > 0.
La fonction g(x) ci-dessus est considérée d'inclure le phénomène de saturation. Capasso et Serio [START_REF] Capasso | A generalization of the Kermack-McKendrick Deterministic Epidemic Model[END_REF] ont également mentionné g(x) = τ x 1 + κx comme un exemple de leurs recherches. An d'étudier le comportement dynamique du modèle épidémique (0.0.3), de nombreux auteurs ont modié (0.0.3) d'une certaine manière. En 1986, Liu et al.

[42] a brièvement examiné l'un des modèles modiés de (0.0.3) en utilisant la fonction spécique

g(x) = τ x α 1 + κx β (0.0.4)

Résume

dans laquelle α = β > 0.

D'autres études utilisant la fonction (0.0.4) sont Ruan et al. [START_REF] Shigui Ruan | Dynamical behavior of an epidemic model with a nonlinear incidence rate[END_REF] avec α = β = 2 et Xiao et al. [START_REF] Xiao | Global analysis of an epidemic model with nonmonotone incidence rate[END_REF]

avec α = 1, β = 2.
Au Chapitre 3, nous considérons le système (0.0.3) discuté dans Magal et al. [START_REF] Magal | Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico[END_REF] avec la fonction (0.0.4) comme suit

           S (t) = - τ S(t)I α (t) 1 + κI β (t) , I (t) = τ S(t)I α (t) 1 + κI β (t) -νI(t), t ≥ 0 R (t) = νI(t) (0.0.5) avec la condition initiale S(0) = S 0 > 0, I(0) = I 0 > 0, R(0) = R 0 ≥ 0 , α, β sont donnés et β + 1 ≥ α ≥ 1.
On voit que la troisième équation de (0.0.5) est juste une conséquence des deux premières équations de (0.0.5), alors (0.0.5) peut être lu comme suit

       S (t) = - τ S(t)I α (t) 1 + κI β (t) , I (t) = τ S(t)I α (t) 1 + κI β (t) -νI(t), t ≥ 0 (0.0.6) où τ S(t)I α (t) 1 + κI β (t)
est appelé un taux d'incidence non linéaire, ν est le taux d'élimination de tous les individus infectieux d'une maladie. En réalité, ce taux ne peut pas être connu car de nombreux individus retirent la maladie sans le signaler. Nous ne pouvons avoir que le nombre d'individus infectieux y(t) = µI(t) avec 0 < µ < ν déclarés par les responsables de la santé publique (voir [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], [START_REF] Ducrot | Identifying the Number of Unreported Cases in SIR Epidemic Models[END_REF]). Supposons que τ , κ, ν, µ, S 0 , I 0 soient les paramètres inconnus de (0.0.6). L'un des problèmes les plus importants concernant le modèle (0.0.6) est le suivant Problem 0.0.2. Considérons le système (0.0.6), si nous avons les données réelles des cas déclarés comme une fonction y(t) = µI(t), peuvent-ils les paramètres de (0.0.6) être déterminés de manière unique à partir de celle-ci ?

Considérons le système (0.0.6), le nombre

R 0 = τ I α-1 0 S 0 ν(1 + κI β 0 )
est appelé le nombre de reproduction de base de (0.0.6) [START_REF] Magal | Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico[END_REF]. Si le R 0 < 1, alors I(t) tombe à zéro, et l'épidémie s'atténue. Désormais, nous considérerons (0.0.6) seulement lorsque l'épidémie se déclenchera avec R 0 > 1. Une autre question est la suivante :

Problem 0.0.3. Le nombre de reproduction de base de (0.0.6) est-il déterminé uniquement à partir de la fonction y(t) = µI(t) lorsque l'épidémie se déclanche ?

Les deux problèmes 0.0.2, et 0.0.3 peuvent être compris comme une sorte de problème concernant l'identiabilité de (0.0.6). Comme nous l'avons mentionné cidessus, il est très important de comprendre l'identiabilité d'un système dynamique pour l'estimation des paramètres à partir des données réelles. Il existe certaines approches pour tester l'identiabilité d'un système dynamique, telles que l'approche des séries de Taylor, l'approche de la transformation de similitude, l'approche de l'algèbre diérentielle, ... [START_REF] Audoly | Global identiability of nonlinear models of biological systems[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Bellu | DAISY: A new software tool to test global identiability of biological and physiological systems[END_REF], [START_REF] Boulier | Dierential Elimination and Biological Modelling[END_REF], [START_REF] Ducrot | Identifying the Number of Unreported Cases in SIR Epidemic Models[END_REF], [START_REF] Evans | Identiability of uncontrolled nonlinear rational systems[END_REF], [START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF], [START_REF] Ljung | On global identiability for arbitrary model parametrizations[END_REF], [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Nemcova | Structural identiability of polynomial and rational systems[END_REF], [START_REF] Pohjanpalo | System identiability based on the power series expansion of the solution[END_REF], [START_REF] Maria | Parameter identiability of nonlinear systems: the role of initial conditions[END_REF], [START_REF] Vajda | Similarity transformation approach to identiability analysis of nonlinear compartmental models[END_REF], [START_REF] Denis-Vidal | Some eective approaches to check the identiability of non-linear systems[END_REF], [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Denis-Vidal | System identiability (symbolic computation) and parameter estimation (numerical computation[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF], [START_REF] Walter | Identiability of State Space Models, with Applications to Transformation Systems[END_REF], [START_REF] Yates | Structural identiability analysis via symmetries of dierential equations[END_REF]. Dans chaque approche, nous pouvons obtenir certains avantages ou inconvénients pour certains systèmes dynamiques spéciques. An de résoudre les problèmes 0.0.2, et 0.0.3, nous choisissons l'approche de l'algèbre différentielle. L'avantage de cette approche est que nous pouvons extraire une équation ne concernant aucun état du système mais seulement la fonction de sortie que nous connaissons dans la réalité. En combinant d'autres hypothèses du problème, nous pouvons obtenir la relation entre deux paramètres lorsque le système dynamique a la même sortie. Ainsi, nous pouvons tester l'identiabilité du système à partir de la sortie et également obtenir la relation entre les paramètres. Cette relation nous aide à déterminer les combinaisons identiables de paramètres, il est donc très important d'estimer les paramètres. Le Chapitre 2 de cette thèse fournit une méthode pour trouver les combinaisons identiables de paramètres. Dans ce chapitre, nous présentons quelques préliminaires en algèbre diérentielle et fournissons un schéma pour trouver des combinaisons identiables de paramètres, puis nous appliquons ce schéma pour le modèle SIR. Le Chapitre 3 vise à trouver les combinaisons de paramètres de (0.0.6) qui peuvent être déterminées de manière unique à partir de la fonction y(t) = µI(t) dans laquelle µ est le taux de retrait des personnes infectées déclarées. La première section du Chapitre 3 fournit la condition susante pour deviner que les combinaisons de paramètres peuvent être déterminées de manière unique à partir de la fonction y(t) = µI(t). Les sections suivantes sont les réponses aux problèmes 0.0.2, 0.0.3 dans certains cas particuliers {α = 1,

β = 2}, {α = 1, β = 1}, {α = 2, β = 2}, {α = 2, β = 1}, {α = 1, β = 3}.
Ces réponses sont réalisées en appliquant l'approche de l'algèbre diérentielle présentée dans les Sections 2.2 et 2.3 du Chapitre 2.

Introduction

An epidemic is understood as an infectious disease that aects a large number of people during a certain period of time. Along with the history of human beings, people witnessed so many epidemics that caused a lot of deaths. About 5000 years ago, an epidemic swept a prehistoric village in China. The bodies were piled up in a house and then burned down. The epidemic does not rule out anyone. The bones found included adults, teenagers and children [77]. Up to now, when mankind's knowledge of epidemic is increasing, there have been still new epidemics that spread worldwide. These epidemics are called pandemics, for instance, COVID-19 pandemic. The COVID-19 also called the coronavirus pandemic, is a pandemic by the transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [76]. The COVID-19 was rst identied in Wuhan, China in December 2019 and up to the 25th of October 2020, COVID-19 has caused 1,160,416 deaths all over the world [78]. In the past, the bubonic plague that is called Black Deaths caused about onethird of the population of Europe from 1346 to 1350. Another epidemic that caused more than 50,000,000 deaths all over the world in the period 1918-1919 is "Spanish" inuenza [START_REF] Brauer | Mathematical epidemiology: Past, present, and future[END_REF].

Since epidemic is the cause of so many deaths, it becomes a problem that humans should concern carefully. Epidemics should be investigated in various ways, from the infectious mechanisms to their inuences. One of the ways that mathematicians have been trying is modeling the epidemic by mathematical dynamical systems. This can explain why mathematical epidemiology has become a very active eld of research. The goal of mathematical epidemiology is setting up mathematical models giving us an understanding of the mechanisms of disease transmission and spread. Mathematical models also help us determine the main factors of the disease transmission process, give us suggestions for eectively controlling and methods of prevention, and provide an estimation for the severity and the size of an epidemic. Roughly speaking, mathematical models can be a part of the toolbox of public health research and decision making [26]. One of the rst mathematical models of epidemics is the work entitled An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it of Daniel Bernoulli (1700-1782) [START_REF] Bacaër | A short history of population dynamics[END_REF]. In his work, Daniel Bernoulli tried to nd a comparison between the benet of inoculation and the immediate risk of dying by setting up a model under the form of ordinary dierential equations. After that, he applied this model with Halley's life table. This life table is introduced in a book "Natural and Political Observations Made upon the Bills of Mortality", published in 1662 in London, and known as a kind of table of data. Nowadays, combining mathematical models and data for studying epidemics is a common way in mathematical epidemiology [START_REF] Bacaër | A short history of population dynamics[END_REF].

From the work in 1760 of Daniel Bernoulli to the late nineteenth century, mathematical models for epidemics didn't have any signicant development. In 1906, for studying the measles epidemics, Hamer suggested using the mass action law to formulate a discrete-time model [START_REF] Hethcote | The mathematics of infectious diseases[END_REF]. Hamer supposes that the number of new cases per unit time of an epidemic depends on the number of susceptible individuals and the number of infected individuals. Hamer's model can be considered as the rst idea of compartmental models, and then so many studies about mathematical models of epidemics also were carried out this way [START_REF] Bacaër | A short history of population dynamics[END_REF]. One of the most famous compartmental models describing the spread of the epidemic is the SIR model that is rstly introduced by Kermack and McKendrick in 1927 [START_REF] Kermack | Contributions to the mathematical theory of epidemics, part I[END_REF], [START_REF] Kermack | Contributions to the mathematical theory of epidemics, part II[END_REF]. Both Kermack and McKendrick were public health physicians but they spent a lot of time studying mathematics in epidemiology. The model introduced by them is now extended in dierent ways for the deterministic systems and stochastic systems. In Kermack and McKendrick's model, a population is divided into three compartments: susceptible individuals abbreviated by the symbol S, the infected individuals abbreviated by the symbol I, the removed individuals abbreviated by the symbol R. Accordingly, some assumptions should be made to set up this model [26].

• The number of population is always a constant. New births or deaths from other reasons can be neglected.

• All the individuals can be susceptible individuals equally. Every infected individuals can transmit disease to susceptible individuals.

• Removed individuals cannot become infected individuals and transmit the disease.

S (t) = -τ S(t)I(t), I (t) = τ S(t)I(t) -νI(t), t ≥ 0 R (t) = νI(t) (0.0.7)
with the initial S(0) = S 0 > 0, I(0

) = I 0 > 0, R(0) = R 0 > 0.
Considering (0.0.7) as a dynamical system, there are so many problems from the qualitative properties to computations. In reality, for studying an epidemic, we have Introduction only the data given by a table containing the number of reported infected individuals.

To understand the causes of an epidemic and estimate its size, we have to nd an appropriate model that matches the data in some sense. It is naturally derived one of the most important problems in mathematical epidemiology: tting data with the model. Suppose that we have the weekly reported case data from an epidemic as the following table

Week (t) Reported Cases 1 N 1 2 N 2 ... ... k N k
Table 2: Weekly reported case data for an epidemic from the week 1 to the week k.

To t the data, we rst determine the parameters τ, ν, S 0 , I 0 , R 0 of (0.0.7) so that we can have the appropriate SIR model. There are some works trying to estimate these parameters by least square method, likelihood method,...such as [START_REF] King | Introduction to inference: parameter estimation, ICTP Workshop on Mathematical Models of Climate Variability[END_REF], [START_REF] Tchavdar | Inverse problem for coecient identication in SIR epidemic models[END_REF], [START_REF] Capaldi | Parameter Estimation And Uncertainty Quantication For An Epidemic Model[END_REF], [START_REF] Hadeler | Parameter identication in epidemic models[END_REF], [START_REF] Capistran | Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus[END_REF], [START_REF] Hadeler | Parameter estimation in epidemic models: simplied formulas[END_REF]. The disadvantage of these methods is that they cannot provide a general view of the parameters of (0.0.7). It means that neither we can nd all the parameters of (0.0.7) nor they mention the identiability of parameters of (0.0.7). In 2018, Pierre Magal and Glenn Webb [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] provided a new approach to estimate the parameters of (0.0.7) by using the turning point. In their work, they consider the unreported cases by adding the new parameter ν 1 , the removed rate of infected individuals due to the reported case data. The parameter ν is considered as a removed rate of infected individuals of all cases including reported cases and unreported cases, then another new parameter appears ν 2 = ν -ν 1 , the removed rate of infected individuals due to unreported cases. The second column of the table of data above is now the values of the function ν 1 I(t), and by building the function CR(t) = t 0 ν 1 I(s)ds, the cumulative number of reported cases at time t, Pierre Magal and Glenn Webb provided an algorithm to compute the parameters numerically. However, the numerical computation in their works requires to know the values of S 0 and I 0 , and once again the identiability of the parameters is neglected. The identiability of parameters of dynamical systems is an active eld studied by many mathematicians, for instance [START_REF] Evans | Identiability of uncontrolled nonlinear rational systems[END_REF], [START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Ljung | On global identiability for arbitrary model parametrizations[END_REF], [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Denis-Vidal | Some eective approaches to check the identiability of non-linear systems[END_REF], [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Denis-Vidal | System identiability (symbolic computation) and parameter estimation (numerical computation[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF]. For understanding the important role of identiability of a dynamical system to the parameter estimation we present here the denition for (0.0.7). Denition 0.0.4. Consider the system (0.0.7) with the output function CR(t) = t 0 ν 1 I(s)ds. The parameters τ ,ν, ν 1 , S 0 , I 0 , R 0 are supposed to be positive and let p = (τ, ν, ν 1 , S 0 , I 0 , R 0 ), p = (τ , ν, ν 1 , S 0 , I 0 , R 0 ). If we have p = p from the condition CR(t, p) = CR(t, p) for every t ≥ 0, the (0.0.7) is called to be identiable from the output function CR(t).

From the weekly reported case data, the output function CR(t) is considered to be determined. If the system (0.0.7) is identiable, then we can arm that the parameters can be uniquely determined from the the output function CR(t). Unfortunately, by the result that we present in Chapter 1, (0.0.7) is not identiable . Moreover, we have the relationship

ν = ν, τ ν 1 = τ ν1 , τ S 0 = τ S 0 , τ I 0 = τ I 0 . (0.0.8)
if CR(t, p) = CR(t, p) for every t ≥ 0. It means that from the weekly reported case data, there are so many parameter vectors p giving the same output function CR(t). This analysis let us see the important role of considering the identiability before trying to estimate the parameters from the data. In Chapter 1, by the view point of [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] and considering the identiability of (0.0.7), we construct a scheme that we nd an appropriate model from the real data of an epidemic. This scheme is applied for some real epidemics happened in New York, Bombay, and France. The key of constructing this scheme is the turning point equation, the values of the turning point t p , CR(t p ), lim t→∞ CR(t), and CR (t p ). Certainly, these values can be taken from the weekly reported case data. After that we can determine all the values of the parameters p if we vary just only the value of S 0 , and estimate the nal size of the epidemic depending on S 0 . Since Kermack and McKendrick published their works on epidemic models, there is a large number of publications trying to extend this model for studying epidemiology. In 1978, after studying cholera epidemic spread in Bari, Capasso and Serio [START_REF] Capasso | A generalization of the Kermack-McKendrick Deterministic Epidemic Model[END_REF] generalized the model (0.0.7) as follows      S (t) = -S(t)g(I(t)), I (t) = S(t)g(I(t)) -νI(t), R (t) = νI(t) (0.0.9) with the same initial condition of (0.0.7). The function g : [0, +∞) → [0, +∞) is assumed to satisfy the following conditions i. g(0) = 0. ii. There exists c > 0 such that g(x) ≤ c for all x ∈ [0, +∞).

iii. The derivative of g exists and is bounded on any compact interval of [0, +∞), with g (0) > 0.

The function g(x) above is considered to include the saturation phenomenon. Capasso and Serio [START_REF] Capasso | A generalization of the Kermack-McKendrick Deterministic Epidemic Model[END_REF] also mentioned g(x) = τ x 1 + κx as an example of their researches.

Introduction

in which α = β > 0.

Some other studies using the function (0.0.11) are Ruan et al. [START_REF] Shigui Ruan | Dynamical behavior of an epidemic model with a nonlinear incidence rate[END_REF] with α = β = 2 and Xiao et al. [START_REF] Xiao | Global analysis of an epidemic model with nonmonotone incidence rate[END_REF] with α = 1, β = 2.

In Chapter 3, we consider the system (0.0.9) discussed in Magal et al. [START_REF] Magal | Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico[END_REF] with the function (0.0.10) as follows

           S (t) = - τ S(t)I α (t) 1 + κI β (t) , I (t) = τ S(t)I α (t) 1 + κI β (t) -νI(t), t ≥ 0 R (t) = νI(t) (0.0.11)
with the initial condition S(0) = S 0 > 0, I(0

) = I 0 > 0, R(0) = R 0 ≥ 0 , α, β are given and β + 1 ≥ α ≥ 1.
It is seen that the third equation of (0.0.11) is just a consequence of the rst two equations of (0.0.11), then (0.0.11) can be read as follows

       S (t) = - τ S(t)I α (t) 1 + κI β (t) , I (t) = τ S(t)I α (t) 1 + κI β (t) -νI(t), t ≥ 0 (0.0.12)
where τ S(t)I α (t) 1 + κI β (t) is called a nonlinear incidence rate, ν is the removed rate of all infectious individuals of a disease. In reality, this rate cannot be known because many individuals remove the disease without report. We can only have the number of infectious individuals y(t) = µI(t) with 0 < µ < ν reported from the public health ocials (see [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], [START_REF] Ducrot | Identifying the Number of Unreported Cases in SIR Epidemic Models[END_REF]). Suppose that τ , κ, ν, µ, S 0 , I 0 are the unknown parameters of (0.0.12). One of the most important problems concerning the model (0.0.12) is the following Problem 0.0.5. Consider the system (0.0.12), if we know the real reported case data as a function y(t) = µI(t), is the parameter of (0.0.12) uniquely determined?

Consider the system (0.0.12), the number

R 0 = τ I α-1 0 S 0 ν(1 + κI β 0 )
is called the basic reproduction number of (0.0.12) [START_REF] Magal | Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico[END_REF]. If the R 0 < 1, then I(t) decreases to zero, and the epidemic subsides. From now on, we will only consider (0.0.12) when the epidemic outbreaks with R 0 > 1. Another question is the folowing Problem 0.0.6. Is the basic reproduction number of (0.0.12) uniquely determined from the function y(t) = µI(t) when the epidemic outbreaks?

The two problems 0.0.5, and 0.0.6 can be understood as a kind of the problem about the identiability of (0.0.12). As we mentioned above, understanding the identiability of a dynamical system is very important for parameter estimation from the real data. There are some approaches in testing the identiability of a dynamical system such as Taylor series approach, similarity transformation approach, dierential algebra approach, ... [START_REF] Audoly | Global identiability of nonlinear models of biological systems[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Bellu | DAISY: A new software tool to test global identiability of biological and physiological systems[END_REF], [START_REF] Boulier | Dierential Elimination and Biological Modelling[END_REF], [START_REF] Ducrot | Identifying the Number of Unreported Cases in SIR Epidemic Models[END_REF], [START_REF] Evans | Identiability of uncontrolled nonlinear rational systems[END_REF], [START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF], [START_REF] Ljung | On global identiability for arbitrary model parametrizations[END_REF], [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Nemcova | Structural identiability of polynomial and rational systems[END_REF], [START_REF] Pohjanpalo | System identiability based on the power series expansion of the solution[END_REF], [START_REF] Maria | Parameter identiability of nonlinear systems: the role of initial conditions[END_REF], [START_REF] Vajda | Similarity transformation approach to identiability analysis of nonlinear compartmental models[END_REF], [START_REF] Denis-Vidal | Some eective approaches to check the identiability of non-linear systems[END_REF], [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Denis-Vidal | System identiability (symbolic computation) and parameter estimation (numerical computation[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF], [START_REF] Walter | Identiability of State Space Models, with Applications to Transformation Systems[END_REF], [START_REF] Yates | Structural identiability analysis via symmetries of dierential equations[END_REF]. In each approach, we can obtain some advantages or disadvantages for some specic dynamical systems. To solve the problems 0.0.5, and 0.0.6, we choose the dierential algebra approach. The advantage of this approach is that we can extract one equation not concerning any state of the system but only the output function that we know in reality. Combining other assumptions from the problem, we can obtain the relationship between two parameters when the dynamical system has the same output. Hence, we can test the identiability of the system from the output and also obtain the relationship between parameters. This relationship helps us determine identiable combinations of parameters so it is very important to estimate parameters. Chapter 2 of this thesis is providing a method for nding the identiable combinations of parameters. In this chapter, we present some preliminaries in dierential algebra and provide a scheme to nd identiable combinations of parameters, then we apply this scheme for SIR model. Chapter 3 aims to nd the combinations of the parameters of (0.0.12) that can be uniquely determined from the function y(t) = µI(t) in which µ is the removed rate of reported infected individuals. The rst section of Chapter 3 provides the sucient condition to guess the combinations of parameters that can be uniquely determined from the function y(t) = µI(t). The next sections are the answers for the problems 0.0.5, 0.0.6 in some particular cases {α = 1, β = 2}, {α = 1, β = 1}, {α = 2, β = 2}, {α = 2, β = 1}, {α = 1, β = 3}. These answers are performed by applying the dierential algebra approach presented in sections 2.2 and 2.3 of Chapter 2.

Chapter 1

Identifying the number of unreported cases in SIR epidemic models

In this chapter, we consider the classical SIR epidemic model with its parameters and initial values, based upon reported case data from public health sources. The objective of this chapter is to provide a method for numerical computing the parameters of the SIR model from the real reported case data. The method here is based on the fact that the unreported cases are counted in the model. In many epidemic diseases, the reported cases are a small fraction of the unreported cases. We compute this fraction numerically by determining identiable combinations of parameters of the model from reported case data. The numerical calculations in this chapter are applied to the Hong Kong seasonal inuenza epidemic in New York City in 1968-1969, the inuenza epidemic in France for the six consecutive years from 1988-1989 to 1993-1994, and some other epidemics.

Introduction

Mathematical models of epidemics have a long history [START_REF] Anderson | Infective Diseases of Humans: Dynamics and Control[END_REF][START_REF] Bailey | The Mathematical Theory of Epidemics[END_REF][START_REF]Mathematical epidemiology[END_REF][START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology[END_REF][START_REF] Busenberg | Vertically Transmitted Diseases: Models and Dynamics[END_REF][START_REF] Diekmann | Mathematical Tools for Understanding Infectious Disease Dynamics[END_REF][START_REF] Hethcote | Qualitative analyses of communicable disease models[END_REF][START_REF] Hethcote | The mathematics of infectious diseases[END_REF][START_REF] Keeling | Modeling infectious diseases in humans and animals[END_REF][START_REF] Murray | Mathematical Biology[END_REF][START_REF] Thieme | Mathematics in Population Biology[END_REF]. One of the most important considerations of epidemic models is the identication of parameters needed for applications. The parameter identication problem for the SIR model has been investigated by many researchers, including [START_REF] Andreasen | The nal size of an epidemic and its relation to the basic reproduction number[END_REF][START_REF] Arino | A nal size relation for epidemic models[END_REF][START_REF] Capistran | Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus[END_REF][START_REF] Chowell | Modelling the transmission dynamics of acute haemorrhagic conjunctivitis: Application to the 2003 outbreak in Mexico[END_REF][START_REF] Chowell | Estimation of the reproduction number of dengue fever from spatial epidemic data[END_REF][START_REF] Diekmann | On the denition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations[END_REF][START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF][START_REF] Grassly | Seasonal infectious disease epidemiology[END_REF][START_REF] Hadeler | Parameter identication in epidemic models[END_REF][START_REF] Hadeler | Parameter estimation in epidemic models: simplied formulas[END_REF][START_REF] Hethcote | Modeling heterogeneous mixing in infectious disease dynamics[END_REF][START_REF] Hooker | Parameterizing state space models for infectious disease dynamics by generalized proling: measles in Ontario[END_REF][START_REF] Hsieh | On epidemic modeling in real time: An application to the 2009 Novel A (H1N1)inuenza outbreak in Canada[END_REF][START_REF] Lange | Reconstruction of disease transmission rates: Applications to measles, dengue, and inuenza[END_REF][START_REF] Li | Characteristics of an epidemic outbreak with a large initial infection size[END_REF][START_REF] Ma | Generality of the nal size formula for an epidemic of a newly invading infectious disease[END_REF][START_REF] Mummert | Studying the recovery procedure for the time-dependent transmission rate(s) in epidemic models[END_REF][START_REF] Pellis | Threshold parameters for a model of epidemic spread among households and workplaces[END_REF][START_REF] Pollicott | Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem[END_REF][START_REF] Roeger | Modeling TB and HIV coinfections[END_REF][START_REF] Van Den Driessche | Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission[END_REF]. Our objective here is to continue the investigation in [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] of the parameter identication problem for the standard SIR ordinary dierential equations model of an outbreak epidemic:

S (t) = -τ S(t)I(t), I (t) = τ S(t)I(t) -νI(t). (1.1.1)
Here S(t) and I(t) denote the number of susceptible and infected individuals, respectively, at time t > 0. The parameter τ > 0 corresponds to the disease transmission rate and the parameter ν > 0 corresponds to the removal rate of infected individuals.

Introduction

The initial conditions of the model are

S(0) = S 0 > 0 and I(0) = I 0 > 0. (1.1.2)
For specic applications, the parameters τ , ν, as well as the initial conditions, S 0 , I 0 , are usually unknown. Our objective here is to determine these values from specic time data of reported infected cases. Typically, the reported cases are only a small fraction of the total number of cases, since only the most severe symptomatic cases are reported. Our approach is based on knowledge of the data of newly reported cases (typically weekly) over the time course of the epidemic. This known data consists of the cumulative reported cases at time t, denoted by CR(t), that correspond to the total number of reported infected cases up to time t. To handle these data we assume that the removal rate ν takes the following form ν = ν 1 +ν 2 , where ν 1 is the removal rate of reported infected individuals, and ν 2 is the removal rate of infected individuals due to all other causes, such as mortality, recovery, or other reasons. With this assumption and notation, the cumulative reported cases are related to the number of infected by the following formula

CR(t) = ν 1 t 0 I(s)ds, (1.1.3)
where ν 1 > 0 is an unknown parameter. We formulate our problem as follows Problem 1.1.1. How can we identify the parameter set Θ = {(τ, ν, ν 1 , S 0 , I 0 )} ⊂ (0, ∞) 5 if we know the cumulative reported cases CR(t) for all time t > 0?

We will show that the knowledge of the cumulative reported cases CR(t) is not sucient to recover the parameter set Θ. Roughly speaking, this set is dened up to one degree of freedom. More precisely, under suitable hypotheses on the cumulative reported case data CR(t), t ≥ 0, only the following combination of parameters and initial values can be reconstructed

I 0 S 0 , S 0 τ, S 0 ν 1 and ν. (1.1.4)
As a consequence, the knowledge about the value of S 0 , the number of susceptible people in the population before the epidemic outbreak, allows us to obtain precise information about the values I 0 , τ , and ν 1 . Then the basic reproduction number of the epidemic

R 0 = S 0 τ ν , (1.1.5)
can be obtained from (1.1.4). The interpretation of R 0 is that if R 0 < 1, then the epidemic subsides, and if R 0 > 1, then the epidemic outbreaks. We will describe a simple method to compute the parameter set in (1.1.4), and apply this method to specic epidemic data.

1. Identifying the number of unreported cases in SIR epidemic models

Identiability of SIR model

In this section we show that the parameter set Θ is not identiable from the reported case data CR(t), t ≥ 0. We refer to Evans et al. [START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF] for more results on this topic. Here we perform a simple proof for the SIR model. Consider the parameter p := (τ, ν, ν 1 , S 0 , I 0 ) ∈ (0, +∞) 5 . Dene (S(t, p), I(t, p)), as the unique solution of (1.1.1)-(1.1.2), and CR(t, p) as the output function (1.1.3), for a given value of the parameter set p ∈ (0, +∞) 5 . We have the following proposition Proposition 1.2.1. Suppose that (S(t, p), I(t, p)) and (S(t, p), I(t, p)) are the two solutions of (1.1.1)-(1.1.2) for the parameter p = (τ, ν, ν 1 , S 0 , I 0 ) and the parameter p = τ , ν, ν 1 , S 0 , I 0 , respectively. Then

CR(t, p) = CR(t, p) (1.2.1)
for every t ≥ 0, if and only if

ν = ν, τ ν 1 = τ ν1 , τ S 0 = τ S 0 , τ I 0 = τ I 0 . (1.2.2)
Proof. (⇒) Assume rst that (1.2.1) holds and dierentiate both sides for t to obtain

CR (t, p) = CR (t, p), ∀t ≥ 0, which is equivalent to ν 1 I(t, p) = ν1 I(t, p), ∀t ≥ 0. (1.2.3)
Dierentiate both sides of (1.2.3) with respect to t, to obtain

ν 1 I (t, p) = ν1 I (t, p), ∀t ≥ 0.
Replacing I (t) by its formula in (1.1.1) on both sides of the above equality, we obtain

ν 1 I (t, p) = ν1 I (t, p) ⇔ ν 1 τ S(t, p)I(t, p) -νI(t, p) = ν1 τ S(t, p)I(t, p) -νI(t, p) ⇔ ν 1 I(t, p) τ S(t, p) -ν = ν1 I(t, p) τ S(t, p) -ν , and (1.2.3) implies that τ S(t, p) -ν = τ S(t, p) -ν, ∀t ≥ 0. (1.2.4)
Dierentiating both side of (1.2.4) with respect to t, we obtain

τ S (t, p) = τ S (t, p) ⇔ τ -τ S(t, p)I(t, p) = τ -τ S(t, p)I(t, p) ⇔ τ 2 S(t, p)I(t, p) = τ 2 S(t, p)I(t, p)

Identiability of SIR model

and by using again (1.2.3), we obtain

ν1 ν 1 τ 2 S(t, p) = τ 2 S(t, p), ∀t ≥ 0. (1.2.5)
Then, by using (1.2.4), and replacing S(t, p) in (2.4.1), we obtain

ν1 ν 1 τ 2 S(t, p) = τ 2 ( ν -ν + τ S(t, p) τ ).
Therefore,

( ν1 ν 1 τ -τ )S(t, p) = τ τ (ν -ν), ∀t ≥ 0. (1.2.6)
Dierentiating both sides of equation (1.2.3) with respect to t, we obtain

( ν1 ν 1 τ -τ )S (t, p) = 0, which implies that ( ν1 ν 1 τ -τ )(-τ S(t, p)I(t, p)) = 0, ∀t ≥ 0. (1.2.7)
Setting t = 0 in equations (1.2.3), (1.2.4), (1.2.6) and (1.2.7) we obtain the following system of equations

               I 0 ν 1 = I 0 ν1 τ S 0 -ν = τ S 0 - ν ( ν1 ν 1 τ -τ )S 0 = τ τ (ν -ν) -τ S 0 I 0 ( ν1 ν 1 τ -τ ) = 0 (1.2.8)
and (1.2.2) follows.

( ⇐) To prove the converse implication, let (S(t), I(t)) be a solution of (1.1.1)-(1.1.2).

Let S 0 > 0, I 0 > 0, and set

S(t) := S 0 S 0 S(t) and I(t) := I 0 I 0 I(t).
Since (S(t),

I(t)) satises (1.1.1)-(1.1.
2), we obtain by replacing S(t) and I(t) with the above formulas,

               S 0 S 0 S (t) = -τ S 0 S 0 S(t) I 0 I 0 I(t) I 0 I 0 I (t) = τ S 0 S 0 S(t) I 0 I 0 I(t) -ν I 0 I 0 I(t) S(0) = S 0 I(0) = I 0 .

Identifying the number of unreported cases in SIR epidemic models

After simplifying,

               S (t) = -τ I 0 I 0 S(t)I(t) I (t) = τ S 0 S 0 S(t)I(t) -νI(t) S(0) = S 0 I(0) = I 0 (1.2.9)
and by using (1.2.2) we deduce that τ and by using (1.2.2) we deduce that

I 0 I 0 = τ S 0 S0 = τ and ν = ν.
τ ν 1 = τ ν 1 , τ I 0 = τ I 0 . Therefore, CR(t, p) = CR(t, p), ∀t ≥ 0.
Remark 1.2.2. Suppose that p = (τ, ν, ν 1 , S 0 , I 0 ) ∈ (0, +∞) for every a > 0.

Computation the combinations of the parameters of SIR model 1.3.1 System of equations to identify the parameters

In this section, we consider the SIR model (1.1.1)-(1.1.2) when the epidemic outbreaks with the basic reproduction number R 0 = τ S 0 /ν > 1. The cumulative reported case function CR(t) := ν 1 t 0 I(s)ds, t ≥ 0 is assumed to be known. We aim to provide a simple method to identify the parameters (1.1.4).

Recall that, since R 0 > 1, the solutions of (1.1.1)-(1.1.2) have a typical outbreak behavior [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] as follows (i) The function t → S(t) is not increasing on [0, ∞) with S(0) = S 0 and S(∞) > 0.

Computation the combinations of the parameters of SIR model

(ii) There exists a unique turning point t p > 0 such that I (t p ) = 0, and I(t) is not decreasing on [0, t p ] and not increasing on [t p , ∞). Moreover I(∞) = 0 and t → I(t) is integrable on [0, ∞).

In addition to the turning point t p , the above properties allow us to dene several important quantities related to the function CR:

CR(t p ), CR (t p ) and CR(∞). (1.3.1)
As it will be seen later, these quantities will be sucient to compute the combinations of parameters in (1.1.4). To compute these four combined parameters we will provide four independent equations. Three of them is derived in [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]. Following the notations introduced in [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], we set

c := CR(∞), r := CR(t p ) CR(∞) .
Next, by setting

X := c τ ν 1 , (1.3.2)
and then by multiplying both sides by S 0 ν 1 we deduce that

X × (S 0 ν 1 ) = cS 0 τ. (1.3.3)
Moreover, by using respectively, equations (3.3), (3.7) and (3.9) in [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], we derive the three following independent equations e -X + Xe -rX = 1 + I 0 S 0 , (1.3.4)

(S 0 ν 1 ) × 1 + I 0 S 0 -e -rX (1 + rX) = CR (t p ) (1.3.5) and ν = (S 0 τ ) × e -rX . (1.3.6)
We recall rst, that by Proposition 3.1 in [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], equation (1.3.4) implies the following compatibility condition with the data: r ∈ (0, 1/2) . Thus, for the model (1.1.1)-(1.1.2) more than half of all cases occur after the turning point t p . As noted in [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], some outbreak epidemics have more than half of all cases occurring before the turning point, and the model (1.1.1)-(1.1.2) does not apply these examples.

Derivation of the equation for the turning point

To dene an equation for the turning point, we rst introduce the function

F (X) := e -X + Xe -rX -1. Lemma 1.3.1. Suppose that 1 > r ≥ 1 2 ⇒ F (X) < 0, ∀X > 0.
1. Identifying the number of unreported cases in SIR epidemic models Proof. We have F (0) = 0 and F (X) = e -rX G(x), with

G(X) := 1 -rX -e -(1-r)X . Then G(0) = 0 and G (X) = -r + (1 -r)e -(1-r)X . So if 1 > r ≥ 1/2 we have G (X) < 0, ∀X > 0. The result follows. Lemma 1.3.2. Assume that r ∈ (0, 1 2
). There exists a unique strictly positive solution X(r) > 0 of equation

F (X) = 0 ⇔ e -X + Xe -rX -1 = 0. (1.3.7)
Moreover, there exists X max ∈ (0, X(r)), such that the function F (X) is strictly increasing on (0, X max ) and strictly decreasing on (X max , X(r)). Furthermore,

F (X) > 0, if X ∈ (0, X(r)), F (X) < 0, if X ∈ (X(r), ∞). (1.3.8)
Proof. We have F (0) = 0 and F (X) = e -rX G(x), with

G(X) := 1 -rX -e -(1-r)X .
Then G(0) = 0 and G (X) = -r + (1 -r)e -(1-r)X . Moreover, we have

G (X) = 0 ⇔ X = 1 1 -r ln 1 r -1 := X * > 0.
Thus, G (X) > 0 for X ∈ (0, X * ) and G (X) < 0 for X > X * . We also have

lim X→∞ G(X) = -∞. Let X max > X * be the unique value in (0, +∞) such that G(X max ) = 0. Moreover, F (X) > 0 on (0, X max ), F (X) < 0 on (X max , ∞), and
F (0) = F (X max ) = 0. Hence, F (X max ) > 0 is the maximum of F . Next, we observe that lim X→∞ F (X) = -1
Therefore there exists a unique X(r) ∈ (X max , ∞) such that F (X(r)) = 0.

The above results are summarized in Figure 1.1.

Computation the combinations of the parameters of SIR model

Figure 1.1: The function X → F (X) when r = 0.1, 0.2, 0.3, 0.4 and 0.49 respectively in red, green, blue, orange, black.

Next, we derive an additional independent equation involving the turning point t p . To that aim, recall that

CR (t) = ν 1 I(t), ∀t > 0 and CR(0) = 0.
As a consequence, we obtain from (1.1.1)-(1.1.2),

d dt S(t) + I(t) + ν ν 1 CR(t) = 0, ∀t > 0, so that S(t) + I(t) + ν ν 1 CR(t) = S 0 + I 0 , ∀t ≥ 0.
Hence, we obtain

CR (t) = ν 1 I(t) = ν 1 S 0 + I 0 -S(t) - ν ν 1 CR(t) , ∀t ≥ 0.
However, by using (1.1.1), we may eliminate S(t), since S(t) = S 0 e -τ ν 1 CR(t) . As a consequence, CR(t) satises the equation

CR (t) = ν 1 I(t) = S 0 ν 1 1 + I 0 S 0 -e -τ CR(t) - ν S 0 ν 1 CR(t) , ∀t ≥ 0.
Now note that since CR (t) = ν 1 I(t) > 0, the function t → CR(t) must be increasing on (0, ∞). We thus have

1 + I 0 S 0 -e -τ ν 1 y - ν S 0 ν 1 y > 0, ∀y ∈ [0, CR(∞)).
1. Identifying the number of unreported cases in SIR epidemic models Remark 1.3.3. The above inequality gives a condition on the nal number of cumulative reported cases c = CR(∞).

Integration of the dierential equation for CR(t) above, from t = 0 to t = t p , yields tp 0 CR (t)

1 + I 0 S 0 -e -τ ν 1 CR(t) -ν S 0 ν 1 CR(t) dt = (S 0 ν 1 )t p .
Set s = CR(t) and we obtain

CR(tp) 0 1 1 + I 0 S 0 -e -τ ν 1 s -ν S 0 ν 1 s ds = (S 0 ν 1 )t p .
Now recalling that CR(t p ) = rc, the change of variable s = cσ yields

r 0 1 1 + I 0 S 0 -e -cτ ν 1 σ -cν S 0 ν 1 σ dσ = (S 0 ν 1 ) t p c . By (1.3.2), that is X = cτ ν 1 , we deduce that r 0 1 1 + I 0 S 0 -e -Xσ -ν τ S 0 Xσ dσ = (S 0 ν 1 ) t p c .
By (1.3.6), that is ν = S 0 τ e -rX , we have

r 0 dσ 1 + I 0 S 0 -e -Xσ -Xe -rX σ = (S 0 ν 1 ) t p c .
By (1.3.5), that is S 0 ν 1 1 + I 0 S 0 -e -rX (1 + rX) = CR (t p ), we obtain

1 + I 0 S 0 -e -rX (1 + rX) × r 0 dσ 1 + I 0 S 0 -e -Xσ -Xe -rX σ = CR (t p ) t p c .
Finally, by (1.3.4), that is e -X + Xe -rX = 1 + I 0 S 0 and Lemma 1.3.2, we deduce that

0 < X < X(r) (1.3.9)
and that X must satisfy the turning point equation

T (X, r) = CR (t p )t p c , (1.3.10)
where the mapping T is dened by

T (X, r) := r 0 H(X, r) H(X, σ) dσ, (1.3.11)
where the right hand side is an improper integral and the function H is dened by H(X, σ) := e -X + Xe -rX -e -Xσ -Xe -rX σ.

(1.3.12) Remark 1.3.4. We observe that H(X, 0) = e -X + Xe -rX -1 = F (X). ), and X ∈ (0, X(r)), then

0 < F (X) = H(X, 0) H(X, σ) H(X, r)
for every σ ∈ [0, r].

Proof. Suppose that r ∈ (0, 1 2 ), X ∈ (0, X(r)), and consider

∂ σ H(X, σ) = X(e -σX -e -rX ) 0, ∀σ ∈ [0, r],
which means that H(X, σ) is increasing with respect to σ.

Lemma 1.3.6. The function X → T (X, r) is well-dened on the open interval (0, X(r)). Moreover,

lim X→X(r) - T (X, r) = ∞, (1.3.13)
and lim

X→0 + T (X, r) = r -1 2 ln(1 -2r). (1.3.14)
Proof. Proof of (1.3.13): By Lemma 1.3.5, T (X, r) is well-dened on the interval (0, X(r)). Since X(r) is the unique positive solution of the equation (1.3.7), we have

H(X(r), σ) = 1 -e -σX(r) -σ(1 -e -σX(r) ). Let k(x, σ) = 1 -e -σx -σ(1 -e -x ) on [0, ∞). Then, ∂ x k(x, σ) = σ(e -σx -e -x ) > 0 for every x ∈ [0, ∞), and σ ∈ (0, 1 2
). This means that k(x, σ) > k(0, σ) = 0 for every x ∈ (0, ∞), and σ ∈ (0, 1 2 ).

Therefore, H(X(r), σ) = 1 -e -σX(r) -σ(1 -e -σX(r) ) > 0 for every σ ∈ (0, 1 2 ). Moreover,

lim σ→0 + H(X(r), σ) σ = lim σ→0 +
1 -e -σX(r) -rX(r)e -rX(r) σ σ

= lim σ→0 + 1 -e -σX(r) σ -rX(r)e -rX(r) =X(r) -X(r)e -rX(r) = X(r)(1 -e -rX(r) ) > 0.
This means that

r 0 lim X→X(r) - H(X, r) H(X, σ) dσ = r 0 H(X(r), r) H(X(r), σ) dσ = ∞,
and by Fatou's Lemma, we have lim

X→X(r) - T (X, r) = ∞.
Proof of (1.3.14): Next, taking the Taylor's expansions of the functions e -X , e -rX , e -hX at X = 0, and σ ∈ [0, r] ⊂ (0, 1 2 ), we obtain

e -σX = 1 -σX + 1 2 σ 2 X 2 + o((σX) 3 ) = 1 -σX + 1 2 σ 2 X 2 + o(X 3 ), e -rX = 1 -rX + 1 2 r 2 X 2 + o(X 3 ), and e -X = 1 -X + 1 2 X 2 + o(X 3 ),
where o(X 3 ) does not depend on σ. Thus,

H(X, σ) = 1 -2r -σ 2 + 2rσ 2 X 2 + o(X 3 ), H(X, r) = (1 -r) 2 2 X 2 + o(X 3 ).
Hence,

H(X, r) H(X, σ) = (1 -r) 2 2 X 2 + o(X 3 ) 1 -2r -σ 2 + 2rσ 2 X 2 + o(X 3 ) = (1 -r) 2 + o(X) 1 -2r -σ 2 + 2rσ + o(X) .
Since o(X) does not depend on σ, when X tends to 0 + the function

H(X, r) H(X, σ) is uniformly convergent to h(σ) = (1 -r) 2 1 -2r -σ 2 + 2rσ on [0, r]. Thus, lim X→0 + T (X, r) = r 0 (1 -r) 2 1 -2r -σ 2 + 2rσ dσ = (1 -r) 2 r 0 1 (2r -1 -σ)(σ -1) dσ = (1 -r) 2 ln( 1 1 -2r
).

In Figure 1.2 we plot the mapping r → X(r), where X = X(r) is the solution of (1.3.7), as r varies in (0, 1 2 ). In Figure 1.3 we plot the mapping x → T (xX(r), r), where T is dened by (1.3.11), as x varies in (0, 1), for dierent values of r. From Figure 1.3 we observe that numerically, the mappings X → T (X, r) are all monotone increasing for each value of r. As a consequence we can conclude (numerically) that equation (1.3.10) has a unique solution X ∈ (0, X(r)).

Remark 1.3.7. From Figure 1.3 we can also visualize the minimum value for T (X, r), which is given by (1.3.14). By using (1.3.10) we deduce that we must have

CR (t p )t p CR(∞) > r -1 2 ln(1 -2r).
where r = CR(t p ) CR(∞) . Therefore, we obtain a new relationship between the values t p , CR(t p ), CR (t p ), CR(∞). Then the basic reproduction number R 0 is uniquely determined by the turning point t p , CR(t p ), CR (t p ) and CR(∞).

Note that, if the function T (X, r) is strictly decreasing on (0, X(r)) then the equation (1.3.10) has a unique solution on (0, X(r)). About the monotony of the function T (X, r) on (0, X(r)), we have the following theorem Theorem 1.3.10. Suppose that r ∈ (0, 1 2

), and X(r) is the unique positive solution of the equation e -X + Xe -rX -1 = 0, then there exists

1 2 > α > 0 such that for every r ∈ ( 1 2 -α, 1 2
) the function

T (X, r) = r 0 H(X, r) H(X, σ)
dσ is strictly increasing on (0, X(r)).

Firstly, the derivative of T (X, r) is computed as follows

T X (X, r) = r 0 H X (X, r)H(X, σ) -H X (X, σ)H(X, r) H 2 (X, σ) dσ.
It is seen that T X (X, r) is a complicated expression, then we hope we can prove that H X (X, r)H(X, σ) -H X (X, σ)H(X, r) > 0 for every X ∈ (0, X(r)), σ ∈ (0, r) for concluding the monotony of T (X, r). However, taking some computations by Maple, the result is not as our expectation. For instance, if r = 25 54 , and σ = 0.01, there is an interval so that T X (X, r) is negative. It is the reason why the study of the monotony of T (X, r) becomes dicult.

Computation the combinations of the parameters of SIR model

Figure 1.4: The graph of H X (X, r)H(X, σ) -H X (X, σ)H(X, r) where r = 25 54

, and σ = 0.01 and X varies in (0, X(r)). Now we study the monotony of T (X, r) when r is closed enough to 1 2 . Suppose that X(r) is the unique positive solution of the equation F (X) = e -X +Xe -rX -1 = 0 as Lemma 1.3.2, it is seen that X(r) is a positive function of the variable r on (0, 1 2 ). We have the following lemma Lemma 1.3.11. lim

r→ 1 2 - X(r) = 0. Proof. Let f (X) = 1 X ln( Xe X e X -1
). It is seen that

X 2 (e X -1)f (X) = (1 -e X ) ln X e X -1 + x + e X -1 -X.
We have

1 -e X = -X - X 2 2 - X 3 6 + o(X 4 ) and ln X e X -1 = - X 2 - X 2 24 + o(X 4 ). Therefore X 2 (e X -1)f (X) =(-X - X 2 2 - X 3 6 + o(X 4 ))(- X 2 - X 2 24 + o(X 4 )) + ( X 2 2 + X 3 6 + o(X 4 )) = - X 3 24 + o(X 4 )
1. Identifying the number of unreported cases in SIR epidemic models

It means that e X -1 X f (X) = - 1 24 + o(X) for X > 0. Since lim X→0 + o(X) = 0, there exists η > 0 such that |o(X)| < 1 24
for every X ∈ (0, η). It implies that X 2 (e X -1)f (X) < 0 for every X ∈ (0, η). Then f (X) < 0 for every X ∈ (0, η), so f is strictly decreasing on X ∈ (0, η).

It is clear that e -X + Xe -rX -1 = 0 ⇔ f (X) = r. So X(r) is the unique positive solution of the equation f (X) = r. Since f is strictly decreasing on X ∈ (0, η), f is continuous and lim

r→ 1 2 - X(r) = 0 then f is a bijection from (0, η) to (f (η), 1 2
). It can be rewritten X(r) = f -1 (r), and

X(r) = f -1 (r) is strictly decreasing on (f (η), 1 2
).

Moreover, X(r) > 0 for every r ∈ (f (η), 1 2 ), then there exists the limit lim

r→ 1 2 - X(r) = X 0. Since X(r)
is the unique positive solution of e -X + Xe -rX -1 = 0, we have

lim r→ 1 2 - e -X(r) + X(r)e -rX(r) -1 = 0. It implies that e -X + Xe -1 2 X -1 = 0. Let p(X) = e -X + Xe -1 2 X -1 for every X ∈ [0, η). We have p (X) = -e -X + e -1 2 X -X 2 e -1 2 X = e -1 2 X (1 -X 2 -e -1 2 X ). Dene q(X) = 1 -X 2 -e -1 2 X , then q (X) = 1
2 (e -X 2 -1) < 0 for every X ∈ [0, η). It means that q(X) is decreasing on [0, η), so q(X) < 0 for every X ∈ (0, η). It follows that p (X) < 0, therefore p is strictly decreasing on (0, η). Moreover p(X) < 0 for every X ∈ (0, η), and p(0) = 0. It means that X = 0 is the unique solution of the equation p(X) = 0, so X = 0, and lim

r→ 1 2 - X(r) = 0.
To conclude the monotony of T (X, r). We need the following lemmas Lemma 1.3.12. Taylor's expansion of

N (X) = H X (X, r)H(X, σ)-H X (X, σ)H(X, r) at X = 0 is (1 -σ)(σ -r) 2 (1 -r) 2 12 X 4 + o(X 5 ).
Proof. We have

H(X, σ) = e -X + Xe -rX -e -Xσ -Xe -rX σ
and H(X, r) = e -X + Xe -rX -e -rX -Xe -rX r.

So H X (X, σ) = -e -X + e -rX -rXe -Xr + σe -σX -σe -rX + rσXe -rX and H X (X, r) = -e -X + e -rX -rXe -Xr + r 2 Xe -rX .

Computation the combinations of the parameters of SIR model

We take the Taylor's expansion of the functions e -X , e -rX , e -σX at X = 0

e -σX = 1 -σX + 1 2 σ 2 X 2 - 1 6 σ 3 X 3 + 1 24 σ 4 X 4 - 1 120 σ 5 X 5 + o(X 6 ) so e -rX = 1 -rX + 1 2 r 2 X 2 - 1 6 r 3 X 3 + 1 24 r 4 X 4 - 1 120 r 5 X 5 + o(X 6
)

and e -X = 1 -X + 1 2 X 2 - 1 6 X 3 + 1 24 X 4 - 1 120 X 5 + o(X 6 )
.

Replacing these above Taylor's expansion to the expression H(X, σ), we have

H(X, σ) = (1 -σ)(σ -2r + 1) 2 X 2 + (1 -σ) 2 r 2 + σ 3 -1 6 X 3 + (σ -1) 6 r 3 - σ 4 -1 24 X 4 + (1 -σ) 24 r 4 + σ 5 -1 120 X 5 + o(X 6 )
and

H(X, r) = (1 -r) 2 2 X 2 + - 1 6 + 1 2 r 2 - 1 3 r 3 X 3 + 1 24 - 1 6 r 3 + 1 8 r 4 X 4 + - 1 120 + 1 24 r 4 - 1 30 r 5 X 5 + o(X 6 ). So H X (X, σ) =(1 -σ)(σ -2r + 1)X + 3(1 -σ) 2 r 2 + σ 3 -1 2 X 2 + 2(σ -1) 3 r 3 - σ 4 -1 6 X 3 + 5(1 -σ) 24 r 4 + σ 5 -1 24 X 4 + o(X 5 )
and

H X (X, r) =(1 -r) 2 X + - 1 2 + 3 2 r 2 -r 3 X 2 + 1 6 - 2 3 r 3 + 1 2 r 4 X 3 + - 1 24 + 5 24 r 4 - 1 6 r 5 X 4 + o(X 5 ).
Finally,

H X (X, r)H(X, σ) -H X (X, σ)H(X, r) = (1 -σ)(σ -r) 2 (1 -r) 2 12 X 4 + o(X 5 ). Lemma 1.3.13. M (r) = r 0 (1 -h)(h -r) 2 (1 -r) 2 dh > 0.0069 for every r ∈ [ 2 5 , 1 2
] .

1. Identifying the number of unreported cases in SIR epidemic models Proof. We have

M (r) = r 0 (r -1) 2 (1 -h)(h -r) 2 dh = (r -1) 2 r 0 (1 -h)(h -r) 2 dh = (r -1) 2 r 0 (1 -r + r -h)(h -r) 2 dh = (r -1) 2 (1 -r) r 0 (h -r) 2 dh + r 0 (h -r) 3 dh = (r -1) 2 -(1 -r) (-r) 3 3 + r 4 4 = (r -1) 2 r 3 (4 -r) 12 .
It implies that By Lemma 1.3.12 and Lemma 1.3.5, we have

I (r) = (1 -r)r 2 (r 2 -4r + 2) 12 . Note that (r 2 -4r + 2) = 2r -2 < 0 for every r ∈ [ 2 5 , 1 2 ], so r 2 -4r + 2 1 2 2 -4 1 2 + 2 > 0 for every r ∈ [ 2 5 , 1 2 ]
T X (X, r) = r 0 H X (X, r)H(X, σ) -H X (X, σ)H(X, r) H 2 (X, σ) dσ = r 0 (1 -σ)(σ -r) 2 (1 -r) 2 12 X 4 + o(X 5 ) H 2 (X, σ) dσ = X 4 12 r 0 (1 -σ)(σ -r) 2 (1 -r) 2 + o(X) H 2 (X, σ) dσ X 4 12H 2 (X, r) r 0 (1 -σ)(σ -r) 2 (1 -r) 2 + o(X) dσ 29 1.
4. An identication method and applications to some outbreak epidemics Using Lemma 1.3.13, we also have

r 0 (1 -σ)(σ -r) 2 (1 -r) 2 + o(X) dσ = r 0 (1 -σ)(σ -r) 2 (1 -r) 2 dσ + r 0 o(X)dσ >0.0069 + r 0 o(X)dσ for every r ∈ [ 2 5 , 1 2
], X ∈ (0, X(r)).

By Lemma 1.3.11, there exists

α > 0 such that if r ∈ ( 1 2 -α, 1 2
), then |o(X)| < 0.12 (note that, we can choose α

1 10 so that ( 1 2 -α, 1 2 ) ⊂ [ 2 5 , 1 2 ]). It implies that | r 0 o(X)dσ| r 0 |o(X)|dσ r 0 0.12dσ = 0.12r 0.006. It follows that r 0 (1 -σ)(σ -r) 2 (1 -r) 2 + o(X) dσ 0.0069 -0.006 > 0.
Therefore T X (X, r) > 0 for every X ∈ (0, X(r)), and T (X, r) = r 0 H(X, r) H(X, σ) dσ is strictly increasing on (0, X(r)).

Remark 1.3.14. Suppose that r ∈ (0, 1 2 ), and X(r) is the unique positive solution of the equation e -X + Xe -rX -1 = 0, then there exists

1 2 > α > 0 such that for every r ∈ ( 1 2 -α, 1 2
) the function T (X, r) = r 0 H(X, r) H(X, σ) dσ is strictly increasing on (0, X(r)). It implies the uniqueness of solution of (1.3.10), and then the function CR(t, p) is uniquely determined by the turning point t p , CR(t p ), CR (t p ) and CR(∞) in which p = (τ, ν, ν 1 , S 0 , I 0 ) is the arbitrary parameter of 1.1.1-1.1.3.

1.4 An identication method and applications to some outbreak epidemics 2) are uniquely determined by the cumulative reported cases function CR(t) for the parameter p = (τ, ν, ν 1 , S 0 , I 0 ). Moreover, the analysis in Section 1.3 allows us to derive a method to compute this combination by the following three steps:

1. Identifying the number of unreported cases in SIR epidemic models

Assume that the values of t p , CR(t p ), CR (t p ), and CR(∞) are known, and set c = CR(∞) and r = CR(t p ) CR(∞) .

Step 1: Solve the equation (1.3.7), e -X + Xe -rX -1 = 0, to obtain the unique positive solution X(r).

Step 2: Solve the turning point equation (1.3.10)

r 0 e -X + Xe -rX -e -rX -rXe -rX e -X + Xe -rX -e -σX -σXe -rX dσ = CR (t p )t p c ,
with the condition (1.3.9), 0 < X < X(r), to obtain the value X = c τ ν 1 .

Step 3: Remark 1.4.1. By the fact that ν 1 < ν, we obtain the following evaluations:

S 0 > S 0 ν 1 ν = a 3 a 6 I 0 > I 0 ν 1 ν = a 5 a 6 τ < ν τ ν 1 = a 1 a 6 .
(1.4.1)

Moreover, the basic reproduction number is

R 0 = τ S 0 ν = a 4 a 6 . (1.4.2)
Remark 1.4.2. Recall from [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF] that

CU (t) = ν 2 t 0 I(s) ds 1.
4. An identication method and applications to some outbreak epidemics is the cumulative unreported infected cases at time t > 0, and

C(t) = ν t 0 I(s) ds
is the cumulative total cases at time t > 0. If S 0 is known, the epidemic nal size of the epidemic is written as follows:

C(∞) = CR(∞) + CU (∞) = CR(∞) + ν 2 ν 1 CR(∞) = ν ν 1 CR(∞) = ν S 0 ν 1 S 0 CR(∞) = a 6 c a 3 S 0 .
(1.4.3)

Denote by N = S 0 +I 0 , the number of the individuals involved in the epidemic,which is typically smaller than the total number of the population, since some people have immunity. Then S 0 satises

a 3 a 6 < S 0 ≤ N -I 0 = N -S 0 I 0 S 0 , which implies a 3 a 6 < S 0 ≤ N 1 + I 0 S 0 = N 1 + a 2 . (1.4.4)
Moreover, the number of susceptible individuals at the end of epidemic can be computed by the following formula To illustrate the method we performed in the previous section, we compute the parameters of SIR model for Hong-Kong inuenza in New York City during 13 weeks of the winter of 1968-1969. The data of this epidemic given in the following table are weekly reported numbers of inuenza deaths (see [START_REF] Smith | The SIR model for spread of disease -background: Hong Kong u[END_REF]). In this application, we have the values CR(∞) = 1080, CR(t p ) = 500, t p = 6.15, CR (t p ) = 190 (see [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]). The total population of New York City in 1968 is 7, 900, 000. Consider the equation 1.4. An identication method and applications to some outbreak epidemics ) and Y = tpCR (tp) CR(∞) . When r = 500 1080 we obtain X(r) = 0.89478, X = 0.7869 and

S(∞) = S 0 + I 0 -C(∞) = S 0 1 + a 2 -a 6 c a 3 . ( 1
I 0 S 0 = e -X + Xe -500 1080 X -1 = 0.0019 .
Finally, by applying Step 3 of the method described above we obtain the following If the value of the initial susceptible individuals S 0 is given, then all the parameters can be obtained. The following table gives these values when S 0 = 1, 976, S 0 = 4, 000, 000, and S 0 = 7, 885, 047 [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]). In Figure 1.6 we compare the model output to the reported case data (see [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]). In Figure 1.7 we illustrate the epidemic nal size as a function of the initial number of susceptibles S 0 . In Figure 1.8 we illustrate the epidemic nal size as a function of the turning point of the epidemic. To run this simulation we x S 0 = 4, 000, 000.

Moreover, the basic reproduction number is

R 0 = τ S 0 ν = a 4 a 6 = 1.4. (1.4.8)
From Remark 1.4.2 whenever S 0 is known the nal size of the epidemic C(∞) is 1.4. An identication method and applications to some outbreak epidemics expressed linearly in function of S 0

C(∞) = a 6 CR(∞) a 3 S 0 , (1.4.9)
and we have the following upper and lower bounded for S 0

a 3 a 6 < S 0 ≤ N 1 + a 2 .
(1.4.10)

Figure 1.7: The relationship between the total case number at the end of epidemic C(∞) and S 0 . Here S 0 varies from 1, 976 (which is strictly larger than the minimal value a 3 a 6 = 1, 975) up to the maximal value S 0 = N 1+a 2 = 7, 885, 047 which corresponds to I 0 = 14, 953.

Application to the plague epidemic in Bombay, India in 1906

The weekly reported case data of the plague epidemic in Bombay, India in 1906 is obtained from the website [75]. The data is taken from the rst 30 weeks of the year 1906 when the plague epidemic broke out in Bombay. We have the table of data as follows 1.4. An identication method and applications to some outbreak epidemics In this application, we get from the data the values (see [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]) CR ∞ = 8840, CR(tp) = 4330, tp = 13.5, CR (tp) = 770. The total population of Bombay in 1906 is approximately 1, 000, 000.

Consider the equation where X ∈ (0, X(r)) with X(r) is the unique positive solution of the equation e -X + Xe -4330 8840 X -1 = 0.

(1.4.12)

1. Identifying the number of unreported cases in SIR epidemic models Solving the equation (1.4.12), we obtain the value of X(r) = 0.2444655268. This value corresponds to the positive zero of the function in the following gure 1.9 -left side.

Next, applying

Step 3 of the method described in Section 1. [START_REF] Audoly | Global identiability of nonlinear models of biological systems[END_REF] The value of the initial susceptible individuals S 0 = 100, 000 is taken as the paper [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], and then all the parameters can be obtained in the following table. 1.6 (see [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]). In Figure 1.10 we compare the model output to the reported case data (see [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF]). In Figure 1.11 we illustrate the epidemic nal size as a function of the initial number of susceptibles S 0 . In Figure 1.12 we illustrate the epidemic nal size as a function of the turning point of the epidemic. To run this simulation we x S 0 = 100, 000.

An identication method and applications to some outbreak epidemics

1. Identifying the number of unreported cases in SIR epidemic models Figure 1.11: The relationship between the total case number at the end of epidemic C(∞) and S 0 . Here S 0 varies from 1, 976 (which is strictly larger than the minimal value a 3 a 6 = 1, 975) up to the maximal value S 0 = N 1+a 2 = 7, 885, 047 which corresponds to I 0 = 14, 953. In this part, we study the inuenza epidemics in France in the consecutive six years from week 37 of the year 1988 to week 15 of the year 1994 with the data from 1.4. An identication method and applications to some outbreak epidemics According to the gure 1.13, it is seen that in the period from 1988 to 1994 there are three times the reported infected cases over 1500 individuals over 100, 000 inhabitants. It means that the three highest peaks of these inuenza epidemics in France took place from 1988 to 1994. That is why we study the inuenza epidemics in France in this period. What we do next is choosing the starting point and the endpoint of each inuenza epidemic in France for every year according to the data from [73], and applying the method in section 1.4.1 to investigate the epidemics. Note that the incidence rate from the data is the reported case per 100, 000 inhabitants for each epidemic.

I. Period from week 43 of the year 1988 to week 8 of the year 1989

From the data in [73](see Figure 1.15), we obtain the values CR ∞ = 9000, CR(tp) = 4400, tp = 7.8, CR (tp) = 1810. Next, applying the method described in section 1. [START_REF] Audoly | Global identiability of nonlinear models of biological systems[END_REF] Next, applying the method described in section 1. Next, applying the method described in section 1. The relationship between the total case number at the end of epidemic C(∞) and S 0 for the inuenza-like illness in France in 1991w2-1991w15 1.4. An identication method and applications to some outbreak epidemics Next, applying the method described in section 1. 4. An identication method and applications to some outbreak epidemics Figure 1.33: The relationship between the total case number at the end of epidemic C(∞) and S 0 for the inuenza-like illness in France in 1991w45-1992w10 Next, applying the method described in section 1. The weekly reported cumulative case data(blue dots) and model output graph of CR(t) for the inuenza-like illness in France in 1992w51-1993w17

Identifying the number of unreported cases in SIR epidemic models

1. Identifying the number of unreported cases in SIR epidemic models Figure 1.38: The relationship between the total case number at the end of epidemic C(∞) and S 0 for the inuenza-like illness in France in 1992w51-1993w17 1.4. An identication method and applications to some outbreak epidemics Next, applying the method described in section 1.

Some preliminaries of dierential algebra and application to system identiability

In this chapter we present some necessary preliminaries of dierential algebra and dierential elimination for our work. What we present here is based on the inputoutput approach from the papers [START_REF] Ljung | On global identiability for arbitrary model parametrizations[END_REF], [START_REF] Denis-Vidal | Some eective approaches to check the identiability of non-linear systems[END_REF], [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF], [START_REF] Bellu | DAISY: A new software tool to test global identiability of biological and physiological systems[END_REF] [9], [START_REF] Boulier | Computing representations for radicals of a nitely generated dierential ideals[END_REF], [START_REF] Boulier | Dierential Elimination and Biological Modelling[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Ollivier | Le problème de l'identiabilité structurelle globale: étude théoritique, méthodes eecitves et bornes de complexité[END_REF], [START_REF] Maria | Parameter identiability of nonlinear systems: the role of initial conditions[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF], [START_REF] Ovchinnikov | Dierential Algebra[END_REF].

Some basic denitions of dierential algebra

The denitions below are based on the papers [START_REF] Boulier | Representation for the radical of a nitely generated dierential ideal[END_REF], [START_REF] Boulier | Computing representations for radicals of a nitely generated dierential ideals[END_REF], [START_REF] Boulier | Dierential Elimination and Biological Modelling[END_REF], the books [START_REF] Ritt | Dierential Algebra[END_REF], [START_REF] Kolchin | Dierential Algebra and Algebra Groups[END_REF], the lecture [START_REF] Ovchinnikov | Dierential Algebra[END_REF], and Maple Help [74]. Denition 2.1.1. Let R be a commutative ring with identity. R is called a differential ring if there exists a nite set of maps {∂ 1 , ..., ∂ n } in which

∂ i : R → R, i ∈ {1, 2, ..n}, satised i. ∂ i (a + b) = ∂ i (a) + ∂ i (b) for all a, b ∈ R, ii. ∂ i (ab) = b∂ i (a) + a∂ i (b) for all a, b ∈ R, iii. ∂ i ∂ j (a) = ∂ j ∂ i (a) for all a ∈ R, 1 ≤ i, j ≤ n.
The map ∂ i here is called a derivation of the ring R. If a set of all derivations of a dierential ring R has only one element, we call R an ordinary dierential ring.

From now on, for simplicity the term ring R is always understood as a commutative ring with identity. Denition 2.1.2. If a dierential ring R is a eld, it is called a dierential eld.

Let K be a eld of characteristic zero, we denote by K[X] the ring of all polynomials that X is the set of all variables and the coecients are in K.

Some basic denitions of dierential algebra

Denition 2.1.3. Suppose that {∂ 1 , ..., ∂ n } is the set of all the derivations of the dierential eld R, and X := {x 1 , x 2 , ..., x k }, K is a eld eld of characteristic zero.

Let D = {∂ i 1 1 ∂ i 2 2 ...∂ in n : i 1 , ..., i n ∈ N}, then K[D(X)
] is called the ring of dierential polynomials with coecients in K where D(X) = {θx i |θ ∈ D, x i ∈ X, i = 1, 2..., k}. We denote K[D(X)] by K{X}. Every element of K{X} is called a dierential polynomial and x 1 , x 2 , ..., x k are called the indeterminates of K{X}.

Example 2.1.4. Consider C ∞ [0, ∞) with the derivation ∂(x) := x for every x ∈ C ∞ [0, ∞). Suppose that x 1 , x 2 , ..., x n are the elements of C ∞ [0, ∞) and K is a subeld of R. The ring K[x 1 , x 2 , ..., x n , x 1 , ..., x n , ..., x " 1 , .
..] is a ring of dierential polynomial in indeterminates x 1 , ..., x n with coecients in K and denoted by

K{x 1 , x 2 , ..., x n }. Every element of K{x 1 , x 2 , ..., x n } is a dierential polynomial. For instance, a 3 x 1 +a 1 x 1 x 2 2 x 2 3 -a 2 x 3 1 (x 2 ) 2 x 3 is a dierential polynomial of K{x 1 , x 2 , ..., x n } with a 1 , a 2 , a 3 ∈ K. Denition 2.1.5. Given a dierential ring R, an ideal I of R is called the dierential ideal of R if ∂(a) ∈ I for all the derivations ∂ of R and a ∈ I.
Let R be a commutative ring and S ⊂ R, the smallest ideal containing S denoted by (S) is called an ideal generated by S. The smallest dierential ideal containing S denoted by [S] is called a dierential ideal generated by S Remark 2.1.6. Suppose that {∂ 1 , ..., ∂ n } is the set of all the derivations of the ring

R. Let D = {∂ i 1 1 ∂ i 2 2 ...∂ in n : i 1 , ..., i n ∈ N}, then [S]
is the ideal generated by θ(S) with θ ∈ D. Denition 2.1.7. Let R be a commutative ring and I is an ideal of R. The radical of I denoted by √ I is the set {a ∈ R : a n ∈ I for some n ∈ N}. An ideal I of R is said to be radical if it coincides with its radical. An ideal I of a dierential ring R is called a radical dierential ideal if I is a dierential ideal and is also radical.

Let R be a dierential ring, and S ⊂ R, the smallest radical dierential ideal containing S denoted by {S} is called a radical dierential ideal generated by S. Denition 2.1.8. Let R be a dierential ring, the dierential ideal P of R is called a prime dierential ideal if ab ∈ P then a ∈ P or b ∈ P A relation ≤ over a set A is called a total order over A if the following conditions hold i. a ≤ a for every a ∈ A. Suppose that ≤ is a total order over A. A strict total order < over A is a relation over A such that for every a, b ∈ A, a < b if a ≤ b and a = b.

Introduction to Rosenfeld-Gröbner algorithm implemented in Maple

Let A be the set of three dierential polynomials

q 1 = p 4 x 1 + p 1 x 1 x 2 x 3 , q 2 = x 2 -p 1 x 1 x 2 x 3 + p 3 x 2 , q 3 = p 2 4 x 3 + 2p 1 p 2 x 1 x 2 2 x 3 3 -2p 2 p 3 x 2 2 x 2 3 of Q(p){x 1 , x 2 , x 3 }.
The leaders of q 1 , q 2 , q 3 are respectively x 1 , x 2 , x 3 . q 1 does not contain any derivative of x 2 , x 3 so that q 1 is partial reduced w.r.t q 2 , q 3 , and then q 1 is reduced w.r.t q 2 , q 3 . The initials and separants of q 1 , q 2 , q 3 are (respectively p 4 , 1, p 4 ) in Q(p), then A is an autoreduced and orthonomic set.

Introduction to Rosenfeld-Gröbner algorithm implemented in Maple

What we present in this section is based on the papers [START_REF] Boulier | Representation for the radical of a nitely generated dierential ideal[END_REF], [START_REF] Boulier | Computing representations for radicals of a nitely generated dierential ideals[END_REF], [START_REF] Boulier | Dierential Elimination and Biological Modelling[END_REF], and [74] Rosenfeld-Gröbner algorithm works with a list of dierential polynomials and a ranking on a dierential ring of dierential polynomials that is understood as input. Suppose that we have a system of dierential equations in which every left-hand side of each equation can be considered as a dierential polynomial when the right-hand side is zero. Rosenfeld-Gröbner algorithm splits the system into cases and in each case, there is no one equation that is the consequence of the others. To understand precisely, we present here the Ritt's reduction with pseudodivision of the dierential polynomials that is a generalization of the Euclidean division.

Let f and g be two polynomials in one variable x with coecients in a ring.

Denote by deg(f, x) the degree of f in x. The pseudodivision of f by g is given by the formula c d f = gq + r in which c is the coecient of the leader x of g, d = deg(f, x) -deg(g, x) + 1, and deg(r, x) < deg(g, x). We call the polynomial r pseudoremainder of f by g, and denote r := prem(f, g, x).

Let A = {f 1 , f 2 , ..., f n } be the set of dierential polynomials of K{X}\K in which K is a eld and X is a set of indeterminates , and g be a dierential polynomial of K{X}\K. Suppose that x i is the leader of f i for i ∈ {1, 2, ..n}. Now we perform a procedure reducing g w.r.t every element of A that we it denote by Rittreduction(g, A). This procedure generate a sequence g 0 , g 1 , ..., g s as follows -First step, we set g 0 = g.

-Next steps, we compute g k+1 from g k in the following cases Case 1. If g k does not depend on any proper derivative of x i and the deg(g k , x i ) < deg(f i , x i ) for every i ∈ {1, 2..., n} then we stop the procedure and assign

g s := g k . Case 2. If deg(g k , x i ) ≥ deg(f i , x i ) for some index i ∈ {1, 2.
.., n} then we assign

g k+1 := prem(g k , f i , x i ).
Case 3. If there is some index i ∈ 1, 2..., n such that g k depends on some proper derivative x (m) i of x i then we assign g k+1 := prem(g k , f

-Last step, when the procedure nishes we obtain the dierential polynomial g s and we dene Rittreduction(g, A) to be g s .

The Rosenfeld-Gröbner algorithm -The input of the algorithm is a list P of dierential polynomials and a ranking.

Dierential algebra approach for system identiability

Equations and Inequations. One and only one of these cases is the general case, and the rest are singular cases that may exist only when sys is nonlinear. In each of the cases returned, each equation is not the consequence of the other equations and all the integrability conditions are taken into account. The relation between the system sys and the systems of each of these cases is the solutions, general and singular, of sys, are given by the union of the general solutions of each of the returned systems (see [74]).

Finally, after calling the function I=RosenfeldGroebner(sys, R), we can test if a dierential polynomial p of the dierential ring R belongs to the radical ideals generated by the lists of I or not by calling the function BelongsTo(p, I). This function returns true if p belongs to the dierential ideals represented by I, else it returns false ( see [74]).

Dierential algebra approach for system identiability

Consider the system as follows

     x (t, p) = f (x(t, p), p), x(0, p) = x 0 (p), t ≥ 0 y(t, p) = h(x(t, p), p). (2.3.1)
where p ∈ Ω, with Ω is open subset of R s , is a parameter vector, x(t, p) ∈ R n is a state variable and y(t, p) ∈ R r is the output function.

Assuming that, for each p ∈ Ω, (2.3.1) has a unique solution, and there exists an open connected subset M (p) of R n such that x(t, p) ∈ M (p) for all t ≥ 0 and f (•, p), h(•, p) are analytic on M (p).

The problem that we want to solve is the following Problem 2.3.1. Consider the system (2.3.1), then what is the relationship between p, p ∈ Ω from the condition y(t, p) = y(t, p)?

The following proposition gives us an easy way to predict the relation between p, p ∈ Ω from the condition y(t, p) = y(t, p), and we can have similar results from the papers of Evans et al. [START_REF] Evans | Identiability of uncontrolled nonlinear rational systems[END_REF], [START_REF] Evans | The structural identiability of the susceptible infected recovered model with seasonal forcing[END_REF]. Proposition 2.3.2. Given a system (2.3.1), and parameters p, p ∈ Ω. Suppose that there exists = 0 such that

x ∈ M (p), (2.3.2)
x 0 (p) = x 0 (p), (2.3.3)
f ( x, p) = f (x, p), (2.3.4)
h( x, p) = h(x, p), (2.3.5)
for all x ∈ M (p), then y(t, p) = y(t, p) for all t ≥ 0. 66 2. Some preliminaries of dierential algebra and application to system identiability Proof. Dene that z(t) = x(t, p) for all t ≥ 0. Since x(t, p) ∈ M (p) then z(t) =

x(t, p) ∈ M (p) for all t ≥ 0.

z (t) = x (t, p) = f (x(t, p), p) (since (2.3.1)) = f ( x(t, p), p) (since (2.3.4)) = f (z(t), p). Moreover, z(0) = x 0 (p) = x 0 (p) (since (2.3.3)).
It means that z(t) is a solution of (2.3.1) with respect to p. By the uniqueness of solution of (2.3.1), z(t) = x(t, p) for all t ≥ 0. Therefore,

y(t, p) = h(x(t, p), p) = h(z(t), p) = h( x(t, p), p) = h(x(t, p), p) (since (2.3.5))
= y(t, p).

Consider the system (2.3.1) in which f i is a rational function for all i = 1, ..., n and h j is a polynomial function for all j = 1, ..., r . Now we consider Problem 2.3.1 with r = 1.

If there is some i ∈ {1, ..., n} such that f i (x, p) = q i (x,p) q(x,p) where q i (x, p), q(x, p) are polynomials, by introducing new variable x n+1 (t, p) = 1 q(x,p) , we have

x n+1 (t, p) = -1 q 2 (x,p) n i=1 ∂q ∂x i x i (t). It derives that x n+1 (t, p) = -x 3 n+1 (x, p) n i=1 ∂q ∂x i q i (x(t, p), p)
in which the right hand side is a polynomial function in x 1 , ..., x n , x n+1 (see [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF]) Hence we can assume that from (2.3.1), by introducing a new state variable z = (x 1 , ..., x n , x n+1 ) we can obtain a new system

     z (t, p) = F (z(t, p), p), z(0, p) = z 0 (p), t ≥ 0 y(t, p) = H(z(t, p), p). (2.3.6)
where F i , H are polynomial functions for all i = 1, ..., n, n + 1.

Next, we present an approach by using dierential algebra and dierential elimination for nding identiable parameters for the system (2.3.1). This part is based on the input-output approach from the papers [START_REF] Ljung | On global identiability for arbitrary model parametrizations[END_REF], [START_REF] Denis-Vidal | Some eective approaches to check the identiability of non-linear systems[END_REF], [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Margaria | Differential algebra methods for the study of the structural identiability of rational function state-space models in the biosciences[END_REF], [START_REF] Bellu | DAISY: A new software tool to test global identiability of biological and physiological systems[END_REF] [9], [START_REF] Boulier | Computing representations for radicals of a nitely generated dierential ideals[END_REF], [START_REF] Boulier | Dierential Elimination and Biological Modelling[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Ollivier | Le problème de l'identiabilité structurelle globale: étude théoritique, méthodes eecitves et bornes de complexité[END_REF], [START_REF] Maria | Parameter identiability of nonlinear systems: the role of initial conditions[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF].

Dierential algebra approach for system identiability

Step 1. The system (2.3.1) is rewritten as follows z (t, p) -F (z(t, p), p) = 0, t ≥ 0 y(t, p) -H(z(t, p), p) = 0.

(2.3.7)

Here we ignore the initial condition for simplicity. However, the initial condition will be used later for another analysis of the method.

Consider the dierential ring Q(p){z 1 , ..., z n+1 , y} where Q(p) is the eld of the rational expressions of p 1 , ..., p s . From (2.3.7), we make some arrangements on the polynomials of the left hand side of (2.3.7) to obtain new polynomials such that they form an autoreduced and orthonomic set, and we call P the prime dierential ideal generated by this set. We use the ranking in Q(p){z 1 , ..., z n+1 , y} to eliminate the state variable z that is described as follows

y < y < ... < y (n) < ... < z 1 < ... < z (n) 1 < ... < z 2 < ... or simply [y] < [z 1 ] < ... < [z n+1 ].
Step 2. Using the Rosenfeld-Gröbner algorithm implemented in Maple (see [START_REF] Boulier | Representation for the radical of a nitely generated dierential ideal[END_REF], [START_REF] Boulier | Computing representations for radicals of a nitely generated dierential ideals[END_REF]) to decompose the prime ideal P. Rosenfeld-Gröbner algorithm with the elimination ranking above decomposes P as an intersection of radical dierential ideals. Concretely, Rosenfeld-Gröbner algorithm splits (2.3.7) into cases, each one specied by their equations and inequations. In each of the cases returned, no one equation is the consequence of the others (see [74]). Among these cases, only one is the general solution and others are singular solutions of (2.3.7). Consider the initial condition given in (2.3.6) for analyzing the singular cases. If these cases cannot describe the complete system, the system (2.3.7) with the initial condition, then the general case is corresponding to (2.3.7).

Assume that (2.3.7) is corresponding to the general case. It means that in the decomposition of P, all the singular cases are redundant. Note that the Rosenfeld-Gröbner algorithm is implemented in Maple, and the general case after using Rosenfeld-Gröbner algorithm is just a list of the generators of a dierential ideal. This set of generators is called a characteristic presentation of P G (see [START_REF] Boulier | Computing representations for radicals of a nitely generated dierential ideals[END_REF]). One of the generators of this characteristic presentation is a dierential polynomial concerning only the indeterminate y with the ranking we used above, and it can be written by the form φ γ+1 (y) + γ i=1 c i (p)φ i (y) where φ i (y) is the dierential polynomial w.r.t the indeterminate y for all i = 1, ..., γ, γ + 1, and c i (p) is the rational function of p 1 , ..., p s for all i = 1, ..., γ. We call this polynomial an output polynomial of (2.3.7). We have

φ γ+1 (y(t, p)) + γ i=1 c i (p)φ i (y(t, p)) = 0. (2.3.8)
It is called an output equation of (2.3.7) ( see [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Meshkat | An algorithm for nding globally identiable parameter combinations of nonlinear ODE models using Gröbner Bases[END_REF], [START_REF] Maria | Parameter identiability of nonlinear systems: the role of initial conditions[END_REF] [69]).

2. Some preliminaries of dierential algebra and application to system identiability (2.3.9)

We have the following proposition that mentioned in [START_REF] Denis-Vidal | An algorithm to test identiability of uncontrolled nonlinear systems[END_REF], [START_REF] Daniel | The input-output relationship approach to structural identiability analysis[END_REF], [START_REF] Verdière | A strategic algorithmic tool for doing an a priori identiability study of dynamical nonlinear models[END_REF] Proposition 2.3.3. Assume that p, p are the parameters of the system (2.3.7). If the Wronskian (2.3.9) does not belong to the ideal P, then from the condition y(t, p) = y(t, p) we have c i (p) = c i (p) for all i = 1, ..., γ.

Proof. The Wronskian does not belong to the ideal P implies that this determinant is not identically zero. It means that if y(t) is the solution of (2.3.7) such that the Wronskian at this solution is not identically zero. It derives the linear independence of the functions φ 1 (y(t, p)), ..., φ γ (y(t, p)).

If we have y(t, p) = y(t, p) for all t ≥ 0, then y (i) (t, p) = y (i) (t, p) for all i ∈ N and for all t ≥ 0. It implies that φ j (y(t, p)) = φ j (y(t, p)) since φ j is a dierential polynomial w.r.t the indeterminate y for all j = 1, ..., γ, γ + 1.

From

(c i (p) -c i (p))φ i (y(t, p)) = 0.
By the linear independence of φ 1 (y(t, p)), ..., φ γ (y(t, p)), we obtain c i (p) = c i (p) for all i = 1, ..., γ. Now the set {c 1 (p), c 2 (p), ..., c γ (p)} is called an exhaustive summary of the output equation (2.3.8). (Ollivier [53]).

Step 3. Solving the equations c i (p) = c i (p) for i = 1, ..., γ, we obtain the relationships between p and p. Note that the initial values are concerning the parameters so that we take the initial conditions and other generators of the general case to obtain the remaining relations between p and p. with the initials S(0) = S 0 > 0, I(0) = I 0 > 0, and the basic production number

R 0 = τ S 0 ν > 1.
Denote by p = (τ, ν, ν 1 , S 0 , I 0 ), p = (τ , ν, ν 1 , S 0 , I 0 ) the unknown parameter vectors, and the set of possible p is given by Ω = {p ∈ R 5 :

p i > 0, i = 1..5}. Let x(t, p) = x 1 (t, p) x 2 (t, p) = S(t) I(t) then x (t, p) = -τ S(t)I(t) τ S(t)I(t) -νI(t) = -τ x 1 (t)x 2 (t) τ x 1 (t)x 2 (t) -νx 2 (t) = f (x(t, p), p), x 0 (p) = x 1 (0) x 2 (0) = S 0 I 0 = p 4 p 5
, and y(t, p) = ν 1 I(t) = p 3 x 2 (t) = h(x(t, p), p).

As we know from Chapter 1 and [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], S(t), I(t) > 0 for all t ≥ 0, so we choose If the condition y(t, p) = y(t, p) for all t 0 implies p = p then (1.

M (p) = R 2 + for every p ∈ Ω.
ν = ν, τ ν 1 = τ ν1 , τ S 0 = τ S 0 , τ I 0 = τ I 0 , (2.4.1)
then y(t, p) = y(t, p) for all t ≥ 0.

Proof. For every

x ∈ M (p) = R 2 + , let f (x) = -τ x 1 x 2 τ x 1 x 2 -

Some preliminaries of dierential algebra and application to system identiability

To show the necessary condition of the above proposition we will apply the tools of dierential algebra approach presented in the previous section.

The system (1.1.1) now can be rewritten as follows

     x 1 (t) = -p 1 x 1 (t)x 2 (t), x 2 (t) = p 1 x 1 (t)x 2 (t) -p 2 x 2 (t), t ≥ 0 y(t) = p 3 x 2 (t) (2.4.2) with x 1 (0) = p 4 , x 2 (0) = p 5 .
Substituting x 2 (t) = y(t) p 3 into (2.4.1) we obtain

p 3 x 1 (t) + p 1 x 1 (t)y(t) = 0, y (t) -p 1 x 1 (t)y(t) + p 2 y(t) = 0, t ≥ 0, (2.4.3)
with x 1 (0) = p 4 , x 2 (0) = p 5 , y(0) = p 3 p 5 .

Consider the dierential polynomials on the left hand sides of (2.4.3) f 1 = p 3 x 1 + p 1 x 1 y and f 2 = y -p 1 x 1 y + p 2 y of dierential ring Q(p){x 1 , y} with the elimination ranking y < y < ... < y (n) < ... < x 1 < x 1 < ...x

(n) 1 < ... The leaders of f 1 , f 2 are respectively x 1 , y . Since f 1 , f 2 do not contain any derivatives of the leaders of each other then f 1 is reduced w.r.t f 2 . It means the set A = {f 1 , f 2 } is an autoreduced set in the dierential ring Q(p){x 1 , y} . Moreover, the initials and separants of these polynomials are respectively p 3 , 1 in Q(p), then A is an orthonomic set of Q(p){x 1 , y}. Let P is a dierential ideal generated these polynomials.

Next, we decompose the ideal P by using Rosenfeld-Gröbner algorithm implemented in Maple. This decomposition returns two lists of dierential polynomials. Each list forms the set of generators of a radical ideal corresponding to either the singular solutions or general solution of (2.4.3). We show here the lists [x 1 (t), y(t)],

[p 1 x 1 (t)y(t) -y (t) -p 2 y(t), p 3 y(t)y"(t) -p 3 (y (t)) 2 + p 1 y 2 (t)y (t) + p 1 p 2 y 3 (t)].

The meaning of the lists is that (2.4.3) is equivalent to

x 1 (t) = 0, y(t) = 0, t ≥ 0, or p 1 x 1 (t)y(t) -y (t) -p 2 y(t) = 0, p 3 y(t)y"(t) -p 3 (y (t)) 2 + p 1 y 2 (t)y (t) + p 1 p 2 y 3 (t) = 0, t ≥ 0,
It can be seen that the rst of the lists above are not corresponding to the complete system including (2.4.3) and the initial condition since the rst list requires y(t) = 0 for all t ≥ 0, contradicting to y(0) = p 3 p 5 > 0.

In the second list (the general case), there is an expression concerning only the indeterminate y, and it is called the output polynomial of (2.4.3) Now we check the linear independence of two polynomials q 1 = y 2 (t)y (t),

q 2 = y 3 (t).
Next, we compute the Wronskian W of q 1 , q 2 and the result is that W = -y 5 (t)y"(t) -y 4 (t)(y (t)) Chapter 3

Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data

In parameter estimation of a given system model from real experimental data, the identiability analysis is the most important prerequisite. There are so many papers mentioning the criterion for testing the identiability, however, when a given system is not identiable there is not a general way for showing the combinations of parameters that are identiable. Thank to dierential algebra and dierential elimination, the output equation of a system model can be given easier for approaching the identiable combinations of parameters. This chapter uses the tools of dierential algebra for studying the identiability of the SIR model with a nonlinear incidence rate. The approach here can be easily applied for some other biological models because some of the tools of dierential algebra were implemented in Maple.

The sucient condition of identiability of SIR model with nonlinear incidence rate

Consider the SIR model with nonlinear incidence rate as follows

           S (t) = - τ S(t)I α (t) 1 + κI β (t) , I (t) = τ S(t)I α (t) 1 + κI β (t) -νI(t), t ≥ 0 y(t) = µI(t) (3.1.1)
with S(0) = S 0 > 0, I(0) = I 0 > 0, and β + 1 ≥ α ≥ 1.

Denote by p = (τ, κ, ν, µ, S 0 , I 0 ), p = (τ , κ, ν, µ, S 0 , I 0 ) the unknown parameter vectors, and the set of possible p is given by Ω = {p ∈ R 6 : p i > 0, i = 1..6}.

The sucient condition of identiability of SIR model with nonlinear incidence rate

Let x(t, p) = x 1 (t, p) x 2 (t, p) = S(t) I(t) , f (x(t, p), p) =     - -τ S(t)I α (t) 1 + κI β (t) τ S(t)I α (t) 1 + κI β (t) -νI(t)     , x 0 (p) = x 1 (0)
x 2 (0) = S 0 I 0 , and y(t, p) = h(x(t, p), p) = µI(t).

By the Theorem 1 of the paper [START_REF] Magal | Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico[END_REF], we have S(t), I(t) > 0 for all t ≥ 0.

For every p ∈ Ω, we choose

M (p) = R 2 + , then x(t, p) = x 1 (t, p) x 2 (t, p) = S (
(p) = x 0 (p) ⇔ S 0 I 0 = S 0 I 0 ⇔ S 0 = S 0 (3.1.a) I 0 = I 0 (3.1.b) Next, (2.3.4) is read as follows f ( x, p) = f (x, p) for all x ∈ M (p) = R 2 + ⇔      - τ α+1 x 1 x α 2 1 + β κx β 2 τ α+1 x 1 x α 2 1 + β κx β 2 -νx 2      =     -τ x 1 x α 2 1 + κx β 2 τ x 1 x α 2 1 + κx β 2 -νx 2     ⇔          -τ α+1 x 1 x α 2 1 + β κx β 2 = -τ x 1 x α 2 1 + κx β 2 τ α+1 x 1 x α 2 1 + β κx β 2 -νx 2 = τ x 1 x α 2 1 + κx β 2 -νx 2 ⇔ τ α + τ α κx β 2 = τ + τ β κx β 2 (3.1.c) νx 2 = νx 2 (3.1.d) Since (3.1.c)-(3.1.d) hold for all x ∈ M (p) = R 2 + , then (3.1.c)-(3.1.d) is equivalent to      τ α = τ (3.1.e) τ α κ = τ β κ (3.1.f ) ν = ν (3.1.g)
= ν, τ τ β = κ κ α , τ τ = µ µ α , µ µ = S 0 S 0 = I 0 I 0 = .
Hence, for each p ∈ Ω, there are innite parameters p ∈ Ω such that y(t, p) = y(t, p) for all t ≥ 0.

As a consequence of Proposition 2.3.

ν = ν, τ τ β = κ κ α , τ τ = µ µ α , µ µ = S 0 S 0 = I 0 I 0 , we have µI(t, p) = µI(t, p).
3.2 SIR model with nonlinear incidence rate (3.1.1) in the case α = 1, β = 2

Consider the system

           S (t) = - τ S(t)I(t) 1 + κI 2 (t) , I (t) = τ S(t)I(t) 1 + κI 2 (t) -νI(t), t ≥ 0 y(t) = µI(t) (3.2.1)
with S(0) = S 0 > 0, I(0) = I 0 > 0, and

R 0 = τ S 0 ν(1 + κI 2 0 ) > 1.
Denote by p = (τ, κ, ν, µ, S 0 , I 0 ), p = (τ , κ, ν, µ, S 0 , I 0 , µ) the unknown parameter vectors, and the set of possible p is given by Ω

= {p ∈ R 6 : p i > 0, i = 1..6}. Let x(t, p) = x 1 (t, p) x 2 (t, p) = S(t) I(t) , f (x(t, p), p) =     - -τ S(t)I ( t) 1 + κI 2 (t) -τ S(t)I ( t) 1 + κI 2 (t) -νI(t)     , x 0 (p) = x 1 (0) x 2 (0) = S 0 I 0 = p 4 p 5
, and y(t, p) = h(x(t, p), p) = µI(t) = p 6 x 2 (t).

the system (3.2.1) can be rewritten as follows

           x 1 (t) = - p 1 x 1 (t)x 2 (t) 1 + p 2 x 2 2 (t) , x 2 (t) = p 1 x 1 (t)x 2 (t) 1 + p 2 x 2 2 (t) -p 3 x 2 (t), t ≥ 0 y(t) = p 4 x 2 (t) (3.2.2) with x 1 (0) = p 5 , x 2 (0) = p 6 .
We introduce here a new state variable

x 3 (t) = 1 1 + p 2 x 2 2 (t) , then x 3 (t) = - 2p 2 x 2 (t)x 2 (t) (1 + p 2 x 2 2 (t)) 2 = -2p 2 x 2 (t)x 2
3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data It can be seen that the rst three lists above are not corresponding to the complete system including (3.2.3) and the initial condition. For instance, the third list requires x 1 (t) = 0 for all t ≥ 0, contradicting to x 1 (0) = p 5 > 0.

In the fourth list, there is an expression that concerning only the indeterminate y, and it is written as follows with S(0) = S 0 > 0, I(0) = I 0 > 0, and

R 0 = τ S 0 ν(1 + κI 0 ) > 1.
With the same notations as in Section 3.2, (3.3.1) can be read as follows

           x 1 (t) = - p 1 x 1 (t)x 2 (t) 1 + p 2 x 2 (t) , x 2 (t) = p 1 x 1 (t)x 2 (t) 1 + p 2 x 2 (t) -p 3 x 2 (t), t ≥ 0 y(t) = p 4 x 2 (t) (3.3.2) with x 1 (0) = p 5 , x 2 (0) = p 6 . Let x 3 (t) = 1 1 + p 2 x 2 (t)
, then

x 3 (t) = - p 2 x 2 (t) (1 + p 2 x 2 (t)) 2 = -p 2 x 2 3 (t) (p 1 x 1 (t)x 2 (t)x 3 (t) -p 3 x 2 (t)) .

Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data

We write (3.3.2) as follows

     p 4 x 1 (t) + p 1 x 1 (t)y(t)x 3 (t) = 0, y (t) -p 1 x 1 (t)y(t)x 3 (t) + p 3 y(t) = 0, t ≥ 0 p 4 x 3 (t) + p 1 p 2 x 1 (t)y(t)x 3 3 (t) -p 2 p 3 y(t)x 2 3 (t) = 0 (3.3.3) with x 1 (0) = p 5 , y(0) = p 4 p 6 , x 3 (0) = 1 1 + p 2 p 6 .
The dierential polynomials on the left hand sides of (3.3.3) form an autoreduced set in the dierential ring Q(p){x 1 , x 3 , y} with the elimination ranking (that we mentioned in Chapter 2) [y] < [x 1 ] < [x 3 ]. Moreover, the initials and separants of these polynomials are in Q(p), then these polynomials form an orthonomic set of Q(p){x 1 , x 3 , y}. Let P is a prime dierential ideal generated by these polynomials.

Next, using Rosenfeld-Gröbner algorithm implemented in Maple to decompose the ideal P. This decomposition returns ve lists of dierential polynomials. Each list forms the set of generators of a radical ideal corresponding to either the singular solutions or general solution of (3.3.3). We show here the rst four lists

[x 3 (t), x 1 (t), y (t) + p 3 y(t)], [x 3 (t), x 1 (t), y(t)], [p 4 x 3 (t) -p 2 p 3 y(t)x 2 3 (t), x 1 (t), y (t) + p 3 y(t)], [p 1 p 2 x 1 (t)x 3 (t) + p 1 -p 2 p 3 , p 2 p 4 x 1 (t) -p 1 y(t) + p 2 p 3 y(t), p 2 y (t) + p 1 y(t)],
It is clear that the rst three lists above are not corresponding to the complete system including (3.3.3) and the initial condition. For instance, the third list requires x 1 (t) = 0 for all t ≥ 0, contradicting to x 1 (0) = p 5 > 0.

The fourth list requires p 2 y (t) + p 1 y(t) = 0 for all t 0. It means that y

(t) = - p 1 p 2 y(t) or x 2 (t) = y(t) p 4
is decreasing for all t ≥ 0. However, by the Theorem 1 from the paper of Pierre Magal et al. [START_REF] Magal | Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico[END_REF] and the condition R

0 = τ S 0 ν(1 + κI 0 ) > 1,
the function x 2 (t) is rstly increasing and then decreasing on the interval [0, +∞).

It derives that the fourth list is not corresponding to the complete system (3.3.3). The fth list is now corresponding to the general case, and we obtain the output equation of (3.3.3) as follows 2(y ) 3 (t)y"(t) + y(t)(y ) 2 (t)y (t) -3y(t)y (t)(y") The Wronskian W of q 1 , q 2 is computed by the package VectorCalculus in Maple. Next, the function BelongsTo in the package DierentialAlgebra of Maple shows that W ∈ P.( see Figure 3.4).

Figure 3.4: The Wronskian W of q 1 , q 2 . By Proposition 2. with S(0) = S 0 > 0, I(0) = I 0 > 0, and R 0 = τ S 0 I 0 ν(1 + κI 0 ) > 1.

Using the same notations as in Section 3.2, (3.4.1) can be rewritten as follows The rst part of Wronskian W of q 1 , q 2 , q 3 , q 4 .

           x 1 (t) = - p 1 x 1 (t)
3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data The second part of Wronskian W of q 1 , q 2 , q 3 , q 4 . The third part of Wronskian W of q 1 , q 2 , q 3 , q 4 . 3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data Figure 3.9: The rst part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . The second part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data The third part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data Figure 3.13: The fth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . The sixth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data The seventh part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . The eighth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data Figure 3.17: The ninth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . The tenth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . The eleventh part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data The twelfth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . The thirteenth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

Conclusion

In this thesis, we consider the identiability of SIR model and SIR model with nonlinear incidence rate. There are so many problems concerning these models and one of the most important problems is the study of identiability and its applications. The identiability analysis of these systems is the most prerequisite for parameter estimation from the real data of an epidemic. However, parameter estimation from the real data of an epidemic is one kind of converse problem and there is not a complete answer. This work aims to continue the ideas about the turning point of Pierre Magal and Glenn Webb in the paper [START_REF] Magal | The Parameter Identication Problem for SIR Epidemic Models: Identifying Unreported Cases[END_REF], and nd the identiable combinations of parameters of SIR model with nonlinear incidence rate.

In chapter one, we use the ideas about the turning point of the model from the real data of an epidemic and the identiability of SIR model for providing a scheme to estimate the parameters of SIR model. Using this scheme we apply for some real epidemics in New York, Bombay, and France. The results that we obtain can be divided into two parts. The rst part includes the analysis of SIR models, the identiability analysis of SIR model, and the scheme to compute the parameters of SIR model. The second part is the applications of the rst part for the epidemics in New York, Bombay, and France. The rst part and the application to Hong Kong inuenza in New York in 1968-1969 is published in [START_REF] Ducrot | Identifying the Number of Unreported Cases in SIR Epidemic Models[END_REF]. Moreover, while we analyze the SIR model with the turning point for building the above scheme we obtain the result about the uniqueness of the solution of the turning point equation (1.3.10)( Theorem 1.3.10). It indicates that when r is closed to 1 2 enough, the identiable combination of parameters is uniquely determined from the value t p , CR(t p ), CR (t p ) and CR(∞). It helps us ensure the uniqueness of the combinations of parameters that we compute in the applications. What we obtain in this chapter is providing the complete scheme to recover the parameters of SIR models from the weekly reported case data by nding the identiable combinations of parameters and parameters estimation.

In chapter two and chapter three, we continue the idea about the scheme in chapter one by studying SIR model with nonlinear incidence rate. The problem becomes more complicated and dicult with the nonlinear incidence rate. Firstly, we provide the method for nding identiable combinations of parameters for general nonlinear models in chapter two and apply for the simple SIR model. The method in chapter two is based on the results in dierential algebra and the package implemented in Maple. Some parts of the computations for nding the identiable combinations of parameters of SIR model with nonlinear incidence rate in chapter three that we conduct are based on the codes we write in Maple. The next part is nding the

Introduction 1

 1 Identifying the number of unreported cases in SIR epidemic models 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Identiability of SIR model . . . . . . . . . . . . . . . . . . . . . . . 1.3 Computation the combinations of the parameters of SIR model . . . 1.3.1 System of equations to identify the parameters . . . . . . . . 1.3.2 Derivation of the equation for the turning point . . . . . . . . 1.3.3 Analysis of the turning point equation . . . . . . . . . . . . . 1.4 An identication method and applications to some outbreak epidemics 1.4.1 Description of the method . . . . . . . . . . . . . . . . . . . . 1.4.2 Application to Hong-Kong inuenza in New York City in 1968-1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Application to the plague epidemic in Bombay, India in 1906 1.4.4 Application to the inuenza epidemic in France in the consecutive six years 1988-1994 . . . . . . . . . . . . . . . . . . . . .

  (3.1.1) in the case α = 1, β = 2 3.3 SIR model with nonlinear incidence rate (3.1.1) in the case α = 1, β = 1 3.4 SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 1 vii CONTENTS 3.5 SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 2 3.6 SIR model with nonlinear incidence rate (3.1.1) in the case α = 1, β = Conclusion Bibliography viii

Figure 1 :S

 1 Figure 1: William Kermack et Anderson McKendrick

Figure 2 :

 2 Figure 2: Flow diagram of SIR model

I 0 t 0 I

 0 Therefore, (S(t), I(t)) satises (1.1.1)-(1.1.2) with the new parameter set p. By the uniqueness of the solution of (1.1.1)-(1.1.2) with the parameter set p, (S(t), I(t)) = (S(t, p), I(t, p)). It means that CR(t, p) := ν1 t 0 I(s, p) ds = ν1 t 0 I(s) ds = ν1 I 0 (s) ds, ∀t ≥ 0,

1. 3 .

 3 Computation the combinations of the parameters of SIR model 1.3.3 Analysis of the turning point equation Lemma 1.3.5. If r ∈ (0, 1 2

1. 3 .Figure 1 . 2 :

 312 Figure 1.2: The mapping r → X(r) where r varies in (0, 1 2 ).

Figure 1 . 3 :

 13 Figure 1.3: The mapping x → T (xX(r), r) where x varies in (0, 1). The red, green, blue, orange, black curves correspond (from the bottom to the top) to r = 0.1, r = 0.2, r = 0.3, r = 0.4 and r = 0.49, respectively.

  . It shows that I (r) > 0 for every r ∈ [ M (r) I( 2 5 ) > 0.0069. Now we show the Proof of Theorem 1.3.10: Proof. (Proof of Theorem 1.3.10)

1. 4 . 1

 41 Description of the method By Proposition 1.2.1, the combination of parameters I 0 S 0 , ν 1 S 0 , S 0 τ , ν 1 I 0 , and ν from the system (1.1.1)-(1.1.

  i. Compute the value of a 1 := τ ν 1 = X c by the formula (1.3.2); ii. Compute the value of a 2 := I 0 S 0 = e -X + Xe -rX -1 by the formula (1.3.4); iii. Compute the value of a 3 := ν 1 S 0 = CR (t p ) 1 + a 2 -e -rX (1 + rX) by the formula (1.3.5); iv. Compute the value of a 4 := τ S 0 = a 1 a 3 ; v. Compute the value of a 5 := ν 1 I 0 = a 2 a 3 ; vi. Compute the value of a 6 := ν = a 4 e -rX by the formula (1.3.6).

.4. 5 ) 1 . 4 . 2

 5142 Application to Hong-Kong inuenza in New York City in 1968-1969

  + Xe -500 1080 X -e -500 1080 X + 500 1080 Xe -500 1080 X e -X + Xe -500 1080 X -e -σX -σXe -X ∈ (0, X(r)) where X(r) = 0.89478 is the positive solution of the equation e -X + Xe -500 1080 X -1 = 0.(1.4.7)First, solve the equation (1.4.7), and obtain the value of X(r) = 0.89478. This value corresponds to the positive zero of the function in Figure1.5 -left side.

Figure 1 . 5 :

 15 Figure 1.5: Left side: The graph of Y = e -X + Xe -500 1080 X -1. Right side: The intersection of Y = T (X, 500 1080) and Y = tpCR (tp) CR(∞) . When r = 500 1080 we obtain X(r) = 0.89478, X = 0.7869 and

Figure 1 . 6 :

 16 Figure 1.6: Hong Kong inuenza epidemic in New York City in 1968-1969. The weekly reported mortality case data and cumulative reported case data , and the model output graph CR (t).To run this simulation we x S 0 = 4, 000, 000.

Figure 1 . 8 :

 18 Figure 1.8: In this gure we x S 0 = 4, 000, 000, vary the turning point t p and plot the nal size C(∞) of the epidemic as a function of the turning point.

  + Xe -4330 8840 X -e -4330 8840 X + 4330 8840 Xe -4330 8840 Xe -X + Xe -4330 8840 X -e -σX -σXe -

Figure 1 . 9 :

 19 Figure 1.9: Left side: The graph of Y = e -X + Xe -4330 8840 X -1. Right side: The intersection of Y = T (X, 4330 8840 ) and Y = tpCR (tp) CR(∞) where X(r) = 0.2444655268, X = 0.1398251831.

Figure 1 . 10 :

 110 Figure 1.10:The plague epidemic in Bombay, India in 1906. The weekly reported mortality case data and cumulative reported case data , and the model output graph CR (t). To run this simulation we x S 0 = 100, 000.

Figure 1 . 12 :

 112 Figure 1.12: In this gure we x S 0 = 4, 000, 000, vary the turning point t p and plot the nal size C(∞) of the epidemic as a function of the turning point.

1. 4 . 4

 44 Application to the inuenza epidemic in France in the consecutive six years[1988][1989][1990][1991][1992][1993][1994]

[Figure 1 . 13 :

 113 Figure 1.13: This gure indicates the outbreak of the inuenza epidemics from 1985 to 2020 (This gure is from [73])

Figure 1 . 14 :

 114 Figure 1.14: This gure indicates the outbreak of the epidemics during the period 1988-1994(This gure is from [73])

Figure 1 . 15 :

 115 Figure 1.15: This gure indicates the outbreak of the epidemics during the period from week 43 of the year 1988 to week 8 of the year 1989 (This gure is from [73])

Figure 1 . 18 :Figure 1 . 19 :

 118119 Figure 1.18: The relationship between the total case number at the end of epidemic C(∞) and S 0 for the inuenza-like illness in France in 1988w43-1989w8

1. 4 .

 4 Figure 1.20: This gure indicates the outbreak of the epidemics during the period from week 45 of the year 1989 to week 10 of the year 1990 (This gure is from [73])

Figure 1 . 23 :Figure 1 . 24 :

 123124 Figure 1.23: The relationship between the total case number at the end of epidemic C(∞) and S 0 for the inuenza-like illness in France in 1989w45-1990w10

Figure 1 . 29 :

 129 Figure 1.29: The relationship between the total case number at the end of epidemic C(∞) and the turning point t p for the inuenza-like illness in France in 1991w2-1991w15

Figure 1 . 30 :

 130 Figure 1.30: This gure indicates the outbreak of the epidemics during the period from week 45 of the year 1991 to week 10 of the year 1992 (This gure is from [73])

Figure 1 . 34 :

 134 Figure 1.34: The relationship between the total case number at the end of epidemic C(∞) and the turning point t p for the inuenza-like illness in France in 1991w45-1992w10

Figure 1 . 39 :

 139 Figure 1.39: The relationship between the total case number at the end of epidemic C(∞) and the turning point t p for the inuenza-like illness in France in 1992w51-1993w17

Figure 1 . 40 :

 140 Figure 1.40: This gure indicates the outbreak of the epidemics during the period from week 45 of the year 1993 to week 5 of the year 1994 (This gure is from [73])

  ii. If a ≤ b and b ≤ a then a = b for every a, b ∈ A.

  iii. If a ≤ b and b ≤ c then a ≤ c for every a, b, c ∈ A iv. a ≤ b or b ≤ a for every a, b ∈ A.

  It means that x(t, p) ∈ M (p) for all p ∈ Ω. Denition 2.4.1. Consider the system (1.1.1) with the parameters p = (τ, ν, ν 1 , S 0 , I 0 ) and p = (τ , ν, ν 1 , S 0 , I 0 ).

Finally, the equation ( 2 . 3 . 5 ) 3 .

 2353 is equivalent to h( x, p) = h(x, p) for all x ∈ M (p) = R 2 + . It implies that (2.3.5) is equivalent to µ x 2 = µx 2 Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data ⇔ µ = µ (3.1.h) Since (3.1.a) -(3.1.b), and (3.1.e) -(3.1.h), we obtain ν

Figure 3 . 1 :

 31 Figure 3.1: Maple code for decomposing the ideal generated by the left hand side of (3.2.4)

3. 2 . 2 Figure 3 . 3 :

 2233 Figure 3.3: Maple code for justifying that W is not in the ideal P. This code must be added at the end of the code of Figure 3.1.

3. 4 . 1 Figure 3 . 5 :

 4135 Figure 3.5:The rst part of Wronskian W of q 1 , q 2 , q 3 , q 4 .

Figure 3 . 6 :

 36 Figure 3.6:The second part of Wronskian W of q 1 , q 2 , q 3 , q 4 .

3. 4 . 1 Figure 3 . 7 :

 4137 Figure 3.7:The third part of Wronskian W of q 1 , q 2 , q 3 , q 4 .

. 1 +

 1 The following is the corollary of the above theorem Theorem 3.4.2. The basic production number R 0 of the SIR model with nonlinear incidence rate (3.4.1) is uniquely determined from the output function y(t) = µI(t). Proof. By Theorem 3.4.1, from the output function y(t) = µI(t) we have ν, τ S 0 I 0 , κI 0 are uniquely determined. It derives that R 0 = τ S 0 I 0 ν(1 + κI 0 ) is uniquely determined. 3.5 SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 2 κI 2 (t) -νI(t), t ≥ 0 y(t) = µI(t) (3.5.1) with S(0) = S 0 > 0, I(0) = I 0 > 0, and R 0 = τ S 0 ν(1 + κI 2 0 ) > 1.

3. 5 . 2 Figure 3 .

 523 Figure 3.10:The second part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

Figure 3 .

 3 Figure 3.11:The third part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. 5 . 2 Figure 3 .

 523 Figure 3.14:The sixth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

Figure 3 .

 3 Figure 3.15:The seventh part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. 5 . 2 Figure 3 .

 523 Figure 3.16:The eighth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. 5 . 2 Figure 3 .

 523 Figure 3.18:The tenth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. 5 . 2 Figure 3 .

 523 Figure 3.20:The eleventh part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

Figure 3 .

 3 Figure 3.21:The twelfth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

3. 5 . 2 Figure 3 .

 523 Figure 3.22:The thirteenth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

1 +

 1 also have the following theorem Theorem 3.5.2. The basic production number R 0 of the SIR model with nonlinear incidence rate (3.5.1) is uniquely determined from the output function y(t) = µI(t). Proof. By Theorem 3.5.1, from the output function y(t) = µI(t) we have ν, τ S 0 I 0 , κI 2 0 are uniquely determined. It derives that R 0 = τ S 0 I 0 ν(1 + κI 2 0 ) is uniquely determined. 3.6 SIR model with nonlinear incidence rate (3.1.1) in the case α = 1, β = 3 κI 3 (t) -νI(t), t ≥ 0 y(t) = µI(t) (3.6.1) with S(0) = S 0 > 0, I(0) = I 0 > 0, and R 0 = τ S 0 ν(1 + κI 3 0 ) > 1.

  2.1 Some basic denitions of dierential algebra . . . . . . . . . . . . . . 2.2 Introduction to Rosenfeld-Gröbner algorithm implemented in Maple 2.3 Dierential algebra approach for system identiability . . . . . . . . 2.4 Finding identiable combinations of parameters of SIR model by differential approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data 3.1 The sucient condition of identiability of SIR model with nonlinear incidence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 SIR model with nonlinear incidence rate

  [START_REF] Anderson | Infective Diseases of Humans: Dynamics and Control[END_REF], nous construisons un schéma qui nous permet de trouver un modèle approprié à partir des données réelles d'une épidémie. Ce schéma est appliqué pour certaines épidémies réelles survenues à New York, Bombay et en France. La clé de

la construction de ce schéma est l'équation du tournant, les valeurs du tournant t p , CR(t p ), lim t→∞ CR(t), et CR (t p ). Ces valeurs peuvent certainement être tirées des données sur les cas déclarés chaque semaine. Ensuite, nous pouvons déterminer toutes les valeurs des paramètres p si nous ne faisons varier que la valeur de S 0 , et estimer la taille nale de l'épidémie en fonction de S 0 .

Depuis que Kermack et McKendrick ont publié leurs travaux sur les modèles d'épidémie, un grand nombre de publications tentent d'étendre ce modèle pour étudier l'épidémiologie. En 1978, après avoir étudié la propagation de l'épidémie de choléra à Bari, Capasso et Serio

  1. Identifying the number of unreported cases in SIR epidemic modelsAs a consequence of Proposition 1.2.1, we have the following theorem. Then the function t → CR(t, p) is uniquely determined by the turning point t p , CR(t p ), CR (t p ) and CR(∞) in which p = (τ, ν, ν 1 , S 0 , I 0 ) is the arbitrary parameter of (1.1.1)-(1.1.3). Remark 1.3.9. Assume that the equation (1.3.10) has a unique solution X in (0, X(r)).

	Theorem 1.3.8. Assume that the equation (1.3.10) has a unique solution X in
	(0, X(r)).

Table 1 .

 1 1. Identifying the number of unreported cases in SIR epidemic models 1: Reported cases data for Hong-Kong inuenza in New York in 1968-1969.

	Week (t) Reported Cases (ν 1 I(t))
	1	14
	2	28
	3	50
	4	66
	5	156
	6	190
	7	156
	8	108
	9	68
	10	77
	11	33
	12	65
	13	24

Table 1 .

 1 2: List of a combination of parameters obtained for Hong-Kong inuenza in New York in 1968-1969. Remark 1.4.3. From the reported case data for the Hong Kong inuenza in New York City in 1968-1969, there are at least three infected individuals at the beginning of the epidemic.

	table of values
	Variable Description Estimated value
	X	cτ /ν 1	0.79
	a 1	τ /ν 1	7.3 × 10 -4
	a 2	I 0 /S 0	0.002
	a 3	ν 1 S 0	3509.1
	a 4	τ S 0	2.56
	a 5	ν 1 I 0	6.65
	a 6	ν	1.78

Table 1 .

 1 1. Identifying the number of unreported cases in SIR epidemic models Variable Estimated value 1 Estimated value 2 Estimated value 3 3: List of parameters obtained for Hong-Kong inuenza in New York in 1968-1969. In this table we vary the value of S 0 between the minimal value 1976 up to the maximal value 7, 885, 047 and compute the corresponding estimated parameters values. In Figures 1.6, 1.7, and 1.8 we provide model (1.1.1)-(1.1.2) output for the Hong Kong inuenza epidemic in New York City in 1968-1968 for the parameters in Table 1.4.2 and the values under S 0 = 4, 000, 000 in Table 1.3 (see

	S 0	1976	4, 000, 000	7, 885, 047
	I 0	3.7472	7, 586	14, 953
	τ	1.3 × 10 -3	6.4 × 10 -7	3.2 × 10 -7
	ν 1	1.78	0.88 × 10 -3	4.5 × 10 -4
	ν 2	3.1 × 10 -4	1.78	1.78
	C(∞)	1080	2.19 × 10 6	4.31 × 10 6
	S(∞)	899	1.82 × 10 6	3.59 × 10 6

Table 1 .

 1 4: Reported cases data for the plague in Bombay, India in 1906.

	Week (t) Reported Cases (ν 1 I(t))
	01	16
	02	24
	03	48
	04	51
	05	92
	06	124
	07	178
	08	280
	09	387
	10	442
	11	644
	12	779
	13	702
	14	695
	15	870
	16	925
	17	802
	18	578
	19	404
	20	296
	21	162
	22	106
	23	64
	24	46
	25	35
	26	27
	27	28
	28	24
	29	26
	30	29

Table 1 .

 1 6: List of parameters obtained for the plague epidemmic in Bombay in 1906. In this table, we take S 0 = 100, 000 and compute all other parameters' values. the plague epidemic in Bombay, India in 1906 for the parameters in Table 1.4.3 and the values under S 0 = 100, 000 in Table

	Variable	Estimated value 1
	S 0	100, 000
	I 0	7.950100000
	τ	0.5248614094 × 10 -4
	ν 1	3.318268395
	ν 2	1.582906876
	C(∞)	13056.92736
	S(∞)	86951.02274

In Figures 1.10, 1.11, and 1.12 we provide model (1.1.1)-(1.1.2) output for

  .1 we obtain the following table with S 0 = 89200

	Variable S 0 I 0 τ ν 1 ν 2 C ∞ S ∞ Table 1.7: List of parameters obtained for the inuenza-like illness in France in 1988w43-Estimated value 89200 3.756390400 0.8185035674 × 10 -4 2.967019577 3.499492216 19615.17429 69588.58210 1989w8. In Figures 1.16, 1.17, 1.18, and 1.19 we provide model (1.1.1)-(1.1.2) output for the inuenza epidemic in France during the period from week 43 of the year 1988 to the week 8 of the year 1989 for the parameters in Table 1.4.4. In Figure 1.16, 1.17 we compare the model output to the reported case data. In Figure 1.18 we illustrate the epidemic nal size as a function of the initial 2 4 6 8 10 12 14 16 18 week CR'(t) Figure 1.16: The weekly reported case data(blue dots) and model output graph of ν 1 I(t) for the inuenza-like illness in France in 1988w43-1989w8 2 4 6 8 10 12 14 16 18 week number of susceptibles S 1.4. An identication method and applications to some outbreak epidemics CR(t)

0 . In Figure

1

.

[START_REF] Chowell | Modelling the transmission dynamics of acute haemorrhagic conjunctivitis: Application to the 2003 outbreak in Mexico[END_REF]

we illustrate the epidemic nal size as a function of the turning point of the epidemic. Figure 1.17: The weekly reported cumulative case data(blue dots) and model output graph of CR(t) for the inuenza-like illness in France in 1988w43-1989w8 1. Identifying the number of unreported cases in SIR epidemic models

Table 1 .

 1 9: List of parameters obtained for the inuenza-like illness in France in 1991w2-1991w15. France during the period from week 2 of the year 1991 to week 15 of the year 1991 for the parameters in Table1.4.4. In Figure1.26, 1.27 we compare the model output to the reported case data. In Figure1.28 we illustrate the epidemic nal size as a function of the initial number of susceptibles S 0 . In Figure1.29 we illustrate the epidemic nal size as a function of the turning point of the epidemic.

	Variable	Estimated value
	S 0	89200
	I 0	2.545054400
	τ	0.2095686786 × 10 -3
	ν 1	4.806980959
	ν 2	13.11249984
	C ∞	7343.773041
	S ∞	81858.77201
	In Figures 1.26, 1.27, 1.28, and 1.29 we provide model (1.1.1)-(1.1.2) output
	for the inuenza epidemic in

4.1 we obtain the following table with S 0 = 89200 Figure 1.27: The weekly reported cumulative case data(blue dots) and model output graph of CR(t) for the inuenza-like illness in France in 1991w2-1991w15 1. Identifying the number of unreported cases in SIR epidemic models Figure 1.28:

Table 1 .

 1 10: List of parameters obtained for the inuenza-like illness in France in 1991w45-1992w10. France during the period from week 45 of the year 1991 to week 10 of the year 1992 for the parameters in Table1.4.4. In Figure1.31, 1.32 we compare the model output to the reported case data. In Figure1.33 we illustrate the epidemic nal size as a function of the initial number of susceptibles S 0 . In Figure1.34 we illustrate the epidemic nal size as a function of the turning point of the epidemic.

	Variable	Estimated value
	S 0	89200
	I 0	96.62144000
	τ	0.1983750434 × 10 -4
	ν 1	0.05103676158
	ν 2	0.8469815172
	C ∞	70505.24235
	S ∞	18791.37909
	In Figures 1.31, 1.32, 1.33, and 1.34 we provide model (1.1.1)-(1.1.2) output
	for the inuenza epidemic in

4.1 we obtain the following table with S 0 = 89200 Figure 1.32: The weekly reported cumulative case data(blue dots) and model output graph of CR(t) for the inuenza-like illness in France in 1991w45-1992w10 1.

Table 1 .

 1 11: List of parameters obtained for the inuenza-like illness in France in 1992w51-1993w17. France during the period from week 51 of the year 1992 to week 17 of the year 1993 for the parameters in Table1.4.4. In Figure1.36, 1.37 we compare the model output to the reported case data. In Figure1.38 we illustrate the epidemic nal size as a function of the initial number of susceptibles S 0 . In Figure1.39 we illustrate the epidemic nal size as a function of the turning point of the epidemic.

	Variable	Estimated value
	S 0	89200
	I 0	35.99505440
	τ	0.4420270591 × 10 -4
	ν 1	0.6102895730
	ν 2	2.820697355
	C ∞	22375.16185
	S ∞	66860.83320
	In Figures 1.36, 1.37, 1.38, and 1.39 we provide model (1.1.1)-(1.1.2) output
	for the inuenza epidemic in

4.1 we obtain the following table with S 0 = 89200 Figure 1.37:

Table 1 .

 1 12: List of parameters obtained for the inuenza-like illness in France in 1993w45-1994w5. France during the period from week 45 of the year 1993 to week 5 of the year 1994 for the parameters in Table1.4.4. In Figure1.41, 1.42 we compare the model output to the reported case data. In Figure1.43 we illustrate the epidemic nal size as a function of the initial number of susceptibles S 0 . In Figure1.44 we illustrate the epidemic nal size as a function of the turning point of the epidemic.

	Variable	Estimated value
	S 0	89200
	I 0	14.80889480
	τ	0.7255641309 × 10 -4
	ν 1	0.9864734585
	ν 2	4.308166928
	C ∞	30432.25414
	S ∞	58782.55475
	In Figures 1.41, 1.42, 1.43, and 1.44 we provide model (1.1.1)-(1.1.2) output
	for the inuenza epidemic in

4.1 we obtain the following table with S 0 = 89200 Figure 1.42: The weekly reported cumulative case data(blue dots) and model output graph of CR(t) for the inuenza-like illness in France in 1993w45-1994w5 Chapter 2

  Consider the Wronskian of the polynomials φ i (y), ..., φ γ (y) as follows

		φ 1 (y)	φ 2 (y)	...	φ γ (y)
		φ 1 (y)	φ 2 (y)	...	φ γ (y)
		...		...	...	...
	φ	(γ-1) 1	(y) φ

  (2.3.8), substituting p by p we have

	γ
	φ γ+1 (y(t, p)) +	c i (p)φ i (y(t, p)) = 0.
	i=1
	Then, substituting φ(y(t, p)) by φ(y(t, p)), we have
	γ
	φ γ+1 (y(t, p)) +	c i (p)φ i (y(t, p)) = 0.
	i=1
	Subtracting this equation by (2.3.8), we obtain
	γ
	i=1

  νx 2 , and h(x) = p 3 x 2 . It means that SIR model (1.1.1) is not identiable.

	Choose = Proposition 2.3.2 hold for all x ∈ M (p) = R 2 S 0 > 0, by the condition (2.4.1) we easily check that (2.3.2)-(2.3.5) from S 0 + . It implies that y(t, p) = y(t, p) for all
	t ≥ 0.

  2.4. Finding identiable combinations of parameters of SIR model by dierential approachp 3 y(t)y"(t) -p 3 (y (t)) 2 + p 1 y 2 (t)y (t) + p 1 p 2 y 3 (t)

	We rewrite this dierential polynomial, then the output equation is obtained as
	follows
	p 3 y(t)y"(t) -(y (t)) 2 + p 1 y 2 (t)y (t) + p 1 p 2 y 3 (t) = 0, t ≥ 0,	(2.4.4)
	Dividing (2.4.4) by p 3 we obtain
	y(t)y"(t) -(y (t)) 2 +	p 1 p 3	y 2 (t)y (t) +	p 1 p 2 p 3	y 3 (t) = 0, t ≥ 0,	(2.4.5)

2 .

 2 By proposition 2.3.3 we have to justify that W is not in the ideal P. This is a testing membership problem of an ideal in Dierential Algebra and we can conduct to check if W is not in the ideal P by the function called BelongsTo in the package DierentialAlgebra of Maple. The result in Maple returned is "false". It means that , p 1 p 4 = p 1 p 4 , p 1 p 5 = p 1 p 5 . Together with Proposition

	2.4. Finding identiable combinations of parameters of SIR model by dierential
				approach
	It implies that
			p 1 p 4 = p 1 p 4 .	(2.4.8)
	Solving algebraic system (2.4.6)-(2.4.8), we have the relationships between p and
	p as follows p 2 = p 2 , 2.4.2, we obtain Proposition 1.2.1 as a consequence. It means that ν, p 1 p 3 p 1 = p 3 are the identiable combinations of parameters of (1.1.1) from the output function τ , τ S 0 , τ I 0 ν 1
	y(t) = ν 1 I(t).
	The following is a consequence of proposition 1.2.1
	Proposition 2.4.3. The basic production number of the SIR model (1.1.1) is uniquely determined from the output function y(t) = ν 1 I(t).
	Proof. From the output function y(t) = ν 1 I(t) we have ν, τ S 0 are uniquely deter-
	mined. It implies that R 0 =	τ S 0 ν	is uniquely determined.
	W ∈ P.

  3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case datato check if W is not in the ideal P by the function called BelongsTo in the package DierentialAlgebra of Maple. The result in Maple returned is "false". It means that

	W ∈ P. By Proposition 2.3.3 and (3.2.6), if y(t, p) = y(t, p), then
			  -2p 3 = -2p 3 ,          p 1 p 4 p 2 = p 1 p 4 -p 1 p 3 p 4 p 2 = -p 1 p 3 p 4 p 2 p 2 ,		(3.2.7)
	Now, consider the initial condition of (3.2.3), (3.2.4), we have y(0, p) = y(0, p) then
			p 4 p 6 = p 4 p 6		(3.2.8)
	Next, we consider the two other equations of the fourth list
	y the fact x 3 (t) = follows p 2 p 2 4 x 3 (t, p) = 1 + p 2 x 2 1 2 (t) y 2 (t, p) 2y (t, p) + . Therefore, we can rewrite the two above equations as y(t, p)y"(t, p) -(y ) 2 (t, p) p 2 (y (t, p) + p 3 y(t, p)) p 1 p 4 , (3.2.11)      S (t) = -τ S(t)I(t) , 1 + κI(t)      y(t) = µI(t)  I (t) = τ S(t)I(t) 1 + κI(t) -νI(t), t ≥ 0 (3.3.1)
	and	p 4 x 1 (t, p) =	y(t, p) 2y (t, p) + p 1 p 4 p 2 (y(t, p)y"(t, p) -(y ) 2 (t, p)) p 1 p 4 (y (t, p) + p 3 y(t, p)) p 2	.	(3.2.12)
	By (3.2.7) and y (i) (t, p) = y (i) (t, p) for all i ∈ N, we have
			   p 4 x 1 (0, p) = p 4 x 1 (0, p), p 2 p 2 4 x 3 (0, p) = p 2 4 p 2 x 3 (0, p),	.	(3.2.13)
	It implies that
			   p 4 p 5 = p 4 p 5 , p 2 p 2 4 1 1 + p 2 p 2 6 =	p 2 p 2 4	1 1 + p 2 p 2 6	,	.	(3.2.14)

2 (t) 2p 2 y (t) + p 1 p 4 y (t) + p 3 y(t) x 3 (t) = p 2 4 y(t)y"(t) -(y ) 2 (t) , (3.2.9) and p 1 p 2 4 y(t)y"(t) -(y ) 2 (t) x 1 (t) = y(t) 2p 2 y (t) + p 1 p 4 y (t) + p 3 y(t) 2 . (3.2.10) We show that y(t), 2p 2 y (t) + p 1 p 4 , y (t) + p 3 y(t), and y(t)y"(t) -(y ) 2 (t) are not generically zeros. Indeed, by the Theorem 1 in [45] and the condition R 0 > 1, the function x 2 (t) is rstly increasing and then decreasing on the interval [0, +∞). It implies that y(t), 2p 2 y (t) + p 1 p 4 , y (t) + p 3 y(t) are not generically zeros. If y(t)y"(t) -(y ) 2 (t) is generically zero then x 3 (t) is generically zero. This contradicts

  1 x 1 (0, p)x 3 (0, p) -p 3 = p 1 x 1 (0, p)x 3 (0, p) -p 3 , ), (3.3.10),(3.3.14) give us the relationships between p and p as follows Together with Proposition 3.1.1, we have the following Theorem 3.3.1. The SIR model with nonlinear incidence rate (3.3.1) is not identiable, and we have µI(t, p) = µI(t, p) if and only if ν = ν, The following is the consequence of Theorem 3.3.1 Theorem 3.3.2. The basic production number R 0 of the SIR model with nonlinear incidence rate (3.3.1) is uniquely determined from the output function y(t) = µI(t). Proof. By Theorem 3.3.1, from the output function y(t) = µI(t) we have ν, τ S 0 , κI 0 are uniquely determined. It derives that R 0 = τ S 0 ν(1 + κI 0 ) is uniquely determined. 3.4 SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 1 In the case α = 2, β = 1 we rewrite (3.1.1) as follows

	3. Determining the identiable combinations of parameters of SIR model with
	nonlinear incidence rate from the reported case data
											p 1 x 1 (t, p)x 3 (t, p) + p 3 =	y (t) y(t)	(3.3.12)
	By (3.3.8), (3.3.9) and y (i) (t, p) = y (i) (t, p) for all i ∈ N, we have
			 	p 2 p 4	x 3 (0, p) =	p 2 p 4	x 3 (0, p),					.	(3.3.13)
			
	It implies that						    	p 2 p 4 p 1 p 5 1 + p 2 p 6 1 1 + p 2 p 6 -p 3 = = p 2 p 4	1 1 + p 2 p 6 1 + p 2 p 6 p 1 p 5	, -p 3	.	(3.3.14)
	(3.3.7p 3 = p 3 ,	p 1 p 1	=	p 2 p 2	=	p 4 p 4	=	p 5 p 5	=	p 6 p 6	. τ τ	=	κ κ	=	µ µ	=	S 0 S 0	=	I 0 I 0	.
														    	p 1 p 3 p 2 p 1 p 2 + p 3 = = p 1 p 3 p 2 p 2 , p 1	+ p 3	(3.3.7)
	(3.3.7) is equivalent to
															 	p 1 p 2	=	p 1 p 2	,	(3.3.8)
															 p 3 = p 3
	or By the initial condition of (3.3.3), y(0, p) = y(0, p) if      p 1 p 2 = p 3 , p 3 = p 1 p 2      S (t) = -τ S(t)I 2 (t) , 1 + κI(t)      y(t) = µI(t)  I (t) = τ S(t)I 2 (t) 1 + κI(t) -νI(t), t ≥ 0	(3.3.9) (3.4.1)
																p 4 p 6 = p 4 p 6	(3.3.10)
	Next, by the similar arguments as in the section 3.2, we obtain the two other
	equations from the fth list
						p 2 p 4	p 2 x p 1	y(t)	,	(3.3.11)

3.3 and (3.3.4)

, if y(t, p) = y(t, p), then 3 (t, p) = y(t)y"(t) -(y ) 2 (t) y(t) (y (t) + p 3 y(t)) y (t) + p

  3.5. SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 2 By (3.4.10) and y (i) (t, p) = y (i) (t, p) for all i ∈ N, we have The algebraic system (3.4.10), (3.4.11), (3.4.15) derives the relationships between p and p as follows p 3 = p 3 , Theorem 3.4.1. The SIR model with nonlinear incidence rate (3.4.1) is not identiable, and we have µI(t, p) = µI(t, p) if and only if ν = ν,

	and		p 1 p 4	x 1 (t, p)x 3 (t, p) =	y (t) + p 3 y(t) y 2 (t)	(3.4.13)
	    	p 2 p 4 p 1 p 4	x 3 (0, p) = x 1 (0, p)x 3 (0, p) = p 2 x 3 (0, p), p 4 p 4 x 1 (0, p)x 3 (0, p), p 1	.	(3.4.14)
	It implies that		    	p 2 p 4 p 1 p 4	1 1 + p 2 p 6 p 5 1 + p 2 p 6	= =	p 2 p 4 p 4 p 1	1 1 + p 2 p 6 1 + p 2 p 6 p 5	,	.	(3.4.15)
	we have the following			p 1 p 1	=	p 4 p 4	2	,	p 2 p 2	=	p 4 p 4	=	p 5 p 5	=	p 6 p 6	. By Proposition 3.1.1,
															τ τ	=	µ 2

  3.6. SIR model with nonlinear incidence rate (3.1.1) in the case α = 1, β = 3 By (3.5.11) and y (i) (t, p) = y (i) (t, p) for all i ∈ N, we have we solve the algebraic system (3.5.11), (3.5.12),(3.5.16), and obtain the relationships between p and p as follows p 3 = p 3 , Together with Proposition 3.1.1, we have the following Theorem 3.5.1. The SIR model with nonlinear incidence rate (3.5.1) is not identiable, and we have µI(t, p) = µI(t, p) if and only if ν = ν,

	and		p 1 p 4	x 1 (t, p)x 3 (t, p) =	y (t) + p 3 y(t) y 2 (t)	.	(3.5.14)
	    	p 2 p 2 4 p 1 p 4	x 3 (0, p) = x 1 (0, p)x 3 (0, p) = p 2 x 3 (0, p), p 2 4 p 4 x 1 (0, p)x 3 (0, p), p 1	.	(3.5.15)
	It implies that		    	p 2 p 2 4 p 1 p 4	1 1 + p 2 p 2 6 p 5 1 + p 2 p 2 6	= =	p 2 p 2 4 p 4 p 1	1 1 + p 2 p 2 6 6 1 + p 2 p 2 p 5	,	.		(3.5.16)
									p 1 p 1	=	p 2 p 2	=	p 4 p 4	2	,	p 4 p 4	=	p 5 p 5	=	p 6 p 6	.
														τ τ	=	κ κ	=	µ 2

(t) (p 1 x 1 (t)x 2 (t)x 3 (t) -p 3 x 2 (t)),and (3.2.2) can be

3.5. SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 2 Figure 3.12:The fourth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

Figure 3.19:The fourth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 .

 Figure 1.43:The relationship between the total case number at the end of epidemic C(∞) and S 0 for the inuenza-like illness in France in 1993w45-1994w5 Figure 1.44: The relationship between the total case number at the end of epidemic C(∞) and the turning point t p for the inuenza-like illness in France in 1993w45-1994w5 2. Some preliminaries of dierential algebra and application to system identiability Denition 2.1.9. Suppose that {∂ 1 , ..., ∂ n } is the set of all the derivations of the dierential eld R, and X := {x 1 , x 2 , ..., x k } . Let D = {∂ i 1 1 ∂ i 2 2 ...∂ in n : i 1 , ..., i n ∈ N}. A strict total order < over the set of the derivatives D(X) = {θx i |θ ∈ D, x i ∈ X, i = 1, 2..., k} is called to be a ranking if for every δ ∈ {∂ 1 , ..., ∂ n } i. u < δu, ii. u < v then δu < δv, for all u, v ∈ D(X).

A ranking such that u < v implies δu < φv for all δ, φ ∈ D is said to be an elimination ranking and written

Denition 2.1.10. Let q be a polynomial of K{x 1 , ..., x k } and < is a ranking on the D(X). The leader u of q is the largest derivative ∂ i 1 1 ∂ i 2 2 ...∂ in n x j of D(X) with respect to the ranking < which appear in q. Let d be the degree of u in q, then the coecient of u d is called the initial of q, denoted by I q . The derivative S q = ∂q ∂u is called the separant of q, and u d is called the rank of q.

Example 2.1.11. We consider again the example 2.1.4, suppose that < is the elimination ranking on K{x 1 , x 2 , ..., x n } such that x 1 < x 2 < ... < x n . It means that x 1 < x 1 < x 1 " < ... < x 2 < x 2 < ... < x n < x n < .... It is clear that x 1 " is the leader of the dierential polynomial a 1 x 1 x 2 2 x 3 3 -a 2 (x 1 ") 2 + a 3 x 1 x 2 . The initial of this polynomial is -a 2 and the separant of this polynomial is -2a 2 x 1 " where the rank is (x 1 ") 2 . Denition 2.1.12. Let q 1 and q 2 be two polynomials of K{u 1 , ..., u k }, and u d is the rank of q 1 . q 2 is said to be partially reduced w.r.t (with respect to) q 1 if q 2 does not contain any proper derivative of u. q 2 is said to be reduced w.r.t q 1 if it is partially reduced w.r.t q 1 and its degree in u is less than d. A set of polynomials A is called an autoreduced set if its elements are pairwise reduced.

Denition 2.1.13. An autoreduced set C of a dierential ideal I is said to be a characteristic set if I does not contain any non-zero element reduced w.r.t all the elements of C. Denition 2.1.14. A set of polynomials A is called an orthonomic set if the initial and separants of its elements are in K.

For an ordinary dierential ring K{u 1 , ..., u k }, an ideal of K{u 1 , ..., u k } generated by an autoreduced and orthonomic set is a prime ideal (see Boulier et al. [9]).

Example 2.1.15. Let Q(p) be the eld of the rational expressions of p 1 , ..., p 4 ∈ R, consider the ring Q(p){x 1 , x 2 , x 3 } of dierential polynomials in indeterminates x 1 , x 2 , x 3 with the coecients in Q(p). We equip on Q(p){x 1 , x 2 , x 3 } the elimination ranking such that

Some preliminaries of dierential algebra and application to system identiability

-The output of the algorithm is a nite family A 1 , A 2 , ..., A s of nite subsets of K{X}\K in which K is a eld and X is a set of indeterminates. Denote by A i the dierential radical ideal for every i ∈ {1, 2..., s} such that for any dierential polynomial p ∈ K{X}, p ∈ A i if and only if Rittreduction(p, A i ) = 0.

Denote by P the radical ideal generated by P , Rosenfeld-Gröbner algorithm gives us the relationship between P and A i as follows

We have P = K{X} when s = 0. Moreover, an algorithm for testing membership problem in P is obtained as follows:

Given a dierential polynomial p ∈ K{X}, we have p ∈ P if and only if Rittreduction(p, A i ) = 0 for every i ∈ {1, 2..., s}.

Example 2.2.1. Let Q(p) be the eld of the rational expressions of p 1 , p 2 , p 3 ∈ R, consider the ring Q(p){x 1 , y} of dierential polynomials in indeterminates x 1 , y with the coecients in Q(p). We equip on Q(p){x 1 , y} the elimination ranking such that

Let P be the list of two dierential polynomials p 3 x 1 + p 1 x 1 y, y -p 1 x 1 y + p 2 y. Applying Rosenfeld-Gröbner algorithm with the elimination ranking above, we obtain a family of two lists A 1 = {x 1 , y} and A 2 = {p 1 x 1 y -y -p 2 y, p 3 yy"-p 3 (y ) 2 + p 1 y 2 y + p 1 p 2 y 3 }. We conduct the procedure Rittreduction(y + 1, A 1 ) as follows:

-Firstly, we assign g 0 := y + 1.

-Secondly, we see that the leaders of f 1 := x 1 and f 2 := y are respectively x 1 and y. Since deg(g 0 , y) = deg(f 2 , y) then we assign g 1 := prem(g 0 , f 2 , y) = 1.

-Finally, Rittreduction(y + 1, A 1 ) := g 1 = 1.

Since Rittreduction(y + 1, A 1 ) = 0 then y + 1 ∈ A 1 , then y + 1 ∈ P where P is the dierential radical ideal generated by P .

The Rosenfeld-Gröbner algorithm is implemented in Maple. To use Rosenfeld-Gröbner algorithm we have to call the package " DierantialAlgebra" in Maple.

Suppose that we have a list of dierential polynomials and a ranking on them.

Firstly, we have to build the dierential ring of polynomials specifying the variables and a ranking for these polynomials by calling the function DierentialRing (derivations = ..., blocks = ..., arbitrary = ...) in which "derivations = ..." is a list of independent variables specifying the possible dependency of the dependent variables, "blocks = ..." is a list of the dependent variables, possibly including sublists of them; only one level of nested sublists is allowed, and "arbitrary = [...]" is a list of algebraically independent arbitrary objects; these can be dependent variables (functions of one or many variables specied in derivations) or other variables (not those specied in derivations)( see [74]).

Secondly, after we have R = DierentialRing(derivations = ..., blocks = ..., arbitrary = ...), we can call Rosenfeld-Gröbner algorithm by the function Rosenfeld-Groebner(sys, R) where "sys" is a list or a set of dierential equations or inequations; all rational in the independent and dependent variables and their derivatives. Rosen-feldGroebner(sys, R) splits the given system sys into cases, each one specied by their 2. Some preliminaries of dierential algebra and application to system identiability Figure 2.1: Maple code for decomposing the ideal generated by the left hand side of (2.4.3) and checking if W is not in the ideal P By proposition 2.3.3 and (2.4.5), if y(t, p) = y(t, p), then

Now, consider the initial condition of (2.4.3), we have y(0, p) = y(0, p) then

(2.4.7)

Next, we consider another equation of the second list p 1 x 1 (t)y(t) -y (t) -p 2 y(t) = 0. We show that y(t) are not generically zeros. Indeed, by the result in [START_REF] Ducrot | Identifying the Number of Unreported Cases in SIR Epidemic Models[END_REF](Chapter 1) and the condition R 0 > 1, the function x 2 (t) is rstly increasing and then decreasing on the interval [0, +∞). It implies that y(t) are not generically zeros. Therefore, we can rewrite the above equation as follows

By (2.4.6) and y (i) (t, p) = y (i) (t, p) for all i ∈ N , we have

3.2. SIR model with nonlinear incidence rate (3.1.1) in the case α = 1, β = 2 written as follows

Consider the dierential polynomials on the left-hand sides of (3.2.4). These polynomials form an autoreduced set in the dierential ring Q(p){x 1 , x 3 , y} with the elimination ranking (that we mentioned in Chapter 2)

. Moreover, the initials and separants of these polynomials are in Q(p), then these polynomials form an orthonomic set of Q(p){x 1 , x 3 , y}. Let P is a prime dierential ideal generated by these polynomials.

Next, the ideal P is decomposed by using Rosenfeld-Gröbner algorithm implemented in Maple. The result is the four lists of dierential polynomials. Each list forms the set of generators of a radical ideal corresponding to either the singular solutions or general solution of (3.2.4). We show here the lists

SIR model with nonlinear incidence rate (3.1.1) in the case

Then the output equation is obtained as follows

Dividing both side of (3.2.5) by p 2 we obtain 2y (3) (t)(y ) 2 (t)y 2 (t) -6y"(t)(y ) 3 (t)y 2 (t) + 2y"(t)(y ) 3 (t)y(t) + 2(y ) 5 (t)

Now we check the linear independence of three polynomials

In order to check that, we compute the Wronskian W of q 1 , q 2 , q 3 . It is easy to obtain this evaluation by the package VectorCalculus in Maple( see gures 3.2, 3.3).

3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data Figure 3.2: Maple code for computing the Wronskian W of q 1 , q 2 , q 3 .

The following is the consequence of Theorem Introducing a new variable

, and (3.4.2) can be written as follows

Consider the dierential polynomials on the left-hand sides of (3.4.3). It is seen that the set of these polynomials is an autoreduced set in the dierential ring

. Moreover, the initials and separants of these polynomials are in Q(p), then these polynomials form an orthonomic set of Q(p){x 1 , x 3 , y}. Let P is a prime dierential ideal generated by these polynomials.

Next, we use Rosenfeld-Gröbner algorithm implemented in Maple for decomposing the ideal P, and obtain four lists of dierential polynomials. Each list is the set of generators of a radical ideal corresponding to the solution of (3.4.3). The lists are shown as follows The rst three lists above are not corresponding to the complete system including (3.4.3) and the initial condition. For instance, the third list requires x 1 (t) = 0 for all t ≥ 0, contradicting to x 1 (0) = p 5 > 0.

Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data

The fourth list is now corresponding to the general case, and the last polynomial in the fourth list forms the output equation of (3.4.3) as follows y 2 (t)(y ) 2 (t)y (3) (t) -3y 2 (t)y (t)(y") 2 (t) + 4y(t)(y ) 3 (t)y"(t) -2(y ) 5 Next, we check the linear independence of four polynomials

q 2 = y 3 (t)y (t)y (3) (t) -y 3 (t)(y") 2 (t) (3.4.6)

q 4 = 3y 4 (t)(y ) 2 (t) -y 5 (t)y"(t).

(3.4.8)

The Wronskian W of q 1 , q 2 , q 3 , q 4 is computed by the package VectorCalculus in Maple. Next, the the function BelongsTo in the package DierentialAlgebra of Maple shows that W ∈ P.( see gures 3.5, 3.6, 3.7, 3.8)

Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data

Figure 3.8: The fourth part of Wronskian W of q 1 , q 2 , q 3 , q 4 . By Proposition 2.3.3 and (3.4.4), if y(t, p) = y(t, p), then

Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data

While the same notations are using as in Section 3.2, (3.5.1) can be rewritten as follows

As mentioned in Chapter 2, we let

, then

Therefore, (3.5.2) can be written as follows

The set of all dierential polynomials on the left hand sides of (3.5.3) is an autoreduced set in the dierential ring Q(p){x 1 , x 3 , y} with the elimination ranking

. Moreover, the initials and separants of these polynomials are in Q(p), then these polynomials form an orthonomic set of Q(p){x 1 , x 3 , y}. Let P is a prime dierential ideal generated by these polynomials.

Next, we decompose the ideal P by using Rosenfeld-Gröbner algorithm implemented in Maple. The four lists of dierential polynomials is returned by this decomposition. Each list forms the set of generators of a radical ideal corresponding to either the singular solutions or general solution of (3.5.3). We show here the lists

3.5. SIR model with nonlinear incidence rate (3.1.1) in the case α = 2, β = 2 +6p 2 (y ) 3 (t)y"(t) -4p 1 y(t)(y ) 2 (t)y"(t) -2p 2 p 3 y(t)(y ) 2 (t)y"(t) -7p 1 p 3 y 2 (t)y (t)y"(t) -p 1 p 2 3 y 3 (t)y"(t) + 6p 1 (y ) 4 (t) + 6p 2 p 3 (y ) 4 (t)

It is seen that the rst three lists above are not corresponding to the complete system including (3.5.3) and the initial condition. For instance, the third list requires x 1 (t) = 0 for all t ≥ 0, contradicting to x 1 (0) = p 5 > 0.

The fourth list is now corresponding to the general case, and the last polynomial in the fourth list forms the output equation of (3.5.3) as follows

Next, we check the linear independence of ve polynomials q 1 = 6(y ) 4 (t) -4y(t)(y ) 2 (t)y"(t) + y 2 (t)y (t)y (3) (t) -y 2 (t)(y") 2 (t), (3.5.5)

)

q 5 = y(t)(y ) 3 (t).

(3.5.9)

We compute the Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 by the package VectorCalculus in Maple. Next, the function BelongsTo in the package DierentialAlgebra of Maple is used to show that W ∈ P. ( see gures from 3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data Figure 3.23: The fourteenth part of Wronskian W of q 1 , q 2 , q 3 , q 4 , q 5 . By Proposition 2.3.3 and (3.5.4), if y(t, p) = y(t, p), then

(3.5.10)

By the initial condition of (3.5.3), y(0, p) = y(0, p) if p 4 p 6 = p 4 p 6 .

(3.5.12)

By the similar arguments from Section 3. 3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data

Using the same notation as in Section 3.2, then (3.5.1) can be rewritten as follows

, then

(3.6.2) can be written as follows

Consider the dierential polynomials on the left-hand sides of (3.6.3). These polynomials form an autoreduced set in the dierential ring Q(p){x 1 , x 3 , y} with the elimination ranking [y] < [x 1 ] < [x 3 ]. Moreover, the initials and separants of these polynomials are in Q(p), then these polynomials form an orthonomic set of Q(p){x 1 , x 3 , y}. Let P is a prime dierential ideal generated by these polynomials.

Next, using Rosenfeld-Gröbner algorithm implemented in Maple to decompose the ideal P. This decomposition returns four lists of dierential polynomials. Each list forms the set of generators of a radical ideal corresponding to either the singular solutions or general solution of (3.6.3). We show here the lists It can be seen that the rst three lists above are not corresponding to the complete system including (3.6.3) and the initial condition. For instance, the third list requires x 1 (t) = 0 for all t ≥ 0, contradicting to x 1 (0) = p 5 > 0.

The fourth list is now corresponding to the general case, and the last polynomial in the fourth list forms the output equation of (3.6.3) as follows 3y(t)y (t) 2(y ) 4 (t) + y 2 (t)y (t)y (3) Next, we check the linear independence of three polynomials q 1 = 3y 2 (t) 4(y ) 4 (t) -4y(t)(y ) 2 (t)y"(t) + y 2 (t)y (t)y (3) (t) -y 2 (t)(y") 2 (t) , (3.6.5)

q 3 = y(t) 3(y ) 3 (t) -4y(t)y (t)y"(t) + y 2 (t)y (3) (t) .

(3.6.7)

Using the package VectorCalculus in Maple to compute the Wronskian W of q 1 , q 2 , q 3 and the function BelongsTo in the package DierentialAlgebra of Maple shows that W ∈ P.( see Figure 3.24) 3. Determining the identiable combinations of parameters of SIR model with nonlinear incidence rate from the reported case data Figure 3.24: The Wronskian W of q 1 , q 2 , q 3 . By Proposition 2.3.3 and (3.6.4), if y(t, p) = y(t, p), then x 3 (0, p), p 1 x 1 (0, p)x 3 (0, p) = p 1 x 1 (0, p)x 3 (0, p), . The answer for the above problem for the general case α, β ∈ N such that β + 1 ≥ α ≥ 1 is also another work of this study.