Fatherland

O F Dschang

Université De Dschang

École Doctorale

REPUBLIC OF CAMEROON PEACE-WORK

Keywords: Cholera, Phage-bacteria infection, Bifurcation, Reaction-diffusion, Optimal control, Lyapunov function, NSFD. vi

It is a great pleasure to indicate that a number of people have encouraged, guided

Résumé

Une phage est un virus qui infecte les bactéries. Cette infection peut d'une part purifier un environement ou l'organisme d'un individu infecté de bactéries, et d'autre part déclencher la propagation des maladies bactériennes. L'objectif de cette thèse est d'étudier l'impact de l'interaction phage-bactérie sur la transmission du choléra et investiguer sur les possibilités d'utiliser les phages pour un but thérapeutique. Pour ce faire, (i) nous formulons un modèle de proie-prédateur pour décrire et étudier l'interaction phage-bactérie prenant en compte les cycles lytique et lysogénique d'une phage.

En utilisant le seuil écologique du modèle, nous déterminons les conditions sous lesquelles la présence des phages peut détruire la population des bactéries et les situations où elle déclenche les maladies bactériennes. Les modèles couplés de choléra et phage-bactérie sont utilisés pour étudier l'influence des phages et bactéries sur la transmission spatiotemporelle du choléra. Le taux de reproduction de base qui représente le nombre moyen d'infectés humains produits par les bactéries infectieuses au cours de leur vie est calculé. Nous montrons à travers ces modèles que lorsque l'infection phage-bactérie est prise en compte dans un modèle de transmission du choléra, la condition classique qui consiste à baisser le seuil épidémiologique en dessous de un n'est pas suffisante pour éradiquer la maladie. Nous déterminons le seuil écologique nécéssaire pour l'éradication du choléra. En effet nos résultats prédisent que les éfforts doivent être faits pour baisser les deux seuils écologique et épidémiologique en dessous de un.

Pour réduire le nombre d'infectés humains de choléra, nous proposons deux moyens de contrôle. La première stratégie consiste à introduire les phages virulents sélectionnées pour infecter et détruire la population de vibrions cholériques dans l'environement. La deuxième stratégie consiste à consommer ces phages sélectionnées pour éliminer les vibrions cholériques de l'organisme humain. Ce dernier moyen de contrôle est la phagothérapie. Utiliser les phages pour un but thérapeutique a plusieurs applications en médécine humaine, médécine vétérinare et en agriculture.

Mots clés: Choléra, Infection Phage-bactéria, Bifurcation, Réaction-diffusion, Contrôle Optimal, Fonction de Lyapunov, Schéma numérique non standard. viii

General Introduction

Cholera is usually known as the "disease of dirty hands". It is an infection of the small intestine caused by some strains of the bacteria called vibrio cholerae. Symptoms may not show up, but when they do, one notices high dehydration of the infected person through watery diarrhea that lasts a few days. This may results in sunken eyes, cold skin, decreased skin elasticity, and wrinkling of the hand and feet. Symptoms start two hours to five days after exposure. Cholera affects an estimated 3-5 million people worldwide and causes 28,800-130,000 deaths a yearly.

Although it is classified as a pandemic since 2010, it is rare in developed countries. Children are mostly affected especially in Africa and Southeast Asia. The cholera fatal rate is usually less than 5%, but may assume 50% in some areas where access to treatment is unavailable.

Vibrio Cholerae can survive in some aquatic environment for more than three months up to two years living in association with zoo-plankton, phytoplankton and the aquatic organisms such as bacteriophages. The two ecological serogroups (Vibrio cholerae 01 and Vibrio cholerae 0139) have the ability to colonize the hosts small intestine.

Bacteriophages or phages are actually viruses that infect bacteria. They are obligate intracellular parasites which rely on the bacteria hosts in order to replicate. Phages are an essential part of the aquatic biology because of their omnipresence in the aquatic ecosystem. They are closely linked to the bacterial population. Phages exhibit three different life cycles based on their survival strategies which are: lytic, lysogenic and pseudo-lysogenic [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Miller | Contribution, pseudlysogeny and starvation to phage ecology[END_REF]. In the lytic cycle, the phage injects bacterium cell, multiplies and progeny phages burst from the cell killing the bacterium. In the lysogenic cycle, the phage does not replicate but its genome goes into a quiescent condition where it is called prophage and where it is usually integrated into the host genome or alternatively it may be maintained as an extra chromosomal plasmid [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Miller | Contribution, pseudlysogeny and starvation to phage ecology[END_REF].

The lysogenic cycle allows the host cell to continue to survive and reproduce, the virus is reproduced in all of the cell's offspring. In the pseudo lysogenic life cycle, the phage does not undergo lysogeny nor does it show lytic cycle but it remains in a non active state. There are some phages which can enter either the lytic or lysogenic cycle. Phages that replicate only via the lytic cycle are known as virulent phages, while phages that replicate using both lytic and lysogenic cycles are known as temperate phages. In the lysogenic cycle, upon detection of cell damage, such as UV radiation light or certain chemical, the prophage is extracted from the bacterial chromosome in a process called prophage induction [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. After induction, viral replication begins via the lytic cycle.

The presence of phages in an environmental reservoir plays an essential role in the evolution of bacterial species. Thus, the interaction between phages and bacteria can contribute to trigger some environmental indirectly transmitted diseases by enabling the emergence of new clones of virulent pathogenic bacteria. For instance, Vibrio cholerae, the causative agent of cholera epidemics represents a paradigm for this process. In fact, the latter organism evolves from environmental non-pathogenic strains to highly pathogenic species by acquisition of virulent genes through the lysogenic life cycle in the phage-bacteria interactions [START_REF] Faruque | Phage-bacterial interactions in the evolution of toxigenic vibrio cholerae[END_REF]. The major virulence factors of V. cholerae which are cholera toxin (CT) and toxin coregulated pilus (TCP) are encoded by a lysogenic phage (CTXφ) and a pathogenicity island, respectively [START_REF] Faruque | Phage-bacterial interactions in the evolution of toxigenic vibrio cholerae[END_REF].

Hence, the importance of incorporating the lysogenic life cycle in the models that describe the interactions between phages and bacteria in the environmental reservoir with the ultimate aim to explain the triggering of bacterial related disease outbreaks. On the other hand, the presence of phages in an environmental reservoir of bacteria can purify this environment by driving the population of bacteria to extinction. It is also possible to control the proliferation of vibrio cholerae in the small intestine by ingesting the selected lytic/virulent phages. This last therapy is known as phage therapy. Phage therapy or phagotherapy is a therapeutic use of bacteriophages to treat pathogenic bacterial infection [START_REF] Sulakvelidze | Bacteriophage Therapy[END_REF]. Therefore, six main research questions come into play:

1. Under which conditions can the presence of phages purify a bacterial polluted environment?

2. In which situations can the presence of phages triggers virulent pathogenic bacterial disease outbreaks? 3. How does the phage-bacteria interaction influences the dynamic of bacteria-borne disease such as cholera? 4. How does the spatial distribution of phages and bacteria influence the reaction-diffusion spread of cholera? 5. Can the release of virulent/lytic phages be an important biological control to stop the spread of a water-borne disease such as cholera? [START_REF] Bakare | Optimal control analysis of an SIR epidemic model with constant recruitment[END_REF]. Is phage therapy be an effective strategy to control the spread of cholera?

This dissertation is a contribution toward finding possible answers to the above-mentioned research questions. To that end, it is organized as follows:

The first chapter is devoted to the biology of cholera and phage bacteria ecology. In chapter 2, we derive a model which takes into account both the lytic and lysogenic life cycles of phages, as well as the prophage induction. More precisely, since the genetic material of phages (called prophage) can be transmitted to bacterial daughter cells at each subsequent cell division, we propose a mathematical model that additionally takes into account the fact that in the lysogenic life cycle, the virus reproduces in all the cell's offsprings. The propounded model is a predatorprey like system with Holling type II functional response. We use it to provide possible responses to the above-mentioned research questions [START_REF] Abboubakar | Backward bifurcation and control in transmission dynamics of arboviral diseases[END_REF] and [START_REF] Abboubakar | Bifurcation thresholds and optimal control in transmission dynamics of arboviral diseases[END_REF]. The basic offspring number N 0 is computed and used to examine the global dynamics and perform an in-depth bifurcation analysis of the system and the three equilibria exhibited are topologically classified as follows:

An unstable environment-free equilibrium (EFE), a globally stable phage-free equilibrium (PFE) whenever N 0 < 1, and a unique locally stable environment-persistent (EPE) equilibrium which exists when N 0 > 1. We use a suitable Lyapunov function to estimate the basin of attraction of EPE. The model undergoes a trans-critical forward bifurcation at N 0 = 1 and a Hopf bifurcation around the EPE. Precisely, we show that when N 0 > 1, there is a critical value N c 0 such that for N 0 ≥ N c 0 , the EPE loses its stability through the appearance of a Hopf bifurcation, given rise to periodic solutions.

Chapter 3 is devoted to the coupled models to assess the impact of the phage-bacteria infection and spatial movement of phages, bacteria and humans on the spread of cholera.

The first model is a system of an ordinary differential equations (ODE) with the following assumptions: (i) bacteria interact with two types of phages (lytic and temperate), (ii) the phage-bacteria functional response similar to the function proposed by Smith in [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF]. The basic reproduction number R 0 is computed, existence and stability of equilibria is investigated.

We prove that the disease free equilibrium (DFE) is locally asymptotically stable whenever R 0 < 1. The system exhibits a bistability phenomenon via the existence of backward bifurcation, which implies that the classical epidemiological requirement for effective elimination of cholera, R 0 < 1, is no longer sufficient, even though necessary. Due to the existence of backward bifurcation, another threshold N 0 is determined, such that the DFE is globally asymptotically stable when both R 0 and N 0 are less than one, irrespective of their order of comparison. On the other hand, based on the range of R 0 and N 0 , the proposed model can exhibits one or more endemic equilibria. The presence of ecological threshold N 0 and existence of bistability scenario highlight the impact of phage-bacteria infection on the dynamic of cholera and provide an answer to research question [START_REF] Alexander | A Vaccination Model for Transmission Dynamics of Influenza[END_REF]. The phage absorption rate is identified to be the cause of backward bifurcation and in its absence the model exhibits a trans-critical forward bifurcation at R 0 = 1. Precisely, it is proven that there is no endemic equilibrium whenever R 0 < 1, and there exists a unique globally asymptotically stable endemic equilibrium whenever R 0 > 1.

To stress on the impact of phage-bacteria infection on the spread of spatiotemporal cholera dynamics and provide a possible response to research question (4), we extend the previous ODE model to a reaction-diffusion model. We first analyze the PDE model without absorption rate. Suitable Lyapunov functionals are constructed to prove the global stability of the constant steady solutions of our PDE. It is shown that the DFE E 0 is globally asymptotically stable whenever R 0 ≤ 1. On the contrary, whenever R 0 > 1, there exists an unique globally asymptotically stable endemic equilibrium E * . Secondly, we consider the full system with positive phage absorption rate. It is proved that the condition R 0 ≤ 1 is not sufficient, to eliminate cholera, however, by introducing a second threshold N 0 , we derived another Lyapunov functional to prove that the DFE is GAS whenever R 0 ≤ 1 and N 0 ≤ 1. Moreover, the DFE is locally asymptotically stable if R 0 ≤ 1 and the inequality R 0 > 1 gives rise to the uniform persistence of the full model. The discrete counterpart of the continuous model is developed to numerically support the theoretical results. It is built based on nonstandard finite difference scheme (NSFD) rules proposed by Mickens [START_REF] Mickens | Application of Nonstandard Finite Difference Schemes[END_REF][START_REF] Mickens | Advance in the Application of Nonstandard Finite Difference Schemes[END_REF]. The discretized model preserves the positivity, boundedness of the solutions of the continuous model. Moreover, by constructing discrete Lyapunov functionals, we showed that it preserves the global stability of equilibria as well.

We extend ODE model to controlled system with the use of virulent vibriophages (phages that prey on vibrio cholerae) to reduce or eliminate the vibrio cholerae population. The controlled model is used to set an optimal control problem and a response to research question [START_REF] Asheshov | from 1st January to 1st September[END_REF]. Using optimal control theory tools, we establish the conditions under which the spread of cholera can be curtained. Furthermore, we examine the impact of control measures on the transmission dynamic of cholera. The Pontryagin's maximum principle is used to characterize the optimal control. Numerical simulations suggest that, the release of lytic vibriophages can significantly reduce the spread of the disease. The investigations in work offer opportunities to phage therapy, which contrary to antibiotic therapy is a treatment of some bacterial-borne diseases without side effects. We formulated a controlled within-host cholera model with the ingestion of virulent vibriophages for therapeutic purpose. Our finding is that a polluted small intestine can be purified in the presence of virulent vibriophages. The latter provides an answer to research question [START_REF] Bakare | Optimal control analysis of an SIR epidemic model with constant recruitment[END_REF].

Chapter 1 Biological and mathematical backgrounds 1.1 Biological background

Cholera is a historically important disease causing many deaths globally and remains a very significant public health concern in several developing countries. Along with other old world bacterial diseases such as tuberculosis, typhoid, plague etc that have costed millions of lives, cholera is even today a considerable risk to public health. Though the number of deaths for tuberculosis seems higher they were coupled with HIV (Human Immunodeficiency Virus) disease, while typhoid shows similar deaths and impact as like cholera but doesn't take pandemic proportion. Thus, cholera remains a bacterial disease with pandemic potential. The world has witnessed several cholera pandemic with the major focus being the Indian subcontinent and with the current pandemic affecting Central and South America with rapid epidemic spread [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Roy | Epidemic cholera spreads like wildfire[END_REF]. The disease is endemic in Africa, Asia and South America, with the occurrence of outbreaks usually coinciding with and/or resulting from conflict and natural disasters. The disease shows seasonality in South Asia with most outbreaks being reported just before and after the monsoon i.e. April and September to December, respectively, with case numbers peaking during the summer season i.e. from January to February. In South America and in the African countries the outbreaks occur following summer rains or floods [START_REF] Emch | Seasonality of cholera from 1974 to 2005: a review of global patterns[END_REF]. The incidence of cholera cases in recent years is shown in the Fig 1 .1. The sudden surge of cases in the year 2011 is attributed to the Haiti epidemic following a devastating earthquake. Haiti still continues to contribute most of the cholera cases reported globally and it was estimated that 45.000 cases to be reported for the year 2014 [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. The most recent epidemic outbreak occurred in South Sudan with 1.812 cholera cases reported and 38 deaths (WHO 2014). Thus, in spite of extensive efforts by governmental and non-governmental agencies over the decades, cholera has not been eradicated as a major cause of epidemic disease and death. The CFR (Crude Fatality Rate i.e. percentage of people who die after being diagnosed positive for cholera) can be as high as 50% without treatment [START_REF] Sack | Suwanvanichkij V. Antimicrobial resistance in shigellosis, cholera and campylobacteriosis[END_REF][START_REF] Sack | [END_REF][START_REF] Sack | Experimental cholera. A canine model[END_REF]. Moreover, many cholera deaths remain unreported, owing to the remoteness of the communities in developing countries and a lack of communication and reporting infrastructure. Developed countries in Europe and North America usually have imported cases, reported from travelers visiting disease-prone areas and returning back with the disease. With appropriate sanitation measures in place, the developed countries are able to prevent cholera but there is a real need to address the possible measures for prevention and effective cholera treatment in developing countries.

Pathogenesis of cholera

Vibrio cholerae (V. cholerae) is the agent of cholera which is contracted through consumption of water or food contaminated with the organism mainly arising directly or indirectly from human faeces. V. cholerae produces an enterotoxin (CTX) which leads to acute, copious, watery diarrhea with as much as 20 liters of fluid loss per day and death due to circulatory failure owing to dehydration, if not treated in time. V. cholerae is a curved and motile Gram negative rod shaped organism with a single polar flagellum. The strains of disease-causing V. cholerae are grouped into two serogroups O1 and O139, based on their lipopolysaccharide (LPS) O antigens. The earlier pandemic were caused by members of the O1 serogroup, but the eighth pandemic was reported to be caused by O139 which originated from the Bay of Bengal [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF].

The major virulence genes needed for pathogenesis are clustered as two genetic elements viz: the genes encoded by the lysogenic filamentous phage CTX ϕ for cholera toxin and the genes encoded for the Toxin Co-regulated Pilus (TCP) [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Faruque | Phage-bacterial interactions in the evolution of toxigenic vibrio cholerae[END_REF]. Cholera toxin CT and TCP are controlled by a regulatory protein, ToxR which co-regulates their expression. Intestinal colonization by V. cholerae is mediated by fimbriae, which are filamentous protein structures.

TCP is essential for the colonization process. TCP attaches to receptors present on the mucous of upper small intestine and helps in colonization [START_REF] Faruque | Phage-bacterial interactions in the evolution of toxigenic vibrio cholerae[END_REF]. Other fimbriae (i.e. Type B and Type C) are non-adhesive and do not play a role in colonization of V. cholerae. After attachment the organism produces CT which causes secretory diarrhea. The CT causes increased chloride ion secretion and net water flow in to the gut lumen and decreased sodium ion absorption into the tissues via the blood stream. As a result there is rapid loss of water into the lumen along with chloride ions causing massive diarrhea and electrolyte imbalance [START_REF] Nakasone | Pili of a Vibrio cholerae O139[END_REF].

Treatment or prophylaxis of cholera 1.1.2.1 Rehydration therapy

Rehydration therapy is effective if replacement of fluids lost due to severe diarrhea are compensated as quickly as they are lost [START_REF] Sack | Suwanvanichkij V. Antimicrobial resistance in shigellosis, cholera and campylobacteriosis[END_REF][START_REF] Sack | Experimental cholera. A canine model[END_REF]. WHO recommends that oral re-hydration salts (ORS) (WHO 2002) which come in the standard sachets are useful but for severely dehydrated patients intravenous fluid replacements are needed.

Antibiotic therapy

Though re hydration therapy is the mainstay of treatment for cholera, oral antibiotics are given to dehydrated patients as soon as possible after vomiting stops. They are given to shorten the duration of illness and also to reduce the diarrhea fluid output. The combination of antibiotics gives a synergistic effect in treatment of cholera [START_REF] Sack | Suwanvanichkij V. Antimicrobial resistance in shigellosis, cholera and campylobacteriosis[END_REF][START_REF] Sack | [END_REF][START_REF] Sack | Experimental cholera. A canine model[END_REF]. The different types of antibiotics used and antibiotic resistance in vibrio cholerae are discussed further.

Vaccination

Soon after discovery of V. cholerae as an etiology of cholera, injectable parenteral vaccines (killed whole cell) were developed but these provided short lived immunity of just 6 months and frequently involved painful local inflammatory reactions. Later oral whole-cell/recombinant-B-subunit cholera vaccines (e.g. Dukoral) were developed which provide protection for up to 2 years (WHO 2010) for cholera and they also provide cross protection for enterotoxigenic E.coli (ETEC) for up to 6 months and are recommended by WHO. Though recent oral cholera vaccines are better than earlier parenteral vaccines they do not confer 100% protection but they reduce the risk by 80% and the immunity can be overcome by a high inoculum of infective organisms [START_REF] Sack | Suwanvanichkij V. Antimicrobial resistance in shigellosis, cholera and campylobacteriosis[END_REF][START_REF] Sack | [END_REF]. The oral cholera vaccines should not be taken as sole preventive measures in control of cholera disease in isolation from other measures.

Ecology of phages and bacteria

Bacteriophages and bacteria relationship

Bacteriophages are viruses that infect bacteria. They are obligate intracellular parasites which rely on the host bacterium in order to replicate. Bacteriophages enclose their nucleic acid in a protein coat (capsid), which may be further surrounded by a lipid layer [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. In addition to the capsid (head), tailed bacteriophage (members of the Caudovirales order), possess a tail which may either be contractile (e.g. T4 phage) or non-contractile (e.g. phage λ). They may also possess additional structures such as a collar, basal plate, spikes and tail fibres, which are involved in attachment to the bacterium and injection of the nucleic acid into the cell (Fig. 1.2).

Phages are ubiquitous on earth and are found in large numbers in the environment (i.e. water, soil, sewage etc.), wherever their hosts are present. Phages can remain viable under adverse conditions [START_REF] Guttman | Basic phage Biology and Applications[END_REF].

Life cycles of a bacteriophage

Phages exhibit three different lifestyles based on their survival strategies: lytic, lysogenic and pseudo-lysogenic [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Faruque | Phage-bacterial interactions in the evolution of toxigenic vibrio cholerae[END_REF]. In a lytic life cycle, the phage infects the cell, multiplies and progeny phages burst from the cell killing the bacterium. In a lysogenic life cycle, the phage does not replicate but its genome goes into a quiescent condition where it is called prophage and where it is usually integrated into the host genome or alternatively it may be maintained as an extrachromosomal plasmid [START_REF] Guttman | Basic phage Biology and Applications[END_REF]. There are some phages which can enter either the lytic cycle or the lysogenic cycle and they are called the temperate phages, while some phages are strictly lytic. The lysogenic life cycle allows the bacterial host cell to continue to survive and reproduce, the phage is reproduced in all of the cell's offspring. In the course of this division, the effect of UV radiations or the presence of certain chemicals can lead to the release of prophage causing proliferation of new phages through the prophage induction (Fig. 1.3). In a pseudo-lysogenic life cycle, the phage does not undergo lysogeny nor does it show a lytic response but it remains in a non-active state. Pseudo-lysogeny occurs during starvation conditions and when nutrient supplies are available again the phage can either enter the lysogenic or lytic life cycle. Apart from these three generally described life cycles, a carrier state life cycle (CSLC) is reported as an alternative phage life cycle in which bacteria and phages are in an equilibrium state with some bacteria resistant to phage but some of them sensitive to phage and thus allowing both of them to sustain.

Phage therapy or phagotherapy 1.3.1 History of phage therapy

The first antibacterial activity of phage was noticed against V. cholerae by a British bacteriologist Hankin in the year 1896, when he filtered water from the rivers in India to find that a substance which was heat labile could cause lysis of V. cholerae and he thought that it could have limited the cholera epidemic. Later, bacteriophages were discovered independently by Twort and d'Herelle. The term bacteriophage meaning bacteria eater was coined by d'Herelle.

As reviewed by Summers (Summers 2005), the therapeutic potential of phages to treat bacterial He also conducted experiments with claims of effectiveness against Salmonella pullorum and Pasteurella infections of animals. In 1926 d'Herelle self administered phage preparations and gave them to his colleagues and family members to ascertain their safety before administering them to human patients in treating bacillary dysentery. After he treated four patients of bubonic plague in Egypt, the British Government invited him to the Haffkine Institute, Bombay, India for Asiatic cholera treatment studies. Later, phages were commercially exploited by pharmaceutical companies for some time with exaggerated claims and marketing e.g. the Enterophagos preparation was marketed for treatment of herpes infections, urticaria and eczema which was clearly not possible. By the end of the 1940, the advent of antibiotics accompanied by the poor quality of some of the scientific studies arising from an inadequate understanding of the phage-host interactions led to phage therapy being discontinued in the western world.

Nevertheless, phage therapy continued to be used in Russia and Eastern Europe [START_REF] Guttman | Basic phage Biology and Applications[END_REF].

Reappraisal of phage therapy

Phage therapy was again revived during the 1980's. Rigorous clinical phage therapy experiments were carried out by H. Williams Smith and his colleagues on oral infections of enterotoxigenic Escherichia coli (ETEC) diseases of neonatal animals and on systemic infection caused by E. coli in mice [START_REF] Keen | Phage Therapy: concept to cure, front[END_REF][START_REF] Sack | Suwanvanichkij V. Antimicrobial resistance in shigellosis, cholera and campylobacteriosis[END_REF][START_REF] Sack | Experimental cholera. A canine model[END_REF][START_REF] Sulakvelidze | Bacteriophage Therapy[END_REF]. They successfully treated experimental systemic E. coli infection in mice using bacteriophage comparing the effect with antibiotics. They tested 15 phages isolated from sewage, out of which 9 were anti-K1 (K1 is an antigen which is an important surface virulence factor of E. coli strain O18ac : K1 : H7ColV + ). Administered intramuscularly, the most effective phage was also the one found to be most rapidly lytic in vitro. Phage prevented death and illness in mice inoculated intramuscularly (in a different muscle) or intracranially with the bacterium. There was evidence of phage multiplication in vivo and a single dose of phage was more effective than 8 doses of streptomycin [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. Some resistant mutants arose but these were largely K1-negative mutants and thus of reduced virulence.

Similar studies have been performed with E.coli septicaemia and meningitis in chickens and colostrum-deprived calves. As evidenced to be effective for treatment of meningitis, the phage therapy can be used in central nervous system diseases as phages can cross the blood brain barrier. Smith's studies on enteritis involved enterotoxigenic E. coli infection in neonatal calves, pigs and sheep. The pathogenesis of ETEC is virtually identical to that of cholera with adhesion to the small intestinal mucosa and production of a toxin affecting cAMP levels. In these cases, phages were sought which would attach to the surface virulence determinants K88 and K99, but without success, and phages attaching to LPS were used. Phages were used singly and in combination, the latter to ensure ability to control phage resistant mutants that arose against single phage use. Phages were used successfully prophylactically and also therapeutically such that administration could be delayed until the onset of diarrhea. They could also be used to spray bedding which was also effective in preventing clinical disease after administration of the pathogen. Although, the interest in phage therapy in the West was reinvigorated by these very rigorous experiments, this approach had not taken hold as might have been expected given its experimental success but there is now a resurgent interest owing to the emergence of antibiotic resistance in recent years [START_REF] Chopra | The search for antimicrobial agents effective against bacteria resistant to multiple antibiotics[END_REF].

In recent years phage therapy was attempted for several other bacterial species. Salmonella spp, Campylobacter spp, Listeria monocytogenes, P. aeruginosa, Staphylococcus aureus, Klebsiella pneumonia, Clostridium difficile, Enterococcus faecalis, etc. There are increasing reports on the therapeutic utility of phages leading to development of a substantial knowledge bank for harnessing the true potential of phages in curing bacterial diseases [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF].

Phage therapy of cholera in human

Fluid replacement therapy is often effective for cholera, if given in the earlier stages. However, due to poor infrastructure and facilities in many countries where cholera is endemic, many patients during cholera outbreaks present with advanced stages of the disease. In these cases the case fatality rate is usually high and also there are limitations on the use of vaccines for cholera to prevent such situations. In addition, fluid replacement therapy results in extensive shedding of the pathogens during treatment and simultaneous use of antibiotics such as tetracycline can increase the risk of development of resistance. Thus, there is a need for a rapid and more effective alternative treatment for cholera and that could be the use of phage therapy [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF].

The first study for phage therapy against cholera disease was reported by Felix d'Herelle.

During this study when the cholera patients were treated with oral doses of bacteriophage the mortality rate was 8.1%, while in the controls with other medicines it was 62.9%. The mortality rate in the phage treated group was zero if treatment occurred within 6 hours of appearance of the first symptoms [START_REF] Herelle | Studies on Asiatic cholera[END_REF]. It was also reported that between 1928 and 1931 phage was used successfully to treat cholera cases in the North Eastern region of India. During the same period of time Asheshov and colleagues reported successful treatment of patients in one location although their treatment with phage was unsuccessful in another location. The authors mentioned that although the phage was able to arrest the progress of disease it was more effective used as a prophylactic rather than a therapeutic [START_REF] Asheshov | from 1st January to 1st September[END_REF]. In the years 1958 and 1960, animal passaged phage preparations were successfully used in treating cholera patients in Afghanistan. An initial intravenous or intramuscular phage administration with saline followed by oral administration for three days gave satisfactory results.

The WHO reported the studies which assessed the effectiveness of phage therapy for cholera.

Monsur and coworkers treated eight patients of cholera with large doses of bacteriophage with high titre phage (10 1 2PFU/ml). They compared their results with 50 patients as control group treated with intravenous fluid alone and also with 18 patients treated with tetracycline. Four out of eight phage treated patients showed a rapid decline of V. cholerae numbers in stool samples as observed by means of dark field microscopy. The same four patients also showed appreciable reduction in the stool output and duration of diarrhea compared to the control group. In another four patients treated with phage, V. cholerae numbers declined more slowly and the volume of the stool as well as the duration of diarrhea was greater than the control group. Overall, the treatment with bacteriophage was partly (50%) effective based on the rate of decline of bacterial numbers, the stool output and duration of diarrhoea but was not as effective as the treatment with tetracycline [START_REF] Asheshov | from 1st January to 1st September[END_REF][START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF] Marcuk and colleagues performed studies using phage preparations of between 10 8 and 10 9 PFU/ml given both orally and intramuscularly to adult as well as pediatric patients. They also compared their results with tetracycline treatment and with a placebo control. The posttreatment stool output (9.3 and 2.4 litres in phage treated; while 1.7 and 1.2 litres in tetracycline treated for adults and children, respectively), the duration of diarrhea (76 and 52 h in phage treated; while 34 and 39 h in tetracycline treated for adults and children, respectively) and the duration of positive culture from stool samples (4.3 and 3.2 days in phage treated; while 0.7 and 0.8 days in tetracycline treated for adults and children, respectively) were all significantly less in the tetracycline treated group as compared to the phage treated and placebo groups.

The authors mentioned that although phage therapy was promising, it did not work in their clinical trial. They reasoned that this might be due to relatively low dosage used in their trial compared to the earlier work of Monsur (1970) [START_REF] Asheshov | from 1st January to 1st September[END_REF][START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. Until Smith's group carried out his animal studies, there was very little known about the phage host interactions. Smith suggested that the phages of highest virulence in vitro should be used for in vivo studies. Previously, phage therapy experiments were poorly designed and in some cases phages were poured in to drinking water wells for control of cholera. Also, the use of phage prophylaxis led to neglect of basic hygiene measures and thus, when antibiotics were successful in treating cholera, phage investigations were discontinued [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF].

Ideal properties of phage for phage therapy

In order to maximize the potential success of bacteriophage therapy, the phages selected should fulfil a number of criteria. First, the phage should have the ability to infect a wide range of susceptible host bacteria (broad host range/lytic profile) and replicate only through the lytic life cycle. In addition, the phage should have a short latent period and high burst size. The phage should not carry genes which encode virulence factors or are involved in antibiotic resistance. Thus, temperate phages which possess many of such characteristics should be avoided for phage therapy [START_REF] Gill | Phage choice, isolation, and preparation for phage therapy[END_REF]. Phages should ideally be selected which attach to specific surface virulence determinants such that any resistant mutants that emerge would show reduced virulence or a virulence [START_REF] Smith | Successful treatment of experimental Escherichia coli infections in mice using phage: its general superiority over antibiotics[END_REF].

Characterization of bacteriophage 1.3.2.1 Biological characterization

To identify the ideal properties of phage for phage therapy, the host range, latent period and burst size of the candidate phage are needed. The host range profile of the phage is defined by lysis of a range of bacterial host strains. The host growth curve provides a mid exponential phase and generation time which is further utilized to devise a protocol for phage one step growth curve which will then provide an estimation of the burst size and latent period.

Physical characterization

Although bacteriophage can be seen under ordinary and phase microscope, an electron microscopy is used and can magnify the image up to 400,000 times. Electron microscopy allows the morphological classification of bacteriophage. As per the International Committee on Taxonomy of Viruses (ICTV), the phages are classified as one order, ten families and forty genera (ICTV 2011). The symmetries of the phage classified are binary (having two divisions), cubical, helical or pleomorphic. Most of the phage genome may have double stranded DNA but a few may contain single stranded DNA and even double or single stranded RNA as their genome. A few phage may have a lipid envelope surrounding the capsid. Most of the phage are classified in to the order Caudovirales which have binary symmetry and are tailed phage. There are three phylogenetically related families in the order Caudovirales. The contractile-tailed Myoviridae, the long non-contractile tailed Siphoviridae and the short tailed Podoviridae [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Gill | Phage choice, isolation, and preparation for phage therapy[END_REF].

Genomic characterization

In order to differentiate phages, the genome size is estimated by pulsed field gel electrophoresis (PFGE) and restriction analysis is done using different restriction enzymes [START_REF] Gill | Phage choice, isolation, and preparation for phage therapy[END_REF]. Using genome sequencing, the potential phage therapy candidates can be screened for harmful genes associated with virulence, antimicrobial resistance or lysogeny-related genes. Pulse field gel electrophoresis is useful for sizing large DNA fragments; in this technique, the direction of an electric field is periodically switched which helps in separation of DNA fragments of up to 5Mb [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Gill | Phage choice, isolation, and preparation for phage therapy[END_REF].

The sequencing of microbial genomes has become more efficient and cost-effective over the past decade, resulting in a significant increase in the number of published genomes in online databases such as Gen bank. Notwithstanding these advances, the number of bacteriophage genomes available in such databases is approximately 750, as compared with over 5.000 indi-vidual phages which have been isolated. High throughput sequencing platforms have been introduced since 2005. After next generation sequencing, genome annotation is done to identify predicted open reading frames (ORFs) or CDSs (CoDing Sequences) which can code for particular proteins. By assigning the functions to different phage genes through CDSs provides an insight in the phage type which we are dealing with so that we can decide whether to use it for phage therapy purpose or not [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF][START_REF] Gill | Phage choice, isolation, and preparation for phage therapy[END_REF].

Mathematical background 1.4.1 Dynamical system (ODE)

Let Ω ⊂ R 1+n be an open connected set. We will denote points in Ω by (t, x) where t ∈ R and x ∈ R n . Let f : Ω → R n be a continuous map. In this context, f (t, x) is called a vector field on Ω.

Given any initial point (t 0 , x 0 ) ∈ Ω, we wish to construct a unique solution to the initial value problem:

dx dt = f (t, x(t)), x(t 0 ) = x 0 . (1.4.1)
In order for this to make sense, x(t) must be a C 1 function from some interval I ⊂ R containing the initial time t 0 into R n such that the solution curve satisfies

{(t, x(t)) : t ∈ I} ⊂ Ω.
Such a solution is referred to as a local solution when I R. When I = R, the solution is called global.

Theorem 1.4.1 (Peano)

If f : Ω → R n is continuous, then for every point (t 0 , x 0 ) ∈ Ω the initial value problem (1.4.1) has local solution.

Proof:[85]

The problem with this theorem is that it does not guarantee uniqueness. We will skip the proof, except to mention that it is uses a compactness argument based on the Arzela-Ascoli Theorem.

Theorem 1.4.2 (Picard)

Let Ω ⊂ R 1+n be open. Assume that f : Ω → R n is continuous an that f (t, x) is locally Lipschitz continuous in x. let K ⊂ Ω be any compact set. Then there is a number δ > 0 such that for every Proof: Suppose first that t ∈ [t 0 , β). Define

(t 0 , x 0 ) ∈ K,
F(t) = c 0 + t t 0 ϕ(s) f (s)ds.
Then F is C 1 and

F (t) = ϕ(t) f (t) ≤ ϕ(t)F(t), for t ∈ [t 0 , β), since f (t) ≤ F(t
f (τ) ≤ F(τ) ≤ c 0 exp τ t 0 ϕ(s)ds,
for τ ∈ [t 0 , β). For (α, t 0 ], perform the analogous argument to the function

G(t) = c 0 + t 0 t ϕ(s) f (s)ds.

Stability

Let Ω = R × O for some set O ⊂ R n and suppose that f : Ω → R n satisfies the hypothesis of the Picard theorem.

: U → R such that (i) E ∈ C(U) ∩ C 1 (U \ x), (ii) E(x) > 0 for x ∈ (U \ x) and E( x) = 0, (iii) DE(x) f (x) ≤ 0 for x ∈ (U \ x).
If strict inequality holds in (iii), then E is called a strict Lyapunov function.

Theorem 1.4.9 If an equilibrium point x of f has a Lyapunov function, then it is stable. If x has a strict Lyapunov function, then it is asymptotically stable.

Proof: Suppose that E is a Lyapunov function for x. Choose any ε > 0 such that Bε ( x) ⊂ U.

Define

m = min{E(x) :|| x -x ||= ε} and U ε = {x ∈ U : E(x) < m} ∩ Bε ( x). Notice that U ε ⊂ U is a neighborhood of x.
The claim is that for any x 0 ∈ U ε , the solution x(t) = x(t, 0, x 0 ) of the IVP, x = f (x), x(0) = x 0 is defined for all t ≥ 0 and remains in U ε . By the local existence theorem and continuity of x(t),

we have that x(t) ∈ U ε on some nonempty interval of the form [0, t). Let [0, T) be the maximal such interval. The claim amounts to showing that T = ∞. On the interval [0, T), we have that

x(t) ∈ U ε ⊂ U and since E is a Lyapunov function, d dt E(x(t)) = DE(x(t)).x (t) = DE(x(t)). f (x(t)) ≤ 0.
From this it follows that We now use the claim to establish stability. Let ε > 0 be given. Without loss of generality, we may assume that Bε ( x) ⊂ U. Choose δ > 0 so that B δ ( x) ⊂ U. Then for every x 0 ∈ B δ ( x), we have that x(t) ∈ U ε ⊂ B ε ( x), for all t > 0. Suppose now that E is a strict Lyapunov function, and let us prove asymptotic stability.

E(x(t)) ≤ E(x(0)) = E(x 0 ) < m, on [0, T). So, if T < β, we would have E(x(t)) ≤ E(x(0)) = E(x 0 ) < m,
The equilibrium x is stable, so given ε > 0 with B ε ( x) ∈ U, there is a δ > 0 so that x 0 ∈ B ε ( x) implies x(t) ∈ B ε ( x), for all t > 0.

Let x 0 ∈ B δ ( x). We must show that x(t) = x(t, 0, x 0 ) satisfies lim t→∞ x(t) = x. We may assume that x 0 x, so that, by uniqueness, x(t) x, on [0, ∞). Since E is strict and x(t) x, we have that

d dt E(x(t)) = DE(x(t)).x (t) = DE(x(t)). f (x(t)) < 0.
Thus, E(x(t)) is a monotonically decreasing function bounded below 0. Set E * = inf{E(x(t)) : t > 0}. Since the solution x(t) remains in the bounded set U ε , it has a limit point. That is, there exist a point z ∈ B ε ⊂ U and a sequence of times t k → ∞ such that x(t k ) → z. We have, moreover, that

E * = lim k→∞ E(x(t k )) = E(z). Let s > 0.
By the properties of autonomous flow, we have that

x(s + t k ) = x(s + t k , 0, x 0 ) = x(s, 0, x(t k , 0, x 0 )) = x(s, 0, x(t k )).
By continuous dependence on initial conditions, we have that

lim k→∞ x(s + t k ) = lim k→∞ x(s, 0, x(t k )) = x(s, 0, z).
From this and the fact that E(x(s, 0, z)) is nonincreasing, it follows that

E * ≤ lim k→∞ E(x(s + t k )) = E(x(s, 0, z)) ≤ E(x(0, 0, z)) = E * .
Thus, x(s, 0, z) is a solution along which E is constant. But

0 = d dt E(x(t, 0, z)) = DE(x(t, 0, z)). f (x(t, 0, z)).
By assumption, this forces x(t, 0, z) = x for all t ≥ 0, and thus, z = x. We have shown that the unique limit point of x(t) is x, which is equivalent to lim t→∞ x(t) = x.

Center manifolds and bifurcation theory

Definition 1.4.10 Let F : R n → R n , be a C 1 vector with F(0) = 0. A center manifold for F at 0 is an invariant manifold containing 0 which is tangent to and of the same dimension as the center subspace of DF(0).

Assume F : R n → R n is a C 1 with F(0) = 0. Set A = DF(0)
, and let E s , E u and E c be its stable, unstable and center subspaces with their corresponding projections P s , P u and P c . Assume that E c 0, there exists constants C 0 , λ, d ≥ 0 such that

|| expAtP s x ||≤ C 0 e -λt || P s x ||, t ≥ 0 || expAtP u x ||≤ C 0 e λt || P u x ||, t ≤ 0 || expAtP c x ||≤ C 0 1 + |t| d || P s x ||, t ∈ R. Write F(x) = Ax + f (x). Then f : R n → R n is C 1 , f (0) = 0, and D f (0) = 0. Moreover we assume that || f || C 1 = sup x∈R n || f (x) || + || D f (x) || ≤ M.
This restriction will be removed later, at the expense of somewhat weakening the conclusions of the next result. As usual, we denote by x(t, x 0 ) the solution of the initial value problem

x = Ax + f (x), x(0) = x 0 .
Thanks to the strong bound assumed for the nonlinear portion of the vector field, the flow is globally defined for all initial points x 0 ∈ R n .

Theorem 1.4.11 (Center Manifold Theorem)

There exists a function η with the following properties:

(i) η : E c → E s + E u is C 1 , η(0) = 0, and Dη(0) = 0.

(ii) The set W c (0) = {x 0 ∈ R n : P s x 0 + P u x 0 = η(P c x 0 )} is invariant under the flow.

(iii) If x 0 has the property that there exists 0 < α < λ and C > 0 such that

|| P s x(t, x 0 ) ||≤ Ce -αt , ∀t < 0,
and

|| P u x(t, x 0 ) ||≤ Ce αt , ∀t > 0, then x 0 ∈ W c (0).
(iv) If x 0 ∈ W c (0), then w(t) = P c x(t, x 0 ) solves w = Aw + P c f (w + η(w)), w(0) = P c x 0 .

Proof: [START_REF] Sideris | Ordinary Differential Equations and Dynamical Systems[END_REF] Remark 1.4.12 It follows from (i) and (ii) that W c (0) is a center manifold.

We obtain the following result.

Corollary 1.4.13 (Local center manifold theorem)

Supose that f : R n → R n is C 1 with f (0) = 0 and D f (0) = 0. There exists a function η and a small neighborhood U = B r (0) ⊂ R n with the following properties:

(i) η : E c → E s + E u is C 1 , η(0) = 0, and Dη(0) = 0. (ii) The set W loc c (0) = {x 0 ∈ U : P s x 0 + P u x 0 = η(P c x 0 )}
is invariant under the flow in the sense that if x 0 ∈ W loc c (0), then x(t, x 0 ) ∈ W loc c (0) as long as x(t, x 0 ) ∈ U. (iv) If x 0 ∈ W c (0), then w(t) = P c x(t, x 0 ) solves w = Aw + P c f (w + η(w)), w(0) = P c x 0 .

as long as x(t, x 0 ) ∈ U. Definition 1.4.14 A set W loc c (0) which satisfies (i) and (ii) is called local center manifold.

The following theorem guaranteed an approximation of the center manifold. Let U ∈ E c be a neighborhood of the origin. Let h : U → E s + E u be a C 1 mapping with h(0) = 0 and

Dh(0) = 0. If for x ∈ U, Ah(x) + (P s + P u ) f (x + h(x)) -Dh(x)[Ax + P c f (x + h(x))] = O(|| x || k ),
as || x || k → 0, then there is a a C 1 mapping η : E c → E s + E u with e ta(0) = 0 and Dη(0) = 0 such that

η(x) -h(x) = O(|| x || k ),
as || x ||→ 0, and

{x + η(x) : x ∈ U}
is a local center manifold.

Proof: [START_REF] Sideris | Ordinary Differential Equations and Dynamical Systems[END_REF].

The following theorem from [START_REF] Chavez | Dynamical models of tuberculosis and their application[END_REF] is used to determined whether the subcritical and trans-critical bifurcation exist on a center manifold.

Theorem 1.4. [START_REF] Chakraborty | Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting[END_REF] Consider the following general system of ordinary differential equations with a parameter τ.

dx dt = f (x, τ), f : R n × R → R n and f ∈ C 2 (R n × R),
where 0 is an equilibrium point of the system (that is, f (0, τ) = 0 for all τ) and

1. A = D x f (0, 0) = ∂ f i
∂x j (0, 0) is the linearization matrix of the system around the equilibrium 0 with τ evaluated at 0, 2. zero is a simple eigenvalue of A and all other eigenvalues of A have negative real part, 3. matrix A has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.

Let f k be the kth component of f and

a = n k,i,j=1 v k w i w j ∂ 2 f k ∂x i ∂x j (0, 0) b = n k,i=1 v k w i ∂ 2 f k ∂x i ∂τ (0, 0).
The local dynamics of the model around 0 is determined by the signs of a and b i. a > 0; b > 0, when τ < 0 with |τ| 1; 0 is locally asymptotically stable, and there exists a nonnegative unstable equilib-rium; when 0 < τ 1; 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.

ii. a < 0; b < 0, when τ < 0 with |τ| 1; 0 is unstable; when 0 < τ 1; 0, is locally asymptotically stable, and there exists a non-negative unstable equilibrium.

iii. a > 0; b < 0, when τ < 0 with |τ| 1; 0 is unstable, and there exists a locally asymptotically stable negative equi-librium; when 0 < τ 1; 0 is stable and a non-negative unstable equilibrium appears.

iv. a < 0; b > 0, when τ changes from negative to non-negative, 0 changes its stability from stable to unstable. Correspond-ingly, a negative unstable equilibrium becomes non-negative and locally asymptotically stable.

Proof: [START_REF] Chavez | Dynamical models of tuberculosis and their application[END_REF].

The Hopf bifurcation occurs when a pair of distinct complex conjugate eigenvalues ofan equilibrium point cross the imaginary axis as the bifucation parameter is varied. At the critical bifurcation value, there are two (nonzero) eigenvalues on the imaginary axis. So this is an example of a co-dimension two bifurcation. As the bifurcation parameter crosses the critical value, a periodic solution is created.

The following presents a criterion for a class of Hopf bifurcation using the properties of coefficients of characteristic equations instead of those of eigenvalues. It is related to the Routh-Hurwitz criterion and is convenient in many applications.

consider the system dx dt = f µ (x), x ∈ R n , µ ∈ R,
with the equilibrium (x 0 , µ 0 ), and f ∈ C ∞ . Assume that (SH1): the jacobian matrix D x f µ 0 (x 0 ) has a simple pair of purely imaginary eigenvalues and others eigenvalues have negative real parts. Then there is a smooth curve (x(µ), µ) with

x(µ 0 ) = x 0 . The eigenvalues λ(µ), λ(µ) of J(µ) = D x f µ (x(µ)) which are purely imaginary at µ = µ 0 vary smoothly with µ. Moreover, if

(SH2): d(Re(λ(µ 0 ))) dµ 0 (transversality condition)
then there is a simple Hopf bifurcation.

Let us denote the characteristic polynomial of the Jacobian matrix J(µ) by

P(λ, µ) = det λI n -J(µ) = p 0 (µ) + p 1 (µ)λ + ... + p n (µ)λ n ,
where every p i (µ) is a smooth function of µ, p n (1) = 1, and we can restrict ourselves to the case p 0 (µ) > 0 because there is no any non-negative real root. Let

L n (µ) =               p 1 (µ) p 0 (µ) • • • 0 p 3 (µ) p 2 (µ) • • • 0 . . . . . . . . . . . . p 2n-1 (µ) p 2n-2 (µ) • • • p n (µ)              
, where p i (µ) if i < 0 or i > n. The Routh-Hurwitz criterion can be stated as that when p 0 (µ) > 0, the polynomial P(λ, µ) of λ has all roots with negative real parts if and only if the following n polynomial subdeterminants of L n (µ) are positive:

D 1 (µ) = det(L 1 (µ)) = p 1 (µ) > 0, D 2 (µ) = det(L 2 (µ)) = p 1 (µ) p 0 (µ) p 3 (µ) p 2 (µ) , • • • , D n (µ) = det(L n (µ)) > 0. Since D n (µ) = p n (µ)D n-1 (µ)
, and in our case p n (µ) = 1, the Routh-Hurwitz conditions can be expressed as

p 0 > 0, D 1 > 0, D 2 > 0, • • • , D n-1 > 0.
Now we can write the criterion for simple Hopf bifurcations as follows.

Theorem 1.4.17 Assume there is a smooth curve of equilibria (x(µ), µ) with x(µ 0 ) = x 0 for the system.

Conditions (SH1) and (SH2) for a simple Hopf bifurcation are equivalent to the following conditions on the coefficients of the characteristic polynomial P(λ, µ):

(CH1): p 0 (µ 0 ) > 0, D 1 (µ 0 ) > 0, D 2 (µ 0 ) > 0, • • • , D n-2 > 0, D n-1 = 0, and
(CH2): dD n-1 (µ 0 ) dµ 0.

Note on the basic reproduction number R 0

The basic reproduction number, denoted R 0 , is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual's. If R 0 < 1, then on average an infected individual produces less than one new infected individual over the course of its infectious period, and the infection cannot grow. Conversely, if R 0 > 1, then each infected individual produces, on average, more than one new infection, and the disease can invade the population [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF]. For the case of a single infected compartment, R 0 is simply the product of the infection rate and the mean duration of the infection. However, for more complicated models with several infected compartments this simple heuristic definition of R 0 is insufficient. A more general basic reproduction number can be defined as the number of new infections produced by a typical infective individual in a population at a DFE [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF]. In order to compute the threshold R 0 , we use the following method proposed in [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF].

Consider a heterogeneous population whose individuals are distinguishable by age, behavior, spatial position and/or stage of disease, but can be grouped into n homogeneous compartments. A general epidemic model for such a population is developed in this section. Let

x = (x 1 , • • • , x n ) t
with each x i ≥ 0, be the number of individuals in each compartment. For clarity we sort the compartments so that the first m compartments correspond to infected individuals.

The distinction between infected and uninfected compartments must be determined from the epidemiological interpretation of the model and cannot be deduced from the structure of the equations alone, as we shall discuss below. The basic reproduction number can not be determined from the structure of the mathematical model alone, but depends on the definition of infected and uninfected compartments. We define X s to be the set of all disease free states. That is

X s = {x ≥ 0 : x i = 0, i = 1, • • • , m} .
In order to compute R 0 , it is important to distinguish new infections from all other changes in population. Let F i (x) be the rate of appearance of new infections in compartment i, V + i (x) be the rate of transfer of individuals into compartment i by all other means, and V - i (x) be the rate of transfer of individuals out of compartment i. It is assumed that each function is continuously differentiable at least twice in each variable. The disease transmission model consists of nonnegative initial conditions together with the following system of equations:

dx dt = F i (x) -V i (x), i = 1, • • • , n, (1.4.2)
where

V i (x) = V - i (x) -V + i (x)
and and the functions satisfy assumptions (A1)-(A5) described below. Since each function represents a directed transfer of individuals, they are all nonnegative. Thus,

(A1): if x ≥ 0, then F i (x), V + i (x), V - i (x) ≥ 0 for i = 1, • • • , n.
If a compartment is empty, then there can be no transfer of individuals out of the compartment by death, infection, nor any other means. Thus,

(A2): if x i = 0, then V - i (x) = 0. In particular, if x ∈ X s then V - i (x) = 0 for i = 1, • • • , m.
Consider the disease transmission model given by [START_REF] Abboubakar | Backward bifurcation and control in transmission dynamics of arboviral diseases[END_REF]

with f i (x), i = 1; • • • ; n, satisfying condi- tions (A1) and (A2). If x i = 0, then f i (x) ≥ 0 and hence, the nonnegative cone (x i ≥ 0, i = 1; ...; n)
is forward invariant. For each nonnegative initial condition there is a unique, nonnegative solution. The next condition arises from the simple fact that the incidence of infection for uninfected compartments is zero.

(A3): F i (x) = 0 if i > m.
To ensure that the disease free subspace is invariant, we assume that if the population is free of disease then the population will remain free of disease. That is, there is no (density independent) immigration of infective. This condition is stated as follows:

(A4): if x ∈ X s then F i (x) = 0 and V + i (x) = 0 for i = 1, • • • , m.
The remaining condition is based on the derivatives of f near a DFE. For our purposes, we define a DFE of (1.4.2) to be a (locally asymptotically) stable equilibrium solution of the disease free model, i.e., (1.4.2) restricted to X s . Note that we need not assume that the model has a unique DFE. Consider a population near the DFE x 0 . If the population remains near the DFE (i.e., if the introduction of a few infective individuals does not result in an epidemic) then the population will return to the DFE according to the linearized system

dx dt = D f (x 0 )(x -x 0 ), (1.4.3)
where D f (x 0 ) is the derivative ∂ f i /∂x j evaluated at the DFE, x 0 (i.e., the Jacobian matrix). Here,

and in what follows, some derivatives are one sided, since x 0 is on the domain boundary. We restrict our attention to systems in which the DFE is stable in the absence of new infection. That is,

(A5): if F (x 0
) is set to zero, then all eigenvalues of D f (x 0 ) have negative real part.

The conditions listed above allow us to partition the matrix D f (x 0 ) as shown by the following lemma.

Lemma 1.4.18 If x 0 is a DFE of (1.4.2) and f i (x) satisfies (A1)-(A5), then the derivatives DF (x 0 ) and DV(x 0 ) are partitioned as

DF (x 0 ) = F 0 0 0 , DV(x 0 ) = V 0 J 3 J 4 ,
where F and V are the m × m matrices defined by

F = ∂F i ∂x j (x 0 ) V = ∂V i ∂x j (x 0 ) with1 ≤ i, j ≤ m.
Further, F is non-negative, V is a non-singular M-matrix and all eigenvalues of J 4 have positive real part.

Proof: [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF].

We call FV -1 the next generation matrix for the model and set

R 0 = ρ FV -1 , (1.4.4)
where ρ(A) is the spectral radius of a matrix A.

Theorem 1.4. [START_REF] Chopra | The search for antimicrobial agents effective against bacteria resistant to multiple antibiotics[END_REF] Consider the disease transmission model given by (1.4.2) with f (x) satisfying condi-

tions (A1)-(A5). If x 0 is a DFE of the model, then x 0 is locally asymptotically stable if R 0 < 1, but unstable if R 0 > 1, where R 0 is defined by (1.4.

2).

Proof: [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF].

Optimal control theory

The basic principle of optimal control is to apply an external force, the control , to a system of differential equations, the state equations , to cause the solution, the state , to follow a new trajectory and/or arrive at a different final state. The goal of optimal control is to select a particular control that maximizes or minimizes a chosen objective functional, the pay-off; typically a function of the state and the control. The pay-off is chosen such that the new trajectory/final state are preferred to that of the uncontrolled state, accounting for any cost associated with applying the control [START_REF] Sharp | Optimal control of acute myeloid leukaemia[END_REF].

A typical optimal control problem will introduce the state equations as functions of the state x(t) and the control u(t), with initial state x(0) = x 0 .

dx dt = f (t, x(t), u(t)), x(t) ∈ R n . (1.4.5)
It is also necessary to specify either a final time t f with the final state free, or a final state x(t f ), with the final time free.

A pay-off function J is defined as a function of the final state, x(t f ), and a cost function

L(t, x(t), u(t)) integrated from initial time (t 0 ) to final time (t f ).
Through choosing an optimal control u * (t) and solving for the corresponding optimal state x * (t), we seek to maximize or minimize this objective function. Selecting the pay-off enables us to incorporate the context of our application and determine the meaning of optimality. In general, the pay-off function can be written as,

J = φ(x(t f )) + t f t 0 L(t, x(t), u(t)).dt (1.4.6)
Depending on the form of φ , it may be possible to incorporate φ into L by restating the ?nal state constraint in terms of an integral expression using the Fundamental Theorem of Calculus, and noting that φ(x(t 0 )) is constant and hence does not impact the optimal control.

The resulting unconstrained optimal control problem is often more straightforward to solve than the constrained problem.

The optimal control can be found by solving necessary con-ditions obtained through application of Pontryagin's Maximum Principle (PMP) [START_REF] Pontryagin | the mathematical theory of optimal proceses[END_REF], or a necessary and sufficient condition by forming and solving the Hamilton-Jacobi-Bellman partial differential equation, a dynamic programming approach. In this thesis we use the PMP and we construct the Hamiltonian,

H(t, x, u, λ) = L(t, x, u) + λ f , where λ = [λ 1 (t), λ 2 (t), ..., λ n (t)
] are the adjoint variables for an n-dimensional state. The adjoint is analogous to Lagrange multipliers for unconstrained optimisation problems. Through the Hamiltonian, the adjoint allows us to link our state to our pay-off function. The necessary conditions can be expressed in terms of the Hamiltonian.

(1) The optimality condition is obtained by minimizing the Hamiltonian,

∂H ∂u = 0, gives L ∂u + λ ∂ f ∂u = 0.
(2) The adjoint, also referred to as co-state , is found by setting,

∂H ∂x = - dλ dt , giving dλ dt = - ∂L ∂x + λ ∂ f ∂x and
(3) Satisfying the transversality condition,

λ(t f ) = ∂φ ∂x | t=t f .

Reaction diffusion system (PDE)

Let X be a Banach space and let us consider the evolution system defined by the differential

equation        du dt = ψu + F(t, u), t > 0 u(0) = u 0 , (1.4.7)
where, u 0 ∈ X, ψ : D(ψ) ⊂ X → X and F : R + × X → X.

Definition 1.4.20 [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] A family (T(t)) t≥0 of bounded linear operators such that T(t) : X → X for all t ≥ 0, is a strongly continuous semigroup of bounded linear operators if the following conditions hold:

(i) T(0) = id X (ii) T(t + s) = T(t)T(s), ∀t, s ≥ 0 (iii) ∀x ∈ X, → T(t)
x is continuous at 0.

A strongly continuous semigroup of bounded linear operators on X will be called a C 0 -semigroup.

Definition 1.4.21 [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] The linear operator ψ defined by

ψx = lim t→0 + T(t)x -x t , for x ∈ D(ψ),
is called the infinitesimal generator of the semigroup (T(t)) t≥0 where

D(ψ) = x ∈ X : lim t→0 + T(t)x -T(0)x t -0 exists ,
is called the domain of ψ.

Definition 1.4.22 [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] Let u : [0, T] → X be a function.

(i) The function u ∈ C([0, T], X) given by u(t) = T(t)u 0 + t 0 T(t -s)F(s, u(s))ds, 0 ≤ t ≤ T, with x ∈ X and F ∈ L 1 ([0, T]; X) is called mild solution of (??), if u is continuous on [0, T] and u(t) ∈ D(ψ) for 0 < t ≤ T satisfies (1.4.7).
Definition 1.4.23 Let X be an ordered Banach with positive cone X + such that int(X + ) ∅. A linear operator A on X is said to be positive if A(X + ) ⊂ X + , strongly positive if A(X + \ 0) ⊂ int(X + ).

Computation of the basic reproduction number for the reaction diffusion system

We focus in this section the theory of basic reproduction numbers for compartmental epidemic models of parabolic type presented in [START_REF] Zhao | Basic reproduction numbers for reaction-Diffusion Epidemic Models[END_REF]. Consider the reaction-diffusion epidemic model described by

               ∂u i ∂t = ∇.(d i (x)∇u i ) + f i (x, u), 1 ≤ i ≤ n, t > 0, x ∈ Ω, ∂u i ∂ν = 0, ∀1 ≤ i ≤ n with d i > 0, t > 0, x ∈ ∂Ω, (1.4.8)
where u i is the density of a population in compartment i, d i (x) is the diffusion coefficient of population u i , f i is the reaction term in compartment i under the influences of demographic process and epidemic interactions, Ω is the spatial habitat in R l with smooth boundary ∂Ω, ν denotes the unit normal vector on ∂Ω, and the no-flux boundary condition means that no individuals cross the boundary. We emphasize that some diffusion coefficients may be zero on

Ω. Let u = (u 1 , • • • , u n ) t
, with each u i ≥ 0, be the state of individuals in all compartments. We assume that they can be divided into two types: infected compartments, labeled by i = 1, ..., m and uninfected compartments, labeled by i = m + 1, ..., n. Define U s to be the set of all diseasefree states:

U s = {u ≥ 0 : u i = 0 ∀i = 1, • • • , m} .
Let F i (x, u) be the input rate of newly infected individuals in the ith compartment, V + i (x, u) be the rate of transfer of individuals into compartment i by other means (for example, births and immigrations), and V - i (x, u) be the rate of transfer of individuals out of compartment i (for example, deaths and recovery). Thus, the model (1.4.8) can be rewritten as

               ∂u i ∂t = ∇.(d i (x)∇u i ) + F i (x, u) -V i (x, u), 1 ≤ i ≤ n, t > 0, x ∈ Ω, ∂u i ∂ν = 0, ∀1 ≤ i ≤ n with d i > 0, t > 0, x ∈ ∂Ω, (1.4.9)
where

V i (x, u) = V - i (x, u) -V + i (x, u).
Following the setting of the ODE epidemic models, we make the following assumptions:

(A1): For each 1 ≤ i ≤ n, functions F i (x, u), V - i (x, u), V + i (x, u)
, and d i (x) are nonnegative and continuous on Ω × R n + and continuously differential with respect to u. (A2):

If u i = 0, then V - i = 0. In particular, if u ∈ U s , then V - i = 0 for i = 1, • • • , m. (A3): F i = 0 for i > m. (A4): If u ∈ U s , then F i = V + i = 0 for i = 1, • • • , m.
Note that (A1) arises from the simple fact that each function denotes a directed nonnegative transfer of individuals. Biologically, (A2) means that there is no transfer of individuals out of a compartment if the compartment is empty, (A3) indicates that there is no infection for uninfected compartments, and (A4) implies that the population will remain free of disease if it is free of disease at the beginning. We assume that system (1.4.8) admits a disease free state

u 0 = (0, • • • , u 0 m+1 (x), • • • , u 0 n (x)) t ,
where

u 0 i (x) > 0, m + 1 ≤ i ≤ n, for all x ∈ Ω. Set u I = (u 1 , • • • , u m ) t , d I (x) = (d 1 (x), • • • , d m (x)) t , u S = (u m+1 , • • • , u n ) t , d S (x) = (d m+1 (x), • • • , d n (x)) t and ∇.(d I (x)∇u I ) = (∇.(d 1 (x)∇u 1 ), • • • , ∇.(d m (x)∇u m )) t ∇.(d S (x)∇u S ) = (∇.(d m+1 (x)∇u m+1 ), • • • , ∇.(d n (x)∇u n )) t f I (x, u) = ( f 1 (x, u), • • • , f m (x, u)) t f S (x, u) = ( f m+1 (x, u), • • • , f n (x, u)) t . Let M 0 (x) := ∂ f i (x, u 0 (x)) ∂u j m+1≤i, j≤n
.

For the linear reaction-diffusion system

               ∂u i ∂t = ∇.(d S (x)∇u S ) + M 0 (x)u S , t > 0, x ∈ Ω, ∂u i ∂ν = 0, ∀m + 1 ≤ i ≤ n with d i > 0, t > 0, x ∈ ∂Ω.
(1.4.10)

We make the following assumptions that u 0 is linearly stable in the disease free space.

(A5): M 0 (x) is cooperative, ∀x ∈ Ω and λ 0 (M 0 ) := s ∇.(d S ∇) + M 0 < 0.
By assumptions (A1)-(A4), we set

D u F (x, u 0 (x)) = F(x) 0 0 0 1≤i, j≤m D u V(x, u 0 (x)) = V(x) 0 J(x) -M 0 (x) 1≤i,j≤m
where F(x) and V(x) are two m × m matrices defined by

F(x) = ∂F i (x, u 0 (x)) ∂u j 1≤i,j≤m V(x) = ∂V i (x, u 0 (x)) ∂u j 1≤i, j≤m , (1.4.11)
respectively, and J(x) is an (nm) × n matrix. Note that (A1) and (A4) imply that F(x) is nonnegative.

Set X 1 := C( Ω, R m ) and X + 1 := C( Ω, R m + ). Let T(t) be the solution semigroup on X 1 associated with the following linear reaction-diffusion system.

               ∂u i ∂t = ∇.(d I (x)∇u I (x)) -V(x)u I , t > 0, x ∈ Ω, ∂u i ∂ν = 0, ∀1 ≤ i ≤ m with d i > 0, t > 0, x ∈ ∂Ω, (1.4.12)
Note that the internal evolution of individuals in the infectious compartments due to deaths and movements among the compartments is dissipative and exponentially decays in many cases because of the loss of infective members from natural mortalities and disease-induced mortalities. Thus, we assume the following.

(A6): -V is cooperative ∀x ∈ Ω and λ 0 (-V) := s(∇.(d I ∇) -V) < 0.
In order to define the basic reproduction number for model (1.4.8), we assume that the state variables are near the disease-free steady state u 0 . Then we introduce the distribution of initial infection described by f (x). Under the synthetical influences of mobility, mortality, and transfer of individuals in infected compartments, the distribution of those infective members as time evolves becomes T(t)φ(x). Thus, the distribution of new infection at time t is F(x)T(t)φ(x).

Consequently, the distribution of total new infections is

∞ 0 F(x)T(t)φ(x)dt. (1.4.13) Define L(φ)(x) := ∞ 0 F(x)T(t)φ(x)dt = F(x) ∞ 0 T(t)φ(x)dt. (1.4.14)
Then L is a continuous and positive operator which maps the initial infection distribution φ to the distribution of the total infective members produced during the infection period.

Following the idea of next generation operators, we define the spectral radius of L as the basic reproduction number (ii) If R 0 < 1, then u 0 is asymptotically stable for system (1.4.8).

R 0 = r(L) (1
Proof: [START_REF] Zhao | Basic reproduction numbers for reaction-Diffusion Epidemic Models[END_REF].

The following result shows that the reaction-diffusion epidemic model in a spatially homogeneous habitat with the Neumann boundary condition admits the same basic reproduction number as its ODE counterpart.

Theorem 1.4. [START_REF] Freemann | Uniform persistence and flows near a close positively invariant set[END_REF] If each d i is a positive constant for 1 ≤ i ≤ m, and F(x) = F and V(x) = V are independent of x ∈ Ω, then R 0 = r(FV -1 ).

Proof: [START_REF] Zhao | Basic reproduction numbers for reaction-Diffusion Epidemic Models[END_REF].

Lyapunov functionals for reaction diffusion system

Let u be the non-negative vector of concentrations u 1 , ..., u m , and let the reaction be governed by the ordinary differential equation

du dt = f (u) (1.4.16) with f : R n → R n is a C 1 function.
Let Ω be a bounded domain in R n with smooth boundary ∂Ω and D = (d 1 , ..., d m ) with d i = 0.

Suppose u * is a positive equilibrium of (1.4.16). Then u * is also a spatially homogeneous steadystate solution to the following reaction-diffusion system with Neumann boundary condition

               ∂u ∂t = D∆u + f (u) in Ω × (0, ∞), ∂u ∂ν = 0 on ∂Ω × (0, +∞), u(x, 0) = u 0 (x) in Ω.
(1.4.17)

Let V(u) function defined on some domain in R + m . When u(t) is a solution of (1.4.16), it is often necessary to compute the time derivative of V(u(t)). It holds that

dV(u(t)) dt = ∇V(u). f (u).
We assume that the range of u(t) is contained in the domain of V(u). The right hand side is given by the gradient of the function V(u) and the vector field f (u). Thus the right hand side is defined without the fact that u(t) is a solution of (1.4.16), and it is important for our calculation of Lyapunov functionals. Let u(t, x) be a solution of (1.4.17), and we put

W = Ω V(u(t))dt. (1.4.18)
Calculating the time derivative of W along the positive solution of model (1.4.17), we get

dW dt = Ω ∇V(u).(D∆u + f (u))dx = Ω ∇V(u). f (u)dx + Ω ∇V(u).D∆udx = Ω ∇V(u). f (u)dx + m i=1 d i Ω ∂V ∂u i ∆u i dx
We assume the integrand of the first term is already calculated for the ordinary differential equation. The second term is simplified by using Green's formula, and we obtain

Ω ∂V ∂u i ∆u i dx = ∂Ω ∂V ∂u i ∂u i ∂ν dσ - Ω ∇u i .∇ ∂V ∂u i dx Since ∂u ∂ν = 0 on ∂Ω, then Ω ∂V ∂u i ∆u i dx = - Ω ∇u i .∇ ∂V ∂u i dx.
Hence,

dW dt = Ω ∇V(u). f (u)dx - m i=1 d i Ω ∇u i .∇ ∂V ∂u i dx.
So we construct the functionV such that

d i Ω ∇u i .∇ ∂V ∂u i dx ≥ 0 ∀, i = 1, • • • , m. (1.4.19)
We summarize the above in the following proposition.

Proposition 1.4.26 [START_REF] Hattaf | Global stability for reaction-diffusion equations in biology[END_REF] If the Lyapunov function for the ordinary differential equation verifies (1.4.19),

then the function W is a Lyapunov functional for the reaction-diffusion system.

Literature review

Cholera being an ancient disease, many mathematical models have been published to investigate and understand its dynamics. Some of the major challenges that have existed in modeling cholera dynamics have been consideration of all the possible transmission pathways, which include human-to-human transmission and the indirect environment-to-human transmission, at the same time. For the most recent works among others, we refer the reader to [START_REF] Dangbe | Mathematical modelling and numerical simulations of the infuence of hygiene and seasons on the spread of cholera[END_REF][START_REF] Sisodiya | Dynamics of cholera epidemics with impulsive vaccination and desinfection[END_REF][START_REF] Wang | Influence of human behavior on cholera dynamics[END_REF][START_REF] Yang | On the intrinsic dynamics of bacteria in waterborne infections[END_REF][START_REF] Yang | A multi-scale cholera model linking betweenhost and within-host dynamics[END_REF]. Metapopulation models have been developed to describe the spread of cholera between communities connected by migratory movement [START_REF] Berge | Global stability of a two-patch cholera model with fast and slow transmissions[END_REF][START_REF] Njagarah | A metapopulation model for cholera transmission dynamics between communities linked by migration[END_REF].

However, due to the limited considerations of spatial heterogeneity in previous mentioned models, it is not sufficient to understand the spatial spread of cholera infection. In 2010, Bertuzzo et al. in [START_REF] Bertuzzo | On spatially explicit models of cholera epidemics[END_REF] proposed a PDE model to study the spatiotemporal dynamics of cholera.

Later, Wang and her cooperators proposed more general PDE cholera model and investigated the dynamics of cholera [START_REF] Wang | Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment[END_REF][START_REF] Wang | Impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment[END_REF][START_REF] Zhang | Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission[END_REF]. In 2018, Zhou et al. [START_REF] Wang | Global dynamics of a reaction-diffusion waterborne pathogen model with general incidence rate[END_REF] proposed and studied the global dynamics of a reaction-diffusion waterborne model with general incidence rate. In addition, using nonstandard finite difference scheme, they also established a discrete counterpart of the continuous model and derived similar global properties of the discretized model.

Despite the huge number of temporal and spatiotemporal models describing the cholera transmission dynamics, little efforts have been devoted to the influence of the phage bacteria infection. We therefore, review some of the previous work done on modeling of the impact of phage-bacteria infection on cholera dynamics and highlight the key results from the selected models.

Within-host dynamics of cholera

In 2016, Xueying Wang et al. proposed in [START_REF] Wang | Influence of human behavior on cholera dynamics[END_REF] the following deterministic model for the within-host dynamics of cholera, with a focus on the bacteriophage-bacteria interaction,

                           dB dt = Λ -α BV B + k -δ 1 B, dZ dt = g(Z) + θ 1 α BV B + k -δ 2 Z, dV dt = h(V) + θ 2 α BV B + k -δ 3 V. (1.5.1)
Where B and Z represent the concentrations of the environmental vibrios and human vibrios, respectively, and V refers to the concentration of the virus. Λ is the influx rate of the ingested environmental vibrios, α is the contact rate between the environmental vibrios and the phages.

δ i (i = 1, 2, 3
) denotes the per capita death rate of each compartment. θ 1 and θ 2 are re-scaled coefficients that capture the generation rates of Z and V through bacteriophages-bacteria interactions, respectively. The functions g(Z) and h(V) are the intrinsic growth rates of Z and V, respectively. They assumed the following properties

(H1) (a) g(0) = 0; (b) g (Z g ) < d 2 for some Z g ≥ 0; (c) g (Z) ≤ 0; (H2) (a) h(0) = 0; (b) h (V h ) < d 3 for some V h ≥ 0; (c) h (V) ≤ 0.
This interaction in the human small intestine is critical in shaping the evolution of the pathogen directly contributes to the epidemiology of cholera at the population level, since the human vibrios shed out of human body will remain highly infectious for a certain period of time and can be transmitted among human hosts. The derived conditions under which highly infectious vibrios will not grow within the human host and the environmental vibrios ingested into human body will not cause cholera infection. The situation where the human vibrios will grow and persist, and lead to human cholera is presented.

Cholera dynamics with Bacteriophage infection: A mathematical study

In 2016 Misra et al. formulated in [START_REF] Misra | Cholera dynamics with bacteriophage infection: A mathematical study[END_REF] a model to investigate on the cholera dynamics with phage-bacteria infection, with an additional assumption that infected bacteria are also responsible for the disease. They modify the model proposed by Codeço, for the spread of cholera infection in human population and the one proposed by Beretta and Kuang, for the spread of Bacteriophage infection in the bacteria population [START_REF] Harrison | Ecological and Evolutionary Benefits of Temperate Phage: What Does or Doesn't Kill You Makes Stronger[END_REF][START_REF] Codeco | Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir[END_REF]. They considered a region in which a human population live and consumes water from a nearby aquatic reservoir and assume that there is enough bacteria density B in the aquatic reservoir so that the bacteria can transmit the disease to humans. The human population is divided into two mutually exclusive classes, the susceptible S and the infected I. Susceptible individuals can contract the infection through consumption of water contaminated by the bacterium V. cholerae from the aquatic reservoir; in so doing they contract the cholera infection and become infected. The growth of the bacteria can be affected by the Bacteriophage in the reservoir, V. The phage infects the healthy bacterium by injecting in it its genetic material through the wall of the cell. The class of infected bacteria is J. The proposed model is the following,

                                                     dS dt = A -β S(B + hJ) B + hJ + k + νI -dS, dI dt = β S(B + hJ) B + hJ + k -(ν + α + d)I, dB dt = ηI + rB (1 -B/L) -φ BV B + m dJ dt = φ BV B + m -θ 0 J dV dt = θ 1 θ 0 J -φ BV B + m -δV. (1.5.2)
The biological definitions of parameters of model (1.5.2) are summarized in Table 1.1.

From the numerical investigation, they found that the phage adsorption rate φ, plays an important role. Indeed, they proved that there exists a critical value φ = φ c which the model possess Hopf-bifurcation. For lower values than φ c , the endemic equilibrium denoted E * is unstable and periodic solutions are observed, while above φ c , the endemic equilibrium E * is locally asymptotically stable, and further shown to be also globally asymptotically stable.

A reaction-diffusion model for the control of cholera epidemic

Misra et al. extended model (1.5.2) in 2017 to a reaction-diffusion cholera model. In [START_REF] Sisodiya | Pathogen Induced Infection and Its Control by Vaccination: A Mathematical Model for Cholera Disease[END_REF], the model deals with the control and spatial spread of cholera epidemic as well phages and bacteria. The proposed model is the following,

                                       ∂S ∂t = A -β SV V + k + νI -dS + D 1 ∇ 2 S, ∂I ∂t = β SV V + k -(ν + α + d)I + D 1 ∇ 2 I, ∂V ∂t = ηI + rBµV -µ 0 V -φ VP V + m + D 2 ∇ 2 V ∂P ∂t = θφ VP V + m -δP + D 3 ∇ 2 P (1.5.3)
The biological definitions of parameters of model (1.5.3) are summarized in Table 1.2. Moreover, they assumed that D 1 is the diffusion rate coefficient of susceptible and infected individuals, D 2 and D 3 are the diffusion rates coefficients of V. cholerae and bacteriophage in the aquatic reservoirs, respectively. The first study the model without diffusion and derived the conditions under which model exhibits periodic solutions. Secondly they considered the full model where

∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the Laplacian operator.
Some questions to answer, observations from aforementioned models and possible modifications that can be made are highlighted below:

(i) In the aforementioned model (1.5.1), lysogen bacteria population and the prophage induction was neglected. The conditions under which a polluted environment or small intestine can be purified using lytic phages did not derived. To fill this gaps, we formulate a model by considering both lytic and lysogenic life cycles of phages and prophage induction altogether. Moreover using optimal control theory we extend this model to a controlled model to investigate on the use of phages for therapeutic purpose.

(ii) The class of lysogen bacteria is absent in model ( 1 (iii) For the reaction diffusion model (1.5.3), the class of lysogen bacteria do not appears, they neglected infections through infected bacteria and the spatial system is studied in two-dimensional space. We then included the class of lysogen bacteria and the infection throught infected bacteria in a reaction diffusion model to fill the gaps. Moreover, our model is studied in n-dimensional space (n ≥ 2), and the discret counterpart is formulated for numerical simulations.

Chapter 2

Phages and bacteria interactions: a mathematical study

Introduction

We use in this chapter a mathematical model to investigate on the phage-bacteria interactions.

The propounded model is a predator-prey like system with Holling type functional response.

The basic offspring number N 0 is computed and used to examine the global dynamics and perform an in-depth bifurcation analysis of the system and the three equilibria exhibited are topologically classified as follows: An unstable environment-free equilibrium (EFE), a globally stable phage-free equilibrium (PFE) whenever N 0 < 1, and a unique locally stable environmentpersistent equilibrium (EPE) which exits when N 0 > 1. We use a suitable Lyapunov function to estimate the basin of attraction of EPE. The model undergoes a trans-critical forward bifurcation at N 0 = 1 and a Hopf bifurcation around the EPE. Precisely, we show that when N 0 > 1, there is a critical value N c 0 such that for N 0 ≥ N c 0 , the EPE loses its stability through the appearance of a Hopf bifurcation, given rise to periodic solutions. This model allows us to provide conditions under which the presence of phages can purify a bacterial polluted environment and derive situations that the presence of phages triggers virulent pathogenic bacterial disease outbreaks.

The mathematical model

To place our model derivation in a specific context, we provide the main modeling hypotheses.

(H1): Since in the pseudo lysogenic, the phages are inactive, we neglect it and focus only on the lytic and lysogenic life cycles.

(H2):

We assume that there are enough bacteria in the aquatic environment for the phagebacteria interactions to last for sufficiently longer time.

(H3):

We consider the logistic growth for the bacteria species and the Holling-type II functional response for the interactions between phages and bacteria. This is justified by the fact many bacteria are free-living pathogens capable of self multiplication in their biotope. Following assumption (H3) our model introduces a threshold for the susceptible bacteria known as carrying capacity K, so that initially, the population grows exponentially and later stabilizes at a constant level K > 0. Susceptible bacteria acquire infection at rate βB/(B + H b ) known as Holling-type II functional response. In the latter, β stands for the contact between the susceptible bacterial cells and phages/viruses. The bacterial mortality rate in the reservoir is µ. The dynamics of B is therefore modeled by the following equation

dB dt = rB 1 - B K -β BP B + H b -µB. (2.2.1)
For Eq. (2.2.1) to be ecologically meaningful, and the whole interaction system to be mathematically tractable, we assume that r > µ. Otherwise, the bacteria population B will collapse in finite time.

Phages can undergo a lytic or a lysogenic life cycle, whereas few are capable of carrying out both. We denote by π the fraction of infected cells that burst and produce new phages, while

(1 -π) is the fraction of lysogen bacteria. The lysogenic life cycle allows the bacterial host cell to continue to survive and reproduce, the phage is reproduced in all of the cell's offspring and we denote the bacterial cell multiplication size by φ. In the course of this division, the effect of UV radiations or the presence of certain chemicals can lead to the release of prophage causing proliferation of new phages through the prophage induction. Therefore, we denote by α the rate of lysogen bacteria who switch from a the lysogenic life cycle to the lytic life cycle. So the dynamic of lysogen bacteria is:

dV dt = φ(1 -π) βBP B + H b -(α + µ)V. (2.2.2)
In the lytic life cycle, bacteria cells burst (lysed) and destroyed after immediate replication of the new phages; we denote by θ the burst size of the bacteria. Hence, the dynamic of phages is modeled by the following equation.

dP dt = θπβ BP B + H b + θαV -δP. (2.2.3)
The Figure 2.1 describes the interactions between lytic phages, lysogenic phages, bacteria and the prophage induction event. The parameters and variables of the model (2.2.4) are summarized in Table 3.1. Based on the above mentioned formulation and assumptions, we schematically simplify the phage bacteria interactions in Figure 2.1 from which we derive the following system of non-linear differential equations.

                             dB dt = rB 1 - B K -β BP B + H b -µB, dV dt = φ(1 -π)β BP B + H b -(α + µ)V, dP dt = θπβ BP B + H b + θαV -δP. (2.2.4)
It should be noted that the equation for the dynamics of the infected bacteria Z, which actually decouples from (2.2.4) is given by:

dZ dt = πβ BP B + H b + αV -(µ + γ)Z. ( 2

Basic mathematical properties and basic offspring number 2.3.1 Existence, uniqueness and positivity of solutions

For the model (2.2.4) to be ecologically meaningful, it is important to prove that all the state variables are non-negative for all time t. In the other words, the solution of the model (2.2.4)

with non-negative initial data should remain non-negative for all t ≥ 0. Setting

N(t) = φB(t) + V(t), ζ = min(µ, δ)
, and P max = (θπrK + θαφrK)/(4µζ),

we summarize these basic properties in the following result.

Theorem 2.3. [START_REF] Abboubakar | Backward bifurcation and control in transmission dynamics of arboviral diseases[END_REF] The system (2.2.4) is a dynamical system in

Ω = (B, V, P) ∈ R 3 + , 0 ≤ N(t) ≤ φrK 4µ , 0 ≤ P(t) ≤ P max .
Proof : From the first equation of (2.2.4), one has

dB B = ψ(B, P)dt, where ψ(B, P) = r -µ - rB K - βP B + H b .
The integration of the previous equation from 0 to t gives,

B(t) = B(0)e t 0 ψ(B,P)dt > 0 ∀t ≥ 0.
Considering the fact that B(t) ≥ 0, and discarding for now the equation for the susceptible bacteria B(t), the remaining sub-system for the compartments V and P takes the form ẋ = Mx, (

where, x = (V, P), and

M =                 -(µ + α) φ(1 -π)βB B + H b θα θπβ B B + H b -δ                
.

M is a Metzler matrix and the system (2.3.1) is cooperative, so if V(0), P(0) ≥ 0, then V(t), P(t) ≥ 0

∀t > 0. Moreover, Ṅ = φ dB dt + dV dt ≤ φrB 1 - B K -µ(φB + V) + φ(1 -π)β BP B + H b -φβ BP B + H b ≤ φrB 1 - B K -µ(φB + V) = φrB 1 - B K -µN. Knowing that max 0≤B≤K B 1 - B K = K 4 , we have, Ṅ ≤ φr K 4 -µN.
Using a Granwall lemma [START_REF] Lakshmikantham | stability analysis of non linear system[END_REF], we show that

N(t) ≤ N(0)e -µt + φr K 4µ (1 -e -µt ).
Thus, if N(0) ≤ φrK/4µ, then N(t) ≤ φrK/4µ, for all t ≥ 0.

Similarly, if we set Y = θπB + P, then

Ẏ = θπrB 1 - B K -µπθB + θαV -δP, ≤ θπrK + θαφrK 4µ -ζY.
Another application of a Gronwall lemma gives P(t) ≤ P max for all t ≥ 0, if P(0) ≤ P max . Thus Ω is a positively invariant set under the flow of system (2.2.4). Hence it is sufficient to consider the dynamics of the model (2.2.4) in Ω.

Basic offspring number and its sensitivity analysis

For simplicity, denote B 0 = K(r -µ)/r, so that the PFE of model (2.2.4) is E 1 = (B 0 , 0, 0). Note that the disease components in model (2.2.4) are V and P. The matrix for the new offspring and that for the transition between compartments are respectively

F =                0 φ(1 -π)βB 0 B 0 + H b 0 θπβB 0 B 0 + H b                and W =        α + µ 0 -θα δ        .
By the next generation matrix method [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF], the basic offspring number is defined as the spectral radius of the next generation matrix FW -1 of the model (2.2.4) which is given by

FW -1 =                 θαφ(1 -π)βB 0 δ(µ + α)(B 0 + H b ) φ(1 -π)βB 0 δ(B 0 + H b ) θ δ(µ + α) θπβB 0 B 0 + H b 1 δ θπβB 0 B 0 + H b                 .
Since FW -1 is a rank 1 matrix, the basic offspring number of the model (2.2.4) (i.e. the spectral radius of FW -1 ) is given by its trace as follows: N 0 following the approach in [START_REF] Van Den | Reproduction number and subtrheshold endemic equilibria for compartemental models of disease transmission[END_REF], the variable Z should be added to the set of disease components. As expected, the expression for the basic offspring number N 0 remains unchanged. This outcome shouldn't be surprising because the variable Z(t) does not appear in the equations for B(t), V(t) and P(t), and therefore cannot influence their dynamics.

N 0 = ρ(FW -1 ) = θπβB 0 δ(B 0 + H b ) + βθαφ(1 -π)B 0 δ(µ + α)(B 0 + H b ) . ( 2
For the biological interpretation of the basic offspring number N 0 , pose

f (B) = βB B + H b , N 0Z = θπ f (B 0 ) δ , N 0V = θφ(1 -π) f (B 0 ) δ α α + µ .
Thus, with these notations, N 0 reads

N 0 = N 0Z + N 0V . (2.3.3)
Different terms in N 0Z and N 0V given by (2.3.3) can be interpreted as follows:

• θπ f (B 0 ) is the mean number of new phages released after burst of bacteria via the lytic life cycle.

• θφ(1 -π) f (B 0 )
is the mean number of new phages produced through the prophage induction process.

• α/(α + µ) is the probability that a lysogen bacterium undergoes a lytic life cycle through the prophage induction process.

• 1/δ is the average lifespan of phages.

Thus, N 0Z is the mean number of phage offspring produced by a single phage, in the fully uninfected bacteria population via the lytic life cycle of the phage, whereas N 0V gives the mean number of phage offspring generated through the prophage induction process, by a single phage introduced in the fully uninfected bacteria population. The sum N 0 = N 0Z + N 0V is therefore the mean number of phage offspring generated by a single phage, either in the lytic or lysogenic life cycle, introduced into the fully uninfected population of bacteria.

The local sensitivity analysis is based on the normalized sensitivity index of N 0 . The normalized forward sensitivity index of a variable to a parameter is the number of the relative change in the variable to the relative change in the parameter. Since the basic offspring number is a differentiable function with respect to any of its parameters, the sensitivity indices are calculated using partial derivative of N 0 [START_REF] Ray | A methodology for performing global uncertainty and sensitivity analysis in system biology[END_REF] and are displayed in Table 2.2 below.

To this aim, denoting by ψ the generic parameter of system (2.2.4), we evaluate the normalized sensitivity index

S N 0 ψ = ψ N 0 ∂N 0 ∂ψ . (2.3.4)
Mathematically, S N 0 ψ indicates how sensitive N 0 is to the change of parameter ψ. A positive (resp. negative) index indicates that an increase in the parameter value results in an increase (resp. decrease) in the N 0 value.

It is worth noticeable that local sensitivity analysis only assesses the effects of individual parameters at particular point in parameter space, without considering the combined variability resulting from all input parameters simultaneously. To address the latter, we perform a global sensitivity analysis to obtain the model response to parameter variation within a wider range in parameter space. Following the approach in [START_REF] Ray | A methodology for performing global uncertainty and sensitivity analysis in system biology[END_REF], PRCC between N 0 and each parameter are derived. The results of the PRCCs of N 0 are shown in Figure 2 2.2, we observe that the parameters β, α, K, θ and φ have the most positive influence (by augmenting it) on N 0 , while have remarkable negative impact (by reducing it) on N 0 are δ, H b and µ. For instance, the increase of β, α, K, θ and φ, say by 10%, will increase N 0 by 10%, 9.2%, 10% and 9.3%, respectively.

According to the

β S N 0 β +1 µ S N 0 µ -0.5372 δ S N 0 δ -1 α S N 0 α +0.5709 K S N 0 K +0.9248 H b S N 0 H -0.9248 θ S N 0 θ +1 φ S N 0 φ +0.9320
Table 2.2: Normalized sensitivity indexes of N 0 . The phage-bacteria contact rate β, the lysis burst size θ, the phage death rate δ, the bacteria cell division rate φ, in the decreasing order, the susceptible bacteria carrying capacity K, the half-saturation bacteria density H b , are the most influential parameters on N 0 . Note that the baseline values in Table 1 should have been used here to compute the sensitivity indexes. This global sensitivity analysis is consistent with the local sensitivity of N 0 , except that the bacteria burst size θ overcome the phage-bacteria contact rate β. However, as a whole the basic offspring number N 0 is highly robust to its parameters variation. characterizing the prophage induction, one may wish to assess their combined influence on the basic offspring N 0 , with the ultimate aim to control the growth of lysogen bacteria, which are thought to be responsible of severe disease outbreaks. This is investigated in Figure 2.3. More importantly, the matching of the local sensitivity analysis (see Table 2.2) and global sensitivity analysis (see Figure 2.2) of N 0 demonstrates its robustness to the parameters variations. Figure 2.3 precisely shows that, for a large value (say, φ = 90) of the cell division size φ, N 0 is always greater than one, regardless the value of α. However, if the value of φ is halved (say, φ = 45), then one should increase the value of α above 0.15 to bring N 0 above one. This underscores the importance of prophage induction in controlling the bacteria population, since it is only when N 0 > 1 that more phages are produced to destroy more bacteria.

Equilibria and bifurcation analysis 2.4.1 Existence of equilibria and trans-critical forward bifurcation

Clearly, if P = 0, then the model (2.2.4) admits two trivial equilibria: the EFE E 0 = (0, 0, 0) and the PFE E 1 = (B 0 , 0, 0).

Proposition 2.4.1

The following statements hold true:

i) The EFE E 0 is always unstable.

ii) The PFE E 1 of the system (2.2.4) is LAS whenever N 0 < 1, and the stable manifold of the PFE is

W s (E 1 ) = {(B, V, P) ∈ Ω, B = B 0 }.
iii) The PFE is unstable whenever N 0 > 1, and there exists a unique EPE E * of the system (2.2.4).

Proof: i) The Jacobian matrix at E 0 is

J(E 0 ) =           r K B 0 0 0 0 -(µ + α) 0 0 θα -δ           .
Clearly, rB 0 /K is a non-negative eigenvalue of J(E 0 ), thus E 0 is unconditionally unstable.

ii) The Jacobian matrix at the PFE E 1 is

J(E 1 ) =               - r K B 0 0 -f (B 0 ) 0 -(µ + α) φ(1 -π) f (B 0 ) 0 θα θπ f (B 0 ) -δ              
.

It is clear that -rB 0 /K is an eigenvalue of J(E 1 ), the local stability of PFE is completely determined by the determinant and the trace of the following (2 × 2)-matrix.

J 0 =       -(µ + α) φ(1 -π) f (B 0 ) θα θπ f (B 0 ) -δ       .
Straightforward computations show that the determinant and the trace of J 0 are respectively:

det(J 0 ) = δ(µ + α)(1 -N 0 ), tr(J 0 ) = -(µ + α) -δ (1 -N 0V ) . (2.4.1)
If N 0 < 1, then det J 0 > 0 and N 0V < 1, which implies that tr(J 0 ) < 0. Thus the LAS of the PFE. It is not difficult to see that the stable manifold of E 1 is W s (E 1 ) defined above. On the other hand, if N 0 > 1, then det(J 0 ) < 0, and the EFE is unstable.

iii) The EPE E * = (B * , V * , P * ) of model (2.2.4) satisfies the system

                             rB * 1 - B * K -β B * P * B * + H b -µB * = 0, φ(1 -π)β B * P * B * + H b -(α + µ)V * = 0, θπβ B * P * B * + H b + θαV * -δP * = 0. (2.4.2)
From the second equation of (2.4.2), we have

V * = φ(1 -π)β (µ + α) B * P * B * + H b . (2.4.3)
The substitution of the expression for V * into the third equation of (2.4.2) yields

θπβ + θαφ(1 -π)β (µ + α) B * P * B * + H b -δP * = 0.
Since we are looking for the non-negative equilibria, then P * > 0, we are left with the equation

θπβ + θαφ(1 -π)β (µ + α) B * B * + H b -δ = 0. (2.4.4)
After some computations we have the following expression of

B * B * = H b B 0 B 0 (N 0 -1) + H b N 0 . (2.4.5)
From the first equation of (2.4.2),

B * r -µ - rB * K = β B * P * B * + H b . That is r K (B 0 -B * ) = β P * B * + H b .
From equation (2.4.5), we have

B 0 -B * = B 0 (B 0 + H b )(N 0 -1) B 0 (N 0 -1) + H b N 0 . (2.4.6)
We note that, since the basic offspring number N 0 > 1, it is clear from (2.4.6) that B * < B 0 , and

P * = r βK (B 0 -B * )(B * + H b ).
Using the expression of B * given by (2.4.5), and the expression (2.4.6) it is easy to obtain the formula for P * below.

P * = rH b βK B 0 (B 0 + H b ) 2 N 0 (N 0 -1) (B 0 (N 0 -1) + H b N 0 ) 2 . (2.4.7)
After the replacement of B * given by (2.4.5) and P * displayed in (2.4.7), we obtain expressions V * and Z * in terms of N 0 as follows:

V * = φ(1 -π)rH b (µ + α)K B 2 0 (B 0 + H b )(N 0 -1) (B 0 (N 0 -1) + H b N 0 ) 2 (2.4.8)
and

Z * = π(µ + α) + φα(1 -π)rH b (µ + γ)(µ + α)K B 2 0 (B 0 + H b )(N 0 -1) (B 0 (N 0 -1) + H b N 0 ) 2 .
(2.4.9)

Hence the existence of a unique EPE for the model (2.2.4), whenever N 0 > 1.

We now focus on the global stability of the PFE E 1 . To this end, we set:

n(B) = B r -µ - rB K = r K B(B 0 -B) and f (B) = βB B + H b . (2.4.10) Proposition 2.4.2
The PFE is GAS in Ω \ {E 0 } whenever N 0 < 1.

Proof:

We consider the following Lyapunov function

L 0 (B, V, P) = (1 -π)φθα δ(µ + α) + θπ δ B B 0 f (x) -f (B 0 ) f (x) dx + θα δ(µ + α) V + 1 δ P.
(2.4.11)

We now compute the derivative of L along the solution of (2.2.4). One has

dL 0 dt = (1 -π)φθα δ(µ + α) + θπ δ 1 - f (B 0 ) f (B) dB dt + θα δ(µ + α) dV dt + 1 δ dP dt .
Using the expressions of derivatives in (2.2.4) yields

dL 0 dt = (1 -π)φθα δ(µ + α) + θπ δ f (B) -f (B 0 ) f (B) n(B) -f (B)P + θα δ(µ + α) ((1 -π)φ f (B)P -(µ + α)V) + 1 δ (θπ f (B)P + θαV -δP) (2.4.12) = (1 -π)φθα δ(µ + α) + θπ δ f (B) -f (B 0 ) f (B) n(B) + (1 -π)φθα δ(µ + α) + θπ δ f (B 0 )P -P = (1 -π)φθα δ(µ + α) + θπ δ f (B) -f (B 0 ) f (B) n(B) + (1 -π)φθα f (B 0 ) δ(µ + α) + θπ f (B 0 ) δ -1 P = r K (1 -π)φθα δ(µ + α) + θπ δ f (B) -f (B 0 ) f (B) B(B 0 -B) + P(N 0 -1).
Since f is increasing, N 0 < 1 and (B, V, P) ∈ Ω \ {E 0 }, we have dL 0 /dt ≤ 0. Moreover, the set in

Ω \ {E 0 } such that dL 0 /dt ≤ 0 = 0 is E = {(B, V, P) ∈ Ω \ {E 0 }; B = B 0 , P = 0}. Replacing P by zero
in the second equation of (2.2.4) leads to lim t→+∞ V(t) = 0. Thus, largest invariant set contained in E is the singleton PFE, and the application of LaSalle's Invariant Principle [START_REF] Lasalle | The stability of Dynamical systems[END_REF], proves that the PFE is GAS in Ω \ {E 0 }.

Purification of the environment

Purification of the environment is the analysis of the conditions under which pathogen bacteria go to extinction. This will be provided using the ecological threshold. We use the expressions obtained in (2.4.5), (2.4.7) and (2.4.3) to investigate on the behavior of the EPE versus the basic offspring number. First, note that whenever N 0 > 1, the denominators of B * , V * and P * cannot vanish. Recall from (2.4.5) that 2 . Thus, B * is decreasing for N 0 > 1, and tends to zero whenever N 0 tends to +∞.

B * (N 0 ) = H b B 0 B 0 (N 0 -1) + H b N 0 , so that B * (1) = B 0 , lim N 0 →∞ B * (N 0 ) = 0, and
∂B * ∂N 0 = - H b B 0 (B 0 + H b ) ((B 0 + H b )N 0 -B 0 )
Similarly, from (2.4.8), we have

V * (N 0 ) = φ(1 -π)rH b (µ + α)K B 2 0 (B 0 + H b )(N 0 -1) (B 0 (N 0 -1) + H b N 0 ) 2 ,
V * (1) = 0, and lim N 0 →∞ V * (N 0 ) = 0, and

∂V * ∂N 0 = φ(1 -π)rH b B 2 0 (B 0 + H b ) (µ + α)K [-(B 0 + H b )N 0 + B 0 + 2H b ] (B 0 (N 0 -1) + H b N 0 ) 3 .
Hence, V * assumes the maximum value at

N m 0 = 1 + H b B 0 + H b ,
giving by

V * max = φ(1 -π)rH b B 2 0 (B 0 + H b ) 4H b (µ + α)K .
Precisely, V * is increasing on 1 ≤ N 0 < N m 0 , and decreasing whenever N 0 > N m 0 and tends to zero.

Remember that from (2.4.7), one has

P * (N 0 ) = rH b βK B 0 (B 0 + H b ) 2 N 0 (N 0 -1) (B 0 (N 0 -1) + H b N 0 ) 2 . P * (1) = 0, lim N 0 →∞ P * (N 0 ) = rH b B 0 βK , and
∂P * ∂N 0 = rB 0 (B 0 + H b ) 2 βKH b (B 0 + N 0 H b -B 0 N 0 ) (B 0 (N 0 -1) + H b N 0 ) 3 .
Hence, P * assumes the maximum value at

N m 0 = -B 0 H b -B 0 .
If, H b > B 0 , then N m 0 < 1 and P * is an increasing but saturated function in [1, ∞[ which assumes a maximum value

P * max = rH b B 0 βK = (r -µ)H b β .
If H b < B 0 , then N m 0 > 1, P * is an increasing in [1, N m 0 [ and decreasing whenever N 0 > N m 0 and tends to P * max . On the other hand, thanks to (2.4.9),

Z * (N 0 ) = π(µ + α) + φα(1 -π)rH b (µ + γ)(µ + α)K B 2 0 (B 0 + H b )(N 0 -1) (B 0 (N 0 -1) + H b N 0 ) 2 ,
so that, Z * (1) = 0, and lim N 0 →∞ Z * (N 0 ) = 0, and

∂Z * ∂N 0 = π(µ + α) + φ(1 -π)rH b B 2 0 (B 0 + H b ) (µ + α)K [-(B 0 + H b )N 0 + B 0 + 2H b ] (B 0 (N 0 -1) + H b N 0 ) 3 .
Thus, Z * , has a maximum value at N m 0 , giving by

Z * max = π(µ + α) + φα(1 -π) rH b B 2 0 (B 0 + H b ) 4H b (µ + α)K .
Thus, Z * is increasing on 1 ≤ N 0 < N m 0 , and decreasing whenever N 0 > N m 0 , and finally vanishes when N 0 → +∞. From these investigations which are illustrated by Figure 2.4, one can derive the following remark. Remark 2.4. [START_REF] Alexander | A Vaccination Model for Transmission Dynamics of Influenza[END_REF] The threshold value N m 0 is the minimum value of the basic offspring number N 0 required to ensure exponential decrease of the all bacterial populations to zero, thus purifying the environment by keeping alive only the population of phages. Fortunately N m 0 is small enough (1 < N m 0 < 2 ), so that, not too much effort is needed to achieve this value in order to purify the environmental reservoir of bacteria.

The following theorem specifies the type of the PFE based on the values or ranges of the basic offspring number N 0 . Theorem 2.4.4 For the system (2.2.4), the PFE is i) an attracting node if

N 0 < 1, ii) a hyperbolic saddle if N 0 > 1, iii) a saddle-node if N 0 = 1.
Proof : The characteristic polynomial of the Jacobian matrix at the PFE denoted by J(E 1 ) has the form

P(λ) = λ + r K B 0 λ 2 -tr(J 0 )λ + det(J 0 ) (2.4.13)
where tr(J 0 ) and det(J 0 ) are given in Eq. (2.4.1).

i) For N 0 < 1, det(J 0 ) > 0 and using the Descartes's rule sign, the roots of (2.4.13) are reals and negative. Thus the PFE in an attracting node.

ii) For N 0 > 1, det(J 0 ) < 0 , and there exists three real eigenvalues with at least one positive.

Thus the PFE is hyperbolic saddle.

iii) For N 0 = 1, the eigenvalues of the Jacobian matrix of (2.2.4) at PFE are -rK/B 0 , tr(J 0 ) and 0. Recalling that tr(J 0 ) = -(µ + α) -δ 1 -N 0 V < 0, then, PFE is a non-hyperbolic critical point with two negative eigenvalue and one simple zero eigenvalue. Thus, the center manifold theory approximation applies in order to determine its stability.

Let's adopt the change of variables U = B -B 0 , V = V, W = P and set X = (U, V, W) T such that the system (2.2.4) has the form

dX dt = AX + F(U, V, W), (2.4.14)
where,

A =            -rB 0 /K 0 -f (B 0 ) 0 -(µ + α) φ(1 -π) f (B 0 ) 0 θα θπ f (B 0 ) -δ            and
F(U, V, W) =                              -rU 2 /K - βH b U (U + B 0 + H b )(B 0 + H b ) W φ(1 -π) βH b U (U + B 0 + H b )(B 0 + H b ) W θπ βH b U (U + B 0 + H b )(B 0 + H b ) W                              .
Using Taylor expansion around (0, 0, 0), we obtain F as follows

F(U, V, W) =                              -rU 2 /K - βH b (B 0 + H b ) UW + O(U 2 ) φ(1 -π) βH b (B 0 + H b ) UW + O(U 2 ) θπ βH b (B 0 + H b ) UW + O(U 2 )                             
.

In order to diagonalize the linear part of the system (2.4.14), we set

η = βB 0 K rB 0 (B 0 + H b ) -K(α + µ + δ)(B 0 + H b ) + KθπβB 0 ,
and consider the matrix P of the eigenvectors of J(E 1 ) given by

P =                     1 η βK r(B 0 + H b ) 0 - (µ + α) θα φ(1 -π) f (B 0 ) α + µ 0 1 1                     .
Then,

P -1 AP = B = diag -rB 0 /K, -(µ + α) -δ 1 -N 0 V , 0 .
Using the transformation Y = PX, where Y = (B, V, P) T we have dY dt = BY + P -1 FPY and the system (2.4.14) takes the form

                           dB dt = -rB 0 B/K + O(| B, V, P | 2 ), dV dt = -µ + α + δ 1 -N 0 V V + O(| B, V, P | 2 ), dP dt = 0 + g 3 (B, V, P), (2.4.15)
where,

g 3 (B, V, P) = Kβ 2 (µ + α) θα(1 -π)φ + θπ(µ + α) r(B 0 + H b ) θα(1 -π)βB 0 + (α + µ) 2 (B 0 + H b ) + B + βB 0 r K B 0 (B 0 + H b ) -(α + µ) + δ(1 -N 0V ) V + β K r(B 0 + H b ) P (V + P).
Then the application of the center manifold theory yields the following system

                           dB dt = -rB 0 B/K + O(| B, V, P | 2 ), dV dt = -µ + α + δ 1 -N 0 V V + O(| B, V, P | 2 ), dP dt = ν P 2 + O(P 3 ), (2.4.16)
where, Remark 2.4. [START_REF] Asheshov | from 1st January to 1st September[END_REF] The instability of the EFE E 0 = (0, 0, 0) predicts that the phages and bacteria cannot simultaneously face extinction, and justify the fact that phages feed on bacteria only and do not consume other resources. Thus in the absence of bacteria, phages are condemned to elimination.

ν = Kβ 2 (µ + α) θα(1 -π)φ + θπ(µ + α) r(B 0 + H b ) θα(1 -π)βB 0 + (α + µ) 2 (B 0 + H b ) . Since ν > 0,

Existence of Hopf Bifurcation

Our purpose here is to determine the conditions under which model (2.2.4) undergoes a Hopf bifurcation and illustrate it numerically. The Jacobian matrix of (2.2.4) evaluated at E * is

J(E * ) =                         r -µ -2rB * /K - βP * H b (B * + H b ) 2 0 - βB * B * + H b φ(1 -π)βP * H b (B * + H b ) 2 -(µ + α) φ(1 -π) βB * B * + H b θπβP * H b (B * + H b ) 2 θα θπ βB * B * + H b -δ                        
.

Knowing that at the EPE point E * , one has

θπβB * B * + H b -δ = - θα(1 -π)φβB * (µ + α)(B * + H b ) , P * = r βK (B 0 -B * )(B * + H b ), r -µ -2rB * /K = r(B 0 -2B * )/K, J(E * ) becomes, J(E * ) =                         r(B 0 -2B * )/K - r K (B 0 -B * )H b (B * + H b ) 0 - βB * B * + H b φ(1 -π) r K (B 0 -B * )H b (B * + H b ) -(µ + α) φ(1 -π) βB * B * + H b θπ r K (B 0 -B * )H b (B * + H b ) θα - θα(1 -π)φβB * (µ + α)(B * + H b )                        
.

The characteristic polynomial of J(E * ) is

P(λ) = λ 3 + a 2 (B * )λ 2 + a 1 (B * )λ + a 0 (B * ), (2.4.17)
where,

a 2 (B * ) = - r K (B 0 -2B * ) + θα(1 -π)φβB * (µ + α)(B * + H b ) + r K (B 0 -B * )H b (B * + H b ) + (µ + α), a 1 (B * ) = r K (B 0 -2B * ) θαφ(1 -π)βB * (µ + α)(B * + H b ) + (µ + α) + (µ + α + δ) rH b K (B 0 -B * ) (B * + H b ) , a 0 (B * ) = rH b δ(µ + α) K(B * + H b ) (B 0 -B * ). ( 2
Q(B * ) = - r K (B 0 -2B * ) 1 + (1 + µ + α) θα(1 -π)φβB * (µ + α)(B * + H b ) + (µ + α)(1 + µ + α) (2.4.20) + θαφ(1 -π)βB * (µ + α)(B * + H b ) + rH b (B 0 -B * ) K(B * + H b ) 1 + (µ + α + δ) + (µ + α) 2 .
After direct but simple computations, (2.4.18) becomes,

a 2 (B * ) = 2r(µ + α)B * 2 + rH b (µ + α)(N 0 -1) + θα(1 -π)φβ + (µ + α) 2 K B * + (µ + α) 2 KH b K(µ + α)(B * + H b ) , a 1 (B * ) = r K (B 0 -2B * ) θαφ(1 -π)βB * (µ + α)(B * + H b ) + (µ + α) + (µ + α + δ) rH b K (B 0 -B * ) (B * + H b ) , a 0 (B * ) = rH b δ(µ + α) K(B * + H b ) B 0 (B 0 + H b )(N 0 -1) (B 0 (N 0 -1) + H b N 0 ) , (2.4.21)
and

Q(B * ) = B * τ 3 B * 3 + τ 2 B * 2 + τ 1 B * + τ 0 K 2 (µ + α) 2 (B * + H b ) 2 , (2.4.22)
where,

τ 3 = 4r 2 (µ + α) θα(1 -π)φβ + (µ + α) 2 > 0, τ 2 = 2δ(µ + α)r 2 H b θα(1 -π)φβ + (µ + α) 2 -2r θα(1 -π)φβ + (µ + α) 2 τ 0 , τ 1 = 2r 2 (µ + α) 2 δH b B 0 + 2K(µ + α) 2 rH b θα(1 -π)φβ + (µ + α) 2 + r(µ + α)H b + Kθα(1 -π)φβ -r(µ + α)B 0 × r(µ + α) 2 H b -θα(1 -π)φβB 0 -r(µ + α) 2 B 0 -(µ + α)H b , τ 0 = -K(µ + α) 2 H b rH b (µ + α) 2 + rθα(1 -π)φβB 0 + r(µ + α) 2 B 0 + δ(µ + α) 2 r 2 H b B 2 0 (1 -N 0 ) . ( 2
q(B * ) = τ 3 B * 3 + τ 2 B * 2 + τ 1 B * + τ 0 . (2.4.24)
We use the Descartes's rule of signs to investigate the number of non-negative roots of Eq. (2.4.24). We note from Eq.(2.4.20) that Q(B * ) > 0 for all B * ≥ B 0 /2, possible roots B * c of q(B * ) lie on the interval (0, B 0 /2). From Eq. (2.4.24) and the fact that N 0 > 1 we conclude that q(0) = τ 0 < 0. Using Eq. (2.4.20), one has Q (B 0 /2) > 0, the intermediate value theorem guarantees the existence of at least one root of q(B * ) in (0, B 0 /2). Moreover, τ 3 > 0 and τ 0 < 0 imply that τ 2 ≥ 0. Thus, regardless the sign of τ 1 , Descartes's rule of signs guarantees the existence of at most one non-negative root of q(B * ). Hence the existence of a unique root B * c of Eq. (2.4.24). Moreover, note that from Eq. (2.4.5), one has

N 0 = B 0 (B * + H b ) B * (B 0 + H b ) , (2.4.25)
so that

B * = H b B 0 B 0 (N 0 -1) + H b N 0 = B * c ⇐⇒ N 0 = N c 0 ,
where,

N c 0 = 1 + B 0 + H b B * c + H b . (2.4.26)
Furthermore,

N c 0 = 1 + B 0 + H b B * c + H b > B 0 + H b B 0 2 + H b = 1 + B 0 B 0 + 2H b .
Due to the high degree of the polynomial q(B * ), it is difficult to find the explicit value of N c 0 . Thus, for numerical illustrations, it is impossible to use the value of N c 0 to test the stability of EPE. Alternatively and equivalently, we will select appropriate parameter sets to obtain numerically the desired signs of Q(B * ).

From the investigations above, it is obvious that Q(B * ) (or equivalently q(B * )) is non-negative for N 0 < N c 0 and negative for N 0 > N c 0 . Thus, the local stability of the EPE is summarized by the following result: Proposition 2.4. [START_REF] Bakare | Optimal control analysis of an SIR epidemic model with constant recruitment[END_REF] The EPE E * is LAS if and only if, 1 < N 0 < N c 0 .

The fact that we have established the LAS (it is even GAS) of PFE for N 0 < 1, its instability for N 0 > 1 and the existence of a positive LAS equilibrium E * when N 0 > 1, shows that model (2.2.4) presents a trans-critical bifurcation at N 0 = 1 as stated in the following result.

Theorem 2.4. [START_REF] Beretta | Modelling and analysis of a marine bacteriophages infection[END_REF] The model (2.2.4) exhibits a trans-critical bifurcation at N 0 = 1.

In order to prove the occurrence of Hopf bifurcation, we note that a 0 (B * ), a 2 (B * ) and Q(B * ) can be expressed as the functions of N 0 which can be chosen as the bifurcation parameter.

Theorem 2.4.8 Denote

H 0 = -a 2 (N c 0 )a 1 (N c 0 )a 1 (N c 0 ) -a 2 (N c 0 )a 1 (N c 0 ) + a 0 (N c 0 ). (2.4.27)
Then, the model (2.2.4) exhibits a Hopf bifurcation at N 0 = N c 0 around the EPE if H 0 0. Moreover, the Hopf bifurcation is super-critical if H 0 > 0 and sub-critical if H 0 < 0.

Proof : We use the method presented in [START_REF] Wei-Min | Criterion of Hopf bifurcation without using eigenvalues[END_REF][START_REF] Panja | Effect of toxicants on Pytoplankton-zooplankton-fish dynamics and harvesting[END_REF] to find the analytic conditions for the system (2.2.4) to undergo a Hopf bifurcation at N 0 = N c 0 . We set

Q(N 0 ) = a 1 (N 0 )a 2 (N 0 ) -a 0 (N 0 ). (2.4.28)
By the condition Q(N c 0 ) = 0, the characteristic equation (2.4.17) of J(E * ) takes the form

λ 2 + a 1 (N c 0 ) λ + a 2 (N c 0 ) = 0. (2.4.29)
Let ρ 1 , ρ 2 , and ρ 3 , denote the roots of (2.4.29), such that ρ 3 = -a 2 (N c 0 ) < 0 and ρ 1 , ρ 2 = ±i a 1 (N c 0 ). We recall from (2.4.21) that a 0 (N c 0 ) > 0, a 2 (N c 0 ) > 0 and observe that a 1 (N c 0 ) = a 1 (B * c ) > 0 (because B * c < B 0 /2). Thus, for N 0 = N c 0 , there are one negative eigenvalue and two purely imaginary eigenvalues of J(E * ).

The general form of ρ 1 , ρ 2 in the neighborhood of N c 0 is ρ 1 = x + iy, and ρ 2 = xiy. We now check the following transversality conditions.

∂R e ρ j (N 0 ) ∂N 0 | N 0 =N c 0 0 j = 1, 2.
(2.4.30)

Substituting ρ j = x ± iy into (2.4.17) and calculating the derivative gives

L 1 (N 0 ) ∂x ∂N 0 -L 2 (N 0 ) ∂y ∂N 0 + L 3 (N 0 ) = 0, L 2 (N 0 ) ∂x ∂N 0 + L 1 (N 0 ) ∂y ∂N 0 + L 4 (N 0 ) = 0, (2.4.31)
where,

L 1 (N 0 ) = 3(x 2 -y 2 ) + 2a 2 (N 0 )x + a 1 (N 0 ), L 2 (N 0 ) = 6xy + 2a 2 (N 0 )y, L 3 (N 0 ) = ∂a 2 (N 0 ) ∂N 0 (x 2 -y 2 ) + ∂a 1 (N 0 ) ∂N 0 x + ∂a 0 (N 0 ) ∂N 0 L 4 (N 0 ) = 2 ∂a 2 (N 0 ) ∂N 0 xy + ∂a 1 (N 0 ) ∂N 0 y.
(2.4.32) Straightforward computations solve system (2.4.31) and yield

∂Reρ j (N 0 ) ∂N 0 | N 0 =N c 0 = ∂x ∂N 0 (N c 0 ) = - L 2 (N c 0 )L 4 (N c 0 ) + L 1 (N c 0 )L 3 (N c 0 ) L 2 1 (N c 0 ) + L 2 2 (N c 0 ) = - a 2 (N c 0 )a 1 (N c 0 )a 1 (N c 0 ) + a 2 (N c 0 )a 1 (N c 0 ) -a 0 (N c 0 ) a 1 (N c 0 ) + a 2 2 (N c 0 ) = - H 0 a 1 (N c 0 ) + a 2 2 (N c 0 ) . ( 2

.4.33)

Since by hypothesis H 0 0, the transversality conditions (2.4.30) hold. Hence the existence of Hopf bifurcation. This ends the proof of Theorem 2.4.8.

To illustrate the Hopf bifurcation phenomenon in Theorem 2.4.8 above, we choose two suitable sets of parameter values from Table 3.1, such that for the first set, Q(B * ) > 0 or equivalently N 0 < N c 0 (i.e E * is locally stable) and for the second set, Q(B * ) < 0 or equivalently N 0 > N c 0 (i.e E * is unstable and periodic solutions occur, see Figure 2.7). For the first set, the LAS of E * is guaranteed and is illustrated in Remark 2.4.9 The existence of periodic solutions could explain the occurrence of repetitive bacteriaborne disease outbreaks.

Estimation of the basin of attraction of equilibria

In this paragraph, an estimate of the basin of attraction of the EPE E * is provided. Note that the existence of stable periodic solutions for model (2.2.4) precludes the global asymptotic stability of E * in the entire interior of Ω. We are therefore left with the possibility of finding a subset Ω E * of Ω containing E * such that every solution initiated in Ω E * converges to E * . Let's denote the stable manifold of E 1 by W s (E 1 ), define the quantity

B m = B 0 -B * = B 0 (B 0 + H b )(N 0 -1) B 0 (N 0 -1) + H b N 0 ≥ 0, (2.4.34)
and the subset

Ω E * = (B, V, P) ∈ Ω : B m ≤ B(t) ≤ φrK 4µ \ W s (E 1 )
.

Recall that if one assumes 1 < N 0 < N c 0 , so that B * ∈ Ω E * is guaranteed, and Q(B * ) > 0, then the EPE E * is LAS. This suggests that the global asymptotic stability can be investigated in Ω E * . Precisely, we prove in Theorem 2.4.10 below that Ω E * is actually contained in the basin of attraction of E * . Theorem 2.4.10 whenever 1 < N 0 < N c 0 , the EPE E * , of the system (2.2.4) is GAS in Ω E * .

Proof : Note that Ω E * contains, neither the EFE E 0 , nor the PFE E 1 . We follow the works in [START_REF] Berge | A simple mathematical model of Ebola in Africa[END_REF][START_REF] Paul | Global dynamics od a predator-prey model with stage structure for predator[END_REF][START_REF] Xiabong | Global dynamics of a predator-prey system with Holling type II functuinal response[END_REF], to propose the following Lyapunov function candidate for the EPE.

L(B, V, P) = a B B * f (x) -f (B * ) f (B) dx + b V -V * -V * ln V V * + c P -P * -P * ln P P * , (2.4.35)
where a, b, and c are three positive constants to be determined shortly. We now compute the derivative of L along the solutions of (2.2.4).

dL dt = a 1 - f (B * ) f (B) dB dt + b 1 - V * V dV dt + c
dL dt = a 1 - f (B * ) f (B) (n(B) -f (B)P) + b 1 - V * V ((1 -π)φ f (B)P -(µ + α)V) + c 1 - P * P (θπ f (B)P + θαV -δP).

Straightforward calculations give

dL dt = a 1 - f (B * ) f (B) (n(B) -n(B * ) -f (B)P + f (B * )P * ) + b 1 - V * V (1 -π)φ f (B)P -(1 -π)φ f (B * )P * V V * + c 1 - P * P θπ f (B)P + θαV -θπ f (B * )P -θαV * P P * . Further expansions yield, dL dt = a f (B) -f (B * ) f (B) (n(B) -n(B * )) -a f (B)P + a f (B * )P + a f (B * )P * -a f (B * ) 2 f (B) P * + b(1 -π)φ f (B)P -b(1 -π)φ f (B * )P * V V * -b(1 -π)φ f (B)P V V * + b(1 -π)φ f (B * )P * + cθπ f (B)P + cθαV -cθπ f (B * )P -cθαV * P P * P -cθπ f (B)P * -cθα P * P V + cθπ f (B * )P * + cθαV * .
After grouping the terms of the above expression, we have,

dL dt = a f (B) -f (B * ) f (B) (n(B) -n(B * )) + f (B)P(-a + b(1 -π)φ)) + f (B * )P * (a + b(1 -π)φ) -a f (B * ) 2 f (B) P * + -b(1 -π)φ f (B * ) P * V * + cθα V + a f (B * ) -cθα V * P * -cθπ f (B * ) P -cθπ f (B)P * -cθα P * P V + cθπ f (B * )P * + cθαV -b(1 -π)φ f (B)P V V * .
Now, choose a, b, and c such that the expressions in the brackets vanish, that is

a f (B * ) -cθα V * P * -cθπ f (B * ) = 0, -b(1 -π)φ f (B * ) P * V * + cθα = 0.
(2.4.37)

Fix c > 0 and pose

a = θαV * f (B * )P * + θπ c, b = θαV * (1 -π)φ f (B * )P * c.
Then, the derivative of L along the trajectories of (2.2.4) becomes,

dL dt = c θαV * f (B * )P * + θπ f (B) -f (B * ) f (B) (n(B) -n(B * )) -cθαV * f (B * ) f (B) + f (B * )P * θαV * f (B * )P * c + θπc + θαV * f (B * )P * c -cθπ f (B * )P * f (B * ) f (B) -cθπ f (B)P * -cθαV P * P -cθαV * f (B)PV * f (B * )P * V + cθπ f (B * )P * + cθαV * = c θαV * f (B * )P * + θπ f (B) -f (B * ) f (B) (n(B) -n(B * )) + 3cθαV * + 2cθπ f (B * )P * -cθαV * f (B * ) f (B) -cθπ f (B * )P * f (B * ) f (B) -cθπ f (B)P * -cθαV P * P -cθαV * f (B)PV * f (B * )P * V .
Further rearrangements lead us to

dL dt = c θαV * f (B * )P * + θπ f (B) -f (B * ) f (B) (n(B) -n(B * )) + cθπ f (B * )P * 2 - f (B * ) f (B) - f (B) f (B * ) + cθαV * 3 - f (B * ) f (B) - f (B)PV * f (B * )P * V - VP * V * P = c θαV * f (B * )P * + θπ β(B -B * ) 2 (B + H b )(B * + H b ) r -µ - r K (B + B * ) + cθαV * 3 - f (B * ) f (B) - f (B)PV * f (B * )P * V - VP * V * P + cθπ f (B * )P * 2 - f (B * ) f (B) - f (B) f (B * ) .
The gathering of some suitably selected terms yields

dL dt = -c r K θαV * f (B * )P * + θπ β(B -B * ) 2 (B + H b )(B * + H b ) (B -(B 0 -B * )) + cθαV * 3 - f (B * ) f (B) - f (B)PV * f (B * )P * V - VP * V * P + cθπ f (B * )P * 2 - f (B * ) f (B) - f (B) f (B * ) = -c r K θαV * f (B * )P * + θπ β(B -B * ) 2 (B + H b )(B * + H b ) (B -B m ) + cθαV * 3 - f (B * ) f (B) - f (B)PV * f (B * )P * V - VP * V * P + cθπ f (B * )P * 2 - f (B * ) f (B) - f (B) f (B * ) .
Finally, using the arithmetic-geometric means inequality, n -(y 1 + y 2 + ... + y n ) ≤ 0, where y 1 y 2 ...y n = 1, and y 1 , y 2 , ...y n > 0, it follows that dL/dt ≤ 0. Furthermore, dL/dt = 0 is equivalent to (B, V, P) = (B * , V * , P * ). The global asymptotic stability of the EPE E * follows from the classical stability theorem of Lyapunov and the LaSalle's Invariance Principle [START_REF] Lasalle | The stability of Dynamical systems[END_REF]. This result shows that, as long as 1 < N 0 < N c 0 , the set Ω E * will never contain the periodic solutions.

Global sensitivity analysis

We carry out sensitivity analysis to ascertain the uncertainty of the parameters to the model output. This is vital since it enables us to identify critical output parameters. Sensitivity and uncertainty analysis are performed using the Latin hypercube sampling (LHS) scheme, a Monte-Carlo stratified sampling method that allows to obtain an unbiased estimate of the model output for a given set of input parameter value. The parameter space is simultaneously sample is used to compute unbiased estimate of output values for state variables [START_REF] Ray | A methodology for performing global uncertainty and sensitivity analysis in system biology[END_REF]. We use predefined variation of the model parameters at 10% and 50% relative to the referential values. Using algorithm from Ray C. [START_REF] Ray | A methodology for performing global uncertainty and sensitivity analysis in system biology[END_REF] , we compute the partial ranking correlation coefficient (PRCC) of parameters against model's variables B, V and P. We use a sample of size 1000 to identify relationship between parameters and output variables. A positive (negative) correlation coefficient corresponds to an increasing (decreasing) monotonic trend between the model's variable and the parameter under consideration. Note that a parameter is significantly correlate to one state variable if the absolute value of PRCC is greater than 0.5 and p-value less than 0.001.

Remark 2.5.1 From Figure 2.2 and Figure 3.2, we can identify six parameters that strongly influence the population dynamics, namely: the contact rate (β), phage death rate (δ), bacteria carrying capacity (K), induction rate (α), burst size (θ) and cell division size (φ). We can then made the following suggestions:

(i) The use of more UV radiations and chemicals to increase the prophage induction could be an effective control measure against the growth of lysogen bacteria.

(ii) The use of biological control to reduce the bacteria cell division size.

(iii) The implementation of the methods proposed in [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF] in order to identify/select lytic phages and release them in the environment. This leads to the increase of contacts between bacteria and lytic phages which in turn favors the reduction of bacteria.

Conclusion

The objectives of this chapter were to provide possible responses to two research questions, namely: (i) what are conditions under which the presence of phages purifies a bacterial polluted environment? (ii) what are conditions under which this infection triggers spread of bacteria-borne disease such as cholera? We build a mathematical model for the phage-bacteria interaction in the environmental reservoir. We have computed the basic offspring number N 0 and use it to provide answers to the research questions. Indeed, this threshold has been used as bifurcation parameter to establish the local/global stability of equilibria. Lyapunov-LaSalle techniques were used for global asymptotical stability results and to estimate the basin of attraction of the locally asymptotically stable equilibrium point. Based on the value and range of N 0 , all the equilibria were topologically classified and the types of bifurcation were specified accordingly. Precisely, we have shown that the system undergoes a transcritical bifurcation around N 0 = 1 and a Hopf bifurcation around the non-negative equilibrium. Computationally, we have used MatLab platform, to perform global sensitivity analysis of the basic offspring number N 0 and the model variables. The result suggests that, the contact rate β, the induction rate α, the bacteria carrying capacity K, the burst size θ and the cell division size φ are the more influential parameters on the phage-bacteria interaction. Moreover, we have simulated the sys-

Chapter 3

Modeling cholera transmission in the complex ecology of phages and bacteria

Introduction

We consider cholera epidemic models including the phage-bacteria interaction with two types of phages (lytic and temperate), the phage-bacteria functional response similar to the function proposed by Smith in [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF].

• For the ODE model the basic reproduction number R 0 is computed, and the existence and stability of equilibria is investigated. We prove that the disease free equilibrium (DFE) is locally asymptotically stable whenever R 0 < 1. The system exhibits a bistability phenomenon via the existence of backward bifurcation, which implies that the classical epidemiological requirement for effective elimination of cholera, R 0 < 1, is no longer sufficient, even though necessary. Due to the existence of backward bifurcation, another threshold N 0 is determined, such that the DFE is globally asymptotically stable when both R 0 and N 0 are less than one, irrespective of their order of comparison. Based on the range of R 0 and N 0 , the proposed model can exhibits one or more endemic equilibria.

The phage absorption rate is identified to be the cause of backward bifurcation and in its absence the model exhibits a trans-critical forward bifurcation at R 0 = 1. Precisely, it is proven that there is no endemic equilibrium whenever R 0 < 1, and there exists a unique globally asymptotically stable endemic equilibrium whenever R 0 > 1.

• A reaction diffusion model is further used to assess the impact of spatial distribution of phages and bacteria on the indirectly transmitted reaction diffusion cholera model. In order to focus on the impact of the phage-bacteria interaction, we split the system into two sub-systems. (i) We begin by analyzing the PDE model without phage absorption rate and prove using suitable choice of Lyapunov functionals that the model exhibits a threshold dynamics, whereby the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when R 0 is less than one and there exists a unique GAS endemic equilibrium whenever R 0 > 1. (ii) Secondly, we consider the full model with non-negative phage absorption rate. Contrary to case (i), the full model does not exhibit a threshold dynamics, and the condition R 0 ≤ 1 is no longer sufficient for cholera elimination, rather, we derive a second threshold N 0 , such that the DFE is GAS whenever R 0 ≤ 1 and N 0 ≤ 1, and cholera persists uniformly whenever R 0 > 1. Finally, the discrete counterpart of the continuous PDE model is derived by constructing a non standard finite difference scheme (NSFD) which is dynamically consistent with the continuous model. This consistency is shown by building discrete Lyapunov functionals with which we replicate the global stability results for the model in (i). The theoretical results are numerically illustrated using the NSFD scheme.

Ordinary differential equation model

Our model falls into the framework of multi-hosts modeling, where the joint dynamics of three distinct populations, namely bacteria-phages-humans beings is described, with the particularity that phages prey on bacteria. In the presence of phages or viruses P, the bacterium population splits into three classes: Susceptible or uninfected bacteria (not yet attacked by phages) B.

Theses are free-living cholera agents capable of self multiplication in the environment. We assume for simplicity that their growth rate is a constant r. Mimicking [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF] we argue that the phage attack rate or the phage-bacteria functional response and the rate of phage loss due to attachment are distinct. In fact the former involves attachment and injection of the one primary (first to inject) phage while the latter takes account of all secondary phage that attach to a cell, this should not be unexpected. We recall that, the functional response is the number of prey successfully attacked per predator as function of prey density, we model the phage-bacteria functional response by

h(B, P) = ε BP F n (cP) , (3.2.1)
where ε is the adsorption rate and

f (P) = P F n (cP) , (3.2.2)
is the phage-attacked rate by H. Smith [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF]. In (3.2.2), c = ε ρ and 1 ρ represents the injection time, the time between binding of phage to host bacteria and subsequent injection of genetic material into host, n denotes the number of binding sites for phage per host (bacterium).

F n (P) = 1 + n l=1 l i=1 P i + P . (3.2.3)
The following properties of f and F n are derived from lemma 2.1 in [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF]:

1. d dP F n (P) > 0 and d dP f (P) > 0 2. lim P→+∞ f (P) = n 3. F n (P) > 1 4. F ∞ (P) ≤ F n+1 (P) ≤ F n (P) ≤ F 1 (P) = 1 + P.
Later on, for the numerical simulation, we shall choose n = 3. This last choice is not a severe limitation, as F n (P) depends rather weakly on n and F 3 (P) is a good approximation of F 100 (P) on 0 < P < 5 [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF]. We assume that the bacteria population cannot maintain itself trough growth in the environment, thus the decay rate µ b of bacterium is greater than r. Since it is known that multiple phage infection is not possible, susceptible bacterium is infected either by temperate or virulent phage. We denote by π the proportion of lysogen bacteria and 1 -π the proportion of bacteria infected by virulent phage.

Let B T be the lysogen bacteria. The lysogenic cycle allows the host cell to continue to survive and reproduce, the virus is reproduced in all of the cell's offspring. The genetic material of phages called prophage can be transmitted to daughter cells at each subsequent cell division [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. We denote the cell multiplication size by φ. In the course of cell division, the effect of UV radiations or the presence of certain chemicals can lead to the release of prophage causing proliferation of new phages through the process called prophage induction. Therefore, αB T is the number of lysogen bacteria who switch from a lysogenic cycle to a lytic cycle, where α denotes the induction rate.

Let B V be population of bacteria infected by virulent/lytic phage. In the lytic cycle, bacteria cells are broken (lysed) and destroyed after immediate replication of the new phages [START_REF] Bhandare | Biocontrol of V. cholorae using bacteriophages[END_REF]. We denote by θ, the burst size of the bacteria and γ the bacteria death due to lysis.

P is the population of phages and µ P is the phage decay rate. The loss of phage may be significant for phage-bacteria interaction. For example, if we assume that a phage cannot detect the state (uninfected or infected) of the host cell to which it binds, then one should not ignore the loss of the phage due to wasted attacks on already infected hosts [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF]. We take into account that a host cell has a multiplicity of potential phage binding sites on its surface, more than one of which may be simultaneously bound by phage. Thus the rate of phage loss due to attachment can be described by the expression

-ε(B + B T + B V )P. (3.2.4)
As For the human population is concerned, we splits it into susceptible (S) and infected (I), and model the cholera epidemic by an SIS -W system, where W stands for the density of V.

cholerae in the environment. Note that in this setting, V. cholerae play the role of bacteria and phages under consideration are those that infect V. cholerae also know as vibriophages. The total human population at time t is N(t) = S(t) + I(t). The recruitment rate in human population is constant and denoted by Λ. Each individual human compartment goes out from dynamics at natural death rate µ h , the death rate due to cholera is denoted by d and the recovered rate is δ. vibriophages can convert their bacterial host from non pathogenic strain to pathogenic strain through a process called phage conversion, by providing the host with phage-encoded virulence genes. Toxigenic V. Cholerae isolates carry the ctxAB genes encoded by lysogenic phage. Thus, the susceptible human population acquire an infection by consuming the lysogen bacteria, at rate βB T S where β is the contact rate with environment. On the other hand, when the susceptible V. cholera are ingested from the environment and reach the small intestine within the human body, complex biological interaction, chemical reaction, and genetic transduction take place that lead to human cholera [START_REF] Xueyung | Modelling the within-host dynamics of cholera: Bacteria-viral interaction[END_REF]. The ingestion of susceptible V. cholerae can cause infection at rate βkB, where k is the infection rate of susceptible bacteria by temperate phage in the small intestine. The ingestion of infected bacteria B V cannot lead to the infection since they are lysed to produce phage. Note that contrary to [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF] we have neglected the delay between the time V. cholera is infected and the moment it lyses. Thus the infection force is given by

λ = β(B T + kB) (3.2.5)
Human contamination of the water supply through infected feces contributes to bacteria levels and is called shedding. Therefore, the shedding rate of susceptible bacteria (B), lysogen bacteria (B T ) and infected bacteria (B V ) are respectively by ω , η and ν. Table 1 summarizes the descriptions of the model parameters.

The above description is schematized in Figure 3

dN dt = Λ -µ h N -dI ≤ Λ -µ h N
Using a Gronwall lemma [START_REF] Lakshmikantham | stability analysis of non linear system[END_REF], we show that

N ≤ Λ µ h + N(0) - Λ µ h e -µ h t , (3.2.8)
where N(0) is the initial condition. From (3.2.8), we observe that if N(0) ≤ Λ µ h , then N(t) ≤ Λ µ h for all t > 0. Similarly, we show that M(t) ≤ M m and P(t) ≤ P m for all t > 0.

Thus Ω is a positively invariant set under the flow described by (3.2.6) so that no solution path leaves through any boundary of Ω. Hence it is sufficient to consider the dynamics of the model (3.2.6) in Ω. In this region, the model can be considered as been epidemiologically and mathematically well-posed.

Analysis of the model

System (3.2.6) has a disease free equilibrium given by E 0 = (S 0 , 0, 0, 0, 0, 0) with S 0 = Λ/µ h . The basic reproduction number of the model (3.2.6) is

R 0 = ωβkS 0 (µ h + d + δ)(µ b -r) + ηβS 0 (µ h + d + δ)(µ b + αγ) . (3.2.9)
In (3.2.9), ωβkS 0 /(µ h + d + δ)(µ br) is the average number of secondary human infections by susceptible bacteria in their entire lifespan, while ηβS 0 /(µ h +d+δ)(µ b +αγ) is the average number of secondary human infections by lysogen bacteria in their entire lifespan.

Proposition 3.2.3

The diseases free equilibrium E 0 is locally asymptotically stable whenever R 0 < 1 and unstable whenever R 0 > 1.

Proof:

The Jacobian matrix at E 0 is

J(E 0 ) =                      -µ h δ -βkS 0 -βS 0 0 0 0 -(µ h + d + δ) βkS 0 βS 0 0 0 0 ω -(µ b -r) 0 0 0 0 η 0 -(µ b + αγ) 0 0 0 ν 0 0 -(µ b + γ) 0 0 e 0 θαγ θγ -µ P                      .
Clearly, -µ h , -(µ b + γ) and -µ P are eigenvalues of J(E 0 ). Therefore, the local stability of E 0 is completely determined by the following sub-matrix

J 0 =         -(µ h + d + δ) βkS 0 βS 0 ω -(µ b -r) 0 η 0 -(µ b + αγ)        
.

The characteristic polynomial of J 0 is

P(λ) = λ 3 + a 2 λ 2 + a 1 λ + a 0 , (3.2.10)
where

                 a 2 = (µ h + d + δ + µ b -r + µ b + αγ), a 1 = (µ h + d + δ)(µ b -r) + (µ h + d + δ)(µ b + αγ) + (µ b + αγ)(µ b -r) -βωkS 0 -βηS 0 , a 0 = (µ h + d + δ)(µ b -r)(µ b + αγ)(1 -R 0 ). (3.2.11)
It follows from the Routh-Hurwitz criteria [START_REF] Birkhoff | Ordinary Differential Equations[END_REF] that E 0 is locally asymptotically stable if and only if

a 2 > 0, a 0 > 0, and a 1 a 2 > a 0 . (3.2.12)
If R 0 < 1, then a 0 is positive and

a 1 a 2 -a 0 = a 0 + (µ h + d + δ) 2 (µ b -r) 1 - βkωS 0 (µ h + d + δ)(µ b -r) + (µ h + d + δ) 2 (µ b + αγ) 1 - βηS 0 (µ h + d + δ)(µ b + αγ) + (µ h + d + δ) (µ b -r) 2 + (µ b + αγ) 2 + (µ b + αγ)(µ b -r) + (µ b + αγ) 2 (µ b -r) ≥ a 0 + (µ h + d + δ) 2 (µ b -r) + (µ b + αγ) (1 -R 0 ) + (µ h + d + δ) (µ b -r) 2 + (µ b + αγ) 2 + (µ b + αγ)(µ b -r) + (µ b + αγ) 2 (µ b -r) > 0.
Thus the disease free equilibrium E 0 is locally asymptotically stable whenever R 0 < 1. Conversely, if R 0 > 1, then a 0 < 0 and E 0 is unstable.

Remark 3.2.4 Note that, for the Model (3.2.6), E 0 is a saddle-node if R 0 = 1. But due to the complexity of the model, the center manifold theory is very tedious to be applied.

The biological implication of the above proposition is that, a sufficiently small flow of infectious individuals will not generate outbreak of the disease unless R 0 > 1. For a better control on the disease, the global asymptotic stability (GAS) of the DFE is needed. Note that, classically, the basic reproduction number of (3.2.6) R 0 is the average number of secondary human infections through environment transmission caused by infectious bacteria (B, B T ) during their entire lifespan. However, one should notice that R 0 do not depends on the parameters of phagebacteria interaction. This is not surprising, since system (3.2.6) couples an epidemic model (cholera) and a population dynamics model (predator-prey system). Usually, the in-depth asymptotic analysis of such a coupled system involves two thresholds, which for our model maybe: an epidemic threshold R 0 and a coexistence threshold for the predator-prey model which will be intuitively defined later. The existence of the latter threshold for model (3.2.6) is actually expected because the infected human individuals contribute to the growth of bacteria.

Thanks to the above mentioned comments, we define the threshold quantity:

N 0 = ηβS 0 (µ h + d + δ)(µ b + αγ) + βωφπS 0 (µ h + d + δ)(µ b + αγ) . (3.2.13)
In order to give a biological interpretation to N 0 , let's rewrite it in the following form

N 0 = βS 0 µ h + d + δ η + ωφπ 1 µ b + αγ . (3.2.14) In (3.2.14)
, βS 0 /(µ h +d+δ) is the average number of the infected individuals, η + ωφπ is the rate of production lysogen bacteria either by shedding (η) or by cell division (φπω) and 1/(µ b + αγ)

is the lysogen bacteria lifespan. Thus, N 0 is the average offspring number of lysogen bacteria produce by one infected human during the phage-bacteria interaction.

Remark 3.2.5 Ecologically, whenever N 0 < 1, the infected-lysogen bacteria population will goes to extinction. On the contrary, N 0 > 1 the population of lysogen bacteria will persist in the environment and enhance the bacteria borne disease outbreak.

For further investigation, the threshold quantity N 0 will be used for the global stability of DFE.

Theorem 3.2.6

The diseases free equilibrium E 0 is globally asymptotically stable in Ω whenever R 0 ≤ 1 and N 0 ≤ 1.

Proof: The proof is done in two steps

Step 1:

N 0 ≤ R 0 ≤ 1
We consider the following Lyapunov function candidate

L = S -S 0 ln S + I + βkS 0 µ b -r B + βS 0 µ b + αγ B V . (3.2.15)
The derivative of L alongside the trajectories is

dL dt = 1 - S 0 S dS dt + İ + βkS 0 µ b -r dB dt + βS 0 µ b + αγ dB V dt = 1 - S 0 S (Λ -βB T S -βkBS -µ h S + δI) + (βB T S + βkBS -(µ h + d + δ)I) + βkS 0 µ b -r (ωI -(µ b -r)B -εB f (P)) + βS 0 µ b + αγ (ηI + φπεB f (P) -(µ b + αγ)B T )
after some computation

dL dt = - µ h S (S -S 0 ) 2 + βB T S 0 + βkBS 0 -(µ h + d + δ)I + βωKS 0 µ b -r I -βkS 0 B - βkS 0 µ b -r εB f (P) + βηS 0 µ b + αγ I + βS 0 µ b + αγ φπεB f (P) -βS 0 B T = - µ h S (S -S 0 ) 2 + I βωKS 0 µ b -r + βηS 0 µ b + αγ -(µ h + d + δ) + εB f (P) βφπS 0 µ b + αγ - βkS 0 µ b -r . Knowing that βφπS 0 µ b + αγ - βkS 0 µ b -r = (N 0 -R 0 ) µ h + d + δ ω .
We have

dL dt = - µ h S (S -S 0 ) 2 + I(µ h + d + δ) βωkS 0 (µ b -r)(µ h + d + δ) + βηS 0 (µ b + αγ)(µ h + d + δ) - 1
+ εB f (P) βφπS 0 µ b + αγ - βkS 0 µ b -r = - µ h S (S -S 0 ) 2 + I(µ h + d + δ)(R 0 -1) + εB f (P) (N 0 -R 0 ) ≤ 0.
Since N 0 ≤ R 0 ≤ 1, dL/dt ≤ 0 and L is indeed a Lyapunov function. Moreover, the largest invariant set contained in Ω such that dL/dt = 0 is {E 0 }, and the application of LaSalle's Invariant Principle [START_REF] Chopra | The search for antimicrobial agents effective against bacteria resistant to multiple antibiotics[END_REF] proves that the DFE is globally asymptotically stable in Ω.

Step 2: R 0 ≤ N 0 ≤ 1

We consider the following Lyapunov function candidate

dL dt = S -S 0 ln S + I + φπ(µ h + d + δ) ωφπ + η B + (µ h + d + δ) ωφπ + η B V . (3.2.16)
One has

dL dt = 1 - S 0 S dS dt + dI dt + φπ(µ h + d + δ) ωφπ + η dB dt + (µ h + d + δ) ωφπ + η dB V dt = 1 - S 0 S (Λ -βB T S -βkBS -µ h S + δI) + (βB T S + βkBS -(µ h + d + δ)I) + φπ(µ h + d + δ) ωφπ + η (ωI -(µ b -r)B -εB f (P)) + (µ h + d + δ) ωφπ + η (ηI + πεB f (P) -(µ b + αγ)B T ) = - µ h S (S -S 0 ) 2 + B βkS 0 - φπ(µ b -r)(µ h + d + δ) ωφπ + η + B T βS 0 - (µ b + αγ)(µ h + d + δ) ωφπ + η = - µ h S (S -S 0 ) 2 + (µ b -r)φπ(µ h + d + δ) φπω + η B βkωS 0 (µ b -r)(µ h + d + δ) + βηkS 0 φπ(µ b -r)(µ h + d + δ) - 1
+ (µ b + αγ)(µ h + d + δ) φπω + η B T βωφπS 0 (µ b + αγ)(µ h + d + δ) + βηS 0 (µ b + αγ)(µ h + d + δ) -1 R 0 ≤ N 0 ≤ 1 implies that k φπ(µ b -r) ≤ 1 µ b + αγ , hence dL dt ≤ - µ h S (S -S 0 ) 2 + (µ b -r)φπ(µ h + d + δ) φπω + η B βkωS 0 (µ b -r)(µ h + d + δ) + βηS 0 (µ b + αγ)(µ h + d + δ) - 1
+ (µ b + αγ)(µ h + d + δ) φπω + η B T βωφπS 0 (µ b + αγ)(µ h + d + δ) + βηS 0 (µ b + αγ)(µ h + d + δ) -1 = - µ h S (S -S 0 ) 2 + (µ b -r)φπ(µ h + d + δ) φπω + η B (R 0 -1) + (µ b + αγ)(µ h + d + δ) φπω + η B T (N 0 -1) .
Since R 0 ≤ N 0 ≤ 1, dL/dt ≤ 0 and L is indeed a Lyapunov function. Moreover, the largest invariant set contained in Ω such that dL/dt = 0 is {E 0 }, and the application of LaSalle's Invariant Principle [START_REF] Chopra | The search for antimicrobial agents effective against bacteria resistant to multiple antibiotics[END_REF] proves that the DFE is globally asymptotically stable in Ω.

Thus, from step 1 and step 2, the DFE is globally asymptotically stable.

We perform a global sensitivity analysis to examine the model response to parameter variation within a wider range in parameter space. Following the approach in [START_REF] Ray | A methodology for performing global uncertainty and sensitivity analysis in system biology[END_REF], PRCC between R 0 , N 0 and each parameter are derived. The results of the PRCC of R 0 and N 0 are shown in figure 3.2. We observe that the parameters Λ, β, ω and η have the most positive influence on R 0 and N 0 (i.e increasing them), while with most negative impacts on R 0 and N 0 (i.e decreasing them) are µ h , δ, α and µ b .

In order to control the spread of cholera, one may make strategies, for instance to reduce the contact rate β, through information of people and increase the decay rate of bacteria µ b through disinfection of contaminated environment as proposed in [START_REF] Sisodiya | Dynamics of cholera epidemics with impulsive vaccination and desinfection[END_REF]. One may wish to assess how their combined action influence the basic reproduction number R 0 and the threshold N 0 . Figure 3.3 is a bifurcation diagram which use the curves of R 0 = 1 and N 0 = 1 to separate R 2 + into four parts. Figure 3.3 specifically shows that, for a couple (µ b , β) ∈ D 0 , and fixing another parameters cholera is eliminated, persists whenever (µ b , β) ∈ D 1 and otherwise exhibits a bistability phenomenon. ii) There is not endemic equilibrium whenever R 0 ≤ 1, and N 0 ≤ 1.

Existence of endemic equilibria

iii) For the other cases, the model (3.2.6) has one or at least three endemic equilibria.

proof: At E * one has                                            Λ -β(B * T + kB * )S * -µ h S * + δI * = 0, β(B * T + kB * )S * -(µ h + d + δ)I * = 0, ωI * -(µ b -r)B * -εB * f (P * ) = 0, ηI * + φπεB * f (P * ) -(µ b + αγ)B * T = 0, νI * + (1 -π)εB * f (P * ) -(µ b + γ)B * V = 0, eI * + θαB * V + θαγB * T -ε(B * T + B * V + B * )P * -µ P P * = 0.
(3.2.17)

Set λ * = β(B * T + kB * ), from the the second equation of (3.2.17)

I * = λ * S * µ h + d + δ . (3.2.18)
Putting (3.2.18) in the first equation of (3.2.17) yields

S * = Λ(µ h + d + δ) λ * (µ h + d) + µ h (µ h + d + δ) . (3.2.19)
Replacing in (3.2.18), we have the following form

I * = Λλ * λ * (µ h + d) + µ h (µ h + d + δ) . (3.2.20)
From the third equation of (3.2.17),

B * = ωI * µ b -r + ε f (P * )
and using (3.2.20) yields

B * = ωΛλ * µ b -r + ε f (P * ) λ * (µ h + d) + µ h (µ h + d + δ) . (3.2.21)
By the same way, from the fourth equation of (3.2.17),

B * T = ε f (P * )(η + φπω) + η(µ b -r) Λλ * (µ b + αγ)(µ b -r + ε f (P * ))(λ * (µ h + d) + µ h (µ h + d + δ)) . (3.2.22)
Now use the notation of λ * , one has the following

λ * = β(B * T + kB * ) = βkω(µ b + αγ) + βε f (P * )(η + φπω) + βη(µ b -r) Λλ * (µ b + αγ)(µ b -r + ε f (P * ))(λ * (µ h + d) + µ h (µ h + d + δ)) .
Since we are interested in the positive values of λ * , after some computations we have the following form

λ * = µ h (µ h + d + δ) (µ b -r)(R 0 -1) + ε f (P * )(N 0 -1) (µ b -r)(µ h + d) + (µ h + d)ε f (P * ) . ( 3
B * = ωΛ (µ b -r)(R 0 -1) + ε f (P * )(N 0 -1) (µ h + d)(µ b -r + ε f (P * )) (µ b -r)R 0 + ε f (P * )N 0 (3.2.24)
and,

B * T = Λ η(µ b -r) + (η + πω)ε f (P * ) (µ b -r)(R 0 -1) + ε f (P * )(N 0 -1) (µ b + αγ)(µ h + d)(µ b -r + ε f (P * )) (µ b -r)R 0 + ε f (P * )N 0 . (3.2.25)
On the other hand, the fifth and sixth equations of (3.2.17) yield,

B * V = Λ ν(µ b -r) + (ν + (1 -π)ω)ε f (P * ) (µ b -r)(R 0 -1) + ε f (P * )(N 0 -1) (µ b + γ)(µ h + d)(µ b -r + ε f (P * )) (µ b -r)R 0 + ε f (P * )N 0 (3.2.26)
and

P * = θαγB * T + θγB * V + eI * ε(B * + B * T + B * V ) + µ P . (3.2.27)
Plugging the expressions of B * , B * T and B * V into (3.2.27) yields

Ψ(P * ) = Φ(P * ), (3.2.28)
where

Ψ(P * ) = P * εΛω(µ b + αγ)(µ b + γ) + εΛ(µ b -r)[(µ b + γ)η + (µ b + αγ)ν] + eµ P (µ b + αγ)(µ b + γ)(µ b -r) 2 R 0 + [εΛ(µ b + γ(η + φπω) + εΛ(µ b + αγ)(ν + (1 -π)ω) + eµ P (µ b + γ)(µ b + αγ)(µ b -r)(R 0 + N 0 )]ε f (P * ) + eµ P (µ b + αγ)N 0 ε 2 f 2 (P * )
and

Φ(P * ) = θγΛ (µ b + γ)α(ωφπ + η) + (µ b + αγ)(ν + (1 -π)ω) (N 0 -1)ε f 2 (P * ) + θγ(µ b -r)Λ (η(µ b + γ) + ν(µ b + αγ) ε f (P * )(N 0 -1) + θγ(µ b -r) (µ b + γ)α(η + φπω) + (µ b + αγ)(ν + (1 -π)ω) ε f (P * )(R 0 -1) + θγ(µ b -r) 2 Λ(η(µ b + γ) + ν(µ b + αγ))(R 0 -1).
The straightforward calculations give

Ψ(0) = 0, and Φ(0) = θγ(µ b -r) 2 Λ η(µ b + γ) + ν(µ b + αγ) (R 0 -1). equation R 0 = 1 gives β = β * .
The jacobian matrix of (3.2.33) is

J * =                      -µ h δ -β * kS 0 -β * S 0 0 0 0 -(µ h + d + δ) β * kS 0 β * S 0 0 0 0 ω -(µ b -r) 0 0 0 0 η 0 -(µ b + αγ) 0 0 0 ν 0 0 -(µ b + γ) 0 0 e 0 θαγ θγ -µ P                      .
After some computations, the right eigenvector of J * is w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) T , where,

w 1 = - (µ h + d) µ h , w 2 = 1, w 3 = ω µ b -r , w 4 = η µ b + αγ , w 5 = ν µ b + γ and w 6 = θαγη µ P (µ b + αγ) + θγν µ P (µ b + γ) + e µ P (µ h + d + δ)
.

The left eigenvector associated to J * is given by v

= (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 )
where,

v 1 = 0, v 2 = 1, v 3 = β * k µ b -r , v 4 = β * µ b + αγ , v 5 = 0, v 6 = 0. Now using the equality d f dP (0) = F n (0) (F n (0)) 2 = 1,
The nonzero second partial derivatives of F are:

∂ 2 f 2 ∂x 1 ∂x 3 = ∂ 2 f 2 ∂x 3 ∂x 1 = βk, ∂ 2 f 2 ∂x 1 ∂x 4 = ∂ 2 f 2 ∂x 4 ∂x 1 = β, ∂ 2 f 3 ∂x 3 ∂x 6 = ∂ 2 f 3 ∂x 6 ∂x 3 = ε, ∂ 2 f 4 ∂x 3 ∂x 6 = ∂ 2 f 4 ∂x 6 ∂x 3 = φπε ∂ 2 f 2 ∂x 3 ∂β * = βS 0 , ∂ 2 f 2 ∂x 4 ∂β * = S 0 .
Thus, we define and compute the numbers A and B as follows:

A = v 2 6 i, j=1 w i w j ∂ 2 f 2 ∂x i ∂x j (0, 0) + v 3 6 i, j=1 w i w j ∂ 2 f 3 ∂x i ∂x j (0, 0) + v 4 6 i,j=1 w i w j ∂ 2 f 4 ∂x i ∂x j (0, 0) = 2 w 1 (β * kw 3 + β * w 4 ) + εw 6 S 0 ω β * φπωS 0 µ b + αγ - β * ωS 0 µ b -r = -2 µ h + d µ h β * kω µ b -r + β * η µ b + αγ + 2 ε S 0 (µ b -r) θαγη µ P (µ b + αγ) + θγν µ P (µ b + γ) + e µ P (µ h + d + δ) β * φπωS 0 µ b + αγ - β * ωS 0 µ b -r .
We note that

β * φπωS 0 µ b + αγ - β * ωS 0 µ b -r = β * φπωS 0 µ b + αγ + β * ηS 0 µ b + αγ - β * ηS 0 µ b + αγ - β * ωS 0 µ b -r = (µ h + d + δ) (N 0 -R 0 ) with R 0 = 1 = (µ h + d + δ) (N 0 -1) .
Finally A can be rewritten in the following form

A = 2 ε(µ h + d + δ) S 0 (µ b -r) θαγη µ P (µ b + αγ) + θγν µ P (µ b + γ) + e µ P (µ h + d + δ) (N 0 -χ 0 ). (3.2.31)
and

B = 6 k,i=1 v k w i ∂ 2 f k ∂x i ∂τ (0, 0) = ωk µ b -r + η µ b + αγ S 0 > 0.
From (3.2.31) one can easily make the following conclusion i. If N 0 < χ 0 , (or equivalently A < 0), then according to Theorem 4.1 in [START_REF] Chavez | Dynamical models of tuberculosis and their application[END_REF], model (3.2.6) exhibits a forward bifurcation.

ii. If N 0 > χ 0 (or equivalently A > 0), then thanks to Theorem 4.1 in [START_REF] Chavez | Dynamical models of tuberculosis and their application[END_REF], model (3.2.6) exhibits a bistability through a backward bifurcation phenomenon.

This end the proof.

The associated bifurcation diagrams are depicted in Figures 3.5. The bi-stability scenario can also be illustrated in Figure 3.6 for R 0 = 0.1711.

For initial condition1, (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 ) = (1000, 100, 10 3 , 500, 100, 400), the solution approaches the disease free equilibrium.

But for the initial condition 2, (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 ) = (1000, 100, 10 6 , 5 × 10 4 , 100, 400) the solution approaches the endemic equilibrium.

Cause of bistability

We now focus on the cause of bistability. Most of the mathematical models for the environmental transmitted diseases exhibit a forward bifurcation at R 0 = 1 [START_REF] Yaghoub | Reproduction numbers for infections with free-living pathogens growing in the environment[END_REF][START_REF] Njagarah | A metapopulation model for cholera transmission dynamics between communities linked by migration[END_REF][START_REF] Berge | Global stability of a two-patch cholera model with fast and slow transmissions[END_REF][START_REF] Codeco | Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir[END_REF]. Since, in our work, bacteria interact with phages, there are no longer free-living. Therefore, in this new setting, it is not surprising that our model exhibits different dynamics. Here, we identified the absorption rate to be a cause of backward bifurcation, which is the situation where a stable disease-free equilibrium coexists with stable disease-persistent equilibrium when R 0 < 1. This is precisely the bistability phenomenon. Precisely, we show in the following proposition that the necessary condition for the existence of backward bifurcation is ε > 0.

Proposition 3.2.9 If ε = 0, then:

i) The DFE of the system (3.2.6) is globally asymptotically stable whenever R 0 ≤ 1. This rule out the possibiliy of backward bifurcation for (3.2.6).

ii) Moreover, if R 0 > 1 the system (3.2.6) has a unique endemic equilibrium which is globally asymptotically stable when δ = 0.

L 0 = S -S 0 ln S + I + βkS 0 µ b -r B + βS 0 µ b + αγ B V . (3.2.32)
From the proof of the Theorem 3.6.4 one has

dL 0 dt = - µ h S (S -S 0 ) 2 + I(µ h + d + δ)(R 0 -1) ≤ 0.
Moreover, the largest invariant set such that dL 0 /dt = 0 is the DFE (S 0 , 0, 0, 0, 0, 0). Thus, by the classical Lyapunov theorem and the LaSalle's Invariance Principle, the global stability of the disease-free equilibrium E 0 is guaranteed. ii) From the proof of the Theorem 3. With R 0 = 0.1711 and initial condition1 (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 ) = (1000, 100, 10 3 , 500, 100, 400), initial condition 2, (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 ) = (1000, 100, 10 6 , 5 × 10 4 , 100, 400).

have

                                                 S * = Λ(µ h + d + δ)(µ b -r)(µ b + αγ) β(µ h + d) ηΛ(µ b -r) + kω(µ b + αγ) (R 0 -1) + (µ b + αγ)(µ b -r)µ h (µ h + d + δ)
,

I * = βΛ 2 ηΛ(µ b -r) + kω(µ b + αγ) (R 0 -1) β(µ h + d) ηΛ(µ b -r) + kω(µ b + αγ) (R 0 -1) + (µ b + αγ)(µ b -r)µ h (µ h + d + δ) , B * = ωΛ (R 0 -1) (µ h + d)(µ b -r)R 0 , B * T = ηΛ (R 0 -1) (µ h + d)(µ b + αγ)R 0 , B * V = νΛ (R 0 -1) (µ h + d)(µ b + γ)R 0 , P * = Λ θαγη(µ b + γ) + θγν(µ b + αγ) (R 0 -1) η(µ b -r)(µ b + γ) + ν(µ b -r)(µ b + αγ) (R 0 -1) + µ P (µ b -r)(µ h + d)(µ b + αγ)(µ b + γ) .

Cholera transmission control by virulent/lytic phages and phage therapy

This section is devoted to the control strategies to reduce the number of infected human. The first strategy is the release/introduction of the selected vibriophage into environment in order to infect and eliminate the population of pathogen bacteria. The second control is the ingestion of virulent vibriophage by an infected individual as a possible therapy.

In order to control a system of differential equations (to cause the solution, the state, to follow a new trajectory) the basic principle of optimal control is often used. The goal of optimal control is to select a particular control that maximizes or minimizes a chosen objective functional, the pay-off which is typically a function of the state and the control. The pay-off is chosen such that the new trajectory/final state are preferred to that of the uncontrolled state, accounting for any cost associated with applying the control.

Release of virulent phages into environment

Controlled model and optimal problem

There are several possible interventions in order to reduce the proliferation of bacteria in the environmental reservoir such as disinfection and water sanitation [START_REF] Sisodiya | Dynamics of cholera epidemics with impulsive vaccination and desinfection[END_REF]. To control the number of infected human one can use the therapeutic treatment or vaccination [START_REF] Mwasa | Mathematical analysis of a cholera model with public healh intervention[END_REF]. In order to reduce the proportion of lysogen after phage bacteria interaction, one can increase the number of virulent phages. In this section, we formulate an optimal control model by proposing the release of virulent phages as strategy for elimination of cholera. The selection can be done using the methods proposed in [START_REF] Hyman | Phages for Phage Therapy: Isolation, Characterization, and Host Range Breadth[END_REF][START_REF] Plaza | Bacteriophages in the control of pathogenic vibrios[END_REF]. Indeed these methods include desirable characteristics such as a relatively broad host range and a lack of other characteristics such as carrying toxin genes and the ability to form a lysogen. While phages are commonly isolated first and subsequently characterized, it is possible to alter isolation procedures to bias the isolation toward phages with desirable characteristics. We denote by h(t) the population of virulent phages V shed at time t, µ is the decay rates of V. We extend model (3. Then there exists h * that minimizes the cost function J in ∆.

J(h * ) = min h∈∆ J(h).

Characterization of an optimal control

According the Pontryagin's maximum principle (PMP), there exists a non trivial absolutely continuous mapping λ[0, t f ] → R 7 , t → λ(t) = (λ 1 (t), λ 2 (t), λ 3 (t), λ 4 (t), λ 5 (t), λ 6 (t), λ 7 (t)) called the adjoint vector containing the adjoint variables. We define the hamiltonian H by

H = 1 2 AI + b 1 h 2 -b 2 V + λ 1 Λ -β(B T + kB)S -µ h S + δI + λ 2 β(B T + kB)S -(µ h + d + δ)I + λ 3 ωI -(µ b -r)B -εB f (P) -εB f (V) + λ 4 ηI + φπεB f (P) -(µ b + αγ)B T + λ 5 νI + (1 -π)εB f (P) -(µ b + γ)B V + εB f (V) + λ 6 θγB V + θαγB T -ε(B T + B V + B)P -µ P P + eP + λ 7 (h -µV).
Theorem 3.3.2 Given an optimal control h * and the corresponding solutions (S * , I * , B * , B * T , B * V , P * , V * ), there exist adjoint variables λ i (t) for i = 1, 2, 3, 4, 5, 6, 7 satisfying the following system of linear differ-ential equations.

dλ 1 dt = β(B T + kB)(λ 1 -λ 2 ) + λ 1 µ h , dλ 2 dt = -A + δ(λ 2 -λ 1 ) + λ 2 (µ h + d) -λ 3 ω -λ 4 η -λ 5 ν -λ 6 e, dλ 3 dt = βkS(λ 1 -λ 2 ) + λ 3 (µ b -r + ε f (V) + ε f (P)) -ε f (P)(φπλ 4 + (1 -π)λ 5 ) + λ 6 εP -ε f (V), dλ 4 dt = βS(λ 1 -λ 2 ) + λ 4 (µ b + αγ) + λ 6 (εP -θαγ), dλ 5 dt = λ 5 (µ b + γ) + λ 6 (εP -θγ), dλ 6 dt = (λ 3 -φπλ 4 -(1 -π)λ 5 )εB f (P) + λ 6 ε(B + B T + B V + µ P ), dλ 7 dt = λ 3 εB f (V) -λ 5 εB f (V) -µλ 7 -b 2 , (3.3.3)
and the transversality conditions

λ * i (t f ) = 0, i = 1, ..., 7. (3.3.4) Furthermore, h * = min 1, max 0, -λ 7 b 1 . (3.3.5)
Proof: The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then the adjoint system can be written as

dλ 1 dt = - ∂H ∂S , dλ 2 dt = - ∂H ∂I , dλ 3 dt = - ∂H ∂B , dλ 4 dt = - ∂H ∂B T , dλ 5 dt = - ∂H ∂B V , dλ 6 dt = - ∂H ∂P , dλ 7 dt = - ∂H ∂V ,
with zero time condition (transversality). To get the characterization of the optimal control, we solve the interior equation of the control set, ∂H ∂h = 0.

Numerical simulations

The simulations are carried out using a set of parameter values taking in table 1 and t ∈ [0, 20].

We use an iterative scheme to solve the optimality system. We first solve the state equations Initial conditions (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 , V 0 ) = (100, 2, 100, 3, 100, 50, 5) . Effect of the released of virulent vibriophages with initial condition (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 , V 0 ) = (100, 2, 100, 3, 100, 50, 5).

vibriophages is maximized, this number has increase from 5 to 250, that is 500%. On Figure 3.8 one can remark that the control strategy resulted in a significantly decrease in the number of infected humans (I), susceptible bacteria (B), and lysogen bacteria (B T ) while a significantly increase is observed in the number of susceptible humans (S). precisely, at the end of the control period of 20 days, our optimal control reduces the number infected humans from 15 Theorem 3.3.4 Given an optimal control h * and the corresponding solutions (B * , J * , P * , V * ), there exist adjoint variables λ i (t) for i = 1, 2, 3, 4, 5, 6, 7 satisfying the following system of linear differential equations.

dλ 1 dt = βHP (B + H) 2 λ 1 -φπλ 2 -θ(1 -π)λ 3 + βHV (B + H) 2 (λ 1 -θλ 3 ) + λ 1 µ -λ 1 n (B)
,

dλ 2 dt = - 1 2 α 1 + α 2 ) + λ 2 µ -λ 2 f (J) , dλ 3 dt = βB B + H λ 1 -λ 2 φπ -λ 3 θ(1 -π) -λ 3 g (P) + λ 3 δ, dλ 4 dt = λ 1 βB B + H -λ 3 βθπB B + H + λ 4 δ, (3.3.8)
and the transversality conditions

λ * i (t f ) = 0, i = 1, ..., 4. (3.3.9)
Furthermore,

h * = min 1, max 0, -λ 4 b 1 . (3.3.10)

Numerical simulations

For the initial condition, (B 0 , J 0 , P 0 ) = (2, 100, 100) and the application of control in 10 days, Figure 3.9 shows that the number of ingested virulent vibriophages is maximized, this number has increase from 5 to 250, that is 500%. On Figure 3.10 one can remark that the control strategy resulted in a significantly decrease in the number of susceptible bacteria (B), infected bacteria (J), while a significantly increase is observed in the number of phages (P). The main objective of this section was to investigate on the use of phages in order to control the dynamic of cholera.

We have solved theoretically and numerically the optimal control problems to assess the role of the use of virulent vibriophages and on the control of cholera. This result showed that, at the end of control period (20 days), the number of virulent vibriophages is maximized at 500%, the number of infected humans decreased to 66%. Consequently, the number of susceptible bacteria decreased to 100% and the number of lysogen bacteria decreased to 50%. For the phage therapy purpose, we remarked that the ingestion of vibriophages reduce the population of bacteria in the small intestine. Overall the use of phages to control the dynamic of cholera as well as for phage therapy are efficient strategies and we suggest this methods in human medicine, veterinary medicine as well as agriculture. Effect of the ingestion of virulent vibriophages with initial condition (B 0 , J , P 0 ) = (2, 100, 100).

Reaction-diffusion model

Modeling Phage-bacteria interaction/prey-predator PDE model

As far as the PDE model is concern, in the presence of phages P, the bacterium population splits into two classes: Susceptible or uninfected bacteria (not yet infected by phage)B. Let J be the infected bacteria and µ p the phages decay rate. The rate of phages loss due to attachment can reaction-diffusion model for cholera epidemic model is given:

                                   ∂S(x, t) ∂t = Λ -λ(B, J)S -µ h S + δI + D∆S, x ∈ Ω, t > 0, ∂I(x, t) ∂t = λ(B, J)S -(µ h + d + δ)I + D∆I, x ∈ Ω, t > 0, ∂B(x, t) ∂t = ωI + rB -µ b B -εB f (P) + D b ∆B, x ∈ Ω, t > 0, ∂J(x, t) ∂t = ηI + φεB f (P) -(µ b + γ)J + D b ∆J, x ∈ Ω, t > 0, ∂P(x, t) ∂t = θγJ -ε(B + J)P -µ p P + D b ∆P, x ∈ Ω, t > 0 (3.4.3)
which is appended by the initial conditions S(x, 0) = s(x), I(x, 0) = i(x), B(x, 0) = b(x), J(x, 0) = j(x), P(x, 0) = p(x), and the Neumann boundary conditions

∂S ∂z = ∂I ∂z = ∂B ∂z = ∂J ∂z = ∂P ∂z = 0, x ∈ Ω, t > 0,
where ∂/∂z denotes the differentiation along the outward normal z to ∂Ω. These conditions show that across the boundary, no external input and output is imposed from outside on these populations. This is not a severe limitation in the sense that, displacement of populations can be controlled during an epidemic situation. Note that model (3.4.3) may also be used to describe the spatial dynamics of any other water-borne disease where specific phages prey on specific bacteria.

Basic properties and basic reproduction number

In this section, we derive the global well-posedness and the threshold dynamics of PDE model (3.4.3).

Well posedness of the system

Let's adopt the following notations, u(x, t) = (S(x, t), I(x, t), B(x, t), J(x, t), P(x; t)) with the initial conditions u(x, 0) = (s(x), i(x), b(x), j(x), p(x)), X = C(Ω, R 5 ) be a Banach space equipped with the usual sup norm || . || X . Let's define X + = C(Ω, R 5 + ), then (X, X + ) is a strongly ordered space. Assume that γ 1 (t), γ 2 (t), γ 3 (t), γ 4 (t),

γ 1 (t), γ 2 (t), γ 3 (t), γ 4 (t), γ 5 (t) : C(Ω, R) → C(Ω, R)
is compact and strongly positive. From corollary 7.3.2 [START_REF] Smith | Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems[END_REF], it is obvious that for all φ ∈ X + , the system (3.4.3) admits a unique mild solution on the interval of existence [0, τ), with τ = τ(φ). The solution (S, I, B, J, P) remains nonnegative for all t ∈ [0, τ).

The following result establishes the global well-posedness result for (3.4.3). ≤ D∆(S(x, t) + I(x, t)) + Λµ h (S(x, t) + I(x, t)), by the same way,

∂φB(x, t) + ∂J(x, t) ∂t = (φω + η)I(x, t) + D b ∆(φB(x, t) + J(x, t)) -(µ b -r)φB(x, t) -(µ b + γ)J(x, t) ≤ (φω + η)I(x, t) + D b ∆(φB(x, t) + J(x, t)) -(µ b -r)(φB(x, t) + J(x, t))
By comparison principle, S(x, t) and I(x, t), B(x, t), J(x, t) are uniformly bounded. From the the last equation of system (3.4.3), we also obtain that P(x, t) is uniformly bounded. Consequently, the solution Z(., t, ϕ) of system (3.4.3) Thus the positive orbits of bounded subset of X + are bounded and this achieved the proof.

Basic reproduction number

Clearly, the disease free equilibrium of model (3.4.3) is E 0 = (S 0 , 0, 0, 0, 0) with S 0 = Λ/µ h . To compute the basic reproduction number R 0 using the technique presented in [START_REF] Zhao | Basic reproduction numbers for reaction-Diffusion Epidemic Models[END_REF], let's consider the following sub-system:

                           ∂I ∂t = λ(B, J)S -(µ h + d + δ)I + D∆I, x ∈ Ω, t > 0, ∂B ∂t = ωI + rB -µ b B -εB f (P) + D b ∆B, x ∈ Ω, t > 0, ∂J ∂t = ηI + φεB f (P) -(µ b + γ)J + D b ∆J, x ∈ Ω, t > 0, ∂I ∂z = ∂B ∂z = ∂J ∂z = 0, x ∈ ∂Ω, t > 0, I(x, 0) = i(x), B(x, 0) = b(x), J(x, 0) = j(x), x ∈ Ω.
(3.5.1)

Set γ(t) the solution semi-group of the above system. The distribution of new infection at time

t is γ(t)X 0 (x) with X 0 (x) = (i(x), b(x), j(x)) T . Define F(x) ≡ F =         0 βρS 0 /H βαS 0 /H 0 0 0 0 0 0         and V(x) ≡ V =         (µ h + d + δ) 0 0 -ω (µ b -r) 0 -η 0 (µ b + γ)         . Model (3.4.
3) is a reaction-diffusion epidemic model in a spatially homogeneous case with the Neumann boundary condition. According to Theorem 3.5 in [START_REF] Zhao | Basic reproduction numbers for reaction-Diffusion Epidemic Models[END_REF], since the diffusion coefficients do not depend on space, the basic reproduction number is actually defined by

R 0 = r(FV -1 ). (3.5.2) Some computations yield R 0 = βωρS 0 H(µ b -r)(µ h + d + δ) + βηαS 0 H(µ b + γ)(µ h + d + δ) . (3.5.3)

Analysis of the model

We now focus on the asymptotic behavior of the solutions of the PDE (3.4.3). We recall that our main goal is to study the impact of environmental phage-bacteria interaction on the diffusive cholera dynamics. To this end, we first neglect the phage absorption rate. The dynamics of resulted model is completely driven by the threshold number called the basic reproduction number R 0 . Secondly we prove for the full model that R 0 may not be sufficient for the disease elimination.

Model without phage absorption

The phage-bacteria interactions also take place in the small intestine of the infected human host, thus whenever ε = 0, the presence of infected bacteria can be justified by the shedding.

The resulted sub-system is the following: Principle. Inspired by [START_REF] Wang | Impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment[END_REF][START_REF] Wang | Global dynamics of a reaction-diffusion waterborne pathogen model with general incidence rate[END_REF][START_REF] Zhang | Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission[END_REF], we choose the following Lyapunov functional candidate:

                                               ∂S(x, t) ∂t = Λ -λ(B, J)S -µ h S + δI + D∆S, x ∈ Ω, t > 0, ∂I(x, t) ∂t = λ(B, J)S -(µ h + d + δ)I + D∆I, x ∈ Ω, t > 0, ∂B(x, t) ∂t = ωI + rB -µ b B + D b ∆B, x ∈ Ω, t > 0, ∂J(x, t) ∂t = ηI -(µ b + γ)J + D b ∆J, x ∈ Ω, t > 0, ∂P(x, t) ∂t = θγJ -µ p P + D b ∆P, x ∈ Ω, t > 0 S(x, 0) = s(x), I(x, 0) = i(x), B(x, 0) = b(x), J(x, 0) = j(x), P(x, 0) = p(x), ∂S ∂z = ∂I ∂z = ∂B ∂z = ∂J ∂z = ∂P ∂z = 0, x ∈ Ω, t > 0. ( 3
L(x, t) = Ω L 0 (x, t)dx = Ω S(x, t) -S 0 -S 0 ln S(x, t) S 0 + I(x, t) + βρS 0 H(µ b -r) B(x, t) + βαS 0 H(µ b + γ) J(x, t) dx.
Dropping the argument (x, t) for notational simplicity, we have:

∂L 0 ∂t = 1 - S 0 S ∂S ∂t + ∂I ∂t + βρS 0 H(µ b -r) ∂B ∂t + βαS 0 H(µ b + γ) ∂J ∂t = 1 - S 0 S D∆S + Λ -λ(B, J)S -µ h S + δI + D∆J + λ(B, J)S -(µ h + d + δ)I , + βρS 0 H(µ b -r) D b ∆B + ωI -(µ b -r)B + βαS 0 H(µ b + γ) D b ∆J + ηI -(µ b + γ)J , = - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + λ(B, J)S + S -S 0 S δI -(µ h + d + δ)I + D∆I + βωρS 0 H(µ b -r) I - βρS 0 H B + βρS 0 H(µ b -r) D b ∆B + βηαS 0 H(µ b + γ) I - βαS 0 H J + βαS 0 H(µ b + γ) D b ∆J.
Clearly, λ(B, J)S 0 ≤ βα(S 0 /H)J + β(ρS 0 /H)B. After grouping like terms, one has

∂L 0 ∂t ≤ - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + I βωρS 0 H(µ b -r) + βηαS 0 H(µ b + γ) -(µ h + d + δ) + D∆I + βρS 0 H(µ b -r) D b ∆B + βαS 0 H(µ b + γ) D b ∆J, = - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + I(µ h + d + δ) (R 0 -1) + D∆I + βρS 0 H(µ b -r) D b ∆B + βαS 0 H(µ b + γ) D b ∆J. Moreover, Ω ∆Sdx = Ω ∆Idx = Ω ∆ b Bdx = Ω ∆ b Jdx = 0, Ω ∆S S dx = Ω | ∆S | 2 S 2 dx ≥ 0. Finally, dL(x, t) dt = Ω ∂L 0 (x, t) ∂t dx ≤ Ω - µ h S (S 0 -S) 2 + I(µ h + d + δ) (R 0 -1) dx ≤ 0.
The largest invariant set contained in Ω such that dL(x, t)/dt = 0 is {E 0 }, and the application of LaSalle's Invariant Principle proves that the DFE is globally asymptotically stable in Ω.

Theorem 3.6.2 Whenever R 0 > 1: i) the system (3.6.1) has a unique endemic equilibrium E * , ii) the unique endemic equilibrium of (3.6.1) is globally asymptotically stable in Ω for δ = 0.

Proof: i) Let E * = (S * , I * , B * , J * , P * ) be any endemic equilibrium of the model (3.6.1). Then

                           λ * = β (αJ * + ρB * ) αJ * + ρB * + H , Λ -λ * S * -µ h S * + δI * = 0, λ * S * -(µ h + d + δ)I * = 0, ωI * -(µ b -r)B * = 0, ηI * -(µ b + γ)J * = 0, θγJ * -µ p P * = 0. (3.6.2)
From the third equation of (3.6.2)

I * = λ * S * µ h + d + δ . ( 3
I * = Λλ * λ * (µ h + d) + µ h (µ h + d + δ) . (3.6.5)
From the fourth equation of (3.6.2),

B * = ωI * µ b -r
and using (3.6.5) yields

B * = ωΛλ * µ b -r λ * (µ h + d) + µ h (µ h + d + δ) . (3.6.6)
By the same way, from the fifth equation of (3.6.2),

J * = ηΛλ * (µ b + γ)(λ * (µ h + d) + µ h (µ h + d + δ)) . (3.6.7)
Now use the notation of λ * , one has the following

λ * = βρω(µ b + γ) + βηα(µ b -r) Λλ * (µ b + γ)(µ b -r)(λ * (µ h + d) + µ h (µ h + d + δ)) .
Since we are interested in the positive values of λ * , after some computations we obtain

λ * = Hµ h (µ h + d + δ)(µ b + γ)(µ b -r)(R 0 -1) Λ αη(µ b -r) + ρω(µ b + γ) + H(µ b + γ)(µ b -r)(µ h + d) . ( 3

.6.8)

Plugging λ * in the expressions of S * , I * , B * J * and P * and setting

K 1 = Hµ h (µ b -r)(µ b + γ)(µ h + d) (R 0 -1) + µ h Λ K 2 with K 2 = Λ Λ αη(µ b -r) + ρω(µ b + γ) + H(µ b + γ)(µ b -r)(µ h + d) yield S * = K 2 K 1 , I * = ΛH(µ b -r)(µ b + γ)µ h (µ h + d + δ) (R 0 -1) K 1 , B * = ΛH(µ b + γ)µ h (µ h + d + δ) (R 0 -1) K 1 , J * = ΛH(µ b -r)µ h (µ h + d + δ) (R 0 -1) K 1 , P * = θγΛH(µ b -r)µ h (µ h + d + δ) (R 0 -1) µ p K 1 .
It easy to remark that for R 0 = 1, S * = S 0 , I * = 0, B * = 0, J * = 0, P * = 0.

ii) For the global stability of the endemic equilibrium E * , let's choose the following Lyapunov functional candidate:

U(x, t) = Ω L * 0 (x,
J * J ∂J ∂t = 1 - S * S Λ -λS -µ h S + D∆S + 1 - I * I λS -(µ h + d)I + D∆I + βρB * S * (ρB * + αJ * + H)ωI * 1 - B * B ωI -(µ b -r)B + D b ∆B + βαJ * S * (ρB * + αJ * + H)ηI * 1 - J * J ηI -(µ b + γ)J + D b ∆J .
At the endemic equilibrium E * , one has

(µ h + d) = λ * S * , (µ b -r) = ωI * /B * , (µ b + γ) = ηI * /J * .
Thus,

∂L * 0 ∂t = 1 - S * S Λ -λS -µ h S + D∆S + 1 - I * I λS - λ * S * I * I + D∆I + βρB * S * (ρB * + αJ * + H)ωI * 1 - B * B ωI - ωI * B * B + D b ∆B + βαJ * S * (ρB * + αJ * + H)ηI * 1 - J * J ηI - ηI * J * J + D b ∆J .
Expanding and grouping like terms of this last expression gives

∂L * 0 ∂t = - µ h S (S -S * ) 2 + βρB * S * (ρB * + αJ * + H) 3 - S * S + λ 1 (B) λ 1 (B * ) - λ 1 (B)SI * λ 1 (B * )S * I - B * I BI * - B B * + βαJ * S * (ρB * + αJ * + H) 3 - S * S + λ 2 (J) λ 2 (J * ) - λ 2 (J)SI * λ 2 (J * )S * I - J * I JI * - J J * + D 1 - S * S ∆S + D 1 - I * I ∆I + D b βρB * S * (ρB * + αJ * + H)ωI * 1 - B * B ∆B + D b βαJ * S * (ρB * + αJ * + H)ηI * 1 - J * J ∆J
where λ 1 (B) = βρB/(ρB + αJ + H), λ 2 (J) = βαJ/(ρB + αJ + H), are increasing functions of B and J respectively. For more investigations, let's set

h(x) = x -1 -ln x, x > 0, h(x) ≥ 0, ∀x > 0.
Therefore,

dU dt = Ω ∂L * 0 ∂t dx ≤ - µ h S (S -S * ) 2 - βρB * S * (ρB * + αJ * + H) h S * S + h λ 1 (B)SI * λ 1 (B * )S * I + h B * I BI * + h B B * -h λ 1 (B) λ 1 (B * ) - βαJ * S * (ρB * + αJ * + H) h S * S + h λ 2 (J)SI * λ 2 (J * )S * I + h J * I JI * + h J J * -h λ 2 (J) λ 2 (J * ) . Furthermore, h λ 1 (B) λ 1 (B * ) -h B B * ≤ λ 1 (B) λ 1 (B * ) - B B * 1 - λ 1 (B * ) λ 1 (B) = λ 1 (B)(B * -B) (λ 1 (B) -λ 1 (B * )) λ 2 1 (B * )B * - B (λ 1 (B) -λ 1 (B * )) 2 λ 2 1 (B * )B * ≤ 0.
Similarly, h λ 2 (J) λ 2 (J * ) h J J * ≤ 0, and dU/dt ≤ 0. Thus the largest invariant set contained in Ω such that dU/dt = 0 is {E * }, and the application of LaSalle's Invariance Principle proves that the E * of the PDE (3.6.1) is globally asymptotically stable in Ω.

The full model with absorption rate

Stability of disease free equilibrium

The local stability of the disease free equilibrium of the full model is given by the following proposition: Proposition 3.6.3 Whenever R 0 ≤ 1, the DFE of the PDE (3.4.3) is locally asymptotically stable.

Proof: Let 0 = µ 0 < µ 1 < µ i < µ i+1 , i = 1, 2..., be the eigenvalues of -∆ on Ω with homogeneous Neumann boundary condition, E(µ i ) the space of eigenfunctions corresponding to µ i and {ϕ i j :

j = 1, 2..., dim(E(µ i ))} an orthogonal basis of E(µ i )). Then X = C 1 Ω 5 can be decomposed as X = ∞ i=1 X i X i j = dimE(µ i ) i=1 X i j
where X i j = {cϕ i j : c ∈ (R) 3 }.

The linearize system at

E 0 is ∂X(x, t) ∂t = diag(D, D, D b , D b )∆X(x, t) + J(E 0 )X(x, t) with J(E 0 ) =                  -µ h δ -βρS 0 /H -βαS 0 /H 0 0 -(µ h + d + δ) βρS 0 /H βαS 0 /H 0 0 ω -(µ b -r) 0 0 0 η 0 -(µ b + γ) 0 0 0 0 θγ -µ P                  . The characteristic polynomial at E 0 is x + µ h + µ i D x + µ p + µ i D b P(x) with P(x) = x + µ h + d + δ + µ i D x + µ b -r + µ i D b x + µ b + γ + µ i D b - βρωS 0 H x + µ b + γ + µ i D b - βαηS 0 H x + µ b -r + µ i D b .
The expansion of P(x) yields P(x) = x 3 + a 2 x 2 + a 1 x + a 0 where

             a 2 = (µ h + d + δ + µ i D + µ b -r + µ i D b + µ b + γ + µ i D b ), a 1 = (µ h + d + δ + µ i D)(µ b -r + µ i D b ) + (µ h + d + δ + µ i D)(µ b + γ + µ i D b ) +(µ b + γ + µ i D b )(µ b -r + µ i D b ) -βωρS 0 /H -βαηS 0 /H, a 0 = (µ h + d + δ + µ i D)(µ b -r + µ i D b )(µ b + γ + µ i D b )(1 -R), (3.6.10)
with

R = βρωS 0 H(µ h + d + δ + µ i D)(µ b -r + µ i D b ) + βαηS 0 H(µ h + d + δ + µ i D)(µ b + γ + µ i D b ) ≤ R 0 . Clearly, a 0 ≥ (µ h + d + δ + µ i D)(µ b -r + µ i D b )(µ b + γ + µ i D b )(1 -R 0 ) ≥ 0, moreover, a 1 a 2 -a 0 = a 0 + (µ h + d + δ + µ i D) 2 (µ b -r + µ i D b ) 1 - βρωS 0 (µ h + d + δ + µ i D)(µ b -r + µ i D b ) + (µ h + d + δ + µ i D) 2 (µ b + γ + µ i D b ) 1 - βηS 0 (µ h + d + δµ i D)(µ b + γ + µ i D b ) + (µ h + d + δ + µ i D) (µ b -r + µ i D b ) 2 + (µ b + γ + µ i D b ) 2 + (µ b + γ + µ i D b )(µ b -r + µ i D b ) + (µ b + γ + µ i D b ) 2 (µ b -r + µ i D b ) ≥ a 0 + (µ h + d + δ + µ i D) 2 (µ b -r + µ i D) + (µ b + γ + µ i D b ) (1 -R) + (µ h + d + δ + µ i D) (µ b -r + µ i D b ) 2 + (µ b + γ + µ i D b ) 2 + (µ b + γ + µ i D b )(µ b -r + µ i ) + (µ b + γ + µ i D b ) 2 (µ b -r + µ i D b ) ≥ a 0 + (µ h + d + δ + µ i D) 2 (µ b -r + µ i D) + (µ b + γ + µ i D b ) (1 -R 0 ) + (µ h + d + δ + µ i D) (µ b -r + µ i D b ) 2 + (µ b + γ + µ i D b ) 2 + (µ b + γ + µ i D b )(µ b -r + µ i ) + (µ b + γ + µ i D b ) 2 (µ b -r + µ i D b ) > 0.
Thus the DFE of the PDE (3.4.3) is locally asymptotically stable.

We now focus on the global asymptotic stability (GAS) of DFE of the PDE (3.4.3) which is needed for the possible elimination of the disease. Our model couples epidemic model (cholera) and a population dynamics model (predator-prey system), and one should notice that the defined reproduction number R 0 does not depends on the parameters of the phagebacteria infection. Moreover the concentration of susceptible bacteria decrease following this interaction, and the infected bacteria concentration increases. Thus for the control of the cholera epidemic, another threshold is needed. To achieved this goal, we derive the following threshold:

N 0 = βαφωS 0 H(µ b + γ)(µ h + d + δ) + βηαS 0 H(µ b + γ)(µ h + d + δ) (3.6.11)
The global stability the DFE is then summarized in the following theorem.

Theorem 3.6. [START_REF] Anderson | Infectious disease of human: Dynamics and control[END_REF] The DFE is globally asymptotically stable whenever R 0 ≤ 1 and N 0 ≤ 1.

Proof:

The proof of this theorem is done in two steps

Step 1: N 0 ≤ R 0 ≤ 1, we choose the following Lyapunov functional candidate:

V = Ω L 1 dx with L 1 = S -S 0 -S 0 ln S S 0 + I + βρS 0 H(µ b -r) B + βαS 0 H(µ b + γ) J, (3
.6.12)

∂L 1 ∂t = 1 - S 0 S ∂S ∂t + ∂I ∂t + βρS 0 H(µ b -r) ∂B ∂t + βαS 0 H(µ b + γ) ∂J ∂t = 1 - S 0 S D∆S + Λ -λ(B, J)S -µ h S + δI + D∆J + λ(B, J)S -(µ h + d + δ)I + βρS 0 H(µ b -r) D b ∆B + ωI -(µ b -r)B -εB f (P) + βαS 0 H(µ b + γ) D b ∆J + ηI + φεB f (P) -(µ b + γ)J = - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + λ(B, J)S + S -S 0 S δI -(µ h + d + δ)I + D∆I + βωρS 0 H(µ b -r) I - βρS 0 H B - βρS 0 H(µ b -r) εB f (P) + βρS 0 H(µ b -r) B b ∆B + βηαS 0 H(µ b + γ) I + βφαS 0 H(µ b + γ) εB f (P) - βαS 0 H J + βαS 0 H(µ b + γ) D b ∆J.
Clearly, λ(B, J)S 0 ≤ βα(S 0 /H)J + β(ρS 0 /H)B. After grouping like terms, one has

∂L 1 ∂t ≤ - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + I βωρS 0 H(µ b -r) + βηαS 0 H(µ b + γ) -(µ h + d + δ) + εB f (P) βφαS 0 H(µ b + γ) - βρS 0 H(µ b -r) + D∆I + βρS 0 H(µ b -r) D b ∆B + βαS 0 H(µ b + γ) D b ∆J. It is not difficult to remark that, βφαS 0 H(µ b + γ) - βρS 0 H(µ b -r) = (µ h + d + δ) ω (N 0 -R 0 ) , that is ∂L 1 ∂t = - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + I(µ h + d + δ) (R 0 -1) + (µ h + d + δ) ω εB f (P) (N 0 -R 0 ) + D∆I + βρS 0 H(µ b -r) D b ∆B + βαS 0 H(µ b + γ) D b ∆J, this implies that, dV dt = Ω ∂L 1 ∂t dx ≤ - µ h S (S 0 -S) 2 + I(µ h + d + δ) (R 0 -1) + (µ h + d + δ) ω εB f (P) (N 0 -R 0 ) ≤ 0.
Step 2: R 0 ≤ N 0 ≤ 1 we choose the following Lyapunov functional in this case

W = Ω L 2 dx with L 2 = S -S 0 -S 0 ln S S 0 + I + φ(µ h + d + δ) ωφ + η B + (µ h + d + δ) ωφ + η J. (3.6.13) ∂L 2 ∂t = 1 - S 0 S ∂S ∂t + ∂I ∂t + φ(µ h + d + δ) ωφ + η ∂B ∂t + (µ h + d + δ) ωφ + η ∂J ∂t = 1 - S 0 S D∆S + Λ -λ(B, J)S -µ h S + δI + D∆J + λ(B, J)S -(µ h + d + δ)I + φ(µ h + d + δ) ωφ + η D b ∆B + ωI -(µ b -r)B -εB f (P) + (µ h + d + δ) ωφ + η D b ∆J + ηI + φεB f (P) -(µ b + γ)J .
Using again the inequalities λ(B, J)S 0 ≤ βα(S 0 /H)J + β(ρS 0 /H)B and grouping like terms, ∂L 2 /∂t becomes,

∂L 2 ∂t ≤ - µ h S (S 0 -S) 2 + D 1 - S 0 S ∆S + J 1 φω + η βαS 0 φω H + βαS 0 η H -(µ b + γ)(µ h + d + δ) + D∆IB 1 φω + η βρS 0 φω H + βηαS 0 H -φ(µ b -r)(µ h + d + δ) + φ(µ h + d + δ) ωφ + η D b ∆B + (µ h + d + δ) ωφ + η D b ∆J = - µ h S (S 0 -S) 2 + J (µ b + γ)(µ h + d + δ) φω + η βαS 0 φω H(µ b + γ)(µ h + d + δ) + βαS 0 η H(µ b + γ)(µ h + d + δ) -1 + B φ(µ b -r)(µ h + d + δ) φω + η βρS 0 ω H(µ b -r)(µ h + d + δ) + βηαS 0 Hφ(µ b -r)(µ h + d + δ) -1 + φ(µ h + d + δ) ωφ + η D b ∆B + (µ h + d + δ) ωφ + η D b ∆J + D∆I + D 1 - S 0 S ∆S.
Knowing that R 0 ≤ N 0 implies that 1/φ(µ br) ≤ 1/(µ b + γ), we have:

∂L 2 ∂t ≤ - µ h S (S 0 -S) 2 + J (µ b + γ)(µ h + d + δ) φω + η βαS 0 φω H(µ b + γ)(µ h + d + δ) + βαS 0 η H(µ b + γ)(µ h + d + δ) -1 + B φ(µ b -r)(µ h + d + δ) φω + η βρS 0 ω H(µ b -r)(µ h + d + δ) + βηαS 0 H(µ b + γ)(µ h + d + δ) -1 + φ(µ h + d + δ) ωφ + η D b ∆B + (µ h + d + δ) ωφ + η D b ∆J + D∆I + D 1 - S 0 S ∆S = - µ h S (S 0 -S) 2 + J (µ b + γ)(µ h + d + δ) φω + η (N 0 -1) + B φ(µ b -r)(µ h + d + δ) φω + η (R 0 -1) + φ(µ h + d + δ) ωφ + η D b ∆B + (µ h + d + δ) ωφ + η D b ∆J + D∆I + D 1 - S 0 S ∆S then, dW dt = Ω ∂L 2 ∂t dx ≤ - µ h S (S 0 -S) 2 + J (µ b + γ)(µ h + d + δ) φω + η (N 0 -1) + B φ(µ b -r)(µ h + d + δ) φω + η (R 0 -1) ≤ 0.
Thus, from step 1 and step 2, the largest invariant set contained in Ω such that dW/dt = 0 is Let's consider the linear system

{E 0 },
                           ∂I(x, t) ∂t = βρωS 0 H B + βαηS 0 H J -(µ h + d + δ)I + D∆I, x ∈ Ω, t > 0, ∂B(x, t) ∂t = ωI -(µ b -r)B + D b ∆B, x ∈ Ω, t > 0, ∂J(x, t) ∂t = ηI -(µ b + γ)J + D b ∆J, x ∈ Ω, t > 0, ∂I ∂z = ∂B ∂z = ∂J ∂z = 0, x ∈ ∂Ω, t > 0. (3.6.14)
Substituting I(x, t) = e νt γ 2 (x), B(x, t) = e νt γ 3 (x) and J(x, t) = e νt γ 4 (x) into (3.6.14) gives the following eigenvalue problem: (3.6.15)

                   νγ 2 (x) = βρωS 0 H γ 3 (x) + βαηS 0 H γ 4 (x) -(µ h + d + δ)γ 2 (x) + D∆γ 2 (x), x ∈ Ω,
The following lemmas are derived from [START_REF] Zhao | Basic reproduction numbers for reaction-Diffusion Epidemic Models[END_REF].

Lemma 3.6. [START_REF] Asheshov | from 1st January to 1st September[END_REF] The problem (3.6.15) has a principal eigenvalue ν 0 with a positive eigenfunction and ν 0 has the same sign as R 0 -1.

Lemma 3.6.6 Let X(., t, ϕ) be a solution of the model (3.4.3) with initial condition X(., t, ϕ) = ϕ ∈ X + , one has: i) ∀ϕ ∈ X + , we have S(x, t, ϕ) > 0, ∀x ∈ Ω, t > 0 and there is a constant τ such that lim inf t→∞ ≥ τ, uni f ormly f or x ∈ Ω, ii) if there exists t 1 > 0 such that I(., t, ϕ) 0 or B(., t, ϕ) 0 or J(., t, ϕ) 0, then, I(x, t, ϕ) > 0, B(x, t, ϕ) > 0 and J(x, t, ϕ) > 0, x ∈ Ω, t > t 1 .

The following theorem allows us to use R 0 as threshold index for Cholera persistence.

Theorem 3.6.7 Whenever R 0 > 1, there exists ζ > 0 such that for all ϕ ∈ X + , with ϕ 2 0 or ϕ 3 0, or ϕ 4 0, one has lim inf Proof: If R 0 > 1, then ν 0 > 0. For any π ∈ (0, π * ), π * sufficiently small, let ν 0 (π) the principal eigenvalue of the following eigenvalue problem: Clearly, lim π→0 ν(π) = ν 0 , one can fix π 0 ∈ (0, π * ) such that ν(π) > 0. Let W = {ϕ ∈ X + : ϕ 3 0, ϕ 4 0, ϕ 4 0} and ∂W = {ϕ ∈ X + : ϕ 2 ≡ 0, ϕ 3 ≡ 0, ϕ 4 ≡ 0}.

                  
W is a positive invariant set for the solution semiflow φ(t). Define

M ∂ = {ϕ ∈ ∂W : φ(t)ϕ ∈ ∂W, t ≥ 0}.
Let ω(ϕ) be the omega set of the orbit of φ(t) through ϕ ∈ X + . For a φ ∈ M ∂ , we have φ(t)ϕ ∈ ∂W,∀t > 0. Thus I(., t, ϕ) ≡ 0, B(., t, ϕ) ≡ 0 and J(., t, ϕ) ≡ 0 ∀t > 0. From the first equation of (3. Suppose by contradiction that lim sup t→+∞ ||φ(t)ϕ 0 -E 0 || ≤ π 0 , ϕ 0 ∈ W. Then, there exists t * > 0 such that S 0 -π 0 < S(x, t, ϕ 0 ) < S 0 + π 0 , I(x, t, ϕ 0 ) < π 0 , B(x, t, ϕ 0 ) < π 0 , J(x, t, ϕ 0 ) < π 0 , t ≥ t * .

We know that, I(x, t, ϕ 0 ), B(x, t, ϕ 0 ) and J(x, t, ϕ 0 ) satisfy Let ψ ν 0 (π 0 ) = (ψ 2 ν 0 (π 0 ) , ψ 3 ν 0 (π 0 ) , ψ 4 ν 0 (π 0 ) ) be the positive eigenfunction associated with ν 0 (π 0 ). Thus the following linear system has a solution u(x, t) = exp ν 0 (π 0 )t ψ λ 0 (π 0 ) (x), since I(x, t, ϕ 0 ) > 0, B(x, t, ϕ 0 ) > 0, J(x, t, ϕ 0 ) > 0, x ∈ Ω and t > 0; there exists σ > 0 such that I(x, t, ϕ 0 ), B(x, t, ϕ 0 ), J(x, t, ϕ 0 ) ≥ σ exp ν 0 (π 0 )t ψ λ 0 (π 0 ) (x), x ∈ Ω, t > 0.

                           ∂I(x,
                          
Since λ 0 (π 0 ) > 0 then lim t→∞ I(x, t, ϕ 0 ), B(x, t, ϕ 0 ), J(x, t, ϕ 0 ) = ∞. This is a contradiction since the variables I, B, J are bounded.

Next, define the continuous function q : X + → [0, ∞) by q(ϕ) = min{min ϕ 3 (x), min ϕ 4 (x)}, ∀ ϕ ∈ X +

We have obviously that q -1 (0, ∞) ⊆ W and q is a generalized distance function for the semiflow φ(t) : X + → X + . Any forward orbit of φ(t) in M ∂ converges to E 0 . The above claims imply that where

A k =                           
                       - M n=0        βαJ * (ρB * + αJ * + H)         h S * S k+1 n + h         λ 2 J k n S k+1 n I * λ 2 (J * ) S * I k+1 n         + h J * I k+1 n J k+1 n I * + h J k+1 n J * -h         λ 2 J k n λ 2 (J * )                        ≤ 0.
Thus {L k } is a non increasing sequence, there exists, L such that,

Numerical simulations

Here, we simulate the proposed model to support our theoretical results and assess the role the diffusion on the spatiotemporal evolution of the cholera model. We choose m = 3. This last choice is not a strong limitation because F m (P) depends rather weakly on m and F 3 (P) is a good approximation of F 100 (P) on 0 < m < 5 [START_REF] Smith | Models of virulent phage growth with application to phage therapy[END_REF]. Just for illustration purposes, we assume that the spatial domain is the segment Ω = [0, 100] and choose the space step-size ∆x = 0.75, and the time step-size ∆t = 0.5. To numerically illustrate the stability of equilibria of PDE model (3.4.3),

we select three sets of model's parameters as follows.

i) We take ε = 0.71, and other parameters such that R 0 = 0.1097 < 1 and N 0 = 0.1049 < 1. This is used to support numerically Theorem 3.6.4 about the GAS the DFE E 0 = (5000, 0, 0, 0, 0) as displayed in Fig 3 .14.

ii) For ε = 0, we choose a set of parameters such that R 0 = 2.16 > 1. This case numerically illustrates Theorem 3.6.2, that is the equilibrium E * = (5000, 4000, 4 × 10 4 , 6.

Conclusion

The main objective of this chapter was the study of the impact of phages and bacteria interaction on the spatial spread of cholera. We first study the temporal model. We computed the basic reproduction number R 0 which served as threshold for stability of the disease free equilibrium and backward bifurcation. The implication of this last phenomenon is that the classical requirement for effective eradication of cholera R 0 < 1 is no longer sufficient. We found a threshold N 0 , such that the disease free equilibrium is globally stable whenever R 0 ≤ 1, and N 0 ≤ 1 altogether.

We have shown that the model had multiple endemic equilibria depending on the range of R 0 and N 0 . We identified the phage absorption rate as a possible cause of backward bifurcation, since in its absence, the system always exhibits a forward bifurcation at R 0 = 1 suggesting that the condition R 0 < 1 is sufficient for the control of cholera. Secondly we have considered the spatiotemporal model (with diffusion). The threshold R 0 have been used as the threshold index of this model. Indeed, whenever R 0 ≤ 1 the disease free model and its discrete nonstandard finite difference counterpart.

(I) We've first proposed a reaction diffusion model to assess the impact of spatial aspect of phage bacteria infection on the indirectly transmitted cholera. The introduction of the diffusion coefficients is motivated by the human host movement, phage-bacteria movement in the environment, the dispersal of phages and bacteria, the water resource, and position. We explicitly computed the basic reproduction number R 0 and use it as the threshold stability of the existent equilibria. We splits the system into two sub-system:

(i) we first analyze the model without phage absorption. On the one hand, using the techniques by Lyapunov and LaSalle, we have shown that the disease free equilibrium is globally asymptotically stable whenever R 0 ≤ 1. On the other hand, we proved that there exists a unique globally asymptotically stable endemic equilibrium E * whenever R 0 > 1.

(ii) Secondly we have considered the full model (with positive phage absorption rate). The threshold R 0 have been used as the threshold index of this model. Indeed, whenever R 0 ≤ 1 the disease free equilibrium is locally asymptotically stable, and is unstable for R 0 > 1 and the system is shown to be uniformly persistent. Here, the condition R 0 ≤ 1 is not sufficient to achieved the possible elimination of cholera (i.e the GAS of the DFE is no longer achieved under the sole condition of R 0 ≤ 1). However, we derive another threshold N 0 , and proved using the Lyapunov-LaSalle's techniques that the DFE is GAS if R 0 ≤ 1 and N 0 ≤ 1.

(II) After proposing a nonstandard finite difference scheme (NSFD), we've derived the discrete counterpart of the continuous model. The results show that the discrete scheme preserves the main properties of solutions for the original continuous model, including positivity, ultimate boundedness, equilibria and their global stability.

(III) The discrete model was further used to illustrate all the theoretical results of the continuous model with regard to global asymptotic stability and uniform persistence.

In chapter 4, in order to mitigate the dynamics of cholera, we have studied optimal control problems. Our first control strategy consisted in the continuous release of lytic/virulent vibriophages into the contaminated environment. We have showed that an optimal control exists and have characterized using the Pontryagin's maximum principle. We have solved numerically the optimal control problem to assess the role of the use of virulent vibriophages on the control of cholera. This numerical result showed that, at the end of control period (20 days), the number of virulent vibriophages is maximized at 500%, the number of infected humans decreased to 66%. Consequently, the number of susceptible bacteria decreased to 100% and the 128 number of lysogen bacteria decreased to 50%. The second strategy consisted in the ingestion of virulent vibriophages for therapy purpose. Results showed that phage therapy can be an effective strategy to eliminate vibrio cholerae in the small intestine.

This dissertation offers many opportunities for extension.

• we intend later to include the innate immune system in our modeling framework to better investigate the perspective of phage therapy.

• Many authors have considered the environment as a reservoir of bacteria (i.e. the pathogen growth rate is always greater than its decay rate), our next plan is to use this assumption to study the existence of periodic solution of the coupled model.

• For the biological control, the continuous release of the virulent vibriophages while mathematically convenient, is not realistic. In general, releases are periodic and instantaneous, to increase realism, impulsive releases is more convenient to be considered.

• We intent to analyze the coupled PDE model with space-dependent diffusion coefficients.

• The existence of traveling waves solutions is also one of the hot topic we are still investigating for PDE model.

• The incorporation of periodic time-dependent model parameters due to seasonality.

Figure 1 . 1 :

 11 Figure 1.1: The incidence of cholera cases (WHO).

Figure 1 . 2 :

 12 Figure 1.2: Schematic representation of a bacteriophage.

Figure 1 . 3 :

 13 Figure 1.3: Lytic and lysogenic cycles of phages.

  and so, by definition of m, x(T) cannot belong to the set || xx ||= ε. Thus, we would have that x(T) ∈ U ε . But this contradicts the maximality of the interval [0, T). It follows that T = β. Since x(t) remains in U ε on [0, T) = [0, β), it remains bounded. So we have that β = T = ∞.

Theorem 1 . 4 . 15 (

 1415 Approximation of the center manifold)

Theorem 1 . 4 . 24

 1424 .4.15) for the model (1.4.8). Let B := ∇.(d I ∇) -V, then we have the following result. Let (A1)-(A6) hold. Then the following statements are valid: (i) R 0 -1 has the same sign as λ * := s(B + F).

(

  H4): We suppose that in the presence of phages (or viruses) population P, the population of bacteria splits into three different classes: (i) The susceptible bacteria are B (not yet attacked by the phages). (ii) The lysogen bacteria population V (those bacteria infected by lysogenic phages), and recall that during the lysogenic life cycle, instead of killing the host, the phage genome integrates into the bacterial chromosome and becomes part of the host. (iii) The population of infected bacteria Z (those bacteria infected by lytic phages), whose equation decouples from the system as we shall see shortly.

Figure 2 . 1 :

 21 Figure 2.1: Schematic representation of the phage-bacteria interactions.

Figure 2 . 2 ,Parameters

 22 N 0 is highly sensitive to the bacteria cell division size φ. Since the prophage induction parameter α is central in this work, because φ and α are keys parameters

Figure 2 . 2 :

 22 Figure 2.2: PRCC s of N 0

Figure 2 . 3 :

 23 Figure 2.3: Contour plots of N 0 versus induction rate α and cell division size φ.

Figure 2 . 4 :

 24 Figure 2.4: Graphs of B * , V * , Z * and P * versus N 0On the left panel, one can observe that the bacteria subpopulations B * , V * , Z * go extinct, while the population of phages sustains for large value of the basic offspring number N 0 . On the right panel, we zoom the figure in the left panel so that one can numerically see that the populations of V * and Z * assume their maximum values at the same threshold value N m 0 .

Figure 2 . 5 :

 25 Figure 2.5: Global asymptotic stability of PFE E 1 whenever N 0 = 0, 409.

.4. 23 )From ( 2 . 4 .

 2324 [START_REF] Dessaleg | Global asymptotic properties of an SEIRS model with multiple infectious stages[END_REF], a 2 (B * ) > 0, and a 0 (B * ) > 0. The number and the signs of the roots of Q(B * ) depend on the number and the signs of the following polynomial

Figure 2 Figure 2 . 6 :Figure 2 . 7 :

 22627 Figure 2.6: LAS of the EPE. N 0 = 1.56, Q(B * ) = 0.17264514 > 0. The parameter set is r = 0.8, K = 10 6 , µ = 0.5, β = 0.75, H b = 10 7 , θ = 100, φ = 80, π = 0.7, α = 0.04 γ = 1, δ = 0.06, B 0 = 8.8 × 10 7 .

Figure 2 . 8 :

 28 Figure 2.8: Global sensitivity analysis (PRCCs) between B, V, P and each parameter.

Figure 3 . 2 :

 32 Figure 3.2: Sensitivity analysis R 0 and N 0 .

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Contour plot of R 0 and N 0 versus β and µ b .

  Set E * = (S * , I * , B * , B * T , B * V , P * ) any endemic equilibrium of the model (3.2.6). The existence of the endemic equilibrium of the model (3.2.6) is summarized in the following theorem: Theorem 3.2.7 i) the model (3.2.6) has a unique endemic equilibrium whenever R 0 > 1, and N 0 > 1.

Figure 3 . 5 :

 35 Figure 3.5: Bifurcation diagrams

Figure 3 . 6 :

 36 Figure 3.6: Bistability phenomenon.

( 3 . 2 . 33 )L 1 =Remark 3 . 2 . 10

 323313210 For the global stability of the endemic equilibrium, we consider the following Lyapunov function S -S * ln S + I -I * ln I + βkS * B * ωI * (B -B * ln B) + βS * B * T ηI * B T -B * T ln B T . (3.2.34)The straightforward computation of the derivative of L 1 alongside the trajectories of (3.2.6) isdL 1 dt = -µ h (S -S * ) 2 S + βS * B * T 3 -* -B T SI * B * T S * I + βkS * B * 3 -S * S -B * I BI * -BSI * B * S * I ≤ 0. (3.2.35)Finally, using the following relation between the arithmetic and the geometric means, n -(y 1 + y 2 + ... + y n ) ≤ 0, where y 1 y 2 ...y n = 1, and y 1 , y 2 , ..., y n > 0, it follows that dL 1 /dt ≤ 0.Furthermore, the largest invariant such that dL 1 /dt = 0 is the singleton{(S * , I * , B * , B * T , B * V , P * )}.The global stability of the endemic equilibrium E * follows from the classical stability theorem of Lyapunov and LaSalle's Invariance Principle. The global stability of the endemic equilibrium for δ 0 remains an open problem for cholera models.

3 . 3 . 1 below by c 1 h β 2 -c 2

 33122 2.6) to obtain the following controlled model for Theorem If the following hold: 1-the set of controls and corresponding state variables is non empty; 2-the control set ∆ is convex and closed; 3-the right hand side of the state system is bounded by a linear function in the state and control; 4-the integrand of the objective functional is convex; 5-there exist constants c 1 > 0 , c 2 > 0 , and β > 1 such that the integrand of the objective functional is bounded

( 3 . 3 . 1 )Figure 3 . 7 :

 33137 scheme. Then, we use the current iterations solutions of the state equation to solve the adjoint Equations by a backward fourth order Runge-Kutta scheme. The result is displayed on Figures 3.7 and 3.8. For the initial condition, (S 0 , I 0 , B 0 , B T0 , B V0 , P 0 , V 0 ) = (100, 2, 100, 3, 100, 50, 5) and the application of control in 20 days, Figure3.7 shows that the number of released virulent

Figure 3 . 8 :

 38 Figure 3.8: Simulation results of optimal control model (3.3.1).

Figure 3 . 9 :

 39 Figure 3.9: Control function.

Figure 3 . 10 :

 310 Figure 3.10: Simulation results of optimal control model (3.3.6).

  γ 5 (t) : C(Ω, R) → C(Ω, R) are the C 0 -semi-groups associated with D∆S-µ h , D∆I-(µ h +d+δ), D b ∆B-(µ b -r), D b ∆J-(µ b +γ), D b ∆P -µ p subject to Neumann boundary condition. Clearly, ∀ϕ ∈ C(Ω, R), we haveγ 1 (t)ϕ(x) = e -µ h t Ω Φ(x, t, s)ϕ(s)ds, γ 2 (t)ϕ(x) = e -(µ h +d+δ)t Ω Φ(x, t, s)ϕ(s)ds, γ 3 (t)ϕ(x) = e -(µ b -r)t Ω Ψ(x, t, s)ϕ(s)ds, γ 4 (t)ϕ(x) = e -(µ b +γ)t Ω Ψ(x, t, s)ϕ(s)ds, γ 5 (t)ϕ(x) = e -µ P t Ω Ψ(x, t, s)ϕ(s)ds, t > 0where Φ and Ψ are Green functions associated to D∆ and D b ∆ subject to Neumann boundary condition respectively. Thanks to corollary 4[START_REF] Martin | Abstract functional differential equations and reactiondiffusion systems[END_REF]

Theorem 3 . 5 . 1

 351 For any ϕ ∈ X + , system (3.4.3) admits a unique solution u(t, x, ϕ) defined on [0, ∞)×Ω, and every solution semiflow ϕ t : X + → X + are bounded.Proof: Adding the first and the second equation of 3.4.3 yields ∂S(x, t) + ∂I(x, t) ∂t = D∆(S(x, t) + I(x, t)) + Λ -(µ h + d)I(x, t) -µ h S(x, t)

.6. 1 )Theorem 3 . 6 . 1

 1361 Setting I = 0 in (3.6.1) implies that, S = S 0 = λ/µ h , B = J = P = 0. Thus the DFE of (3.6.1) is the same that of the full model and is given E 0 = (S 0 , 0, 0, 0, 0). The following theorem gives the global stability of E 0 of the model (3.6.1). If R 0 ≤ 1, the DFE is globally asymptotically stable in Ω.Proof:The global stability of the spatially homogeneous steady-state solution E 0 of the reactiondiffusion system (3.6.1) subject to Neumann boundary conditions is based on the construction of Lyapunov functional for partial differential equations and the use of LaSalle Invariance

νγ 3 (

 3 x) = ωγ 2 (x) -(µ br)γ 3 (x) + D b ∆γ 3 (x), x ∈ Ω, νγ 4 (x) = ηγ 2 (x) -(µ b + γ)γ 4 (x) + D b ∆γ 4 (x), x ∈ Ω, ∂γ 2 ∂z = ∂γ 3 ∂z = ∂γ 4 ∂z = 0, x ∈ ∂Ω.

  t→∞ S(x, t, ϕ) ≥ ζ, lim inf t→∞ I(x, t, ϕ) ≥ ζ lim inf t→∞ B(x, t, ϕ) ≥ ζ, lim inf t→∞ J(x, t, ϕ) ≥ ζ, lim inf t→∞ P(x, t, ϕ) ≥ ζ,uniformly for x ∈ Ω, where X(x, t, ϕ) is a solution of the system (3.4.3) with initial condition X(., 0, ϕ) = ϕ ∈ X + .

νγ 2 (

 2 x) = βρω(S 0 -π) H γ 3 (x) + βαη(S 0 -π) H γ 4 (x) -(µ h + d + δ)γ 2 (x) + D∆γ 2 (x), x ∈ Ω, νγ 3 (x) = ωγ 2 (x) -(µ br)γ 3 (x) + D b ∆γ 3 (x), x ∈ Ω, νγ 4 (x) = ηγ 2 (x) -(µ b + γ)γ 4 (x) + D b ∆γ 4 (x), x ∈ Ω, x ∈ ∂Ω.

  µ h S + D∆S, x ∈ Ω, t > 0 ∂S ∂z = 0, x ∈ ∂Ω, t > 0.Using the theory of asymptotically semiflow; lim t→∞ S(., t, ϕ) = S 0 uniformly for x ∈ Ω, andω(ϕ) = E 0 , ϕ ∈ M ∂ .Now we show that lim sup t→+∞ ||φ(t)ϕ -E 0 || ≥ π 0 .

3 -

 3 (µ h + d + δ)u 1 + D∆u 1 , x ∈ Ω, t > t * , ∂u 2 (x, t) ∂t = ωu 1 -(µ br)u 2 + D b ∆u 2 , x ∈ Ω, t > t * ∂u 3 (x, t) ∂t = ηu 1 -(µ b + γ)u 3 + D b ∆u 3 , x ∈ Ω, t > t * ∂u 1 ∂z = ∂u 2 ∂z = ∂u 3 ∂z = 0, x ∈ ∂Ω,

4 )Theorem 3 . 7 . 1

 4371 The system (3.7.1) has the same equilibria as(3.4.3). The following theorem guarantees the basic properties of the discrete PDE model (3.7.1). For ∆t > 0, and ∆x > 0, the solution of the system (3.7.1)-(3.7.4) is non-negative and bounded, ∀k ∈ N Proof: We proceed by induction on k. From the first equation of (3.7.1), we haveA k S K+1 = S k + Λ k ∆t

2 +( 3 . 7 . 8 ).

 2378 n dφbb(µ br)B k+1 nd(µ b + γ)J k+1 n + bc B k+1 n -B k n + ed J k+1 ndη -(µ h + d) + εB k+1 n b(µ br) + J k+1 n edb(µ b + γ) .For the case N 0 ≤ R 0 ≤ 1, let's choose b, c, d, e as follows:b = βρS 0 /H(µ br), c = (µ br), d = βαS 0 /H(µ b + γ), e = (µ b + γ). (µ h + d)I k+1 n (R 0 -1) + µ h + d + δ ω εB k+1 n f P k n (N 0 -R 0 ) . For the case R 0 ≤ N 0 ≤ 1, we choose; b = φ(µ h + d)/(φω + η), c = (µ br), d = (µ h + d)/(φω + η), e = (µ b + γ), L k+1 -L k ≤ + γ)(µ h + d) φω + η (N 0 -1) + B k+1 n φ(µ br)(µ h + d) φω + η (R 0 -1) . (3.7.9) Using (3.7.8) and (3.7.9) proves that L k+1 -L k ≤ 0. Hence the sequence {L k } is decreasing and bounded. Thus there exists, a positive function L such that lim t→+∞ L k = L. the global stability, we use the following discrete Lyapunov functional candidate.L k = M n=0 1 ∆t S k n -S * -S * ln S k n S * + βαB * S * (ρB * + αJ * + H)ωI * (1 + (µ br)∆t) B k n -B * -B * ln -I * -I * ln I k n I * + βαJ * S * (ρB * + αJ * + H)ηI * (1 + (µ b + γ)∆t) J k n -J * -J * ln µ h S * -µ h S k+1 n + 1 -S * S k+1 λ (B * , J * ) S * -λ B k n ,Recall the following functions λ 1 (B) = βρB ρB + αJ + H , λ 2 (J) = βαJ ρB + αJ + H , and define the function h(x) = x -1ln x.Then, one hasL k+1 -L k ≤

  = S * , lim k→+∞ I k n = I * , lim k→+∞ B k n = B * , lim k→+∞ J k n = J * , lim k→+∞ P k n = P * .

5 × 10 5 , 6 ×

 556 10 8 ) is globally asymptotically stable as shown in Fig 3.12.iii) To support the uniform persistence, we take ε = 0.71, and select other parameters such that R 0 = 5.027 > 1, the uniform persistence is illustrated in Fig 3.13. iv) To carry out the influence of the diffusion on the system, we fix the values of the coefficients D and D b . In Figs 3.14, 3.15,3.16, the time evolution of the solutions is illustrated for different values of the diffusion coefficients for humans, phages and bacteria (assuming that phages and bacteria diffuse similarly).

Figure 3 . 11 :

 311 Figure 3.11: Global stability of the DFE.

Figure 3 . 12 :

 312 Figure 3.12: Global stability of the endemic equilibrium E * .

Proof:[85] Lemma 1.4.3 (Gronwall)

  Let f (t), ϕ(t) be nonegative continuous functions on an open interval J = (α, β) containing the point t 0 .

	Let c 0 ≥ 0. if
	t
	f (t) ≤ c 0 +	ϕ(s) f (s)ds ,
	t 0
	for all t ∈ J, then
		t
	f (t) ≤ c 0 exp	ϕ(s)ds ,
		t 0
	for all t ∈ J.

the initial value problem (1.4.1) has a unique local solution defined on the interval |t-t 0 | < δ.

Definition 1.4.4

A point x ∈ O is called an equilibrium point if f (t, x) = 0, for all t ∈ R.

Definition 1.4.5

  An equilibrium point x is stable if given any ε > 0, there exists a number δ > 0 such that for all ||x 0 -x|| < δ, the solution of the initial value problem x(t, 0, x0) exists for all t ≥ 0 and || x(t, 0, x 0 )x ||< ε t ≥ 0.

	An equilibrium point x is asymptotically stable if it is stable and there exists a number b > 0 such that
	if ||x 0 -x|| ≤ b then
	lim t→∞	|| x(t, 0, x 0 ) -x ||= 0.
	An equilibrium point x is unstable if it is not stable.
	Theorem 1.4.6 Let A be an n × n matrix over R, and define the linear vector field
		f (x) = Ax.
	The equilibrium x is asymptotically stable if and only if Reλ < 0 for all eigenvalues of A. The equilibrium
	x is stable if and only if Reλ ≤ 0 for all eigenvalues of A and A has no generalized eigenvectors
	corresponding to eigenvalues with Reλ = 0.

Theorem 1.4.7

  Let O ⊂ R n be an open set, and let f : O → R n be C 1 . Suppose that x is an equilibrium point of f and that the eigenvalues of A = D f ( x) all satisfy Reλ < 0. Then x is asymptotically stable.

1.4.3 Lyapunov stability

	Let f (x) be a locally Lipschitz continuous vector field on an open set O ⊂ R n . Assume that f
	has an equilibrium point at x ∈ O.
	Definition 1.4.8 Let U ⊂ O be a neighborhood of x. A Lyapunov function for an equilibrium point x of
	a vector field f is a function E

Table 1 .

 1

	Symbols Biological definitions
	S	Susceptible human
	I	Infected human
	B	Susceptible bacteria
	J	Infected bacteria
	V	Phages
	A	recruitment rate of humans
	β	Ingestion rate
	α	Cholera mortality rate
	ν	Recovery rate
	η	shedding rate
	θ 0	Bacteria death rate due to lysis
	δ	Phages death rate
	θ 1	Bacteria burst rate
	m, k	Half saturation rate
	r	Bacteria growth rate

1: Variables and parameters for model (1.5.2).

Table 1 .

 1

	Symbols Biological definitions
	S	Susceptible human
	I	Infected human
	V	Susceptible bacteria
	P	Phages
	A	recruitment rate of susceptible humans
	β	Ingestion rate
	α	Cholera mortality rate
	ν	Recovery rate
	η	Shedding rate
	δ	Phages death rate
	θ	Bacteria burst rate
	φ	Phage absorption rate
	m, k	Half saturation rate
	µ	Bacteria growth rate
	µ 0	Decay rate of bacteria

.5.2). They did not derived conditions 2: Variables and parameters for model

(1.5.3)

. under which phage-bacteria infection cause a bistability phenomenon on a cholera dynamics. The gaps is filled in a model to show that a coupling model of cholera and phage-bacteria infection can exhibits a backward bifurcation. Moreover, since phages prey on bacteria, we extend this model to a controlled model in order to investigate on the impact of release of virulent phages into environment.

Table 2 .

 2

	.2.5)

1: Variables and parameters for model (2.2.4).

  .2. From Table 2.2 and Figure

	Remark 2.3.3

Table 3 .

 3 .1. Based on the above formulation and assumptions, the model describing the cholera dynamics is given by the following deterministic 1: Variables and parameters for model system(3.4.3). population is clearly not constant. Therefore, the evolution equation for the change in the human population is given by

	Symbols Biological definitions	Range	Source
	S	Susceptible human
	I	Infected human
	B	Susceptible bacteria
	B T	Infected bacteria by temperate phages
	B V	Infected bacteria by virulent phages
	P	Phages
	r	Intrinsic bacteria growth rate	0.3 -14.3	[41].
	µ b	Bacteria decay rate	0-1	assumed.
	ω, η, ν	Bacteria shed rates	10-100	[60].
	ε	Phage absorption rate	0-0.0025	[41].
	α	Prophage induction rate	0.001 -0.99 assumed.
	φ	Cell division size	10 -100	assumed.
	π	Fraction of lysogen bacteria	0 -1	[11].
	γ	Bacteria death rate due to lysis	0.1 -1	assumed.
	θ	Burst size	80 -100	[41].
	e	Phage shed rate	0.1 -0.99	assumed.
	Λ	Human recruitment	1-5000	assumed.
	µ h	Human death rate	0-1	[60].
	d	Human death rate due to cholera	0 -0.99	[54].
	δ	human recovery rate	0-1	assumed.
	k	Infection rate in the small intestine	0.4 -0.99	assumed.

  exists globally on [0, ∞).Thus, for sufficiently χ > 0, there exists t 0 such that ∀t > t 0 Then S(x, t), and I(x, t) are ultimately bounded. There exists t 1 > t 0 such that ∀t > t 1 , φB(., t) + I(., t) ≤ (ωφ + η)Λ µ h (µ br) (1 + χ), and P(., t) ≤ θγ(φω + η)Λ µ h (µ br)µ P (1 + χ), ∀t > t 1 , t 1 > t 0

	By the comparison principle, one has
	lim sup t→+∞ (S(x, t) + I(x, t)) ≤	Λ µ h	,	uniformly, ∀x ∈ Ω.

S(., t) ≤ Λ µ h (1 + χ), and I(., t) ≤ Λ µ h (1 + χ), ∀t > t 0 .

  and the application of LaSalle's Invariance Principle proves that the DFE of the PDE(3.4.3) is globally asymptotically stable in Ω.

	Due to complexity of model (3.4.3), the existence and the global stability of the endemic
	equilibrium have not been investigated for ε > 0. Alternatively, in what follows, we prove the
	uniform persistence of the full model in this case.
	3.6.2.2 Uniform persistence of system (3.4.3)

  (µ h + d + δ)I + D∆I, x ∈ Ω, t > t * , ∂B(x, t) ∂t ≥ ωI -(µ br)B + D b ∆B, x ∈ Ω, t > t * ∂J(x, t) ∂t ≥ ηI -(µ b + γ)J + D b ∆J, x ∈ Ω, t > t *

	t) J -∂I ∂t ≥ βρω(S 0 -π 0 ) H B + βαη(S 0 -π 0 ) H ∂z = ∂B ∂z = ∂J ∂z = 0, x ∈ ∂Ω.

a 2 a 3 a 2 ... 0 0 0 0 a 2 a 3 ... 0 0 0 . . . . . . . . . . . . . . . . . . . . .

d 2 d 3 d 2 ... 0 0 0 0 d 2 d 3 ... 0 0 0 . . . . . . . . . . . . . . . . . . . . .

Acknowledgements

(PFE) have been illustrated and the occurrence of periodic solutions have been depicted. Epidemiologically speaking, the existence of stable periodic solutions could explain the occurrence of repetitive outbreaks of some bacteria-borne diseases such as cholera in Africa and Asia.

Bacteriophages can convert their bacterial host from non pathogenic strain to pathogenic strain through a process called phage conversion, by providing the host with phage-encoded virulence genes. Toxigenic V. Cholerae isolates carry the ctxAB genes encoded by lysogenic phage. Only those strains cause epidemic and pandemic cholera [START_REF] Faruque | Phage-bacterial interactions in the evolution of toxigenic vibrio cholerae[END_REF]. The next chapter deals with the coupled cholera epidemic model with phage-bacteria infection model to study the impact of prophage induction and lysogen bacteria in the cholera transmission. system of non-linear differential equations: Though this model is formulated for cholera epidemics, however, we stress that it can apply to other bacterial-borne diseases, for which the disease pathogen can interact with a specific phage and lyse. For example, E. Coli, Q fever, Pyomysitis, Eurysipelas [START_REF] Hyman | Phages for Phage Therapy: Isolation, Characterization, and Host Range Breadth[END_REF].

Basic properties

Since the model (3.2.6) monitors changes in the populations (humans, phages, bacteria), for it to be epidemiologically meaningful, it is important to prove that the solution with non-negative initial data exists, is unique, bounded and remains non-negative for all t ≥ 0. Set

Lemma 3.2.1 Given that the initial conditions of the system (3.2.6) are S(0) > 0, I(0) > 0, B(0) > 0, B T (0) > 0, B V (0) > 0, P(0) > 0, the solutions ( S(t), I(t), B(t), B T (t), B V (t), P(t) ) of the Cauchy problem associated to the model (3.2.6) are non-negative for all t > 0.

Proof: We have

Therefore, due to Lemma 2 in [START_REF] Yang | permanence and positive periodic solution for the single-species nonautonomous delay diffuse models[END_REF], any solution of system (1) is non-negative for all t > 0. .

Proof:

We first split the model (3.2.6) into three parts, the human population that is S(t), I(t), the pathogen population B(t), B T (t), B V (t), and the phage population P. The total human

and

Recalling that, the Smith attachment function f (P) satisfies f (P) > 0 and lim P→+∞ f (P) = n, we conclude that Ψ (P) > 0, lim P * →+∞ Ψ(P * ) = +∞ and lim P * →+∞ Φ(P * ) = y 0 with

Thus, Ψ is an increasing function and Φ has an horizontal asymptote y = y 0 . Note that the sign of y 0 depends on the values of R 0 and N 0 .

The existence and the number of positive endemic equilibria of (3.2.6) depends on the intersection points of the graphs of Ψ and Φ. We proceed by inspection to investigate the number of the positive roots of equation (3.2.28).

i) R 0 ≤ 1, and N 0 ≤ 1. In this case Φ is decreasing function and Φ(0) ≤ 0. Since Ψ is increasing and Ψ(0) = 0. The graphs of Ψ and Φ do not intersect.

ii) R 0 > 1, and N 0 > 1. In this case Φ is increasing, Φ(0) > 0 and y 0 > 0. Since Ψ is increasing and Ψ(0) = 0 there is only one intersection point of graphs of Ψ and Φ.

iii) Otherwise, there is one or at least three intersections points between Ψ and Φ.

Bifurcation analysis

Backward and forward Bifurcations

To conduct the bifurcation analysis, we define the following thresholds:

and ii) If N 0 > χ 0 then the model (3.2.6) exhibits a backward (sub-critical) bifurcation at R 0 = 1. That is there exits a locally stable endemic equilibrium when R 0 < 1 which coexists with the stable DFE, and when R 0 > 1, there exists a stable endemic equilibrium.

Proof: To explore the possibility of a backward bifurcation in the model (3.2.6), we introduce the following notations, we re-label the variables S = x 1 , I = x 2 , B = x 3 , B T = x 4 , B V = x 5 , P = x 6 . Further, by introducing the vector notation X = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) T (3.2.6) has the form dX dt = F(X), where F = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ), as follows:

(3.2.30) Theorem 4.1 in [START_REF] Chavez | Dynamical models of tuberculosis and their application[END_REF] will be used to determined whether or not the model (3.2.6) exhibits a backward bifurcation at R 0 = 1. We set β as the bifurcation parameter. Solving for β the cholera, for which virulent vibrophages are continuously released into the environment where V. cholerae live and can infect human beings. Let t f the time at the end of the control.

, is an extension of (3.4.3) via the inclusion of the continuous release of virulent vibriophages V. This implies the reduction of the population of B by the quantity εB f (V) which enters the population of B V . The objective of the control is to minimize the number of infected individuals (I) and maximized the number of virulent vibriophages at the end of epidemic period, while keeping the coast of the control as low as possible. To achieve this goal, we incorporate the relative coasts associate with each control or combination of policies directed towards controlling the spread of cholera. We define the objective function

and the control set

In (3.3.2), b 1 is the cost of selection and release of virulent phages and b 2 is the concentration of V, A is the social cost. Social cost depends on the number of cholera infections which are related directly with the number of infectious bacteria (B and B T ). Notice that when we minimize the performance index J, the virulent phage is maximized. It is not difficult to prove that there exists an optimal control h * and a corresponding solution (S, I, B, B T , B V , P, V) of the initial value problem (3.3.1) that minimizes the cost function J in ∆ such that

to 10 individuals (ie 66%). Consequently, the number of susceptible bacteria vanishes (100% decrease) and that of lysogen bacteria drops from 9 to 4 (ie ≈ 50% decrease) cells.

Phage therapy

We recall that phage therapy is the therapeutic use of lytic bacteriophages to treat pathogenic bacterial infections. To investigate on its impact on the cholera dynamic we derive a controlled within-hosts model to describe the bacterial-viral interaction taking place in the small intestine, where the vibrios initially enter the human body from the contaminated environment by ingestion.

Controlled model and optimal problem

Through the interaction with the phages P, the ingested susceptible vibrios B are transformed into infected vibrios J whose are highly toxic. Susceptible bacteria, infected bacteria and phages are ingested from environment at rate n(B), f (J) and g(P) respectively. For phage therapy purpose, an infected human continuously ingests the selected virulent/lytic phages V at rate h. The controlled model representing the within-host cholera dynamics is formulated as the following system of ordinary differential equations:

Where, β is the phage absorption rate, H is the bacteria half saturation, µ is the bacteria death rate, φ is the bacteria division size, π is the proportion of lysogeny, θ is the burst size, and δ is the phage death rate. The following properties of n, f and g are derived from [START_REF] Xueyung | Modelling the within-host dynamics of cholera: Bacteria-viral interaction[END_REF]

It is not difficult to prove the well posedness of system (3.3.6).

The objective of the control is to minimize the number of vibrios and maximized the number of virulent vibriophages at the end of control, while keeping the coast of the control as low as possible. We define the objective function

and the control set

In (3. Then there exists h * that minimizes the cost function J in ∆.

Characterization of an optimal control

We define the hamiltonian H by

be described by the expression

The spatial displacement of bacteria and phages is modeled by diffusion D b ∆, where D b is the common diffusive coefficient for bacteria and phages and ∆ is the Laplace operator.

Diffusive Cholera epidemic model

Let Ω ⊂ R n , n ∈ R, be a spatial habitat where human population lives and consumes water from a nearby aquatic reservoir. Let S and I be the susceptible and infected humans respectively. Vibriophages (phages that infect vibrio cholerae) can convert their bacterial host from non pathogenic strain to pathogenic strain through a process called phage conversion, by providing the host with phage-encoded virulence genes. For instance, toxigenic V. Cholerae isolates carry the ctxAB genes encoded by lysogenic phage. On the one hand, the ingestion of infected bacteria cause cholera at rate βαJ, where β is the contact rate with environment. The constant rate α is the probability that the consumption of infected bacteria leads to human infection. On the other hand, when the susceptible V. cholera are ingested from the environment and reach the small intestine within the human body, complex biological interactions, chemical reactions, and genetic transduction take place that lead to human cholera [START_REF] Xueyung | Modelling the within-host dynamics of cholera: Bacteria-viral interaction[END_REF]. The ingestion of susceptible V.

cholerae can cause infection at rate βρB, where ρ is the infection rate of susceptible bacteria by temperate phage in the small intestine. The infection force is then given by:

where H is the half-saturation bacteria density. Note that λ(B, J) is a Holling type II functional response. Human contamination of the water supply through infected feces contributes to bacteria levels and is called shedding. Therefore, the shedding rate of susceptible bacteria (B)

and infected bacteria (J), are respectively ω and η. We assume that D is the shared diffusive coefficient of susceptible (S) and infected (I). Based on the above description, the following E 0 is isolated in X + and W s ∩ W = ∅, where W s is the stable set of E 0 . It is clear that there is no cycle in M ∂ from E 0 to E 0 . We conclude that there exists ζ > 0 such that min ψ∈ω(ϕ)

Choose ζ small enough such that lim inf t→+∞ S(., t, ϕ) ≥ ζ uniformly for x ∈ Ω. We proceed by the same way to show that lim inf t→+∞ P(., t, ϕ) ≥ ζ uniformly for x ∈ Ω and this achieves the proof.

Discrete model and numerical simulations 3.7.1 Nonstandard finite difference scheme (NSFD)

We construct the discrete model associated to (3.4.3). For this system to be mathematically tractable, and for our scheme to be dynamically consistent with the continuous model, we assume here one and for good that δ = 0 (this is the same as saying that cholera immunity is

where

where n ∈ 0, 1, ..., M and k ∈ N, with the initial conditions

and the boundary condition

and for all

Therefore, it is easy to see that E k is a strictly diagonal dominant matrix. Thus,

Using the fact that B k+1 , J k+1 ≥ 0 and the induction on k, if P k ≥ 0 then P k+1 ≥ 0.

We now focus on the boundedness of the solutions of (3.7.1). Set {N k } a sequence defined by

Adding the first and the second equations of (3.7.1) yields,

By induction we have the following inequality lim sup

Set,

By the third equation we get

The boundedness of the sequence {D k } implies that, the sequence {P k } is bounded as well. Thus the solutions of the system (3.7.1) with positive initial conditions are positive and bounded.

On the other hand, it is not difficult to prove that the discrete model (3.7.1) fixed points correspond to the equilibria of the continuous model.

Moreover the following main result for the global stability of fixed points is in order.

Theorem 3.7.2 For the system (3.7.1)-(3.7.4) with ∆t > 0 and ∆x > 0:

i) the disease free fixed point is globally stable whenever R 0 ≤ 1 and N 0 ≤ 1 ii) the endemic fixed point is globally stable if R 0 > 1 and ε = 0.

Proof: i) We propose the discrete Lyapunov functional candidate that reads as follows:

116 where b, c, d, e are positive constant to be determined shortly.

Using the inequality λ B k n , J k n ≤ βρ/H B k n + βα/H J k n and grouping like terms, one has: equilibrium is locally asymptotically stable, and is unstable for R 0 > 1 and the system is shown to be uniformly persistent. Here, the condition R 0 ≤ 1 is not sufficient to achieved the possible 123 elimination of cholera (i.e the GAS of the DFE is no longer achieved under the sole condition of R 0 ≤ 1). However, we derive another threshold N 0 , and proved using the Lyapunov-LaSalle's techniques that the DFE is GAS if R 0 ≤ 1 and N 0 ≤ 1. The discret counterpart has been formulated for numerical simulations. Overall, the presence of the ecological threshold N 0 highlight the impact of phages and bacteria interactions on the spread of cholera.

Since there are some phages (virulent/lytic phages) that infect and kill bacterium cell's, and there exists biological methods to select them into environment, the next chapter deals with the use of these selected phages in order to control the spread cholera.

General conclusion and perspectives

In this dissertation, we used mathematical models to study the impact of the interactions between bacteriophages and bacteria on the transmission dynamics of cholera with the main view of understanding possible ways through which the disease can be contained or an outbreak prevented.

In chapter 2, From the modeling perspective, we built a mathematical model for the phagebacteria interactions in the environmental reservoir by taking into account the prophage induction process. This work considers three aspects of life cycles of phages and bacteria: (a)

The lytic life cycle: phages infect cells and the progeny phages are produced through lysis killing the bacteria and producing numerous phages. (b) The lysogenic life cycle: phages do not kill cells and support the emergence of new clones of bacteria. (c) The prophage induction event, that is the switching from lysogenic to lytic life cycle. The resulted mathematical model obtained from these processes is a prey-predator like system with Holling type II functional response and logistic growth of free bacteria.

From the theoretical analysis point of view, we did an in-depth investigation of asymptotic behavior and bifurcation analysis of the system. In this regard, we have computed the basic offspring number N 0 and used it as the bifurcation parameter to establish the local/global stability of equilibria. Lyapunov-LaSalle techniques were used for the global asymptotical stability results and for the estimation of the basin of attraction of the locally asymptotically stable EPE.

Based on the values and range of N 0 , all the equilibria were topologically classified using the center manifold approximation, and the types of bifurcation were specified accordingly. Precisely, we have shown that the system undergoes a trans-critical bifurcation around N 0 = 1 and a Hopf bifurcation around the EPE.

Computationally, we used MatLab platform to perform both the global sensitivity analysis of N 0 and the model variables. The result of that sensitivity analysis suggests that, the contact rate β, the induction rate α, the bacteria carrying capacity K, the burst size θ and the cell division size φ are the more influential parameters on the phage-bacteria interactions. Moreover, we have simulated the system to illustrate our theoretical results: namely, the GAS of the PFE has been illustrated, as well as the occurrence of periodic solutions. Epidemiologically speaking, the existence of stable periodic solutions could explain the occurrence of repetitive outbreaks of some bacteria-borne diseases such as cholera in Africa and Asia. Finally, ecologically and epidemiologically speaking, we have provided the following responses to the two research questions raised in the introduction: For small values of the basic offspring number of phages, it is possible that periodic bacterial diseases outbreak occur. On the other hand, for sufficiently large values of the basic offspring number, the total population of bacteria go extinct and the polluted environment is purified.

In chapter 3 we built two mathematical models describing the impact of phage-bacteria infection on the indirectly transmitted cholera. The firs model relies on the fact that: (i) we We found a threshold N 0 , such that the disease free equilibrium is globally stable whenever R 0 ≤ 1, and N 0 ≤ 1 altogether. We have shown that the model without control had multiple endemic equilibria depending on the range of R 0 and N 0 . We identified the phage absorption rate as a possible cause of backward bifurcation, since in its absence, the system always exhibits a forward bifurcation at R 0 = 1 suggesting that the condition R 0 < 1 is sufficient for the control of cholera.