[Non fourni] / Acute Myeloid Leukemia (AML) is a cancer of white cells characterized by a quick proliferation of immature cells, that invade the circulating blood and become more present than mature blood cells. This thesis is devoted to the study of two mathematical models of AML. In the first model studied, the cell dynamics are represented by PDE’s for the phases G₀, G₁, S, G₂ and M. We also consider a new phase called Ğ₀, between the exit of the M phase and the beginning of the G₁ phase, which models the fast self-renewal effect of cancerous cells. Then, by analyzing the solutions of these PDE’s, the model has been transformed into a form of two coupled nonlinear systems involving distributed delays. An equilibrium analysis is done, the characteristic equation for the linearized system is obtained and a stability analysis is performed. The second model that we propose deals with a coupled model for healthy and cancerous cells dynamics in AML consisting of two stages of maturation for cancerous cells and three stages of maturation for healthy cells. The cell dynamics are modelled by nonlinear partial differential equations. Applying the method of characteristics enable us to reduce the PDE model to a nonlinear distributed delay system. For an equilibrium point of interest, necessary and sufficient conditions of local asymptotic stability are given. Finally, we derive stability conditions for both mathematical models by using a Lyapunov approach for the systems of PDEs that describe the cell dynamics.

Dans ce manuscrit, nous étudions le retard distribué et ses applications. Nous présentons la définition du retard distribué et l'étude de ses propriétés. Nous expliquons aussi le problème de la mise en œuvre du retard distribué et donnons une méthode générale pour son approximation. Ensuite nous présentons trois applications à l'aide de retard distribué, qui sont différentes avec les applications apparaissent dans la littérature. Le premier est l'inversion stable et le modèle appariement. Nous introduisons une nouvelle classe d'inversion stable et modèle d'appariement pour les systèmes linéaires de dimension finie invariables dans le temps. L'inversion stable (resp. modèle d'appariement) est une sorte d'inversion (resp. modèle d'appariement) de rapprochement. En fait, nous obtenons l'inversion exacte (resp. exacte modèle d'appariement) après un temps t = h, où le temps t = h peut être choisi arbitrairement. La deuxième application est le contrôle de la stabilité et du pôle placement finie pour une classe de système de dimension infinie. La dernière application du retard distribué est la synthèse de l'observateur pour l'estimation ou la commande de sortie. Nous craignons seulement avec les systèmes linéaires de dimension finie. Nous introduisons une boucle fermée observateurs sans mémoire par injection d'entrée. Convergence asymptotique ainsi que la convergence en temps fini de l'estimation sont analysés par injection de sortie et des informations d'entrée via retard distribué. Enfin, nous introduisons une nouvelle classe de l’approximation des systèmes à paramètres distribués. Nous travaillons sur la topologie du graphe, et montrons que sous certaines hypothèses faibles, une telle approximation peut être réalisé en utilisant retard distribué. / A distributed delay is a linear input-output operators and appears in many control problems. We investigate distributed delay and its applications. After introducing the definition and the main properties of the distributed delay, the numerical implementation problem of distributed delays is analyzed and a general method for its approximation is given. Then three applications are focused on where distributed delay appears. The first application is the stable inversion and model matching. A new class of stable inversion and model matching problem for finite dimensional linear time-invariant systems is defined. The stable inversion (resp. model matching) is an approximation of the inverse of a given model (resp. model matching), where exact inversion (resp. exact matching) is reached after a time $t=h$, which is a parameter of our procedure. The second application is concerned with stabilization and finite spectrum assignment for a class of infinite dimensional systems. The last application concerns observer synthesis for estimation or output control. For linear finite dimensional systems. A closed-loop memoryless observer by input injection is introduced. Asymptotic convergence as well as finite time convergence of the estimation are analyzed by output injection and input information via distributed delay. At last, we introduce a new class for approximation of distributed parameter systems. We work on the graph topology, and show that under some weak assumptions, such an approximation can be realized using distributed delay.

There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.

In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.

<p>The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models.&nbsp / Comparative numerical results are also provided for each model.</p>

<p>The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models.&nbsp / Comparative numerical results are also provided for each model.</p>

There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.

In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.

Philosophiae Doctor - PhD / There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them. / South Africa

Dengue fever is endemic in tropical and sub-tropical countries, and some of the important features of Dengue fever spread continue posing challenges for mathematical modelling. We propose a model, namely a system of integro-differential equations, to study a multi-serotype infectious disease. The main purpose is to include and analyse the effect of a general time delay on the model describing the length of the cross immunity protection and the effect of Antibody Dependent Enhancement (ADE), both characteristics of Dengue fever. Analysing the system, we could find the equilibriums in the invariant region. A coexistence endemic equilibrium within the region was proved, even for the asymmetric case. The local stability for the disease free equilibrium and for the boundary endemic equilibriums were proved. We have also results about the stability of the solutions of the system, that is completely determined by the Basic Reproduction Number and by the Invasion Reproduction Number, defined mathematically, as a threshold value for stability. The global dynamics is investigated by constructing suitable Lyapunov functions. Bifurcations structure and the solutions of the system were shown through numerical analysis indicating oscillatory dynamics for specific value of the parameter representing the ADE. The analytical results prove the instability of the coexistence endemic equilibrium, showing complex dynamics. Finally, mortality due to the disease is added to the original system. Analysis and discussions are made for this model as perturbation of the original non-linear system. / A Dengue é endêmica em países tropicais e subtropicais e, algumas das importantes características da dengue continua sendo um desafio para a modelagem da propagação da doença. Assim, propomos um modelo, um sistema de equações integro-diferenciais, com o objetivo de estudar uma doença infecciosa identificada por vários sorotipos. O principal objetivo é incluir e analisar o efeito de um tempo geral de retardo no modelo descrevendo o tempo de imunidade cruzada para a doença e o efeito do Antibody Dependent Enhancement (ADE). Analisando o sistema, encontramos os equilíbrios, onde a existência do equilíbrio de coexistência foi provado, mesmo para o caso assimétrico. A estabilidade local para o equilíbrio livre de doença e para os equilíbrios específicos de cada sorotipo foi provada. Também mostramos resultados para a estabilidade das soluções do sistema que é completamente determinada pelo Número Básico de Reprodução e pelo Número Básico de Invasão, definido matematicamente como um valor limiar para a estabilidade. A dinâmica global é investigada construindo funções de Lyapunov. Adicionalmente, bifurcações e as soluções do sistema foram mostrados via análise numérica indicando dinâmica oscilatória para específicos valores do parâmetro que representa o efeito ADE. Resultados analíticos obtidos pela teoria da perturbação provam a instabilidade do equilíbrio endêmico de coexistência e apontam para um complexo comportamento do sistema. Por fim, a mortalidade causada pela doença é adicionada ao sistema original. Análises e discussões são feitas para este modelo como uma perturbação do sistema não linear original.