Etude de quelques modèles épidémiologiques de métapopulations: application au paludisme et à la tuberculose

Abstract : The objective of this thesis is first the modeling, the mathematical analysis and numerical simulations of the metapopulation models of infectious diseases based on some modern approaches of the mobility patterns of humans. Secondly to examine the influence of the mobility (movement) of people on the spread of some human infectious diseases. Finally to deal with the difficult question of the existence and stability of endemic equilibria of metapopulation models. For certain diseases such as Malaria, Tuberculosis or some Sexually Transmitted Diseases that do not confer any immunity, we give some metapopulation models that extend to multiple patches the well know epidemiological models in one patch. Our models are based on the mobility patterns of humans wich can take different forms leading to numerous approaches of modeling metapopulations : the Euler approach of the movement of particles (here humans) as in Fluid Mechanics, is used in the first part. The Lagrange approach of the movement of particles (here humans) as in Fluid Mechanics, is used in the second part. The last and more recent approach based on Statistical Mechanics, wich takes into account the degree distribution of the network of the metapopulation is used in the third and last part of this work. For each approach, we build a metapopulation model for a chosen disease, and gve its mathematical analysis. The theoretical framework we use to analyze ou models is that of triangular, monotone or anti-monotone non-linear dynamical systems. We also use some Lyapunov-Lasalle techniques. In the fisrt two parts of our work, we prove that the steady solutions (called equilibria) of the given systems are globally asymptotically stable when the basic reproduction number R0 is less than or equal to the unity (for the disease free equilibria), and when R0 is greater than one (for the endemic equilibria). In the last part, we build a model to describe the spreading of tuberculosis hinging on the two most used forces of infection in mathematical modeling of epidemics : the frequency-dependant transmission and the density-dependant transmission. For each type of trasmission model, we give the explicit formula for the basic reproduction number. We prove for the frequency-dependant transmission model, that the disease free equilibrium is globally asymptotically stable when R0 is less than one. And for the density-dependant transmission model, we prove the existence of an endemic equilibrium when R0 is greater than one. Numerical simulations are performed at the end of each part to examine the influence of human's mobility on the basic reproduction number, as well as on the behavior of the solutions and consequently on the spreading patterns of the diseases under study
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Berge Tsanou. Etude de quelques modèles épidémiologiques de métapopulations: application au paludisme et à la tuberculose. Mathématiques générales [math.GM]. Université de Lorraine, 2012. Français. ⟨NNT : 2012LORR0055⟩. ⟨tel-01749226⟩

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