Analyse de la stabilité des modèles intra-hôtes avec retard : application à des modèles intra-hôtes de paludisme et de V.I.H-1

Abstract : In this thesis, we analyze certain epidemiologic models with delays. The phenomenon of delay is omnipresent in biology. The differential equations with constant discrete delay called by Jacquez "lag"have been much studied. However the distributed delays are adapted to many biology phenomena. By distributed delays, we mean delays described by a continuous probability density function. Indeed for example the incubation time for a germs is not constant. The delays are very important in biology particulary in epidemiologic. For example in the case of infectious disease the time of incubation is very important for the analysis of the transmission. The entrance of a parasite into a targetcell initiates a cascade of events that ultimately lead to the production of new parasites by the infected cell. The production of new parasites is not instantaneous. The models studied in this thesis are within-host models with delays. The objectives of within-host parasite models is to describe the dynamic of various stages of the parasites and their interaction with the host cells like red blood cells and the immunity effectors. the delay used in this thesis is the probability density functions of Erlang. The within-host parasite models with distributed delays can also be see as the systems arising from systems with parallel classes of different length of latently infected target cells SE1... EnI where, Ei are latently infected classes. We used some tools of control theory to "generalize" the "linear chain trick". We computed the basic reproduction ratio R0 for systems under consideration. We established the following results : ? if R< 1, the parasite is cleared and the parasite free equilibrium is glabal asymptotic stability on the positive orthan ; ? if R > 1 and with a sufficient condition is satis fied, we conclude to the global asymptotic stability of the endemic equilibrium. In application, we improve the results of Nelson and Perelson
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Joseph Mbang. Analyse de la stabilité des modèles intra-hôtes avec retard : application à des modèles intra-hôtes de paludisme et de V.I.H-1. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2009. Français. ⟨NNT : 2009METZ002S⟩. ⟨tel-01752615⟩

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