| In this thesis, we study the spread of virus with mutant in two sides, first we concern with the spread of influenza with mutant, the second is the spread of HIV-1 virus with mutant, we develop mathematical models for each side we considered, and dynamic analysis are also given for the two mathematical models.The thesis is composed of three chapters.In the first chapter, the history background, signification and evolving of-In-fectious disease are briefly reviewed, furthermore, we simply introduce the main work in this thesis.In Chapter 2, A mathematical model is proposed to interpret the spread of influenza with mutant. By introducing the basic reproduction number for the epi-demic models and obtain the persistence of the system as well as the globally asymptotical stable conditions for the free equilibrium, the mutant-endemic equi-librium and the full endemic equilibrium. Numerical simulations are also provided to confirm our analytic results. Our mathematical model suggests that the out-break of both common influenza and mutant influenza may occur if the humans do not keep infection rate of common influenza at a low level. In order to prevent the spread of common influenza and mutant influenza, some measures such as quaran-tine have to be taken to keep a lower level of the infection rates for both epidemics when mutant influenza has already occurred.The third chapter investigated a mathematical model for the spread of HIV-1 with mutant. We analysis the existence of the infected-free equilibrium, the single-infected equilibrium and the double-infected equilibrium, and obtain the sufficiently condition for the local asymptotical stable or the global asymptotical stable of the infected-free equilibrium, the single-infected equilibrium and the double-infected equilibrium. Numerical simulation are provided for the global asymptotical stable of the double-infected equilibrium. We give a conjecture for the global asymptotical stable of the double-infected equilibrium.
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