A Qualitative Study Of Simian Immunodeficiency Virus Model

Simian Immunodeficiency Virus(SIV)evolution shows that a relatively slowly replicating and early mild cytopathic virus and a faster replicating and more cytopathic virus late disease infection. In the cell culture experiment, we can see that the early mildly cytopathic variant out-competed the late highly cytopathic strain. In this paper, we consider the effect of immune responses, especially cytotoxic T lymphocyte (CTL) responses.In Chapter 2, we consider the kinetic property of competitive dynamic system of the virus with cytotoxic T lymphocyte (CTL) responses. The disease free equilibrium E0 is globally asymptotically stable if the basic reproduction number R₀₁≤1, R₀₂≤1, and virus stain 1 and virus stain 2 goes extinct. The boundary equilibrium E1 is globally asymptotically stable if the basic reproduction number R₀¹ > 1, R₀²≤1, and virus stain 1 wins and virus stain 2 goes to extinct. The boundary equilibrium E2 is globally asymptotically stable if the basic reproduction number R₀¹≤1, R₀²> 1, and virus stain 2 wins and virus stain 1 goes to extinct. The unique endemic equilibrium E3 is globally asymptotically stable if the basic reproduction number (1) (2)R0 > 1, R0> 1, and virus stain 1 and virus stain 2 coexist.In the Chapter 3, we discuss the global kinetic property of competitive dynamic system of the virus with delay and cytotoxic T lymphocyte (CTL) responses. By using direct constructing Lyapunov functional method and applying classical LaSalle invariance principle, we obtain the global dynamics which depend on the basic reproduction number. The disease free equilibrium F0 is globally asymptotically stable if the basic reproduction number R0 3≤1, R04≤1, and virus 1 and virus 2 goes extinct. The boundary equilibrium F1 is globally asymptotically stable if the basic reproduction number R0 ( 3) > 1, R0(4)≤1 andβ₁ /β 2 =β 3 /β₄,and virus 1 wins and virus 2 goes to extinct. The boundary equilibrium F2 is globally asymptotically stable if the basic reproduction number R0 ( 3)≤1, R0(4)> 1 andβ₁ /β 2 =β 3 /β₄, and virus 2 wins and virus 1 goes to extinct. The unique endemic equilibrium F3 is globally attracting if the basic reproduction number R0 ( 3) > 1, R0(4)> 1 andβ₁ /β 2 =β 3 /β₄, and virus 1 and virus 2 coexist.