Dynamic Behavior Analysis Of Delayed Differential Equations With Immune Response

In this paper,according to the effects of immune responses,we establish HIV dynamics model with immune response function and intracellular delay.We also study the dynamics of model and biological significance.The full paper is divided into five chapters.The first chapter is introduction.We first briefly introduce the virus infection and immune related knowledge and significance of the research,the research status of virus dynamics and the mian work.Then some important definitions and preliminary knowledge of delay differential equations are given.In chapter 2,we study a class of antibody immune response and immune delay.By constructing Lyapunov function and LaSalle invariance principle,we study the asypmtotically stability of the model.We derive the basic reproduction number,and get that the dynamics of the model is given by the basic reproduction number and immune delay τ.If R₀ ≤ 1,then the uninfected steady state E0 is globally asymptotically stable.If R₀ > 1,then the uninfected steady state E0 is unstable.If<1R₀<1+（β²Nb）/（cud）,the infected steady state without immune response E₁ is local asymptotically stable.If R₀>1+（β²Nb）/（cud）,the infected steady state E₁ is unstable.When satisfying（T₁）（T₂）,if τ ∈（0,τ₀）,the infected steady state with immune response E₂ is local asymptotically stable,if τ >（0,τ₀）,the infected steady state with immune response E₂ is unstable,τ = τ₀,the model appears a Hopf bifurcation.This conclusion proves that antibody immune response and immune delay can lead to complex dynamic behavior.In chapter 3,we study a class of HIV viral dynamics models with saturation infection and immune response.Firstly,in this model,the infection-free equilibrium is given.Then the sufficient conditions of the global asymptotic stability of the infection-free equilibrium and the local asymptotic stability of the chronic-infection equilibrium.Finally,the thesis is summarized.In chapter 4,we study a class of delayed viral dynamics models with the functional response of saturation type and continuous-time delay,analysising the global asymptotic stability of the model.We derive the basic reproduction number R₀ and the corresponding immune response reproduction numbers for the viral infection models.By constructing Lyapunov function,we can get that the dynamics of model is completely determined by the basic reproduction numbers.If R₀ ≤ 1,then the uninfected steady state E0 is globally asymptotically.E₁ is globally asymptotically stable.If R_c ≤ 1 < R₀ and R_a ≤ 1,then the infected steady state without immune response E₁ is globally asymptotically stable.If R_a > 1 and R₄ ≤ 1,then the infected steady state with CTL immune response E₂ is globally asymptotically stable.If R_c > 1and R₃ ≤ 1,then the infected steady state with antibody immune response E₃ is globally asymptotically stable.If R₃ > 1 and R₄ > 1,then the infected steady state with two immune response E₄ is globally asymptotically stable.In chapter 5,we summarize the thesis briefly.