Study Of Global Stability And Hopf Bifurcation In Delayed Virus Infection Model

In the thesis,we investigate delayed virus infection dynamic models with adaptive responses by combining Routh-Hurwitz criteria,Lyapunov functional,LaSalle's invariance principle,roots distribution of exponential polynomial equation,the normal form argument and center mainfold theory.The specific research are as follows.In chapter one,the history and development,signification and evolving of delayed virus infection dynamic models are reviewed.Furthermore,we simply introduce the main work in this paper.In chapter two,virus infection models with different delays are studied.And,this chapter is composed of two sections.The first section investigates stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington-DeAngelis incidence and three delays.The second section investigates global stability of delayed virus infection models with nonlinear antibody and CTL immune responses and general incidence rate.In these two sections,the sufficient threshold conditions on the global stability of the equilibria for infection-free,immune-free,antibody response,CTL response and antibody and CTL responses are obtained.Particularly,in the first section,when the intracellular delay and virus replication delay are greater than or equal to zero,the existence of Hopf bifurcation at infection equilibrium with only CTL response and infection equilibrium with CTL and antibody responses with immune delay as a bifurcation parameter is investigated by using the bifurcation theory.The model which includes immune delay represents that the delay can change the stability of the model,that is,the virus infection in clinical is complex.Finally,numerical simulations are validated the correctness of the obtained results.In chapter three,reaction diffusion virus infection models with homogeneous Neumann boundary conditions and different delays are investigated.And,this chapter consists of two sections.The first section investigates global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response.By constructing the suitable Lyapunov functionals,the global stability of the equilibria for infection-free,immune-free,antibody response,CTL response and antibody and CTL responses,respectively,are established.This implies the model on the homogeneous Neumman boundary condition does not exhibit Hopf bifurcation.The intracellular delay and virus replication delay do not affect the stability of equilibria.The second section investigates dynamical analysis of a delayed reaction-diffusion virus infection model with logistic growth and humoral immune impairment.By taking ? as the bifurcation parameter,the model at the antibody-present equilibrium undergoes Hopf bifurcation,and an algorithm determining the direction of spatially Hopf bifurcation and the stability of bifurcated periodic solutions is given.Finally,numerical simulations are validated the correctness of the obtained results.The final chapter establishes diffusion analysis for a stochastic HIV epidemic model.Approximations of quasi-stationary distributions and time to extinction are derived.The approximations are valid for sufficiently large cell sizes.By using Kolmogorov forward equations,we obtain differential expression of the quasi-stationary distributions.Meanwhile,the quasi-stationary distributions can be approximated by Gaussian diffusion according to two cases:?i?R₀> 1 and R₁< 1,?ii?R₁> 1,where R₀ is the basic reproduction ratio and R₁ is the immune response reproductive number.Finally,numerical simulations are validated the correctness of the obtained results.