Dynamic Analysis On Several Delayed HIV-1 Therapy Models

In this thesis, some mathematical models of HIV-1 therapy for controlling viralinfections using genetically modified viruses are discussed. Based on the fact thata latent period between the time target cells are contacted by the virus particlesand the time the virons enter the cells as well as a virus production period for newvirons to be produced within and released from the infected cells, constant delaysand continuously distributed delays are introduced in the systems. Some su?cientconditions ensuring local or global stability of the equilibria are obtained.The paper is composed of four chapters.In the first chapter, the background, signification and evolving of AIDS arebrie?y reviewed. Furthermore, we simply introduce the main work in this paper.In the second and third chapter, using the simply mass action rate and HollingtypeΠas the incidence rates, we study the dynamics of the delay models. The suf-ficient conditions for the local and global stability of the infection-free equilibriumare obtained. We also obtain the su?cient conditions ensuring the local stabilityof the single infection equilibrium.The fourth chapter is concern with the stability of the infection-free equilibriumand the single infection equilibrium of the models of HIV-1 therapy with delays byGamma functions. Specifically, when the basic reproduction number ratio R0 < 1,the infection-free equilibrium is global asymptotically stable; when R0∈(1,1 +δ)whereδis some positive constant which depend on the parameters in the system,there is a single infection equilibrium that is global asymptotically stable; whilewhen R0 > 1 +δ, Es becomes unstable and the double-infection equilibrium Edcomes into existence. Numerical simulations seem to support occurrence of Hopfbifurcation giving rise to periodic solutions.