Dynamics Of Infectious Models With Time Delays And CTL Response

Mathematics models are widely used in many fields. Especially, many infectiousdiseases can be described by mathematics models. Firstly, people establish mathematicsmodel by infectious mechanism. Next, people understand infectious diseases throughstudying dynamics of mathematics models. Many infectious mathematics models aredelayed models. Stability and bifurcation are important problems in delayed models,which are of great significance in both theory and practice.In this thesis, we mainly investigate several kinds of infectious mathematics delayedmodels step by step by applying Lyapunov stability theory, La Salle invariance principle,local Hopf bifurcation and global Hopf bifurcation theorem and some other mathemati-cal methods. We also take advantage of mathematical software–Matlab to do numericalsimulation for supporting our results. The main results are listed in following:(1) A class of HTLV-I infectious models with intracellular delay are studied usingLyapunov functional, La Salle invariance principle, roots distribution of exponential poly-nomial equation, Routh-Hurwitz criteria. We obtain the positive property of solutions andthe global stability of equilibria of the system. Namely, P0 and P1are global asymptoticalstable, P2 is global attractive. Those show the impact of only incorporating intracellulardelay into the system.(2) We study global Hopf bifurcation of a class of HTLV-I infectious models withCTL response delay utilizing global Hopf bifurcation theorem, and obtain global exis-tence of periodic solutions, which consummates the work of Li and Shu in 2012. Somenumerical simulations are carried out to illustrate the analysis results.(3) A HTLV-I infection model with two delays is considered. There are not onlyCTL response delay but also intracellular delay in the infection model. R0 and R1arebasic reproduction numbers for viral infection and for CTL response, respectively. IfR0<1, the boundary equilibrium P0 is globally asymptotically stable. If R1< 1 < R0,the boundary equilibrium P1 is globally asymptotically stable. If R1>1, there exists aunique interior equilibrium P2. And the dynamics of P2 changes when the CTL responsedelay τ2varies. Hopf bifurcations appear and there exist stability switches for the fixedpoind P2, which are illustrated in the numerical simulations. We also study the globalHopf bifurcation of this model. To our knowledge, there are few results about globalHopf bifurcation with two delays.(4) A class of virus infection models with three delays are considered, which includeone immune delay and two intracellular delays. We find incorporating three delays don’tdestroy the globally asymptotical stability of P0 when R0<1 and also don’t destroy theglobally asymptotical stability of P1 when R1<1 < R0. When R1> 1, we can see P2 is still global attractability under only incorporating two intracellular delays τ1and τ2.But P2 can undergo Hopf bifurcation on proper conditions under only incorporating theimmune delay τ3.Furthermore, oscillations and stability switches can appear. The im-mune delay destroys the global attractability of P2. Those show immune delay dominatesintracellular delays in some viral infection models, which indicates human immune sys-tem has special effect in virus infection models with CTLs response and human immunesystem itself is very complicated.