Dynamic Analysis Of Virus Dynamics Model With Three Stages

In the analysis of virus dynamics models,the global stability of the model has always been one of the most important research direction.By analyzing the existence of the equilibrium point,the stability of the equilibrium point can help us intuitively understand the global dynamic behavior of the virus model.It provides some theoretical suggestions for us to prevent and control virus infection.This paper mainly investigated the virus dynamics model with three stages.On this basis,we also considered CTL reaction,the age of infection of target cells along with CTL reaction and infection age of target cells exist at the same time.The global dynamic properties of the model are studied by using Lyapunov theorem and LaSalle invariance principle.In addition,a new idea for constructing Lyapunov function is proposed in this paper.Firstly,a dynamics model of virus infection is proposed and investigated,where it is assumed that a part of infected cells is defective for producing virus,and that,after being infected,the target cells can transport directly to each of three different stages of infected cells.For the model,the basic reproduction number of virus determining the extinction or existence of the infection is obtained.After discussing the existence of feasible equilibria of the model,their global stability is proved by constructing the corresponding Lyapunov functions.That is,the infection-free equilibrium is globally asymptotically stable when R0≤1,and the infection equilibrium is globally asymptotically stable when R0>1.Secondly,based on the characteristics of stage structure of virus infected cells and the response of CTL to infected cells,we established a virus infection model of infected cells with defective virus and CTL response,determines the basic reproduction number to determine whether the virus is finally extinct.By constructing an appropriate Lyapunov function,it is proved that if R0≤1,the infection-free equilibrium is globally asymptotically stable,that is,the virus is finally cleared;whereas if R0>1,the infection equilibrium is globally asymptotically stable.Thirdly,according to the important characteristic that cells have the age of infection,a virus infection dynamics model with defective infected cells and infection age structure was established.On the basis of determining the basic reproduction number of virus,by constructing Lyapunov function,it is proved that the non-infection equilibrium is globally asymptotically stable when R0≤1;When R0>1,the infection equilibrium is globally asymptotically stable.Finally,based on the third model,we further considered CTL response and established a staged viral dynamics model with infection age structure and CTL immune response.The basic reproduction number of virus is obtained by calculation,and it is proved that the global dynamic behavior is determined by the value of the basic reproduction number:if R0≤1,the non-infection equilibrium point is globally asymptotically stable;whereas if R0>1,the infection equilibrium is globally asymptotically stable.