Studies On Viral Infection Models With Immune Impairment And Delayed Immune Response

In this paper, we establish several infection models within host, and analyze the dynamic behaviors as well as the biological meanings of these models. In some virus infections, the presence of the antigen can not only stimulate immunity but also suppress immune responses or even destroy immunity especially when the load of pathogens is too high. Thus, in the first two parts, two chronic viral infection models considering immune impairment are proposed and studied, with cell-mediated immunity and humoral immunity, respectively. In the first part, we consider cell-mediated immunity. It is shown that the virus persists in the host if the basic reproductive ratio of the virus is greater than one. The immune cells persist when there is only one positive equilibrium. The system can exhibit two positive equilibria if the basic reproductive ratio of the virus is above a threshold. This allows a bistable behavior, and the immune cells persist or die out, i.e., infection will result in disease or immune control outcome, depending on the initial conditions. By theoretical analysis and numerical simulations, we show that therapy could shift the patient from a disease progression to an immune control outcome, despite that the therapy is not necessarily lifelong. This would allow the immune response to control the virus in the long term even in the absence of continued therapy. In the second part, we consider humoral immunity. We analyze the local and global stability of the four equilibria as well as the permanence of the system mathematically. The behaviors of the second model is similar to those of the first model. In the third part, a viral infection model with delayed immune response is proposed and studied. In the model, a saturating immune proliferation, instead of linear proliferation, is used to describe the expansion of the CTL response, and the expansion of the target T cells population is in general form. Furthermore, the delay reflecting the retarded immune response is taken into account. It is shown that if the basic reproductive ratio of the virus is less than one, the infection-free equilibrium is globally asymptotically stable. Analytical and numerical results show that if the basic reproductive ratio of the virus is greater than one, the time delay of the immune response is to create a rich dynamics. By taking the discrete time delay as a bifurcation parameter, it is found that this system undergoes a sequence of Hopf bifurcations and chaotic dynamical behavior is found when the time delay is sufficiently large. Finally, mathematical results indicate that the density-dependent T-cell proliferation and the saturating immune expansion can affect a lot on the dynamical behaviours of the system.