Global Properties Of Virus Dynamics Model With Immune Response

In this paper, the global properties of the virus dynamics models with immune response are studied. In the first part of this paper, the global properties of the virus dynamics model with antibody immune response are studied. By means of Lyapunov functions, the global properties of the model are obtained. The virus is cleared if the basic reproduction number R_o≤ 1, and the virus persists in the host if R₀> 1. Further, positive solutions of the model approach to an immune-free steady state if the immune response reproductive number R₁< 1 and to an endemic steady state if R₁ > 1.In the second part, the global properties of two virus dynamics models with CTL immune response are studied. If the susceptible cells proliferate linearly, the virus is cleared if the basic reproduction number R₀≤ 1, and the virus persists in the host if R₀> 1;further, positive solutions of the model approach to an immune-free steady-state if the immune response reproductive number R₁≤ 1 and to an endemic steady-state if R₁> 1. If the susceptible cells proliferate logistically, the virus is cleared if the basic reproduction number R₀≤1. Sufficient conditions for the asymptotical stability of an immune-free equilibrium and an endemic equilibrium are obtained. When the immune-free equilibrium is unstable in the absence of the endemic equilibrium, numeric simulations show that Hopf bifurcation may occur: if the endemic equilibrium is unstable, numeric simulations show that and Hopf bifurcation does not occur but complicated dynamical behavior occurs.In the third part, the global properties of the virus dynamics model with both CTL and antibody immune response are studied. It is shown that the basic reproductive number i?o, CTL immune response reproductive number i?i and antibody immune response reproductive number i?2, CTL immune response competition reproductive number R%, antibody immune response competition reproductive number R4 determine the dynamical behaviors of the model. If Ro < 1, the virus is cleared. For Rq > 1, positive solutions approach to an immune-free equilibrium if Ri < 1 and i?2 < 1, to a CTL dominant equilibrium if i?i > 1 and Ri < 1, to a antibody dominant equilibrium if R2 > 1 and R3 < 1, and to an endemic equilibrium if R3 > 1 and Ri > 1.In the last part, the deterministic and stochastic properties of a cell-to-cell virus dynamics model with immune impairment are studied. The dynamics of the deterministic model is completely determined by a basic reproduction number Rq. If Rq < 1, a disease-free equilibrium is globally stable;while an endemic equilibrium is globally stable if Ro > 1. The stochastic stability of the model suggests that the deterministic model is robust with respect to stochastic perturbations.