Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R0 and for CTL response R1 such that R1 < R0. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R0 ≤ 1, an infected equilibrium without immune response is globally asymptotically stable if R1 ≤ 1 < R0 and an infected equilibrium with immune response is globally asymptotically stable if R1 > 1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R1 > 1.