Global Stability of a Nonlinear Viral Infection Model with Infinitely Distributed Intracellular Delays and CTL Immune Responses

Determining sharp conditions for the global stability of equilibria remains one of the most challenging problems in the analysis of models for the management and control of biological systems. Yet such results are necessary for derivation of parameter thresholds for eradication of pests or clearing infections. This applies particularly to models involving nonlinearity and delays. In this paper, we provide some general results applicable to immune system dynamics: we consider a viral model with general target-cell dynamics, nonlinear incidence functions, state dependent removal functions, infinitely distributed intracellular delays, and the cytotoxic T lymphocyte response (CTL). This general model admits three types of equilibria: infection-free equilibria, CTL-inactivated infection equilibria, and CTL-activated infection equilibria. The model admits two critical values: $R_0$ (the basic reproduction number for viral infection) and $R_{1}$ (the viral reproduction number at the CTL-inactivated infection equilibrium). Under certain assumptions, it is shown that if $R_{0}\leq 1$, then the infection-free equilibrium $E_0$ is globally stable and the viruses are cleared. If $R_{1}\leq 1<R_{0}$, then there exists a unique CTL-inactivated infection equilibrium $E_1$ which is globally stable and the infection becomes chronic with no sustained immune response. If $R_{1}>1$, then there is a unique CTL-activated infection equilibrium, which is globally stable implying persistent immune responses. Our results cover and improve many existing ones and include the case when the nonlinear functions are nonmonotone.