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

TL;DR

This paper analyzes a delay differential equation model of viral infection and immune response, establishing conditions for global stability of different infection states and highlighting immune activation's positive effects.

Contribution

It introduces a model with distributed delays for virus and cell infection times and provides rigorous stability analysis for various equilibrium states.

Findings

Uninfected equilibrium is globally stable if R0 ≤ 1.

Infected equilibrium without immune response is stable if R1 ≤ 1 < R0.

Immune response equilibrium is stable if R1 > 1.

Abstract

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in \textit{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 $R_{0}$ and for CTL response $R_{1}$ such that $R_{1}<R_{0}$. 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 $R_{0}\leq1$, an infected equilibrium without immune response is globally asymptotically stable if $R_{1}\leq1<R_{0}$ and an infected equilibrium with immune response is globally asymptotically stable if $R_{1}>1$. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if $R_{1}>1$.