Dynamics of a delayed cytokine-enhanced diffusive HIV model with a general incidence and CTL immune response

Abstract

In this paper, a new HIV CD4$^{+}$T cells reaction-diffusion model has been introduced, which aims to explore the joint impact of cell-free infection, cytokine-enhanced viral infection, CTL immune response and time delay. Our results show that inflammatory cytokine infection, time delay and reaction-diffusion play an important role in HIV model and can not be ignored. First, we investigate the existence and nonnegativity of the solution of the model. Then, we obtain the existence of three equilibria namely infection-free equilibrium $E_{0}$, immune-free equilibrium $E_{1}$ and immunity-inactivated equilibrium $E_{2}$ respectively. Next, two threshold parameters $R_{0}$ (basic reproduction number) and $R_{CTL}$ (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Using Lyapunov functionals and LaSalle’s invariance principle, we show that $E_{0}$ is globally asymptotically stable when $R_{0}\le 1$; If $R_{0}>1$, the model is uniformly persistent and $E_{1}$ is globally asymptotically stable when $R_{CTL}<1<R_{0}$; $E_{2}$ is globally asymptotically stable when $R_{0}>1$ and $R_{CTL}>1$. Finally, a special case is given to show the global attractiveness of the three equilibria by numerical simulation.