Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response

In this paper, an age-structured HIV-1 infection model with CTL immune response is investigated. In the model, we consider the infection age (i.e. the time that has elapsed since an HIV virion has penetrated the cell) of infected $T$ cells. The asymptotic smoothness of the semi-flow generated by the system is established. By calculation, the immune-inactivated reproduction rate $\mathscr{R}_0$ and the immune-activated reproduction rate $\mathscr{R}_1$ are obtained. By analyzing the corresponding characteristic equations, the local stability of an infection-free steady state and a CTL-inactivated infection steady state of the model is established. By using the persistence theory for infinite dimensional system, the uniform persistence of the system is established when $\mathscr{R}_1>1$. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if $\mathscr{R}_0<1$, the infection-free steady state is globally asymptotically stable; if $\mathscr{R}_1<1< \mathscr{R}_0$, sufficient conditions are derived for the global stability of the CTL-inactivated infection steady state; if $\mathscr{R}_1>1$, sufficient conditions are obtained for the global attractivity of the CTL-activated infection steady state. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.