Unified Lyapunov functional for an age-structured virus model with very general nonlinear infection response

The aim of this paper is to provide a unified Lyapunov functional for an age-structured model describing a virus infection. Our main contribution is to consider a very general nonlinear infection function, gathering almost all usual ones, for the following problem:

$$\begin{aligned} \left\{ \begin{array}{lll} T'(t)=A- dT(t)-f(T(t),V(t)) \ t \ge 0,\\ i_t(t,a)+i_a(t,a)=-\delta (a) i(t,a), \\ V'(t)=\int _0^{\infty } p(a)i(t,a)da-cV(t), \end{array} \right. \end{aligned}$$

(0.1)

where T(t),  i(t, a) and V(t) are the populations of uninfected cells, infected cells with infection age a and free virus at time t respectively. The functions \(\delta (a),\) p(a),  are respectively, the age-dependent per capita death, and the viral production rate of infected cells with age a. The global asymptotic analysis is established, among other results, by the use of compact attractor and strongly uniform persistence. Finally some numerical simulations illustrating our results are presented.