Lyapunov functionals for a general time-delayed virus dynamic model with different CTL responses

A time-delayed virus dynamic model is proposed with general monotonic incidence, different nonlinear CTL (cytotoxic T lymphocyte) responses [CTL elimination function pyg1(z) and CTL stimulation function cyg2(z)], and immune impairment. Indeed, the different CTL responses pose challenges in obtaining the dissipativeness of the model. By constructing appropriate Lyapunov functionals with some detailed analysis techniques, the global stability results of all equilibria of the model are obtained. By the way, we point out that the partial derivative fv(x,0) is increasing (but not necessarily strictly) in x>0 for the general monotonic incidence f(x,v). However, some papers defaulted that the partial derivative was strictly increasing. Our main results show that if the basic reproduction number R0≤1, the infection-free equilibrium E0 is globally asymptotically stable (GAS); if CTL stimulation function cyg2(z)=0 for z=0 and the CTL threshold parameter R1≤1<R0, then the immunity-inactivated infection equilibrium E1 is GAS; if the immunity-activated infection equilibrium E+ exists, then it is GAS. Two specific examples are provided to illustrate the applicability of the main results. The main results acquired in this paper improve or extend some of the existing results.