Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays

Abstract

In this paper, we investigate the dynamics of a five-dimensional virus model with immune responses and an intracellular delay which describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is an improvement of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function f for virus infections, and (iii) antibody responses, which are implemented by the functioning of immunocompetent B lymphocytes, play a critical role in preventing and modulating infections. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we show that the global dynamics of the model is determined by the reproductive numbers for viral infection R0, for CTL immune response R1, for antibody immune response R2, for CTL immune competition R3 and for antibody immune competition R4. The global stability of the model precludes the existence of Hopf bifurcation and other complex dynamical behaviors in long time. Numerical simulations are also performed in order to illustrate the dynamical behavior.