Virus dynamics model with intracellular delays and immune response

In this paper, we incorporate an extra logistic growth term for uninfected CD4+ T-cells into an HIV-1 infection model with both intracellular delay and immune response delay which was studied by Pawelek et al. in [26]. First, we proved that if the basic reproduction number R0 < 1, then the infection-free steady state is globally asymptotically stable. Second, when R0 > 1, then the system is uniformly persistent, suggesting that the clearance or the uniform persistence of the virus is completely determined by R0. Furthermore, given both the two delays are zero, then the infected steady state is asymptotically stable when the intrinsic growth rate of the extra logistic term is sufficiently small. When the two delays are not zero, we showed that both the immune response delay and the intracellular delay may destabilize the infected steady state by leading to Hopf bifurcation and stable periodic oscillations, on which we analyzed the direction of the Hopf bifurcation as well as the stability of the bifurcating periodic orbits by normal form and center manifold theory introduced by Hassard et al. Third, we engaged numerical simulations to explore the rich dynamics like chaotic oscillations, complicated bifurcation diagram of viral load due to the logistic term of target cells and the two time delays.