Dynamics analysis of a delayed HIV-1 model with general incidence rate and immune impairment

This paper explores the dynamics analysis of an HIV-1 model with a general incidence rate, saturated Cytotoxic T Lymphocyte (CTL) immune response and immune impairment. In the model, there are two time delays: intracellular delay τ1, which represents the time required for cell infection, and immune response delay τ2, which represents the time required for immune response to be activated. Firstly, based on the given initial condition, we obtain two key thresholds and three possible equilibria. Secondly, the conditions for the stability of equilibria are provided through constructing Lyapunov functions. When immune delay τ2 is present, the steady state of immune-activated equilibrium is disrupted and leads to a Hopf bifurcation. Furthermore, the direction and stability of the Hopf bifurcation are determined. Finally, we choose the Beddington–DeAngelis infection rate as an example to establish a mathematical model and conduct numerical simulations, investigating the impact of saturated CTL immune response delay on viral infection and revealing the general patterns of dynamic behavior of the model. Furthermore, the impact of certain parameters on the thresholds of model is discussed.