Modeling and bifurcation analysis of a viral infection with time delay and immune impairment

The rich dynamics of a viral infection model is studied under the assumption that the immune response is proposed based on some important biological meanings. In this paper, we study the dynamical behaviour of delayed viral infection with immune impairment model, viewing the infection rate of HIV-1 infection as a saturated function of the viral load, to formulate HIV-1 model dynamics. We explicitly link the individual and the viral population scale, and derive the basic reproduction number R-0 for the coupled system. The effect of time delay on stability and Hopf bifurcation for the infected equilibrium have been studied. By fixing the immune delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. Using center manifold argument and normal form theory, we derive explicit formulae to determine the stability and direction of the limit cycles of the model. To analyze the model and perform a detailed global dynamics analysis, two Lyapunov functionals are constructed to prove the global asymptotical stability of the disease-free and infected equilibria. Theoretical results indicate that R-0 provides a threshold value determining whether or not the disease dies out. Numerical simulations are carried out to explain the mathematical conclusions.