Analysis of time delay in viral infection model with immune impairment

In this paper, a class of virus infection models with CTLs response is considered. We incorporate three delays with immune impairment into the virus infection model. We begin model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By analyzing corresponding characteristic equation, the local stability of each feasible equilibria and the existence of Hopf bifurcation at the infected equilibrium are addressed, respectively. 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. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of steady states at \(\tau _3<\tau _3^*\). By theoretical analysis and numerical simulations, we show that the immune impairment rate has a dramatic effect on the infected equilibrium of the DDE model.