Dynamics of an HIV infection model with two time delays

Abstract

In this paper, a mathematical model for HIV-1 infection and immune response is considered, involving two discrete time delays in the intracellular as well as in activation of immune response. Using a recently developed geometric method for studying a class of transcendental equation with two time delays and delay dependent coefficients, we obtain the stability and bifurcation results at the non-trivial equilibrium. In particular, the crossing curves on the two-delays parameter plane can be completely characterized, on which Hopf and double-Hopf bifurcation will take place. In the case of Hopf bifurcation, there exist stability switches, and the direction and stability of delay induced Hopf-bifurcation can be determined using normal form theory and center manifold theorem. These results imply the model will exhibit complex temporal dynamics, such as period oscillations, quasi-periodic solutions, etc. Numerical examples are also carried out to verify these results.