Dynamics of an HIV-1 infection model with cell mediated immunity

• Improving the existing HIV-1 infection models with cell mediated immunity.
• Showing the necessity to include virus in the study of immunity models.
• Rigorously proving the existence of oscillation due to Hopf bifurcation.
• Numerical simulations supporting the theoretical results.

In this paper, we study the dynamics of an improved mathematical model on HIV-1 virus with cell mediated immunity. This new 5-dimensional model is based on the combination of a basic 3-dimensional HIV-1 model and a 4-dimensional immunity response model, which more realistically describes dynamics between the uninfected cells, infected cells, virus, the CTL response cells and CTL effector cells. Our 5-dimensional model may be reduced to the 4-dimensional model by applying a quasi-steady state assumption on the variable of virus. However, it is shown in this paper that virus is necessary to be involved in the modeling, and that a quasi-steady state assumption should be applied carefully, which may miss some important dynamical behavior of the system. Detailed bifurcation analysis is given to show that the system has three equilibrium solutions, namely the infection-free equilibrium, the infectious equilibrium without CTL, and the infectious equilibrium with CTL, and a series of bifurcations including two transcritical bifurcations and one or two possible Hopf bifurcations occur from these three equilibria as the basic reproduction number is varied. The mathematical methods applied in this paper include characteristic equations, Routh–Hurwitz condition, fluctuation lemma, Lyapunov function and computation of normal forms. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.