Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

In this paper, we propose a mathematical model for HIV-1 infection with three time delays. The model examines a viral-therapy for controlling infections by using an engineered virus to selectively eliminate infected cells. In our model, three time delays represent the latent period of pathogen virus, pathogen virus production period and recombinant (genetically modified) virus production period, respectively. Detailed theoretical analysis have demonstrated that the values of three delays can affect the stability of equilibrium solutions, can also lead to Hopf bifurcation and oscillated solutions of the system. Moreover, we give the conditions for the existence of stable positive equilibrium solution and Hopf bifurcation. Further, the properties of Hopf bifurcation are discussed. These theoretical results indicate that the delays play an important role in determining the dynamic behavior quantitatively. Therefore, it is a fact that delays are very important, which should not be missed in controlling HIV-1 infections.