Impact of delay on HIV-1 dynamics of fighting a virus with another virus.

In this paper, we propose a mathematical model for HIV-1 infection with intracellular delay. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. For this model, the basic reproduction number R0 are identified and its threshold properties are discussed. When R0<1, the infection-free equilibrium E0 is globally asymptotically stable. When R0>1, E0 becomes unstable and there occurs the single-infection equilibrium Es, and E0 and Es exchange their stability at the transcritical point R0=1. If 1<R0<R1, where R1 is a positive constant explicitly depending on the model parameters, Es is globally asymptotically stable, while when R0>R1, Es loses its stability to the double-infection equilibrium Ed. There exist a constant R2 such that Ed is asymptotically stable if R1<R0<R2, and Es and Ed exchange their stability at the transcritical point R0=R1. We use one numerical example to determine the largest range of R0 for the local stability of Ed and existence of Hopf bifurcation. Some simulations are performed to support the theoretical results. These results show that the delay plays an important role in determining the dynamic behaviour of the system. In the normal range of values, the delay may change the dynamic behaviour quantitatively, such as greatly reducing the amplitudes of oscillations, or even qualitatively changes the dynamical behaviour such as revoking oscillating solutions to equilibrium solutions. This suggests that the delay is a very important fact which should not be missed in HIV-1 modelling.