Dynamics of an HIV-1 therapy model of fighting a virus with another virus.

In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio R0<1, then the infection-free equilibrium E0 is globally asymptotically stable; when R0>1, E0 loses its stability and there is the single-infection equilibrium Es. If R0∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium Es that is globally asymptotically stable, while when R0>1+δ, Es becomes unstable and the double-infection equilibrium Ed comes into existence. When R0 is slightly larger than 1+δ, Ed is stable and it loses its stability via Hopf bifurcation when R0 is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory. [ABSTRACT FROM AUTHOR]