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 ℛ(0)<1, then the infection-free equilibrium E (0) is globally asymptotically stable; when ℛ(0)>1, E (0) loses its stability and there is the single-infection equilibrium E (s). If ℛ(0)∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E (s) that is globally asymptotically stable, while when ℛ(0)>1+δ, E (s) becomes unstable and the double-infection equilibrium E (d) comes into existence. When ℛ(0) is slightly larger than 1+δ, E (d) is stable and it loses its stability via Hopf bifurcation when ℛ(0) 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.