Impact of delay on HIV-1 dynamics of fighting a virus withanother virus

In this paper, we propose a mathematical model for HIV-1 infectionwith intracellular delay. The model examines a viral-therapy for controllinginfections through recombining HIV-1 virus with a genetically modifiedvirus. For this model, the basic reproduction number $\mathcal{R}_0$are identified and its threshold properties are discussed.When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globallyasymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomesunstable and there occurs the single-infection equilibrium $E_s$,and $E_0$ and $E_s$ exchange their stability at the transcriticalpoint $\mathcal{R}_0 =1$.If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitlydepending on the model parameters, $E_s$ is globally asymptotically stable,while when $\mathcal{R}_0 > R_1$, $E_s$ loses itsstability to the double-infection equilibrium $E_d$.There exist a constant $R_2$ such that $E_d$ is asymptoticallystable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange theirstability at the transcritical point $\mathcal{R}_0 =R_1$.We use one numerical exampleto determine the largest range of $\mathcal R_0$ for the localstability of $E_d$ and existence of Hopf bifurcation. Some simulationsare performed to support the theoretical results.These results show that the delay plays animportant role in determining the dynamic behaviour of the system.In the normal range of values, the delay may change the dynamic behaviourquantitatively, such as greatly reducing the amplitudes of oscillations,or even qualitatively changes the dynamical behaviour such as revokingoscillating solutions to equilibrium solutions.This suggests that the delay is a very importantfact which should not be missed in HIV-1 modelling.