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 $\mathcal{R}_0$ are identified and its threshold properties are discussed. When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globally asymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomes unstable and there occurs the single-infection equilibrium $E_s$, and $E_0$ and $E_s$ exchange their stability at the transcritical point $\mathcal{R}_0 =1$. If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitly depending on the model parameters, $E_s$ is globally asymptotically stable, while when $\mathcal{R}_0 > R_1$, $E_s$ loses its stability to the double-infection equilibrium $E_d$. There exist a constant $R_2$ such that $E_d$ is asymptotically stable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange their stability at the transcritical point $\mathcal{R}_0 =R_1$. We use one numerical example to determine the largest range of $\mathcal R_0$ for the local stability of $E_d$ 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

Study of an HIV-1 Model with Time Delays

We propose a mathematical model for HIV-1 infection with two time delays, one for the average latent period of cell infection and the other for the average time needed for the virus production after a virion enters a cell. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. When only the intracellular delay is enrolled into model (1.13), the basic reproduction numbers Rq and Rd are identified and their threshold properties are discussed. When Rq \u3c 1, the infection-free equilibrium Eq is globally asymptotically stable. When Rq \u3e 1, Eq becomes unstable and there occurs the single-infection equilibrium Es. If Rq \u3e 1 and Rd \u3c 1, Es is asymptotically stable, while for Rd \u3e 1, Es loses its stability to the double-infection equilibrium. For the double-infection equilibrium Ed, we show how to determine its stability and existence of Hopf bifurcation. Some simulations are presented to demonstrate the theoretical results. Further investigation is carried over by introducing the second time lag into model (2.1). We have identified the new basic reproduction numbers Rqand Rd, and proved that for Rq \u3c 1 the infection-free equilibrium Eq is globally asymptotically stable. If Rq \u3e 1 and Rd \u3c 1, the single-infection equilibrium Es is asymptotically stable. For the double-infection equilibrium Ed, it has been found that there exist both Hopf and N double Hopf bifurcations. These theoretical predictions are verified by using some numerical examples. Evidences indicate that the viral-therapy, of recombining HIV-1 virus with a genetically modified virus may be effective in reducing the HIV-1 load, and larger delays may be able to help eradicate the virus

Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4 +

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of CD4+ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results

Analysis of the dynamics of a delayed HIV pathogenesis model

AbstractIn this paper, considering full Logistic proliferation of CD4+ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results