A mathematical model of the infection of CD4+ T-cells by HIV that includes the effects of treatment by a reverse transcriptase inhibitor (RTI) and a protease inhibitor (PI) is studied. The model includes three populations of CD4+ T-cells (healthy cells, latently-infected cells which cannot produce virus, and productively-infected cells which can produce virus) and two populations of free virus in the blood (infectious virus and non-infectious virus). The model includes a time delay between a T-cell becoming latently infected and productively infected. The model has a virus-free and a chronic infection equilibrium. It is shown that the model has Andronov-Hopf bifurcations leading to limit cycle behavior in the chronic infection region at critical values of the time delays. For three data sets obtained from the work of previous authors, numerical simulations have given critical delay values ranging from approximately 15 days to more than 200 days. This range includes the period of approximately 50 days for intermittent viral blips reported by Rong and Perelson (Plos Comp. Biol. 5(10), 1-18 (2009)). Simple formulas are derived for the sensitivity indices of the equilibrium populations and the basic reproductive number with respect to all parameters in the model. Numerical simulations are carried out to support the analytical results. The numerical results suggest that the most effective methods of reducing both the basic reproductive number and the chronic infection CD4+ T-cell and virus populations are the following: (1) to increase the efficacy of the antiretroviral treatments and (2) to increase virus clearance rate, decrease infection rate, or decrease viral reproduction rate.
In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov-Hopf bifurcations in the differential equations and Neimark-Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark-Sacker bifurcations are less than critical times for Andronov-Hopf bifurcations but converge to them as the time step of the discretization tends to zero.
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