Permanence and extinction of a non-autonomous HIV-1 model with time delays

Xia Wang; Shengqiang Liu; Libin Rong

doi:10.3934/dcdsb.2014.19.1783

Article Contents

2014, Volume 19, Issue 6: 1783-1800. Doi: 10.3934/dcdsb.2014.19.1783

This issue Previous Article On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction Next Article Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian

Permanence and extinction of a non-autonomous HIV-1 model with time delays

1.
School of Mathematics and Information Sciences, Shaanxi Normal University, Xi'an, 710062
2.
Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080
3.
Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309

Received: April 30, 2013

Revised: October 31, 2013

Published: July 2014

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Abstract

The environment of HIV-1 infection and treatment could be non-periodically time-varying. The purposes of this paper are to investigate the effects of time-dependent coefficients on the dynamics of a non-autonomous and non-periodic HIV-1 infection model with two delays, and to provide explicit estimates of the lower and upper bounds of the viral load. We established sufficient conditions for the permanence and extinction of the non-autonomous system based on two positive constants $R^{\ast}$ and $R_{\ast}$ ($R^{\ast}\geq R_{\ast}$) that could be precisely expressed by the coefficients of the system: (i) If $R^{\ast}<1$, then the infection-free steady state is globally attracting; (ii) if $R_{\ast}>1$, then the system is permanent. When the system is permanent, we further obtained detailed estimates of both the lower and upper bounds of the viral load. The results show that both $R^{\ast}$ and $R_{\ast}$ reduce to the basic reproduction ratio of the corresponding autonomous model when all the coefficients become constants. Numerical simulations have been performed to verify/extend our analytical results. We also provided some numerical results showing that both permanence and extinction are possible when $R_{\ast }< 1 < R^{\ast}$ holds.

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Mathematics Subject Classification: Primary: 34K11, 37N25; Secondary: 92D30.

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