Mathematical analysis of a HIV model with quadratic logistic growth term

Xinyue Fan; Claude-Michel Brauner; Linda Wittkop

doi:10.3934/dcdsb.2012.17.2359

Article Contents

2012, Volume 17, Issue 7: 2359-2385. Doi: 10.3934/dcdsb.2012.17.2359

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Mathematical analysis of a HIV model with quadratic logistic growth term

1.
School of Mathematical Sciences, Xiamen University, 361005 Xiamen
2.
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China, and Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence cedex
3.
(MD) INSERM U897, Centre de recherche Epidémiologie et Biostatistique, ISPED, Université de Bordeaux, 33076 Bordeaux Cedex, France, and Centre Hospitalier, Universitaire de Bordeaux, 33076 Bordeaux Cedex

Received: November 30, 2011

Revised: April 30, 2012

Published: September 2012

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Abstract

We consider a model of disease dynamics in the modeling of Human Immunodeficiency Virus (HIV). The system consists of three ODEs for the concentrations of the target T cells, the infected cells and the virus particles. There are two main parameters, $N$, the total number of virions produced by one infected cell, and $r$, the logistic parameter which controls the growth rate. The equilibria corresponding to the infected state are asymptotically stable in a region $(\mathcal I)$, but unstable in a region $(\mathcal P)$. In the unstable region, the levels of the various cell types and virus particles oscillate, rather than converging to steady values. Hopf bifurcations occurring at the interfaces are fully investigated via several techniques including asymptotic analysis. The Hopf points are connected through a "snake" of periodic orbits [24]. Numerical results are presented.

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Mathematics Subject Classification: Primary: 34A34; Secondary: 34D20, 92C50.

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