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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Heterogeneous viral environment in a HIV spatial model

Pages: 545 - 572, Volume 15, Issue 3, May 2011      doi:10.3934/dcdsb.2011.15.545

 
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Claude-Michel Brauner - School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China, and Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence cedex, France (email)
Danaelle Jolly - Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence cedex, France (email)
Luca Lorenzi - Dipartimento di Matematica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, I-43124 Parma, Italy (email)
Rodolphe Thiebaut - (M.D.) Equipe Biostatistique de l'U897 INSERM ISPED, Université de Bordeaux, 33076 Bordeaux cedex, France (email)

Abstract: We consider a basic model of virus dynamics in the modeling of Human Immunodeficiency Virus (HIV), in a two-dimensional heterogenous environment. It consists of two ODEs for the uninfected and infected CD4$^+$ T lymphocytes, $T$ and $I$, and a parabolic PDE for the free virus particles $V$. We introduce a new parameter $\lambda_0$ which is the largest eigenvalue of some Sturm-Liouville problem and takes the heterogenous reproductive ratio into account. For $\lambda_0<0$ the uninfected steady state is the only equilibrium. When $\lambda_0>0$, it becomes unstable and there is a unique positive infected equilibrium. Considering the model as a dynamical system, we prove the existence of a positively invariant region. Finally, in the case of an alternating structure of viral sources, we define a homogenized limiting environment which justifies the classical approach via ODE systems.

Keywords:  Viral dynamics, stability, analytic semigroups, homogenization.
Mathematics Subject Classification:  Primary: 35K55; Secondary: 35B35, 92C50.

Received: February 2010;      Revised: March 2010;      Available Online: February 2011.

 References