Free access article

Issue M2AN Volume 37, Number 4, July-August 2003 Biological and Biomedical Applications Page(s) 709 - 723 DOI 10.1051/m2an:2003045

M2AN, Vol. 37, N°4, pp. 709-723
DOI: 10.1051/m2an:2003045

Global Stability of Steady Solutions for a Model in Virus Dynamics

Hermano Frid¹, Pierre-Emmanuel Jabin² and Benoît Perthame<sup>3

1</sup>  Instituto de Matemática Pura e Aplicada - IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, RJ 22460-320, Brazil. hermano@impa.br.
²  Département de Mathématiques et Applications, École Normale Supérieure de Paris, 45 rue d'Ulm, 75230 Paris Cedex 05, France. jabin@dma.ens.fr.
³  Département de Mathématiques et Applications, École Normale Supérieure de Paris, 45 rue d'Ulm, 75230 Paris Cedex 05, France. perthame@dma.ens.fr.

Abstract
We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence - a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical and theoretical arguments, we also examine how, releasing these assumptions, the system can blow-up.

Mathematics Subject Classification. 34A34, 34G20, 70K20, 92D25, 92D10, 37A60.

Key words: Virus dynamics, population dynamics, genetics, nonlinear integro-differential equations, nonlinear ordinary differential equations, dynamical systems in statistical mechanics, immunology, evolution theory.


© EDP Sciences, SMAI 2003

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