Global asymptotic stability for the disease free equilibrium for epidemiological models

Jean Claude Kamgang; Gauthier Sallet

doi:10.1016/j.crma.2005.07.015

Dynamical Systems/Ordinary Differential Equations

Global asymptotic stability for the disease free equilibrium for epidemiological models
[Stabilité globale et asymptotique de l'équilibre sans maladie des modèles épidémiologiques]

Présenté par : Alain Bensoussan

Jean Claude Kamgang ¹ ; Gauthier Sallet ²

¹ ENSAI, Department of Mathematics, University of Ngaoundéré, PO box 455, Ngaoundéré, Republic of Cameroon
² CONGE Project, INRIA Lorraine, I.S.G.M.P., bâtiment A, Île du Saulcy, 57045 Metz cedex 01, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 433-438.

Résumé
Abstract

Pour une classe générale de modèles, nous prouvons la globale asymptotique stabilité de l'équilibre sans maladie sous des hypothèses générales. Ces conditions sont relatives au nombre de reproduction de base $R_{0}$ . Nous donnons également un algorithme pratique permettant d'établir une condition de seuil équivalente à $R_{0} ⩽ 1$ . Nous montrons que ces deux résultats peuvent être appliqués à de nombreux modèles épidémiologiques de la littérature.

For a general class of models, we prove the global asymptotic stability (GAS) of the disease free equilibrium (DFE) under general assumptions. These conditions are related to the basic reproductive ratio $R_{0}$ . We also give a practical algorithm to compute a threshold condition equivalent to $R_{0} ⩽ 1$ . We show that these two results can be applied to numerous epidemiological models from the literature.

Reçu le : 2004-05-27
Accepté le : 2005-07-19
Publié le : 2005-09-12

DOI : 10.1016/j.crma.2005.07.015

Affiliations des auteurs :

Jean Claude Kamgang ¹ ; Gauthier Sallet ²

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     author = {Jean Claude Kamgang and Gauthier Sallet},
     title = {Global asymptotic stability for the disease free equilibrium for epidemiological models},
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     pages = {433--438},
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     year = {2005},
     doi = {10.1016/j.crma.2005.07.015},
     language = {en},
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Jean Claude Kamgang; Gauthier Sallet. Global asymptotic stability for the disease free equilibrium for epidemiological models. Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 433-438. doi : 10.1016/j.crma.2005.07.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.015/

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