Global asymptotic stability for the disease free equilibrium for epidemiological models

Kamgang, Jean Claude; Sallet, Gauthier

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

Kamgang, Jean Claude ¹ ; Sallet, Gauthier 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, Tome 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 :

Kamgang, Jean Claude ¹ ; Sallet, Gauthier ²

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     title = {Global asymptotic stability for the disease free equilibrium for epidemiological models},
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Kamgang, Jean Claude; Sallet, Gauthier. Global asymptotic stability for the disease free equilibrium for epidemiological models. Comptes Rendus. Mathématique, Tome 341 (2005) no. 7, pp. 433-438. doi : 10.1016/j.crma.2005.07.015. http://archive.numdam.org/articles/10.1016/j.crma.2005.07.015/

Bibliographie
Cité par

[1] Castillo-Chavez, C.; Feng, Z. To treat or not to treat: The case of tuberculosis, J. Math. Biol., Volume 35 (1997), pp. 629-656

[2] Castillo-Chavez, C.; Feng, Z.; Huang, W. On the computation of $R_{0}$ and its role on global stability, Mathematical Approaches for Emerging and Reemerging Infectious Diseases, IMA Vol. Math. Appl., Springer, New York, 2001

[3] Cherry, B.R.; Reeves, M.J.; Smith, G. Evaluation of bovine diarrhea virus control using a mathematical model of infection dynamics, Prev. Vet. Med., Volume 33 (1998), pp. 91-108

[4] Diekmann, O.; Heesterbeek, J.A. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000

[5] Diekmann, O.; Heesterbeek, J.A.; Metz, J.A.J. On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., Volume 28 (1990), pp. 365-382

[6] Jacquez, J.A.; Simon, C.P. Qualitative theory of compartmental systems, SIAM Rev., Volume 35 (1993) no. 1, pp. 43-79

[7] LaSalle, J.P. Stability theory for ordinary differential equations, J. Differential Equations, Volume 4 (1968), pp. 57-65

[8] Li, M.Y.; Fan, M.; Wang, K. Global stability of an SEIS model with recruitment varying total population, Math. Biosci., Volume 170 (2001), pp. 199-208

[9] Li, M.Y.; Smith, H.L.; Wang, L. Global dynamic of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., Volume 62 (2001), pp. 58-69

[10] Liu, W.M.; Hethcote, H.W.; Levin, S.A. Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., Volume 25 (1987), pp. 359-380

[11] Perelson, A.S.; Kirschner, D.E.; De Boer, R. Dynamics of HIV infection of CD4+ T cells, Math. Biosci., Volume 114 (1993), pp. 81-125

Cité par Sources :