Lyapunov exponent of a stochastic SIRS model

Guoting Chen; Tiecheng Li; Changjian Liu

doi:10.1016/j.crma.2012.11.010

Ordinary Differential Equations

Lyapunov exponent of a stochastic SIRS model
[Exposant de Lyapunov dʼun modèle SIRS stochastique]

Présenté par : the Editorial Board

Guoting Chen ¹ ; Tiecheng Li ² ; Changjian Liu ³

¹ UFR de Mathématiques, UMR CNRS 8524, Université de Lille-1, 59655 Villeneuve dʼAscq cedex, France
² Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China
³ School of Mathematics, Soochow University, Suzhou 215006, PR China

Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 33-35.

Résumé
Abstract

Nous considérons un modèle de type SIRS avec perturbation stochastique. Nous démontrons que les solutions sont positives pour des conditions initiales positives et sont définies globalement. Nous présentons des conditions nécessaires et suffisantes pour la stabilité asymptotique presque sûre de la solution triviale du système stochastique.

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.

Reçu le : 2012-09-28
Accepté le : 2012-11-16
Publié le : 2013-01-01

DOI : 10.1016/j.crma.2012.11.010

Affiliations des auteurs :

Guoting Chen ¹ ; Tiecheng Li ² ; Changjian Liu ³

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     author = {Guoting Chen and Tiecheng Li and Changjian Liu},
     title = {Lyapunov exponent of a stochastic {SIRS} model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--35},
     publisher = {Elsevier},
     volume = {351},
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     year = {2013},
     doi = {10.1016/j.crma.2012.11.010},
     language = {en},
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Guoting Chen; Tiecheng Li; Changjian Liu. Lyapunov exponent of a stochastic SIRS model. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 33-35. doi : 10.1016/j.crma.2012.11.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.11.010/

Bibliographie
Cité par

[1] L. Arnold Random Dynamical Systems, Springer Monogr. Math., Springer, 1998

[2] Lyapunov Exponent, Proceedings, Oberwolfach 1990 (L. Arnold; H. Crauel; J.-P. Eckmann, eds.), Lecture Notes in Math., vol. 1486, Springer, 1991

[3] G. Chen; T. Li Stability of stochastic delayed SIR model, Stoch. Dyn., Volume 9 (2009), pp. 231-252

[4] A. Friedman Stochastic Differential Equations and Applications, vol. 1, Academic Press, 1975

[5] W.O. Kermack; A.G. Mckendrick Contributions to the mathematical theory of epidemics, Proc. R. Soc. A, Volume 115 (1927), pp. 700-721

[6] R.Z. Khasʼminskii Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Theory Probab. Appl., Volume 12 (1967), pp. 144-147

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

[8] W.M. Liu; S.A. Levin; Y. Iwasa Influence of nonlinear incidence rate upon the behavior of SIRS epidemiological models, J. Math. Biol., Volume 23 (1986), pp. 187-204

[9] Q. Lu Stability of SIRS system with random perturbations, Phys. A, Volume 388 (2009), pp. 3677-3686

[10] X. Mao Exponential Stability of Stochastic Differential Equations, Dekker, New York, 1994

[11] V.I. Oseledec A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., Volume 19 (1968), pp. 197-231

[12] E. Tornatore; S.M. Buccellato; P. Vetro Stability of a stochastic SIR system, Phys. A, Volume 354 (2005), pp. 111-126

[13] Y. Xiao; L. Chen; F. Ven den Bosch Dynamical behavior for a stage-structured SIR infectious disease model, Nonlinear Anal. Real World Appl., Volume 3 (2002), pp. 175-190

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