Ordinary Differential Equations
Lyapunov exponent of a stochastic SIRS model
Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 33-35.

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.

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.

Published online:
DOI: 10.1016/j.crma.2012.11.010
Chen, Guoting 1; Li, Tiecheng 2; Liu, Changjian 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
     author = {Chen, Guoting and Li, Tiecheng and Liu, Changjian},
     title = {Lyapunov exponent of a stochastic {SIRS} model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--35},
     publisher = {Elsevier},
     volume = {351},
     number = {1-2},
     year = {2013},
     doi = {10.1016/j.crma.2012.11.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.11.010/}
AU  - Chen, Guoting
AU  - Li, Tiecheng
AU  - Liu, Changjian
TI  - Lyapunov exponent of a stochastic SIRS model
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 33
EP  - 35
VL  - 351
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.11.010/
DO  - 10.1016/j.crma.2012.11.010
LA  - en
ID  - CRMATH_2013__351_1-2_33_0
ER  - 
%0 Journal Article
%A Chen, Guoting
%A Li, Tiecheng
%A Liu, Changjian
%T Lyapunov exponent of a stochastic SIRS model
%J Comptes Rendus. Mathématique
%D 2013
%P 33-35
%V 351
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.11.010/
%R 10.1016/j.crma.2012.11.010
%G en
%F CRMATH_2013__351_1-2_33_0
Chen, Guoting; Li, Tiecheng; Liu, Changjian. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.11.010/

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

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

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

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

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

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

[7] 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

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

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

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

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

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

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

Cited by Sources: