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Random perturbations of recursive sequences with an application to an epidemic model

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Daniel Pierre Loti Viaud
Daniel Pierre Loti Viaud*
Affiliation:
Université Paris VI
*
∗Postal address: Université Paris VI, L.S.T.A., T.45-55, E.3, Boite 158, 4, place Jussieu, 75252 Paris Cedex 05, France.
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Abstract

We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.

Keywords

DISCRETE-TIME DYNAMICAL SYSTEMMULTITYPE POPULATION DYNAMICS

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 559 - 578
Copyright
Copyright © Applied Probability Trust 1995 

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References

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and Its Applications. Griffin, London.Google Scholar
Ball, F. G. and O'Neill, P. (1993) A modification of the general stochastic epidemic motivated by AIDS modelling. Adv. Appl. Prob. 25, 39–62.CrossRefGoogle Scholar
Biscarat, J.-C. (1994) Almost sure convergence of a class of stochastic algorithms. Stoch. Proc. Appl. 50, 83–100.Google Scholar
Celeux, G. and Diebolt, J. (1993) A stochastic approximation type EM algorithm for the mixture problem. Stoch. Stoch. Rep. 41, 119–134.Google Scholar
Hale, J. K. (1988) Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs 25. American Mathematical Society, Providence, RI.Google Scholar
Haraux, A. (1991) Systèmes dynamiques dissipatifs et applications. Masson, Paris.Google Scholar
Jacquez, J. A. and O'Neill, P. (1991) Reproduction numbers and thresholds in stochastic epidemic models I. Math. Biosci. 107, 161–186.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, Chichester.Google Scholar
Kesten, H. (1972) Limit theorems for stochastic growth models I and II. Adv. Appl. Prob. 4, 193–232, 393–428.Google Scholar
Pierre Loti Viaud, D. (1993) Large deviations for discrete-time epidemic models. Math. Biosci. 117, 197–210.Google Scholar
Ruelle, D. (1981) Small random perturbations of dynamical systems and the definition of attractors. Commun. Math. Phys. 82, 137–151.Google Scholar