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Random perturbations of recursive sequences with an application to an epidemic model. (English) Zbl 0841.58058

If \(D\) is a closed subset of \(\mathbb{R}^n\), and \(T : D \to D\) a continuous function, a discrete \(D\)-valued dynamical system can be defined by \(x_{n + 1} - T(x_n) = 0\) for \(n = 0, 1, 2, \dots\) A randomly-perturbed version of this system consists of a sequence \(X_0\), \(X_1, \dots\) of \(D\)-valued random variables that satisfy \(X_{n + 1} - T(X_n) \to 0\) almost surely (or, in probability) as \(n \to \infty\). The author shows that if the non-perturbed discrete dynamical system has a finite number of limits sets, then – under suitable conditions – the perturbed dynamical system shares the same property if the perturbation converges to zero. Of primary interest to the author is the case where the unperturbed dynamical system is dissipative (so there exists a compact subset of \(D\) attracting all the bounded subsets of \(D)\). In particular, the author considers the classic situation where the unperturbed dynamical system has a Lyapunov function and \(T\) has only a finite number of fixed points in \(D\). The author applies his results to an epidemic model.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
37N99 Applications of dynamical systems
92D30 Epidemiology
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