Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. (English) Zbl 1263.92042

Summary: We consider a class of multi-group SEIR epidemic models with stochastic perturbations. By the method of stochastic Lyapunov functions, we study their asymptotic behavior in terms of the intensity of the stochastic perturbations and the reproductive number \(R_{0}\). When the perturbations are sufficiently large, the exposed and infective components decay exponentially to zero whilst the susceptible components converge weakly to a class of explicit stationary distributions regardless of the magnitude of \(R_{0}\).
An interesting result is that, if the perturbations are sufficiently small and \(R_{0}\leq 1\), then the exposed, infective and susceptible components have similar behaviors, respectively, as in the case of large perturbations. When the perturbations are small and \(R_{0}>1\), we construct a new class of stochastic Lyapunov functions to show the ergodic property and the positive recurrence, and our results reveal some cycling phenomena of recurrent diseases. Computer simulations are carried out to illustrate our analytical results.


92D30 Epidemiology
34F05 Ordinary differential equations and systems with randomness
60H30 Applications of stochastic analysis (to PDEs, etc.)
68U20 Simulation (MSC2010)
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