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The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence. (English) Zbl 1231.92058
Summary: We include stochastic perturbations into SIR and SEIR epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number \(R_{0}\). The long time behavior of the two stochastic systems is studied. Mainly, we utilize stochastic Lyapunov functions to show under some conditions that the solution has the ergodic property as \(R_{0}\)>1, and exponential stability as \(R_{0}\leq 1\). At last, we make simulations to conform our analytical results.

92D30 Epidemiology
34F05 Ordinary differential equations and systems with randomness
34D10 Perturbations of ordinary differential equations
34A99 General theory for ordinary differential equations
37N25 Dynamical systems in biology
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