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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 $R_{0}\le 1$. At last, we make simulations to conform our analytical results.

MSC:
92D30Epidemiology
34F05ODE with randomness
34D10Stability perturbations of ODE
34A99General theory of ODE
37N25Dynamical systems in biology
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