Dynamics Of A Stochastic SIQRS Epidemic Model With Saturated Incidence Rate And Vaccination

In this paper,we study the dynamical behaviors of the stochastic SIQRS epidemic model with saturated incidence rate and vaccination.Firstly,we investigate a stochastic SIQRS epidemic model with white noise.By constructing a suitable Lyapunov function,we prove the existence and uniqueness of the positive solution,and we obtain the suffi-cient conditions for the existence of a stationary distribution with ergodicity of this model.Besides,under the assumption that the intensity of white noise satisfies a certain condition,we show that when R0s<1,the disease tends to become extinct and when R0s>1,the disease tends to be persistent.According to the corresponding deterministic SIQRS epidemic model,we get its basic reproduction number R0.By R0>R0s,we find that white noise in the environment can inhibit the spread of infectious diseases.Secondly,we establish a stochastic SIQRS epidemic model with white noise and Lévy jumps,and the existence and uniqueness of the positive solution of the model are proved.Furthermore,we prove that when Rjump<1,the disease will become extinct and when Rjump>1,the disease will be persistent.We can see that Lévy jumps will suppress the explosion of diseases from R0>R0s>Rjump.Finally,through numerical simulations,we verify the correctness of theoretical results of the above two stochastic SIQRS epidemic models.