Dynamic Analysis For Several Kinds Of Stochastic SIRS Epidemic Models

The spread and control of diseases are two of the important issues concerned by the world health organization.SIRS models have been widely used and deeply studied in the research of spread and control of diseases.Many results have been obtained in the study of deterministic SIRS epidemic models.In the real world,the spread of disease is often affected by environmental random factors,so the stochastic models can describe the actual problems more accurately.In this paper,white noise is introduced into the classical deterministic epidemic models,and three kinds of stochastic SIRS epidemic models with vertical transmission rate,stochastic natural death rate and nonlinear incidence rate are studied respectively,and some new results are obtained.The main contributions are summarized as follows:1.The following deterministic SIRS epiderImic rImodel with vertical transmission rate is established.(?)The asymptotic stability of disease-free equilibrium and endemic equilibrium in the deterministic model are analyzed,and the threshold which determine the extinction or persistence of the disease is given.Considering the random fluctuations of population contact rate,we establish the following stochastic SIRS epidemic model with vertical transmission rate.(?)Base on the existence of global unique positive solution of the system,we use Fokker-Planck equation and Markov semigroup theory to prove that the Markov semi-group generated by the solution of the model corresponding to the Fokker-Planck equation is asymptotically stable,that is,the model has a stationary distribution.The sufficient conditions for disease extinction and the threshold which determine the ex-tinction or persistent of the disease are obtained through the Ito's formula and the strong law of large numbers for martingales.Studies have shown that Rs<R0,the random fluctuations in the population contact rate can affect the spread of disease.2.Considering the random fluctuations of population natural death rate,we establish the following stochastic SIRS epidemic model with stochastic natural death rate.(?)Base on the existence of global unique positive solution of the system,the stochas-tic ultimately bounded of the solution is proved by using Ito's formula and Chebyshev's inequality.The sufficient conditions of disease extinction and the existence of ergodic stationary distribution are obtained by using Lyapunov function and the strong law of large numbers for martingales of stochastic analysis theory.The threshold Rs which de-termine extinction or persistent of the disease is given.More concretely,when Rs<1,the disease is extinction;When Rs>1,the model has a stationary distribution,this means the disease is persistent.Studies have shown that Rs<R0,the random fluctuations in the population natural death rate can affect the spread of disease.3.Continue to consider the random fluctuations of the population contact rate,we establish the following more general stochastic SIRS epidemic model with nonlinear incidence rate.(?)Base on the existence of global unique positive solution of the system,the stochas-tic ultimately bounded of the solution of the system is proved by using Ito's formula and Chebyshev's inequality.A set of sufficient conditions for extinction and persistence in mean of diseases are obtained by using analy tical skills such as the strong law of large numbers for martingales.Thresholds R1s and R2s which determine the extinction or persistent of diseases are obtained.More concretely,when R1s<1 and R2s<1,the diseases I1 and I2 are extinction;When R1s>1 and R2s<1,the disease I1 is persistent and the disease I2 is extinction;When R1s<1 and R2s>1,the disease I1 is extinction and the disease I2 is persistent;When R1s>1 and R2s>1,the diseases I1 and I2 are persistent.