Stationary Distribution For Stochastic Epidemic Model

In this paper,we mainly discuss the existence and uniqueness of the ergodic stationary distribution of a class of affine nonlinear stochastic epidemic models.Besides of the stochastic analysis theory,the distinctive method that we used in this paper is the state space feedback lin-earization method in the nonlinear control theory.This method has been applied to a stochastic SIRS model with general nonlinear incidence and driven by 1-dimensional and m-dimensional standard Brownian motion respectively.The first chapter introduced the research background and significance of stochastic epi-demic model,and reviewed the research status of ergodic stationary distribution.For the de-generate diffusion case,the shortcomings of existing research methods and the problems to be solved are pointed out.In order to solve these problems,this paper gives a simple and feasi-ble method by means of feedback exact linearization theory:by verifying rank condition,the existence and uniqueness of ergodic stationary distribution of stochastic model can be derived.In the second chapter,the affine nonlinear stochastic epidemic model driven by 1-dimen-sional standard Brownian motion is studied,and the sufficient condition for the existence and uniqueness of the ergodic stationary distribution for the stochastic model is obtained.This con-clusion is applied to a stochastic SIRS epidemic model with general nonlinear incidence and single stochastic perturbation.To begin with,we verify that there is a unique global positive solution starting from the positive initial value by the stopping time technique.When the in-tensity of environmental white noise is large enough,or when the stochastic basic reproduction number R₀^S<1,but the intensity is small,the disease will die out exponentially almost surely.When the disease persists,the distance between the solution of the stochastic model and its quasi-positive equilibrium is estimated.It is proved that as long as the controllability matrix is full row rank and R₀^S>1,the stochastic model has a unique ergodic stationary distribution.Finally,the corresponding Fokker-Planck equation is solved by WKB-approximation method,and the approximate analytical expression of the stationary density for the stochastic model near the quasi-positive equilibrium is derived,and the theoretical results are verified by numerical simulations.The third chapter generalized the method in the second chapter to the affine nonlinear stochastic epidemic model driven by m-dimensional standard Brownian motion,and obtains the sufficient condition for the existence of a unique ergodic stationary distribution.This method is applied to a stochastic SIRS epidemic model with general nonlinear incidence and multiple stochastic perturbations.When the intensity of environmental white noise is strong enough,the disease will die out exponentially almost surely.When the disease persists,the distance between the solution of the stochastic model and the quasi-positive equilibrium is estimated.It is also proved that as long as the controllability matrix is full row rank and the intensity of envi-ronmental white noise is small,the model has a unique ergodic stationary distribution.Finally,the corresponding Fokker-Planck equation is solved by the WKB-approximation method,and the approximate analytical expression of the stationary density for the stochastic model near the quasi-positive equilibrium is obtained,and the theoretical results are verified by numerical simulation.All in all,we point out that the existence and uniqueness of an ergodic stationary distribu-tion depends on two conditions,one is the persistence of system,the other is the rank condition.Interestingly,this result is independent of the pattern of disease incidence.