The Dynamical Behaviors Of Several Epidemic Models Under Stochastic Perturbation

Plenty of random perturbations will occur during the growth process of the biological species.And random perturbations affect the species in any time and everywhere.They are not constant,but random,nonlinear and uncertain.Therefore,the real situation of ecosystems can be properly described by the stochastic biological models.In this paper,some analytical techniques of the stochastic differential equations are adopted to discuss two different kinds of random perturbation.One is the random perturbation which is described as "white noise" in a small but independent way.The other is the perturbation which has somewhat large intensity.In many cases,the large intensity perturbation can be described by the Markov chain.Firstly,a stochastic SIQS epidemic model with the saturated incidence rate in a population of varying size is discussed.By means of Lyapunov function method and some related conclusions of ergodicity theory,under some sufficient conditions,the existence-and-uniqueness of the solution and the exponential stability to the disease free equilibrium(A/d,0,0)are obtained by defining the stopping time.Moreover,the stationary distribution of the solution with ergodicity is investigated,by constructing Lyapunov functions and applying Ito’s formula.Further,the solution of the model obeys a three-dimensional normal distribution in the long time scale by linearization and Fourier transform,the expressions of mean and covariance matrix are obtained.In addition,numerical simulations are presented to illustrate the effectiveness of our conclusions.Secondly,we introduce the perturbations into the death rate and infection force of the disease,a new SIQS epidemic model is considered.And similar conclusions and numerical simulations are carried out.Thirdly,we discuss a general stochastic SEIRS epidemic model with the nonlinear incidence rate in a population of varying size.Our research shows that,the solution of the stochastic model exponentially tends to extinction almost surely if the reproduction number R0≤1,that is to say,the disease will die out.If R0>1,the disease is prevalent.Further,the numerical simulations are carried out to illustrate our main results.Finally,we would like to investigate a general nonautonomous Lotka-Volterra model with infinite delay and Markov chain.Taking the stochastic perturbations into account,a stochastic type Lotka-Volterra model is established in this system.By means of the exponential martingale inequality,the Chebyshev’s inequality,Borel-Cantelli’s lemma and some elementary inequalities,the existence and uniqueness of the global positive solution and stochastic ultimate boundedness are obtained.Further,the moment estimation and stochastic boundedness of the solution are derived by the nonnegative semi-martingale convergence theorem.Our research results show that the stochastic systems will have the similar properties comparing with the deterministic systems if the white noises are controlled in a certain range.If the intensities of the white nosises become large and go out of this range,some properties of the stochastic systems will be completely different.Our results provide a reliable and theoretical reference for supervising the development of the stochastic systems and controlling the spread of some disease.