The Research Of Threshold Problem Of Stochastic Epidemic Models

Epidemic dynamics is an important subject on a quantitative study of infectious diseases. By the study on dynamical behavior and numerical simulation of mathematical models, it analyzes the development of the disease process, reveals its popular rule, predicts trends and provides a theoretical foundation and quantity basis for the prevention and treatment decisions for people. In the realistic ecosystem, environmental white noise is everywhere. Therefore, researching the dynamics behavior of stochastic infectious diseases system can reflect the actual phenomenon more accurately, reveal the influence of random disturbances on infectious diseases system. It is important for predicting the development trend of the disease, epidemic prevention and control scientifically.This paper mainly studies the dynamic behavior of an SIS model with a vaccination e?ect, SIRS models with standard incidence and saturated incidence under random perturbations. Firstly we give the global existence of positive solutions to stochastic systems by using Lyapunov functional method. This is a basic research of random systems’ dynamic behavior. Secondly, we investigate the extinction and persistence of disease system, as well as the asymptotic behavior of solution. Using random inequalities and martingale methods, we obtain the exponential stability and persistence in the time mean of the system solution, analysis to determine the threshold, which reveals when the disease disappears or is popular. Further,in the case of persistence, by Has’ minskii ergodic theory and Markov semigroup theory, we point out the existence of the systems’ stationary distribution and it is ergodicity.The above studies show when the white noise is small, the stochastic systems have the similar properties of the corresponding deterministic system. If the threshold value of stochastic system?R0<1, the disease will disappeared; If?R0>1,then the disease is prevalent. Compared with the corresponding deterministic system, the threshold of stochastic system is related to the intensity of white noise.While when white noise is large, stochastic system will appear this property which is di?erent from the deterministic system’s completely. Even if the deterministic system’s basic reproduction number R0> 1, the disease will disappear. In practice, a large white noise can be considered as a burst and bad weather, severe natural disasters and so on. Finally, numerical simulations verify the above main conclusions.