Dynamic Behavior Of Stochastic Epidemic Models Perturbed By White Noise

Recently, deterministic epidemic models have been studied widely and tremendous results were obtained. Mathematical models have become important tools in analyzing the spread and control of infectious diseases. In reality, environmental noise is ubiquitous. So it is an important and interesting work to study the dynamic behavior of stochastic epidemic models perturbed by environmental noise.The content of this paper is as follows.1. Dynamic behavior of stochastic SIR epidemic models. We consider the stochastic models with infectious transmission rate perturbed by white noise, or with multiple white noise. The su?cient conditions for the existence of stationary distribution and the extinction of infectious disease are presented. When the model is of non-degenerate di?usion term, ergodic theory due to Hasminskii is employed; whereas when the model is of degenerate di?usion term, the tool is Markov semigroup theory.2. Dynamic behavior of stochastic SIS and SISV epidemic models with perturbed infectious transmission rate. Since the obtained stochastic model is of degenerate type, Markov semigroup theory is adopted. For stochastic SIS model, we obtain the threshold behavior: when threshold value is less than one, the disease will extinct in probability; whereas when threshold value is bigger than one, the densities of the distributions of the solution can converge in L1 to an invariant density. For stochastic SISV model, we present the su?cient condition for the existence of stationary distribution.3. Dynamic behavior of stochastic SIR and SIRS epidemic models with periodic coe?cients. First, we present the threshold for epidemic to occur, that is,when threshold value is less than one, the disease will become extinct exponentially; whereas when threshold value is bigger than one, the disease will persist in mean time. In the case of persistence the su?cient condition for periodic solution is given. The tool we used here is the existence theorem of the periodic solution due to Hasminskii.The results obtained in this paper improve the existing results greatly, and can provide a deep insight into the dynamics of stochastic epidemic models. In this sense, our results are interesting.