Study On The Dynamical Behaviors Of Two Classes For SIRI Epidemic Models With Stochastic Effects

In recent years, stochastic phenomenon has become very popular among scholars, and stochastic differential equation has been well developed. It is of great research significance to consider applying stochastic effect and use the fundamental theories and methods of stochastic differential equations to study epidemic models’dynamic behavior. This paper mainly concerns with dynamical behaviors such as the existence and uniqueness of solution, boundedness, stability, ergodicity, and so on. The detailed research of this paper will be introduced in the following two aspects:In the first part, we consider a stochastic SIRI epidemic model when the trans-mission coefficient λ is subjected to stochastic perturbation. Firstly, we show that the existence and uniqueness global positive solutions of the stochastic SIRI epidemic model and boundedness by using Ito formula. Furthermore, the disease-free equilibrium of the stochastic model is globally asymptotically stable in stochastic meaning when the basic reproduction number R0≤1. The solution fluctuates around the endemic equilibrium of the deterministic model and the estimate for the oscillation amplitude is obtained when R0> 1. Finally, some numerical examples and it simulations are provided to illustrate the effectiveness of the obtained results.In the second part, we consider a stochastic SIRI epidemic model when the death rate β is subjected to stochastic perturbation. The existence, uniqueness and bounedness of the positive solution of the stochastic system are derived. By using some novel Lyapunov functions and stochastic analysis tools, we mainly investigate asymptotical behaviors of the SIRI model under stochastic perturbations in terms of the basic reproduction number R0.That is, when R0≤1 (or R0>1) and some conditions on white noise are sat-isfied, the solution of stochastic model will oscillate around the disease-free equilibrium (or endemic equilibrium) of the deterministic model and the estimate for the oscillation amplitude is obtained. Furthermore, the existence of a stationary distribution and the ergodicity of solutions are studied for the stochastic system.Finally, some numerical examples and it simulations are provided to illustrate the effectiveness of the obtained results.