Existence Of Stationary Distribution In Two Types Of Stochastic Epidemic Models

As COVID-19 has been widely spread around the world,people pay more and more attention to the research of epidemic.The way of establishing epidemic model by means of differential equation has shown its important role in the analysis of epidemic propagation law and control strategy.Because there are uncertain factors everywhere in real life,such as environmental noise,four seasons transformation,etc.This paper establishes two kinds of infectious disease models with random interference,and analyzes the existence,uniqueness and ergodicity of stationary distribution,which plays an extremely important guiding role for disease control and prevention.The research contents of this paper are as follows:1.The dynamic behavior of syphilis transmission model under the joint interference of white noises and telegraph noises is studied.Firstly,a random syphilis epidemic model was established by introducing multiple infection and treatment stages.Then we construct a suitable Lyapunov function by using the theory of Has’minskii,and obtain the existence and ergodicity of the stationary distribution and determine the critical value₀R^*corresponding to the basic reproduction number of the constant differential system.When₀R^*（29）1,the system has a unique ergodic stationary distribution.In addition,by using the comparison principle of stochastic differential equations,the sufficient conditions for disease extinction are given.Finally,numerical simulations verify the theoretical results,which show that under certain conditions,strong white noises can lead to the extinction of the diseases,and telegraph noises can enhance the stability of the system.2.Dynamical behaviors of a stochastic SIR model with distributed delay are considered.Firstly,the stochastic SIR model with strong kernel and weak kernel is transformed into equivalent high-dimensional stochastic system by using linear chain technique.Then the global stability of the equilibrium of the corresponding determined system is studied by using eigenvalue analysis and Lyapunov method.Next,an appropriate Lyapunov function is constructed by using the theory of Has’minskii,and the existence of stationary distribution disturbed by white noises is studied.Finally,numerical simulation verifies this paper’s theoretical results.