Dynamic Analysis Of The Stochastic Discrete SIR Epidemic Model

Since ancient times,people have paid great attention to the spread and development of infectious diseases,many scholars have established a large number of mathematical models to understand the transmission laws of infectious diseases,and conducted qualitative and quantitative research on the established models.So far,deterministic infectious disease models have been widely studied.In fact,random factors have an undeniable impact on the transmission of infectious diseases,such as the growth rate of the system,environmental capacity,competition coefficient,and other parameters that are affected to varying degrees by random white noise.With the gradual deepening of the research on infectious diseases,in recent years,continuous infectious disease models have been discretized,and the study of difference schemes for infectious diseases has become one of the hot topics in the field of biomathematics,under certain specific conditions,the difference forms will exhibit richer dynamic properties than continuous models.Therefore,using the stochastic discrete model to describe the spread of infectious diseases is more realistic.In the second part,we construct a stochastic discrete SIR(susceptible-infection-recovery)epidemic model with nonmonotonic incidence.Based on the continuous SIR epidemic model with random noise disturbance and non-monotonic incidence,the Euler-Marryama method is used to discretize it,then a stochastic discrete SIR model is obtained.We use Lyapunov function to prove the sufficient conditions for the stability of the system at the equilibrium point,propose the sufficient conditions for the probabilistic stability of the nonlinear difference equation at the zero solution,and the sufficient conditions for the mean square stability of the linear difference equation at the zero solution.Then the stability of the system at the positive equilibrium point and the boundary equilibrium point is proved.Finally,the conclusions obtained are verified by numerical simulation.In the third part,we first introduce a SIR model with saturation incidence and vaccination rate,considering that random noise has a great impact on disease transmission,and we discretize the model by using the nonstandard finite difference(NSFD)method,finally obtain a stochastic discrete SIR epidemic model.Then we use the Lyapunov function method and the matrix method to prove the stability of the system at the equilibrium point.Finally we use the numerical simulation to verify the conclusions,and the nonstandard finite difference scheme and Euler-Marryama method are compared by numerical simulation.This paper mainly analyzes the stability of stochastic discrete SIR epidemic model.Analyze the dynamic properties of stochastic discrete models with different incidence rates and obtained by different difference methods,proposed the sufficient conditions of the model at the stable point,and the influence of noise intensity on the stability of the system at the equilibrium point is analyzed.