| Infectious diseases seriously affect human health and the development of social econ-omy.Therefore,the main task of human beings is to study the spread law of infectious diseases,find out the key factors of infectious diseases,predict the spread trend of infectious diseases,and formulate prevention and control strategies and measures.The dynamics of in-fectious diseases is mainly based on the dynamic behavior of mathematical model to analyze the development process of the disease,so as to make a quantitative study on the transmis-sion law of infectious diseases.On the one hand,there are a lot of uncertain factors in nature,which seriously affect the spread of infectious diseases.In addition,the infectious disease system is often impacted by external factors such as volcanoes and typhoons,which leads to the change of transmission mode and law of infectious diseases.Therefore,it is necessary to study the infectious disease system driven by white noise.On the other hand,the spread of infectious diseases is not only related to the current state,but also to the historical state.At this time,the original differential equation model can not be used to describe the inter-nal law of infectious disease development.Therefore,it is more realistic to use stochastic differential equation with time delay to describe the transmission mechanism of infectious diseases.In this paper,the dynamics of infectious diseases affected by environmental noise is considered.The main work is as follows:In Chapter 1,the biological background and significance of the infectious disease model are briefly introduced.The stochastic infectious disease model and the related content of the proof are given.In Chapter 2,we study the asymptotic behavior of the recurrent stochastic SIRI epi-demic model with standard incidence and system disturbance.The existence and uniqueness of the global positive solution are proved by using the knowledge of random Lyapunov func-tion.The extinction of the system is proved by using It?o,s formula and large number theorem,and the sufficient conditions for the extinction of the disease are given.By constructing the Lyapunov function and using the ergodicity theory of Has’Minskii,the existence and ergod-icity of the stationary distribution are proved.Finally,the theoretical results are applied to the actual cases.Taking tuberculosis as an example,the simulation results show that reducing the recurrence rate is conducive to the extinction of infectious diseases without considering the impact of white noise.When considering the influence of white noise,compared with the corresponding deterministic model,the threshold value affected by white noise is less than the basic reproduction number of deterministic system R0.In other words,random disturbance can inhibit the spread of SIRI infectious diseases.In Chapter 3,we study the asymptotic behavior of stochastic SIRS epidemic model with distributed delay and temporary immunity.The existence and uniqueness of the glob-al positive solution are proved by using the knowledge of random Lyapunov function.By using the theoretical knowledge of It?o,s formula and large number theorem,we prove the ex-tinction of the system and the persistence in the sense of average time,and give the threshold conditions of the persistence in the sense of disease extinction and average time.By using the ergodicity theory of Lyapunov function and Has’Minskii,it is found that the system has stationary distribution and ergodicity.According to the threshold analysis of disease extinction,random disturbance can inhibit the spread of SIRS infectious diseases. |
