The Dynamics Analysis Of Stochastic Epidemic Models

By establishing the form of diferential equation and diference equation equa-tion and studying dynamics of equation, research on epidemiological dynamics canreveal the evolution law of infectious diseases and provide a theoretical basis forthe prevention and control of the infectious diseases. Considering the infuence ofrandom factors on the transmission of infectious diseases, this thesis establishessome stochastic model of infectious disease by adding random disturbance to adeterministic epidemic model. Dynamics property is analysed and the asymptot-ic relationship the stochastic model and the corresponding deterministic model isobtained.This paper is divided into fve chapters and the dynamics behavior of threestochastic epidemic models are studied.The frst chapter introduces the course of dynamic stochastic epidemic devel-opment and main content of this paper.The second chapter gives some defnitions and preliminary theorems which willbe used in this paper are introduced.The third chapter establishes a stochastic SIRS model by adding random dis-turbance to a deterministic SIRS model. The existence of the unique solution tothe stochastic SIRS model are proved. In addition, we establish the asymptoticrelationship between the stochastic SIRS model and the corresponding determin-istic model by choosing suitable Lyapunov functions. That is, if R₀≤1, thenthe stochastic SIRS model is exponentially stable at the disease-free equilibrium ofthe deterministic SIRS model, and if R₀>1, then the stochastic SIRS model isexponentially stable at the endemic equilibrium of the deterministic SIRS model.The fourth chapter, considering isolation of patients is a common method toprevent infectious disease difusion, we establish a stochastic SIQS model to describethe spread of infectious diseases for some patients after treatment process go backto susceptible population. Similar to the research methods in the third chapter, wediscuss the existence and uniqueness of the solution of the stochastic SIQS modeland obtain the asymptotic relationship between disease-free equilibrium as well asendemic equilibrium of the corresponding deterministic SIQS model.In the ffth chapter, on the basis of the fourth chapter, considering isolation oftreatment for patients and the patients recovering from the disease can obtain permanent immune infectious disease, we establish a stochastic SIQR model.It prove the existence and uniqueness of the solution and the relationship betweencorresponding deterministic SIQR model and stochastic model is obtained.