Dynamical Analysis Of Two Kinds Of Stochastic Infectious Disease Models

Infectious diseases have brought great harm to human beings.Mathematical models have played an important role in the study of the spread of infectious diseases.By establishing suitable models of infectious diseases,analyzing their dynamics and using computer simulations,the spread of infectious diseases can be studied macroscopically.In this thesis,two kinds of stochastic epidemic models are established by introducing stochastic effects into the epidemic models described by ordinary differential equations.The global dynamics of two kinds of stochastic epidemics are investigated by using the theory and method of stochastic differential equations.There are four chapters in this thesis.In Chapter 1,we introduce the research background and current situation,and give the relevant basic knowledge.In Chapter 2,a stochastic SIS epidemic model with nonlinear incidence rate is proposed and analyzed.We assumes that the susceptible may be infected with two different infectious diseases,and the infection rate is affected by stochastic white noise.As a comparison,we first investigate the dynamic properties of the corresponding deterministic model.By analyzing the local asymptotic stability of its equilibrium,we obtain the threshold which can determine the elimination and persistence of the disease.Then,we investigate the global dynamics of stochastic SIS epidemic model.By constructing suitable function and utilizing Ito formula,we obtain a threshold similar to deterministic model.By analyzing the threshold,we find that stochastic white noise has a significant impact on the spread of infectious diseases,that is,the larger stochastic interference will cause disease elimination,which is conducive to the control of infectious diseases.At the same time,by comparing the "threshold" of the stochastic model with the corresponding deterministic model,we also find that the "threshold" of the former is smaller than the latter,which causes a permanent disease in a deterministic model to eventually disappear under stochastic white noise interference.In Chapter 3,based on a deterministic SIR epidemic model with vertical infection and continuous vaccination,considering the impact of stochastic white noise on infection rates,a stochastic SIR epidemic model with vertical transmission and vaccination is established.The global dynamics of the stochastic SIR epidemic model is analyzed by the theory and method of stochastic differential equations.By constructing suitable functions and using the Ito formula,sufficient conditions for disease elimination and persistence are obtained.The results show that stochastic interference has important effect to the spread of infectious diseases and large stochastic interference is more conducive to the control of infectious diseases.In Chapter 4,we summarize the full text and give some questions worthy of further study.