The Analysis Of Stability And Bifurcation Of Epidemic Models With Nonlinear Growth And Time Delay

Infectious disease dynamics model provides important theory value in investigation of the epidemic disease spreading law,the prediction and control of epidemic disease,the results of epidemic disease investigation have great instructive significance in predicting the trend of epidemic spreading,preventing and controlling epidemic disease effectively.This dissertation is devoted to studying epidemic disease theoretically by applying theories and methods of dynamic system,and proving the correctness and effectiveness of the theoretical analysis by simulation and a practical example.Chapter one introduces the background and significance of epidemic disease firstly,as well as the study of the situation at home and abroad.The second chapter introduces the knowledge and theorems introduced in this paper.The third chapter studies an epidemic disease with saturation recovery rate,general nonlinear incidence rate and two delays theoretically.Using the basic reproduction number,we will study the local stability of disease-free equilibrium,by making fitted Lyapunov function,we study the global stability of the disease-free equilibrium,in the same time,By the different values of the defined basic reproduction number and Routh-Hurwitz theorem,we investigate six casesof the local stability of the endemic equilibrium,by Hopf bifurcationg theorem,we study the bifurcation phenomenon.Using the center manifold theorem and the formation theory,we can get the concise expression determining the direction of the Hopf bifurcation and stability of bifurcation periodic solution.Finally,the correctness of the theoretical analysis is verified by numerical simulation.In the fourth chapter,the existing data are used for empirical analysis,we discuss a SEIR epidemic model with Logistic growth,nonlinear incidence rate and two-delays.By Lyapunov function,it is concluded that the disease-free equilibrium is globally asymptotic stable,and by the basic reproduction number,the local asymptotic stability of the disease-free equilibrium is locally stable,by Routh-Hurwitz theorem,we can learn that the endemic equilibrium is locally stable.Finally,Then by means of numerical simulation of the measles model in China from2013 to 2016 in the statistical yearbook,we will prove the accuracy of theoretical analysis and results of this model.The fifth chapter summarizes funding of this thesis systematically,clarifing the latest research results and representative issues in this direction,making it clear what should we do for further study.