Dynamics Of Epidemic Models With Nonlinear Incidence Rate

Mathematical modeling offers an important research tool to study the evolution of infectious diseases.By analyzing of the dynamics of the models,we can find the key factors influencing the spread of infectious diseases and build some theoretical foundation for preventing and controlling of infectious diseases.Based on the different incident rates and the non-uniform mixing of the susceptibles,three classes of epidemiological dynamics models are established in this thesis.By using of the basic theory of differential equations,dynamical systems and numerical simulations,we study the dynamics of these models and obtain some sufficient conditions ensuring the stability of the equilibrium and the uniform persistence.This thesis consists of four chapters.In the first chapter,the significance,background and the main work of this paper are briefly addressed.In the second chapter,considering the psychological effects of groups and precaution of diseases,we establish a SEIRS epidemic model with a non-monotone incidence.The sufficient conditions of global stability of disease-free equilibrium and the uniform persistence are obtained.Also some numerical simulations are given to illustrate the effectiveness of the results.In the third chapter,according to the non-uniform mixing of the susceptibles,we divide the susceptible into some differential susceptibles.An epidemic model with the incidence rate of Beddington-DeAngelis and multi-susceptibles is established.By constructing Lyapunov functional and using the LaSalle invariable principle,some sufficient conditions ensuring the stability of the system are obtained.Also some numerical simulations are presented to illustrate the effectiveness of the results.In the fourth chapter,concerning the non-uniform mixing of the susceptibles,we consider difference of infectivity of the infectives stay on different stages.The groups of the infectives are divided into three stages:the infectives stay on the early stage are not infectious;the infectives stay on the middle stage are infectious to the individuality of the susceptibles;the infectives stay on the last stage cannot contact effectively with the susceptible.An epidemiological dynamics equation with multistage of illness and multi-susceptibile is established.By analyzing the model,some sufficient conditions ensuring the stability of the system are obtained.