Analysis On The Dynamics Of Two Types Of Epidemic Models With Stage Structure

For a long time,the prevention and control of infectious diseases has been the focus of the world.Since the outbreak of infectious diseases will have a great impact on human's production,life and even lives,it is of great practical significance to establish mathematical models to study the transmission dynamics and asymptotic behaviors of contagions.It is well know that the growth and development of human have certain stages,and the growth at various stages shows differences in physiology.For example,measles and varicella always occur in immaturity,while typhus,diphtheria and sexually transmitted diseases always break out in maturity.Based on these,the population are divided into adults and juveniles,and the hypothesis is proposed that only the adult individuals can be infected.In consideration of the latent period of the disease,the fertility of infected people and the influence of media coverage,two kinds of epidemic models with stage structures are established,and their dynamical behaviors and biological significance are analyzed.In Chapter 1,the research background and significance of epidemics are briefly introduced,and research progress of epidemic models with stage structures are described,meanwhile the main work of this thesis and the basic theoretical knowledge that we needed are presented.In Chapter 2,an SIS epidemic model with stage structures and latent timedelay (?) is established and studied,in which the fertility reduction factor is introduced to describe the fertility rate of the infected population.Firstly,positivity of solutions for the system is proved and the basic reproduction number of the system is calculated,then the existence of the equilibria is discussed.Secondly,instability of the population annihilation equilibrium,local stabilities of the disease-free equilibrium and the endemic equilibrium are proved.By constructing the Lyapunov functional,sufficient conditions for global asymptotical stability of the disease-free equilibrium are obtained.Then,it is proved that under certain conditions,(?) can change the stability of the endemic equilibrium and make Hopf bifurcation of the system occur.Finally,the theoretical results that we concluded are proved by numerical simulations,and the effects of fertility reduction factor on disease are illustrated.In Chapter 3,an SEIR epidemic model with stage structures incorporating media coverage is developed and studied.The influence of media coverage on infectious diseases is depicted by the contact rate of disease,and the contact rate of the model gradually decreases with the increase of the number of infected adults.Firstly,positivity and boundedness of the system solutions are proved,the basic reproduction number is calculated and the existence and uniqueness of the equilibria are determined.Then,local asymptotical stabilities of the equilibrium are proved.By constructing suitable Lyapunov functions,global dynamics,which is completely determined by the basic reproduction number,is established.More precisely,when the basic reproduction number R₁< 1,the disease-free equilibrium is globally asymptotically stable,that is,the disease will eventually be eradicated and when R₁> 1,the endemic equilibrium is globally asymptotically stable,that is,the disease will persist and the disease is endemic.Finally,the theoretical results obtained are proved by numerical simulations,and the impact of media coverage on the dynamics of disease is further illustrated.In Chapter 4,the research contents and conclusions are briefly summarized.At the same time,a few disadvantages of the work in this thesis as well as some problems that deserve further research are proposed.