Dynamic Behavior Analysis Of Epidemic Models With Stage Structure And Immune Evasion

In this paper, according to the “compartment model” for epidemics with immune responses, we establish three kinds of SIRS epidemic dynamics models with stage structure and immune evasion. Then we study the dynamical behavior and biological significance of these models.The full paper is divided into five chapters.The first chapter is introduction. First, We introduce the background development and history of infectious diseases study briefly, list the basis of topic selection and the transmission way. Secondly, we summarize the pathogenesis related knowledge of infectious diseases, the research significance, and the research status of epidemic dynamics models related to stage structure and immune evasion, and the main work. Finally, some important definitions and preliminary knowledge of dynamic behavior analysis are given.In chapter 2, according to the paternal-fetal or mother-to-child vertical transmission and immune evasion of Hepatitis B virus, we establish a SIRS epidemic dynamic model by concerning vertical transmission and continuous vaccination. By using Routh-Hurwitz criterion, LaSalle invariant set principle, and the generalized Dulac function, we study the asypmtotically stability of the model. We derive the sufficient condition of global asymptotic stability with disease-free equilibrium point and positive equilibrium point. Finally, we select the appropriate parameters for numerical simulation.In chapter 3, as for the infectious disease spreads trend estimate and the prevention work, it is more important that the estimate of the disease incidence in the research of infectious disease dynamics model.According to the diversity of incidence rate of the epidemic models, we establish a SIRS epidemic dynamic model with nonlinear incidence rate.We get the basic reproduction number R0, and derive the sufficient conditions of the global asymptotic stability of the infection-free equilibrium point and the chronic-infection equilibrium point, by using Routh-Hurwitz criterion, LaSalle invariant set principle, the theory about asymptotically orbital in differential equations and compound matrix.Finally, we select the appropriate parameters for numerical simulation,and study the biological significance of model.In chapter 4, some recent medical studies indicate that there are differences in immune escape due to the different age stage structures of susceptible persons. Therefore, according to the differences of immune escape, we divide the susceptible person into two stage structures:childhood and adulthood, and establish a S1S2IR epidemic dynamic model with stage structure and immune evasion, by considering the infectious disease can spreads in juvenile and adult groups and just considering the continuous vaccination for young children. We establish the basic reproduction number R0, and obtain the sufficient conditions of the global asymptotic stability of the infection-free equilibrium point and the chronic-infection equilibrium point, by applying Routh-Hurwitz criterion and comparison theorem of differential equation. At last, we choose the appropriate parameters for numerical simulation, and study the biological significance.In chapter 5, we summarize the thesis briefly.