Studies On Dynamics Of Several Epidemic Models With Heterogeneity

In this thesis,based on the method of compartment modelling of epidemic dynamics,several epidemic models with heterogeneity are established and studied by considering temporal heterogeneity(seasonal variation),spatial heterogeneity(multi-group,spatial diffusion,environmental heterogeneity)and individual heterogeneity(age structure).By using the stability and uniform persistence theory of dynamical system,theory of parabolic partial differential equation,theory of almost periodic differential equation,graph theory,matrix theory and other mathematical theories and methods,systematic analyses of the dynamic behaviors for these systems are estabilished,such as:global attractivity,stability and persistence,etc.In addition,to validate the conclusions obtained in this thesis,numerical simulations are made by using the Matlab.Thus,fundamental bases for the establishment of infectious disease’s prevention and control strategy are provided.The main contents and results are as follows:In Chapter 1,the background and significance of the research on infectious diseases and their heterogeneity are introduced,and the overseas and domestic research status of infectious disease models with seasonal variation,multi-group,spatial diffusion,environmental heterogeneity and age structure is summarized,then,the main work of this thesis is given.In Chapter 2,a multi-group hepatitis B transmission model with almost periodic infection coefficient is established to analyze the effect of population migration and the almost periodic infection rate on the transmission of hepatitis B virus.First of all,in the case of the infection coefficients are constant,by utilizing next generation matrix approach of epidemic dynamical system and matrix theory,the basic reproduction number of this autonomous system is determined.By means of matrix theory and theory of limit system,the existence,uniqueness and global asymptotic stability of disease-free equilibrium are constructed.According to the stability theory of dynamic system and Routh-Hurwitz criteria,the existence,local asymptotic stability of the endemic equilibrium and uniform persistence of the system are proved.In addition,the global attractivity of the system and the existence of almost periodic solution are studied based on the theory of almost periodic differential equation when the system is non-autonomous,i.e.the infection coefficients are almost periodic functions.Finally,the feasibility of the main theoretical results is verified by numerical simulation.In Chapter 3,a multi-group SEIR infectious disease model with age structure and spatial diffusion is established to study the effects of spatial heterogeneity and individual heterogeneity on the transmission of infectious diseases.Firstly,by constructing Picard sequence,Banach fixed point theorem and partial differential equation theory,the positive,boundedness,existence and uniqueness of the solution and the existence of compact global attractor of the associated solution semiflow are analyzed.Then,by constructing a suitable Lyapunov functional and combining with LaSalle invariant principle,it is proved that the diseasefree equilibrium point is globally asymptotically stable when the parameters meet certain conditions.Next,by using Perron-Frobenius theorem and graph theory,the existence and global stability of endemic equilibrium are proved under appropriate conditions.Finally,a numerical simulation of the model with two groups is carried out to verify the correctness of the main theoretical results.In Chapter 4,a multi-group SEIR infectious disease model with spatial diffusion in heterogeneous environment is established to analyze the influence of environmental heterogeneity and spatial diffusion of population on the transmission of infectious diseases.Firstly,according to the standard parabolic comparison theorem and theory of dissipative system,the positivity and ultimate boundedness of the solution and the existence of the global attractor of the associated solution semiflow are established.Besides,by using the next generation operator method of reaction-diffusion epidemic model and stability theory,the definition of the basic reproduction number is given and its properties are analyzed.Moreover,according to the comparison theorem of parabolic equation and LaSalle invariant principle,the global asymptotic stability of the disability-free equilibrium is analyzed,and the uniform persistence of the system is proved by the strong maximum principle for parabolic equation and the uniform persistence theory of dynamic system.Finally,the feasibility of the main theoretical results is verified by numerical simulation of the model with two groups.In Chapter 5,the main work of this thesis is summarized,and on this basis,the work worthy of further study in the future is proposed.