Spatial And Temporal Transmission Characteristics Of The Diffusive SIS Epidemic Models

Since ancient times,infectious diseases have been one of the major problems to be solved in contemporary society.Re-emergence of existing infectious disease or the outbreak of the new infectious disease have posed a great threat to human life and property safety and global economic development.Even technology and medical development are so mature in modern times,infectious disease is still one of the difficult problems that people need to overcome.Centering on the theme of infectious diseases,this dissertation takes SIS(susceptible-infected-susceptible)type infectious diseases as the research object,and takes different diffusion models as the main line,focusing on the influence of spatial heterogeneity,periodic evolution region,free boundary in local diffusion and nonlocal diffusion on the spread of SIS infectious diseases.Specifically,the dissertation includes the following aspects:Chapter 1 mainly present the background knowledge and significance of the research on infectious diseases,and the main content of our research topic.Chapter 2 considers the dynamic behavior of an SIS infectious disease model with spontaneous infection.In order to study the impact of spontaneous infection on the transmission of infectious diseases,we consider an SIS epidemic reactiondiffusion model with spontaneous infection in addition to contact transmission.The existence of the endemic equilibrium is firstly proved.And then we discuss the asymptotic behavior of the endemic equilibrium when the migration rate of the susceptible or infected population is sufficiently small.Our theoretical results show that spontaneous infection can enhance the persistence of the disease and make the disease more difficult to prevent and control.In order to explore the impact of periodically evolving domain on the transmission of disease,we study a SIS reaction-diffusion model a periodically evolving domain in Chapter 3.The basic reproduction number is given by the next generation infection operator,and relies on the evolving rate of the periodically evolving domain,diffusion coefficient of infected individuals and size of the space.We then discuss the limiting behavior of the basic reproduction number with respect to diffusion coefficient and interval length by using the relevant theories and estimation methods of partial differential equations.The basic reproduction number as threshold can be used to characterize whether the disease-free equilibrium is stable or not.Our theoretical results and numerical simulations indicate that small evolving rate,small diffusion of infected individuals and small interval length have positive impact on prevention and control of disease.In Chapter 4,considering the spatial heterogeneity and free boundary conditions,the SIS reaction diffusion problem with free boundary is proposed.Firstly,the existence and uniqueness of the global solution are obtained by the contraction mapping theorem and the standard theory of parabolic equations.Then the basic reproduction number related to time as well as its analytical properties is defined by variational method.and then a spreading-vanishing dichotomy of infectious diseases is obtained by constructing the appropriate upper and lower solutions.The impacts of the diffusion rate of infected individuals,expanding capability,and the scope and scale of initial infection on the spreading and vanishing of infectious disease are analyzed.Numerical simulations are given to reveal that the large expanding capability is unfavorable to the prevention and control of the disease.In Chapter 5,in order to study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease,we consider a nonlocal SIS reaction-diffusion model with free boundaries.The free boundary describes the moving front of the infected individuals,and the nonlocal diffusion operator characterizes the long-distance spatial movement of individuals.The existence and uniqueness of the global solution are given by using two fixed point theorems.Then,we define the generalized principal eigenvalue of the integral operator,and analyze their properties,and prove that it is principal eigenvalue under some conditions.Next,we discuss the dynamic behavior of the principal eigenvalue with respect to interval length and diffusion coefficient by utilizing comparison principle and the upper and lower solutions method.Sufficient conditions for disease spreading and vanishing are given by the upper and lower solutions method and using the properties of the principal eigenvalue.Finally,the difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.Finally,we mainly make a brief summary of the research work of this dissertation,and make a plan for the future research work.