The Dynamic Behavior Of A Cholera Diffusion Model In A Heterogeneous Space

The dynamics of infectious diseases is an important branch of biological mathematics.We formulate partial differential equations to model the transmission of cholera,and analyze some properties of the threshold systematically.Also we study the global asymptotic behavior of cholera.The main contents are as follows:Chapter 1 introduces the mathematical models of cholera,as well as its related background and research status.Also the main contents of this paper are briefly outlined.Chapter 2 deals with the existence,uniqueness and asymptotic behaviors of equilibria of the ordinary differential model of cholera.First,the basic reproduction number of the model is proposed through the next generation matrix method.Subsequently,combining the method of linearization with spectral analysis,we obtain the following conclusion:The disease free equilibrium is locally stable when R0<1;The disease free equilibrium is unstable,and the model has unique endemic equilibrium which is locally asymptotically stable as R0>1.Finally,the global asymptotic stabilities of the disease free equilibrium and the endemic equilibrium are obtained by constructing the Lyapunov function and the upper and lower solutions respectively.Chapter 3 studies a nonlinear parabolic problem with third boundary conditions in which the diffusion of the bacteria and infected individuals are incorporated.The basic reproduction number of the system is given firstly by relevant eigenvalue problem.Then we use the method of constructing an upper solution and the comparison principle to gain the global asymptotic stability of the disease free equilibrium.The existence and stability of the endemic equilibrium are discussed by utilizing the method of upper and lower solutions.In chapter 4,we further study the reaction diffusion equations in a heterogeneous environment,which aims to explore the reaction diffusion problem with variable coefficients.First,the basic reproduction number of the system is given.Then we prove the local asymptotic property of the disease free equilibrium by contradiction with the aid of relevant eigenfunction.Next,we use the method of constructing an upper solution and the comparison principle to obtain the global asymptotic stability of the disease free equilibrium.The existence and stability of the endemic equilibrium are obtained subsequently by deriving its upper and lower solutions,as well as the iteration of monotonic sequence.Moreover,we consider a special diffusive pattern in which bacteria diffuses and human do not,and then seek sufficient conditions for disease to vanish or spread.Chapter 5 is devoted to numerical simulations of the above problems.Graphs have been plotted to verify the theoretical results obtained.The main work of this paper is summarized in chapter 6,and the future research which deserves further study is discussed.