Traveling Wave Solutions For Several Classes Of Diffusive Epidemic Models

Infectious diseases have became one of the most serious public health problems all over the world.With the development of globalization and the convenience of transportation,the diffusive of population is more and more frequent,which brings great challenges to the prevention and control of infectious diseases.In recent years,the diffusive epidemic models have attracted widespread attention.Generally,the reaction-diffusion equation(system)is used to describe the diffusion and migration of people.For this kind of systems,there is a special solution called traveling wave solution,which moves at a constant speed in a certain direction and keep the shape unchanged.In the infectious disease model,traveling wave solution could be used to describe that the disease starts from the one source and spreads at a constant speed.Through theoretical analysis of the model,the threshold value and the minimum wave speed of traveling wave solution could be found,so as to analyze the prevention and control strategy of infectious diseases.This dissertation will study several kinds of epidemic models with different diffusion,mainly focusing on the existence,boundedness,persistence and asymptotic behavior of traveling wave solutions.Firstly,we establish an SVIR model with infection-age and local diffusion.By solving the problem of corresponding eigenvalue problem of linear subsystem with exponential solution,we obtain the formula of the minimum wave speed when the basic reproduction number is greater that one.The existence and nonexistence of traveling wave solutions are obtained.When the traveling wave solution exists,some boundedness properties of the traveling wave solution are discussed.Moreover,we construct a Lyapunov functional to show the asymptotic behavior of the traveling wave solution.Secondly,we propose a class of discrete diffusive SIR model with general incidence.When the basic reproduction number is greater than one,the existence of traveling wave solution is obtained if wave speed is greater than the minimum wave speed by constructing the sub-and super-solutions and applying the Schauder's fixed point theorem;the nonexistence of traveling wave solution is proved when the wave speed is less than the minimum wave speed.Furthermore,the boundedness and persistence of traveling wave solutions are analyzed.In addition,a new Lyapunov functional applied to the discretediffusion model is proposed,and the convergence of the traveling wave solution at the positive infinity is obtained.Thirdly,we formulate a nonlocal diffusive SVIR epidemic model with bilinear incidence and time delay.The results of nonlocal diffusion linear equation are extended to the case with time delay.According to the basic reproduction number,we found the threshold for the existence of the traveling wave solutions.By using way of contradiction,we obtain the boundedness and persistence of traveling wave solution if it exists.Further,by constructing a suitable Lyapunov functional and using the Lebesgue dominated convergence theorem,we obtain the asymptotic behavior traveling wave solution.Finally,we study a Cholera model with nonlocal diffusion in the host population.By analyzing the corresponding eigenvalue problem of linear subsystem,we obtain the minimum wave speed when the basic reproduction number is greater than one,there exists a traveling wave solution for the model when the wave speed is greater than the minimum wave speed;there exists no traveling wave solution when the wave speed is less than the minimum wave speed.By using the theory of spreading speed of degenerate nonlocal diffusive cooperative system,the permanence of traveling wave solution is obtained,then a suitable Lyapunov functional is constructed and the asymptotic behavior of traveling wave solution is obtained.