The Study On Traveling Wave Solutions Of Diffusive Epidemic Models With Temporal Delay

During the past three decades,a large number of mathematical models have been proposed to discuss the spread of various types of infectious diseases.The qualitative analysis of the epidemic model(especially the analysis of epidemic waves)can reveal the epidemic law of the disease,predict its development trend and provide theoretical basis for the prevention and control of the disease.In this paper,the time-delay reactiondiffusion epidemic model,the time-delay non-local diffusion epidemic model and the time-delay mixed diffusion epidemic model are proposed.Then the existence and nonexistence of traveling wave solutions of these models are proved by upper and lower solution method,Schauder's fixed point theorem,bilateral Laplace transform and other methods.For the delayed reaction-diffusion epidemic model,we will apply the auxiliary system method,the upper-lower solution method,Schauder's fixed point theorem,the squeeze theorem and the bilateral Laplace to prove the existence and nonexistence of the traveling wave solutions.Firstly,we construct the auxiliary system(perturbed system)corresponding to the original traveling wave system.Secondly,we construct the upper and lower solutions of the auxiliary system and define a nonlinear operator corresponding to the auxiliary system and a closed convex cone.Thirdly,the existence of solutions of the auxiliary system is transformed into the existence of fixed point of the nonlinear operator.Then we use Schauder's fixed point theorem to prove that the nonlinear operator has a fixed point.Fourthly,the asymptotic boundary of the solution is obtained by means of the contradictory arguments and squeeze theorem.Fifthly,we obtain the uniformly bounded estimation of the solution by the classical analytical method and derive the existence and asymptotic boundary of the traveling wave solution of the original system by a limit argument.Sixthly,we utilize the bilateral Laplace transform to prove the nonexistence of traveling wave solutions of the system.For the delayed nonlocal diffusion epidemic model,we will apply the upper-lower solution method,Schauder's fixed point theorem,the squeeze theorem,the limit argument and the bilateral Laplace transform to prove the existence and nonexistence of traveling wave solutions.Firstly,we reduce the original system to the traveling wave system.Secondly,the upper and lower solutions of the traveling wave system are constructed,and a closed convex cone is established on the closed interval.Thirdly,an ODE system with initial value is constructed for a traveling wave system and the existence and uniqueness of the solution are obtained by using the ODE theory.Fourthly,we construct a nonlinear operator and use Schauder's fixed point theory to prove the existence of fixed point in the region.Fifthly,we obtain the uniformly bounded estimation for the solution by using the analytical method.Then the limit discussion is used to prove the existence of fixed point on the whole real line of the nonlinear operator.Sixthly,we derive the asymptotic boundary of the traveling wave solution by using the squeeze theorem and the contradictory arguments.Seventhly,the bilateral Laplace transform is utilized to prove the nonexistence of traveling wave solutions of the system.For the delayed mixed diffusion epidemic model,we prove the existence and nonexistence of traveling wave solutions by the upper-lower solutions method,Schauder fixed point theorem,the squeeze theorem and the bilateral Laplace transform.Firstly,we reduce the original system to wave system.Secondly,we construct a pair of upper and lower solutions of the wave system and define a closed convex cone on the whole real line and a nonlinear operator.Thirdly,the existence of solution for wave system is transformed into the existence of a fixed point for the nonlinear operator.Then Schauder fixed point theorem is used to prove the existence of a fixed point for the nonlinear operator.Fourthly,the asymptotic boundary of the solution is obtained by using the contradictory argument and the squezze theorem.Finally,we prove the nonexistence of traveling waves by the bilateral Laplace transform.