The Traveling Wave Solutions Of Some Diffusive Epidemic Models With Delay

The traveling wave solutions of diffusiveepidemic models can describe the spatial transmission of epidemic diseases in the form of waves.Therefore,the research theories of traveling wave solutions in such models have important guiding significance for the prevention,development trend prediction and control strategies formulation of epidemic diseases.This article is concerned with the nonlocal diffusion epidemic model with distributed delay,mixed diffusion epidemic model with distributed delay,and reaction-diffusion epidemic model with saturated incidence rate and discrete delay.By the upper and lower solutionmethod and Schauder's fixed point theorem,two-sided Laplace transform method,we prove the existence and nonexistence of travelling wave solutions of these epidemic models.For the nonlocal diffusion epidemic model with distributed delay,first of all,the eigenvalue information of the linearized system is obtained by traveling wave transformation.Then,the upper and lower solutionsof the traveling wave system are constructed,and a closed convex cone is defined on the finite interval.And thena nonlinear operator is defined.By Schauder's fixed point theory,we prove that the operator has a fixed point on the finite interval,that is,the existence of solutions of traveling wave system in the finite interval is proved.The uniformly bounded estimation of the solutions is obtained by the analytical method,and the existence of the traveling wave solution of the original system on the whole real line is proved.The asymptotic boundary of thetraveling wave solutionsat infinity is obtained by using the squeeze theorem and proof of contradiction.Finally,the two-sided Laplace transform is used to prove the nonexistence of the traveling wave solutions of the system.For the mixed diffusion epidemic model with distributed delay,the eigenvalue information of the linearized system is obtained by traveling wave transformation.Then,the upper and lower solutions of the traveling wave system are constructed,and a closed convex set is defined on the finite interval.Then,a nonlinear operator is defined,and the existence of a fixed point on a finite interval is proved by Schauder's fixed point theory,that is,the existence of a solution for a traveling wave system on a finite interval is proved.The uniformly bounded estimation of the solution is obtained by using the analytical method,and the existence of the original traveling wave solution on the whole real number line is proved.The asymptotic boundary of the traveling wave solution at infinity is obtained by the squeeze theorem and proof of contradiction.Finally,the two-sided Laplace transform is constructed to prove the nonexistence of the traveling wave solution of the model.For the discrete delay reaction-diffusion epidemic model with saturation incidence,the existence of critical eigenvalues of a linearized traveling wave system is discussed.Then,according to the characteristics of critical eigenvalues,the upper and lower solutions of the critical traveling wave system are constructed,and then a closed convex cone and a nonlinear operatorare defined.We apply Schauder's fixed point theorem to prove the existence of fixed point for nonlinear operators,that is,to prove the existence of solutions for critical traveling wave systems.Then,the asymptotic boundary of the critical traveling wave solution is obtained by contradiction and the squeeze theorem.Finally,we prove the nonexistence of the traveling wave solutionof the model by the two-sided Laplace transform.