Traveling Wave Solutions For Several Epidemic Diffusive Models

In recent years,dealing with specie invasions and spread of infectious diseases by mathematical methods has been a hot issue in biological mathematics.In a large number of mathematical models,reaction-diffusion equation is the most common,so it has been paid more and more attention.The traveling wave solution,which can describe many properties of equation itself and explain the biological phenomena such as disease transmission and species invasion,is a special form of the solution of the reaction-diffusion equation.This paper mainly studies the existence of traveling wave solutions for several infectious diseases models and the influence of the parameters on the solutions.Firstly,considering that the non-local dispersal of the disease is more realistic than the local diffusion in some cases,we propose a class of non-local dispersal SIR epidemic models with general incidence rate and time delay.The existence of traveling wave solutions is established by using Schauder's fixed point theorem for a truncation problem and taking the limit,while the non-existence of traveling wave solutions is studied by using the Laplace transform method.Then we analyze the influence of the time delay on the minimum wave velocity of traveling wave solutions,which shows that the increase in the latency period of the disease will reduce the minimum wave velocity of disease transmission.Secondly,because infected individuals have the ability to move freely and spread the disease during the incubation period,we propose a non-local dispersal SIR epidemic model with spatio-temporal delay.By Schauder's fixed point theorem,we prove the existence of fixed points on the invariant cone constructed in the initial function on a bounded region.The non-existence of traveling wave solutions is obtained by Laplace transform method.And the influence of the diffusion coefficient of the infections and the delay on the minimum velocity is discussed.Finally,traveling wave solutions of HBV epidemic model with spatial structure are studied.We prove the existence of traveling wave solutions by using truncation method and obtain the asymptotic behavior of the solution by constructing a (1 function.Lastly,we use the method of two-sided Laplace transform to get the non-existence of traveling wave solutions of the model.