Traveling Wave Solutions For Two Types Of SIR Models With Delay And Nonlocal Dispersal

In infectious disease models,the existence or nonexistence of traveling wave solutions indicates whether the disease spreads or not.The disease will break out quickly or not and whether it will be eliminated eventually are determined by the boundedness and asymptotic behaviors of traveling waves.Therefore,it is of practical significance and theoretical value to discuss the existence,non-existence,boundedness and asymptotic behaviors of traveling waves in infectious disease models for preventing and controlling diseases.Among them,it is relatively difficult to investigate infectious disease systems with(nonlocal)delay and nonlocal diffusion.Firstly,some methods to the classical Laplace equations are not efficient because the model is changed into an integral differential equation by the introduction of nonlocal diffusion.In addition,the smoothness of solutions is lower,and thus many convergence conclusions are not valid any longer for the lack of compactness of nonlocal diffusion operators.For example,the existence of traveling waves under critical wave speed can not be obtained simply by taking limit.Secondly,the common monotonicity methods are not effective because of the loss of monotonicity of equations derived from the appearance of coupling.Finally,the boundedness estimate becomes more difficult because the nonlocal nonlinear terms with respect to space and time contain integral items.Therefore,for the investigation of traveling waves for infectious disease models with(nonlocal)delay and nonlocal diffusion,it not only improves the theory of traveling waves for classical Laplace equations,but also extends the conclusion of traveling waves for nonlocal diffusion equations without delay to nonlocal diffusion equation with nonlocal delay.Based on the above facts,this thesis mainly deals with the existence,non-existence,boundedness and asymptotic behaviors of traveling waves for two types of(nonlocal)delay SIR models with nonlocal diffusion and nonlinear incidence.The main work in this thesis is organized as follows:The existence,nonexistence,boundedness and asymptotic behaviors of traveling waves of a class of delayed SIR models with nonlocal diffusion are investigated.Firstly,the boundedness and asymptotic behaviors of traveling waves under non-critical wave speed(c>c*)are established by analytical techniques,which improves the existence of traveling waves of previous results for the model.Secondly,the existence,boundedness and asymptotic behaviors of traveling waves under critical wave speed(c=c*)are established by a prior estimate,limit theory and analysis techniques.In addition,the non-existence of traveling waves under non-critical wave speed(0<c<c*)is also obtained by using analysis techniques which is different from Laplace transform.Finally,these results of existence,non-existence,boundedness and asymptotic behaviors of traveling wave solutions of the delayed SIR model are extended to the corresponding SIR model with non-local delay.The existence,nonexistence,boundedness and asymptotic behaviors of traveling waves for a class of nonlocal delay SIR models with input terms and nonlocal diffusion are considered.Firstly,the invariant cone on a bounded region is constructed and the existence of traveling waves under non-critical wave speed(c>c*)is established by using Schauder fixed point theorem.Secondly,the asymptotic behaviors of traveling waves for c>c*are established by using analysis techniques.Then,the existence and asymptotic behaviors of the traveling waves under critical wave speed(c=c*)are also obtained by using a prior estimate,limit theory and analysis techniques.Finally,the non-existence of the traveling waves for 0<c<c*is established by using analysis techniques which is different from Laplace transform.