Traveling Wave Solutions For Several Types Of Non-local Epidemic Models

Biomathematics is a relatively new subject,whose aim is to study the practical problems in biology from the perspective of mathematics.Epidemic model is an particularly impor-tant one in biology.Most of the reaction-diffusion systems of infectious diseases are non-monotonic.This thesis mainly studied the traveling wave solutions and related problems in several epidemic models.In chapter 2,a lattice differential model of infectious diseases with saturated incidence rate is proposed and studied.We first prove the existence of traveling wave solutions for c>c*via considering a truncated problem and using Schauder’s fixed point theorem.Then,by virtue of the result on the non-critical traveling waves and using the limit method,the existence of the traveling wave solutions with speed c=c*is proved.Finally,the nonexistence of traveling wave solution is proved for 0<c<c*or c>c*and R0>1(R0 is the basic reproduction number of the corresponding ordinary differentia systems).In chapter 3,a SIR lattice differential model of infectious diseases with standard incidence is studied by using similar method in chapter 2.However,since the model cannot be simplified to a system of two equations,it is relatively difficult to study.In particular,it is more difficult to prove the boundedness of the wave profile R.With more detailed analysis,some sufficient conditions for the boundedness of the wave profile R is given.In chapter 4,we study the existence of traveling waves with the critical speed for a nonlocal dispersal epidemic model with delay and nonlinear incidence.At first,we proved the uni-form boundedness of non-critical waves.Then,the existence of critical waves is proved by analyzing non-critical waves in detail and using the limit method.