Traveling Wave Solutions Of Diffusive Epidemic Models With Delay

A number of reaction-diffusion equations have been proposed to model the propagation dynamics of epidemic disease.In mathematical epidemiology,the study of traveling wave solutions in diffusive epidemic models is of great importance since they can describe the state that a disease propagates spatially with a constant speed.In this paper,we study two diffusive epidemic models with delay.Under certain conditions,we obtain the existence and nonexistence of nontrivial and non-negative traveling wave solutions for these two models.In Chapter 1,we introduce the research background,present the current situation and development trend and outline the arrangement and the novelty of this paper.In Chapter 2,we study the existence and non-existence of traveling wave solutions for a non-local diffusive epidemic model with spatio-temporal delay.When the basic regenerative number₀R?29?1andc?29?c^*?where c is the wave speed,^*c the critical wave speed?,by using the Schauder's fixed point theorem,we prove that this model has a non-trivial and positive traveling wave solution.With the aid of reduction to absurdity and two-sided Laplace transform,we show that if₀R?29?1 and0?27?c?27?c^*,or₀R?1and c?0,this model has no non-trivial and positive traveling wave solutions.In Chapter 3,we study the traveling wave solutions with critical wave speedc?28?c^*for a generalized diffusive epidemic model with discrete delay.The existence of positive traveling wave solutions is solved by constructing upper and lower solutions and applying Schauder's fixed point theorem.In Chapter 4,we summarize the whole work and present the prospect of our future work.