Study On Solutions Of Two Types Of Diffusive Epidemic Models

In real life,infectious diseases usually spread at a certain speed among various groups of people in space.Therefore,diffusive partial differential equations are often used to describe the propagation laws and mechanisms of epidemics.It is of great practical significance to study this kind of infectious disease model.In the first part of this thesis,we investigate the existence and non-existence of traveling wave solutions in a class of epidemic models with nonlocal dispersal and nonlocal delay involving susceptible,asymptomatically infected,symptomatically infected and vaccinated(SAIV)individuals.Constructing a perturbed system,using Schauder’s fixed point theorem and the method of limiting argument,we prove that when the basic reproduction number R0>1,there is critical wave speed c*>0 such that for each c≥ c*,there is a nontrivial,bounded and positive travelling wave solution with wave speed c.Compared with the previous literature,the research strategy adopted in this part is more direct,and the existence results of traveling wave solutions have no constraints on the diffusion rate in the model.In addition,by virtue of two-sided Laplace transform and reduction to absurdity we prove that when R0>1 and 0<c<c*or R0 ≤1 and c ∈ R,this model has no nontrivial,bounded and nonnegative travelling wave solutions.In second part of this thesis,we investigate the asymptotic spreading speed of a diffusive SIR epidemic model with Holling-II incidence.Previous work in the literature has shown that there is a critical traveling wave speed in this model.On this basis,by using the comparison principle and the asymptotic spreading theory of the reaction-diffusion equation,we show that the minimum wave speed of the traveling wave solution in this model equals to the asymptotic spreading speed.