Propagation Phenomena In A Reaction-Diffusion SIR Epidemic Model With Non-Monotone Incidence

The thesis is devoted to study the asymptotic speed of spread and traveling wave solutions for a time-periodic reaction-diffusion SIR epidemic model with non-monotone incidence rate,the non-autonomous and non-quasi-monotone structure of such model systems make it difficult to study spatial propagation dynamics.Firstly,based on the basic reproduction number R₀ and the minimum wave speed c*of spatial homogeneous system,asymptotic speed of this model is established.More precisely,if R₀≤1,the solution of the system converges to the disease-free equilibrium（disease extinction）as t→∞ and if R₀>1,the disease is persistent behind the front and extinct ahead the front.Secondly,in this paper the existence of periodic traveling wave solutions is reduced to fixed point problem by constructing non-monotone operator on the closed convex set of appropriate periodic function.Then,the fixed point is found by using Schauder’s fixed point theorem,and then the existence of periodic traveling wave solutions as c>c*.In addition,when R₀ ≤1 or R₀>1 and 0<c<c*,the wave solutions of the two circumstances does not exist.