Traveling Wave Fronts In A Vector-Disease Model With Nonlocal Dispersal And Spatio-temporal Delay

This paper is concerned with the existence and stability of the traveling wave-fronts of a vector disease model with spatio-temporal delay and nonlocal dispersal.Firstly, we consider the existence of traveling wavefronts for a class of integro-differential equations with a more general nonlinear reaction term. The main meth-ods are to transform the existence of traveling wavefront into the existence of a fixed point of the corresponding integral operator, and then apply the Schauder's fixed point theorem and upper-lower solutions techniques. Under the assumption that the nonlinear reaction term satisfies the so-called quasimonotone condition, we get the existence of traveling wavefronts by a pair of suitable upper-lower solutions. As an application. we establish the existence of traveling wavefronts of a vector-disease model.Then, we study the stability of the traveling wavefronts of a vector disease model with spatio-temporal delay and nonlocal dispersal when the time-delay term is local with respect to the time variable. By investigating the corresponding initial value problem, and then using the comparison principle and the squeezing technique, we establish the stability of the traveling wavefronts of a vector-disease model under the suitable assumptions on initial function.