Propagation Phenomena For An Epidemic Model With Nonlocal Dispersal

Reaction-diffusion equation,whose diffusion operator is classical Laplace operator,has attracted much attention since it can accurately describe such natural phenomena as atmospheric diffusion,the spread of epidemic diseases and Marine pollution.However,the nonlocal diffusion operator described by convolution is more helpful to describe the nonlocal interaction of the population in the space than the classical Laplace operator The traveling wave solutions as well as entire solutions,as special solutions of diffusion equations,can be good descriptions of several practical problemsIn this paper,we investigate traveling wave solutions and entire solutions of an epi-demic model by oral-fecal transmission with asymmetric kernel and bistable nonlinearity First,we consider the existence and asymptotic behavior of traveling wave solutions of the system.We take advantage of the abstract theory for the monotone dynamical system with weak compactness to establish the existence of bistable waves.Furthermore,we use the bilateral Laplace transform and Ikehara theorem to obtain the asymptotic behavior of traveling wave solutions which plays an important role in the proof of the existence of entire solutionsFinally,we explore the entire solutions of this system.By restricting the range of the variable,we can consider the interactions of the monostable and bistable traveling wave solutions.Because of the asymmetry of dispersal kernels,the corresponding nonincreasing and nondecreasing traveling wave solutions may be asymmetry and the minimal wave speeds of monostable waves may be non-positive.Thus,it makes the types of entire solutions more diverse.Under certain conditions on wave speeds,the existence of six types of entire solutions originating from three fronts are established by constructing some suitable pairs of super-and sub-solutions and applying corresponding comparison principle.We also gain their some properties.Furthermore,by introducing the definitions of terminated sequence and non-extendable sequence,we show that there exist no entire solutions originating N fronts if N?5.