Traveling Wave Solutions Of A Nonlocal Delayed SIR Model Without Outbreak Threshold

Tliis paper is concerned with the existence, nonexistence and asymptotic be-havior of traveling wave solutions of a diffusive SIR system with nonlocal delay, which formulate the propagation of disease without outbreak threshold. Moreover, it is proved that at any fixed moment, the faster the disease spreads, the more the individual infects, and the larger the recovery/remove ratio is, the less the individual infects.We firstly study the existence of traveling wave solution for SIR model with nonlocal delays. Through defining a bounded and closed convex set on a suitable Banach space and constructing a completely continuous operator, we can obtain the existence of traveling waves by Schauder’s fixed theorem.We further study the asymptotic behavior of traveling waves by the theory of asymptotic-spreading and monotony, proving the asymptotic behavior of any nontrivial bounded positive traveling wave solution. These conclusions show that the disease portrayed by the model can break out without threshold, which means if only the infective exsits, all individuals in the area will be infected when the basic reproduction ratio is larger than1.Finally, we study the nonexistence of traveling waves by comparison principle and two sided Laplace transform. It is shown that the disease can not spread if the basic reproduction ratio is smaller than l;and the spread speed will be no less than a constant decided by the infectious rate if the basic reproduction ratio is larger than1.