Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

Kuilin Wu,Kai Zhou

doi:10.3390/math7070641

Abstract

In this paper, we study the traveling wave solutions for a nonlocal dispersal SIR epidemic model with standard incidence rate and nonlocal delayed transmission. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding reaction system and the minimal wave speed. To prove these results, we apply the Schauder’s fixed point theorem and two-sided Laplace transform. The main difficulties are that the complexity of the incidence rate in the epidemic model and the lack of regularity for nonlocal dispersal operator.

Highlights

Due to the important significance in modeling the disease transmission, traveling wave solution has been intensively researched in many epidemic models, such as the SIR epidemic models and their various extensions
Li et al [20] have studied diffusive SIR epidemic model with standard incidence rate, which is different from system (2) that they considered the effect of nonlocal delayed transmission
We have studied the existence and nonexistence of nontrivial traveling wave solutions for system (5)

Summary

Introduction

Due to the important significance in modeling the disease transmission, traveling wave solution has been intensively researched in many epidemic models, such as the SIR epidemic models and their various extensions. Wang and Wu [2] considered the existence and nonexistence of traveling wave solution for a diffusive Kermack–McKendrick epidemic model with nonlocal delayed transmission. The traveling wave solution for diffusive SIR models with a standard incidence rate has attracted more and more attention. Li et al [20] have studied diffusive SIR epidemic model with standard incidence rate, which is different from system (2) that they considered the effect of nonlocal delayed transmission. Zhen, Wei, Tian et al [22] have considered the following diffusive SIR epidemic model with standard incidence rate and spatiotemporal delay. Many researchers pay more attention to the study of traveling wave solutions of nonlocal dispersal SIR epidemic models, for instance, in [28,29,30,31,32,33].

Some Preliminaries

Existence of Traveling Waves

Asymptotic Behavior

Nonexistence of Traveling Waves

Conclusions and Discussion