Traveling waves in nonlocal dispersal SIR epidemic model with nonlinear incidence and distributed latent delay

Weixin Wu,Zhidong Teng

doi:10.1186/s13662-020-03073-2

Abstract

This paper studies the traveling waves in a nonlocal dispersal SIR epidemic model with nonlinear incidence and distributed latent delay. It is found that the traveling waves connecting the disease-free equilibrium with endemic equilibrium are determined by the basic reproduction number mathcal{R}_{0} and the minimal wave speed c^{*}. When mathcal{R}_{0}>1 and c>c^{*}, the existence of traveling waves is established by using the upper-lower solutions, auxiliary system, constructing the solution map, and then the fixed point theorem, limiting argument, diagonal extraction method, and Lyapunov functions. When mathcal{R}_{0}>1 and 0< c< c^{*}, the nonexistence result is also obtained by using the reduction to absurdity and the theory of asymptotic spreading.

Highlights

Mathematical models can be a powerful tool for designing strategies to control the spread of diseases
Function f (S, I) denotes the nonlinear incidence rate; the distributed latent delay term τ 0 h(s)f (S(x, t s), I(x, t s)) ds shows that the disease transmission has an incubation period, and the period of incubation is not constant
5 Existence of traveling waves we investigate the existence of traveling waves of system (5)

Summary

Introduction

Mathematical models can be a powerful tool for designing strategies to control the spread of diseases. Zhou et al [23] proposed the following nonlocal dispersal susceptible-infected-removed (SIR) epidemic model with nonlinear incidence f (S)g(I):. Zhang et al [34] established the following SIR epidemic model with nonlocal dispersal and nonlinear incidence:. In this paper, we propose the following nonlocal dispersal SIR epidemic model with the general nonlinear incidence f (S, I) and the distributed latent delay:. Function f (S, I) denotes the nonlinear incidence rate; the distributed latent delay term τ 0 h(s)f s), I(x, t s)) ds shows that the disease transmission has an incubation period, and the period of incubation is not constant. 5, the existence of traveling waves is established firstly for the auxiliary system, and for model (4) by using Schauder’s fixed point theorem, the limiting argument, and the diagonal extraction method.

Preliminary

Numerical examples

Findings

Conclusions