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In this paper, the periodic traveling wave solution for a reaction–diffusion SIR epidemic model with demography and time-periodic coefficients is investigated. Because the traveling wave system of non-autonomous reaction–diffusion model is a partial differential equation system, some traditional methods using only the theory of ordinary differential equations are no longer applicable. To overcome these difficulties, the traditional methods are extended and improved, and some new techniques are introduced. The research results show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R 0 and the minimal wave speed c ∗ . Specifically, when R 0 > 1 and the wave speed c > c ∗ the existence of periodic traveling wave solutions is proved by means of auxiliary system, upper–lower solutions, fixed-point theorems and some limit arguments. Otherwise, when R 0 < 1 , for any wave speed c > 0 the nonexistence of periodic traveling wave solutions is proved. Lastly, the numerical examples are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the periodic traveling wave solutions in a nonautonomous reaction-diffusion susceptible-infected-removed epidemic model with general nonlinear incidence and time-periodic environment. The basic reproduction number R 0 and the critical wave speed c* are defined. By the fixed-point theorem and upper–lower solutions, the sufficient conditions for the existence of traveling waves satisfying some asymptotic boundary conditions are deduced, and the nonexistence of periodic traveling waves is also obtained. Numerical simulations are carried out to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the traveling wave in a nonlocal dispersal susceptible-infected-removed epidemic model with general nonlinear incidence and nonlocal delayed effects. It is shown that the existence and nonexistence of nontrivial traveling waves are fully determined by the basic reproduction number R 0 and critical wave speed c*. When R 0 > 1 and c > c*, the existence of traveling waves is obtained by means of an auxiliary system, the methods of upper-lower solutions, Schauder's fixed point theorem, and some limiting techniques. When R 0 > 1 and 0 < c < c*, the nonexistence of traveling waves is established by the reduction to absurdity and the theory of asymptotic spreading. [ABSTRACT FROM AUTHOR]