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This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed c ∗ and the basic reproduction number R 0 are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder's fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when ℛ 0 > 1 , for every speed c > c ∗ there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any 0 < c < c ∗ when ℛ 0 > 1 or c > 0 when ℛ 0 < 1. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of traveling waves of a virus infection model with humoral immunity, cell‐to‐cell transmission, and general nonlinear incidence. Our results show that the existence of traveling wave solutions is determined not only by the basic reproduction number R0$$ {\mathcal{R}}_0 $$ of virus infection and the antibody response reproduction number R1$$ {\mathcal{R}}_1 $$ but also by the critical wave speed c∗$$ {c}^{\ast } $$. More precisely, we obtain the existence of traveling wave solution connecting infection‐free equilibrium and antibody‐free infection equilibrium for R0>1,R1<1$$ {\mathcal{R}}_0>1,{\mathcal{R}}_1<1 $$, and c>c∗$$ c>{c}^{\ast } $$ and connecting infection‐free equilibrium and antibody‐present infection equilibrium for R0>1,R1>1$$ {\mathcal{R}}_0>1,{\mathcal{R}}_1>1 $$, and c>c∗$$ c>{c}^{\ast } $$. The existence of traveling wave solution connecting infection‐free equilibrium and antibody‐present infection equilibrium is discussed. Some numerical simulations are carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

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differential equation system, some traditional methods using only the theory of ordinary differential equations are no longer applicable.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]

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 R 0 and the minimal wave speed c ∗ . When 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 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. [ABSTRACT FROM AUTHOR]