Traveling waves for a diffusive SEIR epidemic model

Abstract

In this paper, we propose a diffusive SEIR epidemic model with saturating incidence rate. We first study the well posedness of the model, and give the explicit formula of the basic reproduction number R 0. And hence, we show that if R 0 > 1, then there exists a positive constant c * > 0 such that for each c > c * , the model admits a nontrivial traveling wave solution, and if R 0 ≤ 1 and c ≥ 0 (or, R 0 > 1 and c ∈ [0, c *)), then the model has no nontrivial traveling wave solutions. Consequently, we confirm that the constant c * is indeed the minimal wave speed. The proof of the main results is mainly based on Schauder fixed theorem and Laplace transform.

Wave propagation in a diffusive SEIR epidemic model with nonlocal reaction and standard incidence rate

“…On the other hand, there is no nonnegative and nontrivial traveling wave solution for system (7) to (9) if 0 < c < c * and R 0 = ∕ > 1, or R 0 = ∕ ≤ 1. We also refer to Tian 18 and Tian and Yuan, 18,19 and Xu, 20 and references therein for some relevant progress on the existence and nonexistence of traveling wave solutions of SEIR models with or without nonlocal interaction.…”

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

“…Traveling wave solution of spatial epidemic models represents the transition process of outbreak from the initial disease-free equilibrium to another disease-free equilibrium. Recently, it has been drawing much attention for infectious models, and there are many works using spatially dependent models to study the disease transmission (see other works, 2,7,10,11,15,16,22,25,[27][28][29][33][34][35][36][37][38][39][40][41][42][43][44][45]48,49,[51][52][53][54] for example). However, to the best of our knowledge, there is quite a few issues on the existence and nonexistence of traveling waves for epidemic models consisting of four equations.…”

Traveling Wave Solutions of a Diffusive SEIR Epidemic Model with Nonlinear Incidence Rate

“…For further developments, we refer to Tian and Yuan [27,29]. Xu [36] proposed a diffusive SEIR epidemic model with saturating incidence rate which has the form of Sg(I) and g is a continuous function with I and analyzed the existence and nonexistence of traveling wave solutions of the system by using the Schauder fixed point theorem and the Laplace transform. In addition, Xu [37] proposed a simple diffusive SEIR epidemic model where the total population is variable and considered the existence and nonexistence of traveling wave solutions connecting two equilibria which is determined by the basic reproduction number.…”