Abstract
We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R_{0}=partial _{I}F(S_{0},0)/gamma of the corresponding ordinary differential equations and the minimal wave speed c^{*}. The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals.
Highlights
In this paper, we consider the following diffusive SEIR model with nonlocal delayed transmission and a general nonlinear incidence rate: t +∞∂tS(x, t) = d1∂xxS(x, t) – F S(x, t), K(x – y, t – s)I(y, s) dy ds, –∞ –∞∂tE(x, t) = d2∂xxE(x, t) + F S(x, t), K(x – y, t – s)I(y, s) dy ds – αE(x, t), (1.1) (1.2)∂tI(x, t) = d3∂xxI(x, t) + αE(x, t) – γ I(x, t), (1.3)∂tR(x, t) = d4∂xxR(x, t) + γ I(x, t), (1.4)
Traveling wave solution is an important tool used in the study of the spreading speed of infectious diseases; see [3, 11, 15]
By using the Schauder fixed point theorem, the authors proved the existence of traveling wave solution of this system when d1 = 1
Summary
We consider the following diffusive SEIR model with nonlocal delayed transmission and a general nonlinear incidence rate:. Traveling wave solution is an important tool used in the study of the spreading speed of infectious diseases; see [3, 11, 15]. Y, t s)I(y, s) dy ds shows the effects of spatial heterogeneity (geographical movement), nonlocal interaction, and time delay such as latent period on the transmission of diseases They proved the existence and nonexistence of traveling wave solutions for system (1.11)–(1.12) by Schauder’s fixed point theorem and Laplace transform. By using the Schauder fixed point theorem, the authors proved the existence of traveling wave solution of this system when d1 = 1. We construct an invariant convex closed set, apply the Schauder fixed point theorem to prove the existence of traveling wave solutions for an auxiliary system, and extend the result to the original system by a limiting argument. The inverse operator Di–1 is given by the integral representation
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call