Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:505

Article

Adimy, Mostafa¹  Chekroun, Abdennasser²  Kuniya, Toshikazu³ 

[1] Univ Lyon 1, Univ Lyon, INRIA, CNRS,UMR 5208,Inst Camille Jordan, 43 Bd 11 Novembre 1918, F-69200 Villeurbanne, France

[2] Univ Tlemcen, Lab Anal Nonlineaire & Math Appl, Tilimsen 13000, Algeria

[3] Kobe Univ, Grad Sch Syst Informat, Nada Ku, 1-1 Rokkodai Cho, Kobe, Hyogo 6578501, Japan

关键词: Age-space-structured PDF;    Reaction-diffusion system with nonlocal term;    Continuous difference equation;    Traveling wave solution;    SIR epidemic model;   

DOI  :  10.1016/j.jmaa.2021.125464

来源: Elsevier

【 摘 要 】

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

【 授权许可】

Free