带有扩散项和接种的传染病模型的行波解

数学物理学报 ›› 2017, Vol. 37 ›› Issue (6): 1129-1147.

带有扩散项和接种的传染病模型的行波解

郭庭光, 徐志庭

华南师范大学数学科学学院广州 510631

收稿日期:2017-01-18 修回日期:2017-05-21 出版日期:2017-12-26 发布日期:2017-12-26
通讯作者: 徐志庭,E-mail:xuzhit@126.com E-mail:xuzhit@126.com
作者简介:郭庭光,E-mail:27539970@qq.com
基金资助:
广东省自然科学基金（2016A030313426）和华南师范大学高水平大学建设经费（2016YN30）

Existence of Traveling Waves in a Spatial Infectious Disease Model with Vaccination

Guo Tingguang, Xu Zhiting

School of Mathematical Sciences, South China Normal University, Guangzhou 510631

Received:2017-01-18 Revised:2017-05-21 Online:2017-12-26 Published:2017-12-26
Supported by:
Supported by the Natural Science Foundation of Guangdong Province (2016A030313426) and the HLUCF of South China Normal University (2016YN30)

摘要/Abstract

摘要： 该文研究带有扩散项和接种的传染病模型的行波解存在性.首先建立一个带扩散项和接种的具有空间结构的传染病模型，并给出其解适定性.其次，构造一对向量型上、下解，应用Schauder不动点原理和Lyapunov函数方法得到此模型存在连接无病平衡点和有病平衡点的非平凡正行波解.利用稳定流形定理，得到行波指数衰减估计，进而，通过拉普拉斯变换，确定该模型行波解的不存在性.该文的研究技巧对建立高维非合作反应扩散系统行波解存在性提供了有效方法.

关键词: 行波解, 传染病模型, Schauder不动点原理, 向量型上、下解, 拉普拉斯变换

Abstract: The current paper is devoted to investigate the existence of traveling waves in a spatial infectious disease model with vaccination. First, we propose a spatial infectious disease model with vaccination, and then study the well-posedness of it. Second, based on constructing a pair of the vector-value upper and lower solutions and the applications of Schauder's fixed point theorem, we show that the model admits nontrivial and positive traveling waves connecting the disease free equilibrium and the endemic equilibrium. Third, by Laplace transforms, we establish the non-existence of traveling waves for the model, in which the prior estimate of the exponential decay of the traveling wave solutions is obtained by the Stable Manifold Theorem. The approach in this paper provides an effective method to deal with more general high dimensional non-cooperative reaction-diffusion systems.

Key words: Traveling waves, Infectious disease model, Vector-value upper and lower solutions, Schauder's fixed point theorem, Laplace transform

中图分类号:

O175.14

郭庭光, 徐志庭. 带有扩散项和接种的传染病模型的行波解[J]. 数学物理学报, 2017, 37(6): 1129-1147.

Guo Tingguang, Xu Zhiting. Existence of Traveling Waves in a Spatial Infectious Disease Model with Vaccination[J]. Acta mathematica scientia,Series A, 2017, 37(6): 1129-1147.

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