Traveling waves for a nonlocal dispersal vaccination model with general incidence

Jinling Zhou; Yu Yang; Cheng-Hsiung Hsu

doi:10.3934/dcdsb.2019236

Article Contents

2020, Volume 25, Issue 4: 1469-1495. Doi: 10.3934/dcdsb.2019236

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Traveling waves for a nonlocal dispersal vaccination model with general incidence

1.
Department of Mathematics, Zhejiang International Studies University, Hangzhou 310023, China
2.
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
3.
Department of Mathematics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan

^* Corresponding author: Yu Yang
^* Corresponding author: Yu Yang

Received: October 31, 2018

Revised: February 28, 2019

Early access: November 2019

Published: April 2020

The third author was partially supported by the MOST (Grant No. 107-2115-M-008-009-MY3) and NCTS of Taiwan

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Abstract

This paper is concerned with the existence and asymptotic behavior of traveling wave solutions for a nonlocal dispersal vaccination model with general incidence. We first apply the Schauder's fixed point theorem to prove the existence of traveling wave solutions when the wave speed is greater than a critical speed $ c^* $. Then we investigate the boundary asymptotic behaviour of traveling wave solutions at $ +\infty $ by using an appropriate Lyapunov function. Applying the method of two-sided Laplace transform, we further prove the non-existence of traveling wave solutions when the wave speed is smaller than $ c^* $. From our work, one can see that the diffusion rate and nonlocal dispersal distance of the infected individuals can increase the critical speed $ c^* $, while vaccination reduces the critical speed $ c^* $. In addition, two specific examples are provided to verify the validity of our theoretical results, which cover and improve some known results.

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Mathematics Subject Classification: Primary: 37N25, 35K57; Secondary: 92B05.

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Figure 1. Graphs of $ ( \overline{S}(\xi), \overline{V}(\xi),\overline{I}(\xi)) $ and $ ( \underline{S}(\xi), \underline{V}(\xi),\underline{I}(\xi)) $

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