# American Institute of Mathematical Sciences

April  2020, 25(4): 1469-1495. doi: 10.3934/dcdsb.2019236

## 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

Received  November 2018 Revised  March 2019 Published  November 2019

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

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.

Citation: Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
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Graphs of $( \overline{S}(\xi), \overline{V}(\xi),\overline{I}(\xi))$ and $( \underline{S}(\xi), \underline{V}(\xi),\underline{I}(\xi))$