# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021087
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## Critical traveling wave solutions for a vaccination model with general incidence

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

* Corresponding author: Yu Yang

Received  July 2020 Revised  January 2021 Early access March 2021

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 of traveling wave solutions for a vaccination model with general incidence. The existence or non-existence of traveling wave solutions for the model with specific incidence were proved recently when the wave speed is greater or smaller than a critical speed respectively. However, the existence of critical traveling wave solutions (with critical wave speed) was still open. In this paper, applying the Schauder's fixed point theorem via a pair of upper- and lower-solutions of the system, we show that the general vaccination model admits positive critical traveling wave solutions which connect the disease-free and endemic equilibria. Our result not only gives an affirmative answer to the open problem given in the previous specific work, but also to the model with general incidence. Furthermore, we extend our result to some nonlocal version of the considered model.

Citation: Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021087
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