# American Institute of Mathematical Sciences

March  2022, 27(3): 1209-1225. doi: 10.3934/dcdsb.2021087

## 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 Published  March 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (3) : 1209-1225. doi: 10.3934/dcdsb.2021087
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##### References:
 [1] Y.-S. Chen and J.-S. Guo, Traveling wave solutions for a three-species predator-prey model with two aborigine preys, Japan J. Indust. Appl. Math., (2020). doi: 10.1007/s13160-020-00445-9. [2] A. Ducrot, J.-S. Guo, G. Lin and S. X. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), 25 pp. doi: 10.1007/s00033-019-1188-x. [3] S.-C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069. [4] J.-S. Guo, K. I. Nakamura, T. Ogiwara and C.-C. Wu, Traveling wave solutions for a predator-prey system with two predators and one prey, Nonlinear Anal. RWA, 54 (2020), 103111, 13pp. doi: 10.1016/j.nonrwa.2020.103111. [5] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [6] Y. Li, W.-T. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal., 14 (2015), 1001-1022.  doi: 10.3934/cpaa.2015.14.1001. [7] J. D. Wei, J. B. Zhou, Z. L. Zhen and L. X. Tian, Super-critical and critical traveling waves in a two-component lattice dynamical model with discrete delay, Appl. Math. Comput., 363 (2019), 124621. doi: 10.1016/j.amc.2019.124621. [8] J. D. Wei, J. B. Zhou, Z. L. Zhen and L. Tian, Super-critical and critical traveling waves in a three-component delayed disease system with mixed diffusion, J. Comput. Appl. Math., 367 (2020), 112451, 15pp. doi: 10.1016/j.cam.2019.112451. [9] J. D. Wei, J. B. Zhou, W. X. Chen, Z. L. Zhen and L. X. Tian, Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay, Commun. Pure. Appl. Anal., 19 (2020), 2853-2886.  doi: 10.3934/cpaa.2020125. [10] C. F. Wu, Y. Yang, Q. Y. Zhao, Y. L. Tian and Z. T. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068. [11] Z. T. Xu, Y. Q. Xu and Y. H. Huang, Stability and traveling waves of a vaccination model with nonlinear incidence, Comput. Math. Appl., 75 (2018), 561-581.  doi: 10.1016/j.camwa.2017.09.042. [12] L. Zhao and Z.-C. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033. [13] J. B. Zhou, L. Y. Song, J. D. Wei and H. M. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066. [14] J. B. Zhou, L. Y. Song and J. D. Wei, Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay, J. Differ. Equ., 268 (2020), 4491-4524.  doi: 10.1016/j.jde.2019.10.034. [15] J. L. Zhou, Y. Yang and C.-H. Hsu, Traveling waves for a nonlocal dispersal vaccination model with general incidence, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1469-1495.  doi: 10.3934/dcdsb.2019236.
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