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Traveling wave of a reaction–diffusion vector-borne disease model with nonlocal effects and distributed delay

Journal of Dynamics and Differential Equations (2021)Cite this article

Abstract

This paper is devoted to investigate the existence and nonexistence of traveling wave solution for a diffusive vector-borne disease model with nonlocal reaction and distributed delays. We demonstrate that the basic reproduction number \({\mathcal {R}}_0\) of the corresponding ordinary differential equation system as a threshold determines whether the model admits traveling waves or not and there exists a critical wave speed \(c_m^*>0\) when \({\mathcal {R}}_0>1\). Specifically, (i) As \({\mathcal {R}}_0>1\) and the wave speed \(c>c_m^*\), the existence of traveling waves for the system is established with the aid of a perturbed system; (ii) As \({\mathcal {R}}_0>1\) and \(0<c<c_m^*\), the nonexistence of traveling waves is proved via the two-sided Laplace transform; (iii) As \({\mathcal {R}}_0\le 1\) and \(c>0\), the nonexistence is obtained by utilizing the comparison principle. The theoretical results are applied to dengue fever epidemics. We study the effects of geographical movement, nonlocal interaction, incubation period and \({\mathcal {R}}_0\) on the threshold speed \(c_m^*\) for dengue fever.

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Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for helpful suggestions which improved the manuscript. The research is supported by Natural Science Foundation of China (No. 11971013); an NSERC Discovery Grant; Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0169); China Postdoctoral Science Foundation (No. 2021M691577).

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Kai Wang, Hongyong Zhao & Ran Zhang

  2. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China

    Kai Wang, Hongyong Zhao & Ran Zhang

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Hao Wang

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  1. Kai Wang
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Correspondence to Hongyong Zhao.

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This work is supported by Natural Science Foundation of China (No. 11971013); an NSERC Discovery Grant; Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0169); China Postdoctoral Science Foundation (No. 2021M691577)

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Wang, K., Zhao, H., Wang, H. et al. Traveling wave of a reaction–diffusion vector-borne disease model with nonlocal effects and distributed delay. J Dyn Diff Equat (2021). https://doi.org/10.1007/s10884-021-10062-w

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  • DOI: https://doi.org/10.1007/s10884-021-10062-w

Keywords

  • Traveling wave solution
  • Vector-borne disease model
  • Nonlocal reaction and distributed delay
  • Basic reproduction number
  • Critical wave speed

Mathematics Subject Classification

  • 35B40
  • 35C07
  • 35K57
  • 92D30