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Traveling wave solutions in a two-group SIR epidemic model with constant recruitment

Journal of Mathematical Biology volume 77, pages 1871–1915 (2018)Cite this article

Abstract

Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves of a two-group SIR epidemic model with time delay and constant recruitment and show that the existence of traveling waves is determined by the basic reproduction number \(R_{0}.\) More specifically, we prove that (i) when the basic reproduction number \(R_{0}>1,\) there exists a minimal wave speed \(c^*>0,\) such that for each \(c \ge c^*\) the system admits a nontrivial traveling wave solution with wave speed c and for \(c<c^*\) there exists no nontrivial traveling wave satisfying the system; (ii) when \(R_{0} \le 1,\) the system admits no nontrivial traveling waves. Finally, we present some numerical simulations to show the existence of traveling waves of the system.

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Notes

  1. The term “external supplies” was first used by Ducrot and Magal (2011) to describe the situation that the host population is recruited at a constant rate from outside of the compartment. Several authors followed them to use this terminology (see, for example, Li et al. 2014). After discussing with the authors of Ducrot and Magal (2011), we all agreed that “constant recruitment” is a more appropriate term to describe the phenomenon.

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Acknowledgements

The authors are grateful to the two anonymous reviewers and the handling editor (Professor Klaus Dietz) for their helpful comments and suggestions which helped us in improving the paper.

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Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Lin Zhao & Zhi-Cheng Wang

  2. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People’s Republic of China

    Lin Zhao

  3. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Shigui Ruan

Authors
  1. Lin Zhao
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  2. Zhi-Cheng Wang
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  3. Shigui Ruan
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Corresponding author

Correspondence to Shigui Ruan.

Additional information

Dedicated to the memory of Professor Karl Hadeler.

Zhi-Cheng Wang: Research was partially supported by NNSF of China (11371179). Shigui Ruan: Research was partially supported by NSF (DMS-1412454).

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Zhao, L., Wang, ZC. & Ruan, S. Traveling wave solutions in a two-group SIR epidemic model with constant recruitment. J. Math. Biol. 77, 1871–1915 (2018). https://doi.org/10.1007/s00285-018-1227-9

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  • DOI: https://doi.org/10.1007/s00285-018-1227-9

Keywords

  • Two-group epidemic model
  • Basic reproduction number
  • Time delay
  • Constant recruitment
  • Traveling wave solutions

Mathematics Subject Classification

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