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Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary

Abstract

In this paper, a reaction–diffusion system is proposed to model the spatial spreading of West Nile virus in vector mosquitoes and host birds in North America. Transmission dynamics are based on a simplified model involving mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infected region. The spatial-temporal risk index \(R_0^F(t)\), which involves regional characteristic and time, is defined for the simplified reaction–diffusion model with the free boundary to compare with other related threshold values, including the usual basic reproduction number \(R_0\). Sufficient conditions for the virus to vanish or to spread are given. Our results suggest that the virus will be in a scenario of vanishing if \(R_0\le 1\), and will spread to the whole region if \(R_{0}^F(t_0)\ge 1\) for some \(t_0\ge 0\), while if \(R^F_0(0)<1<R_0\), the spreading or vanishing of the virus depends on the initial number of infected individuals, the area of the infected region, the diffusion rate and other factors. Moreover, some remarks on the basic reproduction numbers and the spreading speeds are presented and compared.

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Notes

  1. 1.

    Centers for Diseases Control and Prevention, West Nile Virus: Preliminary Maps & Data for 2014. http://www.cdc.gov/westnile/statsMaps/preliminaryMapsData/index.html.

  2. 2.

    Public Health Agency of Canada: Surveillance of West Nile virus. http://healthycanadians.gc.ca/diseases-conditions-maladies-affections/disease-maladie/west-nile-nil-occidental/surveillance-eng.php.

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Corresponding author

Correspondence to Huaiping Zhu.

Additional information

The work is partially supported by the NSFC of China (Grant Nos. 11371311 and 11171267), the High-End Talent Plan of Yangzhou University, and NSERC and CIHR of Canada.

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Cite this article

Lin, Z., Zhu, H. Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381–1409 (2017). https://doi.org/10.1007/s00285-017-1124-7

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Keywords

  • West Nile virus
  • Vector mosquitoes
  • Host birds
  • Spatial spreading
  • Reaction–diffusion systems
  • Free boundary
  • The basic reproduction number
  • Risk index
  • Spreading speeds

Mathematics Subject Classification

  • Primary 35K55
  • 35R35
  • Secondary 35B40
  • 92D30