Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1381–1409 | Cite as

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

Article

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.

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 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical ScienceYangzhou UniversityYangzhouChina
  2. 2.Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and StatisticsYork UniversityTorontoCanada

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