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Bulletin of Mathematical Biology

, Volume 71, Issue 3, pp 627–647 | Cite as

Threshold Conditions for West Nile Virus Outbreaks

  • Jifa Jiang
  • Zhipeng Qiu
  • Jianhong Wu
  • Huaiping Zhu
Original Article

Abstract

In this paper, we study the stability and saddle-node bifurcation of a model for the West Nile virus transmission dynamics. The existence and classification of the equilibria are presented. By the theory of K-competitive dynamical systems and index theory of dynamical systems on a surface, sufficient and necessary conditions for local stability of equilibria are obtained. We also study the saddle-node bifurcation of the system. Explicit subthreshold conditions in terms of parameters are obtained beyond the basic reproduction number which provides further guidelines for accessing control of the spread of the West Nile virus. Our results suggest that the basic reproductive number itself is not enough to describe whether West Nile virus will prevail or not and suggest that we should pay more attention to the initial state of West Nile virus. The results also partially explained the mechanism of the recurrence of the small scale endemic of the virus in North America.

Keywords

West Nile virus Differential equations Multiple equilibria and stability Backward bifurcation Threshold conditions Dynamics 

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Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  • Jifa Jiang
    • 1
  • Zhipeng Qiu
    • 2
  • Jianhong Wu
    • 3
  • Huaiping Zhu
    • 3
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiPeople’s Republic of China
  2. 2.Department of Applied MathematicsNanjing University of Science and TechnologyNanjingPeople’s Republic of China
  3. 3.Centre for Diseases Modelling, Laboratory of Mathematical Parallel Systems, Laboratory for Industrial and Applied Mathematics, Department of Mathematics and StatisticsYork UniversityTorontoCanada

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