Skip to main content

Threshold Conditions for West Nile Virus Outbreaks

Bulletin of Mathematical Biology volume 71, pages 627–647 (2009)Cite this 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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Price includes VAT (Finland)
Tax calculation will be finalised during checkout.

References

  • Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans. Oxford University Press, London.

    Google Scholar 

  • Bowman, C., Gumel, A.B., Wu, J., van den Driessche, P., Zhu, H., 2005. A mathematical model for assessing control strategies against West Nile virus. Bull. Math. Biol. 67(5), 1107–133.

    Article  MathSciNet  Google Scholar 

  • Brauer, F., Castillo-Chavez, C., 2000. Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics Series, vol. 40. Springer, New York.

    Google Scholar 

  • Campbell, L.G., Martin, A.A., Lanciotti, R.S., Gubler, D.J., 2002. West Nile virus. Lancet Infect. Dis. 2, 519–29.

    Article  Google Scholar 

  • Centers for Disease Control and Prevention, 2003. West Nile virus: fact sheet. http://www.cdc.gov/ncidod/dvbid/westnile/wnv_factSheet.htm.

  • Cruz-Pacheco, G., Esteva, L., Montaño-Hirose, J.A., Vargas, C., 2005. Modelling the dynamics of West Nile virus. Bull. Math. Biol. 67(6), 1157–172.

    Article  MathSciNet  Google Scholar 

  • Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology. Wiley, Chichester.

    Google Scholar 

  • Hayes, C.G., 2001. West Nile virus: Uganda, 1937, to New York City, 1999. Ann. N.Y. Acad. Sci. 951, 25–7.

    Article  Google Scholar 

  • Hethcote, H.W., 2002. The mathematics of infectious diseases. SIAM Rev. 42, 599–53.

    Article  MathSciNet  Google Scholar 

  • Kenkre, V.M., Parmenter, R.R., Peixoto, I.D., Sadasiv, L., 2005. A theoretical framework for the analysis of the West Nile virus epidemic. Math. Comput. Model. 42(3–4), 313–24.

    MATH  Article  MathSciNet  Google Scholar 

  • Komar, N., Langevin, S., Hinten, S., Nemeth, N., Edwards, E., et al., 2003. Experimental infection of North American birds with the New York 1999 strain of West Nile virus. Emerg. Infect. Dis. 9(3), 311–22.

    Google Scholar 

  • Lanciotti, R.S., Rohering, J.T., Deubel, V., Smith, J., Parker, M., et al., 2003. Origin of the West Nile virus responsible for an outbreak of Encephalitis in the Northeastern United States. Science 286, 2333–337.

    Article  Google Scholar 

  • Lewis, M., Rencławowicz, J., van den Driessche, P., 2006a. Traveling waves and spread rates for a West Nile virus model. Bull. Math. Biol. 68(1), 3–3.

    Article  MathSciNet  Google Scholar 

  • Lewis, M., Rencławowicz, J., van den Driessche, P., Wonham, M., 2006b. A comparison of continuous and discrete-time West Nile virus models. Bull. Math. Biol. 68(3), 491–09.

    Article  MathSciNet  Google Scholar 

  • Liu, R., Shuai, J., Wu, J., Zhu, H., 2006. Modeling spatial spread of West Nile virus and impact of directional dispersal of birds. Math. Biosci. Eng. 3(1), 145–60.

    MATH  MathSciNet  Google Scholar 

  • Lord, C.C., Day, J.F., 2001. Simulation studies of St. Louis encephalitis and West Nile virues: the impact of bird mortality. Vector Borne Zoonotic Dis. 1(4), 317–29.

    Article  Google Scholar 

  • Nosal, B., Pellizzari, R., 2003. West Nile virus. Can. Med. Assoc. J. 168(11), 1443–444.

    Google Scholar 

  • Smith, H.L., 1995. Monotone Dynamical Systems: An Introduction to Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., vol. 41. AMS, Providence.

    MATH  Google Scholar 

  • Takac, P., 1992. Domains of attraction of generic -limit sets for strongly monotone discrete-time semigroups. J. Reine Angew. Math. 423, 101–73.

    MathSciNet  Google Scholar 

  • Thomas, D.M., Urena, B., 2001. A model describing the evolution of West Nile-like Encephalitis in New York City. Math. Comput. Model. 34, 771–81.

    MATH  Article  MathSciNet  Google Scholar 

  • Wang, Y., Jiang, J., 2001. The general properties of discrete time competitive dynamical systems. J. Differ. Equ. 176, 470–93.

    MATH  Article  MathSciNet  Google Scholar 

  • Wonham, M.J., de-Camino-Beck, T., Lewis, M., 2004. An epidemiological model for West Nile virus: invasion analysis and control applications. Proc. R. Soc. Lond. Ser. B 1538, 501–07.

    Article  Google Scholar 

Download references

Author information

Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Jifa Jiang

  2. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Zhipeng Qiu

  3. Centre for Diseases Modelling, Laboratory of Mathematical Parallel Systems, Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

    Jianhong Wu & Huaiping Zhu

Authors
  1. Jifa Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhipeng Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianhong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Huaiping Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huaiping Zhu.

Additional information

Supported by the Chinese NSF grants 10531030 and 10671143.

Supported by the Chinese NSF grants 10801074.

Supported by Canada Research Chairs Program, Mathematics for Information Technology and Complex Systems (MITACS), National Microbiology Laboratory, Natural Sciences and Engineering Research Council (NSERC), Canadian Foundation of Innovation (CFI) and Ontario Innovation Trust (OIT), Ontario Ministry of Health and Long-term Care, Peel, Toronto, Chat-Kent Health Units, and Public Health Agency of Canada (PHAC).

Supported by NSERC, MITACS, CFI/OIT a new opportunity fund, Early Research Award of Ministry of Research and Innovation (ERA) of Ontario, Infectious Diseases Branch of Ministry of Health and Long Term Care (MOH) of Ontario and PHAC.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jiang, J., Qiu, Z., Wu, J. et al. Threshold Conditions for West Nile Virus Outbreaks. Bull. Math. Biol. 71, 627–647 (2009). https://doi.org/10.1007/s11538-008-9374-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-008-9374-6

Keywords

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