Threshold Conditions for West Nile Virus Outbreaks

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. 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.