The Complete Classification for Dynamics in a Nine-Dimensional West Nile Virus Model

Abstract

Bowman et al. [Bull. Math. Biol., 67 (2005), pp. 1107–1133] proposed a nine-dimensional system of ordinary differential equations modelling West Nile virus in a mosquito–bird-human community and presented some mathematical analysis and its biological explanation. Jiang et al. [Bull. Math. Biol., 2008, DOI 10.1007/s11538-008-9374-6] continued to study the existence and classification of all equilibria and all their local stability and dealt with saddle-node bifurcation of the system. The previous investigation shows that the unique positive equilibrium is globally asymptotically stable if the basic reproduction number is greater than one and the bird death rate is suitably small, but numerical simulations suggest that the unique endemic equilibrium is globally asymptotically stable even if for a large value of the bird death rate. So, they all leave it an open problem. The present paper is to provide a thorough classification of dynamics for this system. In particular, if the reproduction number is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the feasible region, and the disease persists at an endemic equilibrium if it initially exists, which completely solves the open problem above. Besides, the sufficient and necessary conditions for switch phenomena of the model are obtained if the reproduction number is smaller than one. The results show that the reproduction number alone is not enough to determine whether West Nile virus can prevail or not. Meanwhile, the dynamics of the model for the critical case where the reproduction number is one is also analyzed.