A state-dependent control against transmission of West Nile virus from mosquitoes to birds

In recent decades, West Nile virus (WNV) has become a substantial public health concern in many subtropical and tropical countries throughout the world. WNV is considered significant effects on host populations, but to date some effects have been difficult to measure. Based on a simple and easy operating statistical method, a simpler model describing the transmission of WNV between mosquitoes and birds is proposed. By the linearization method and Bendixson–Dulac theorem of differential equation, we obtain the basic reproductive number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0$$\end{document} for this disease, which illustrates the global asymptotical stability of the disease-free equilibrium and endemic equilibrium. Further, in order to control the spread of WNV, the model is extended to a control model, where state-dependent pulse control strategies are introduced. By the Poincaré map, differential inequality techniques and qualitative theory of ordinary differential equation, the existence and orbital stability of order-1 or order-2 periodic solutions for this control model are obtained. Finally, numerical simulations clearly illustrate the main theoretical results and feasibility of state-dependent pulse control strategies, and also discuss the roles of control parameters in the course of WNV control. Theoretical results and numerical simulation also show that the quantity of infected birds can be kept within a lower level. This provides a new control strategy to against the spread of WNV.