Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate

Abstract

Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction number $R_0$. Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that if $R_0\le 1$, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. If $R_0>1$, a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.