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 numberR0. 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 ifR0≤1, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. IfR0>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.
Highlights
Pine wilt disease (PWD) is caused by the pinewood nematode Bursaphelenchus xylophilus Nickle, which is vectored by the Japanese pine sawyer beetle Monochamus alternatus
The nonlinear incidence term αφShIV/(1 + mIV) denotes the rate at which the pine trees host Sh gets infected by infectious adult beetles IV(t) which do carry pinewood nematode at time t, and γIhSV/(1+nIh) refers to the rate at which the susceptible pine sawyers SV have pinewood nematode when it emerges in the infected pine trees Ih and m, n determine the level at which the force of infection saturates
We prove the global stability of the endemic equilibrium E∗, when the reproduction number R0 is greater than the unity
Summary
Pine wilt disease (PWD) is caused by the pinewood nematode Bursaphelenchus xylophilus Nickle, which is vectored by the Japanese pine sawyer beetle Monochamus alternatus. Since PWD was found in Japan, the pinewood nematode has spread to Korea, Taiwan, and China and has devastated pine forests in East Asia. Lee and Kim [4] introduced a model of a pine wilt disease with nonlinear incidence rate. Their model does not include an exposed class for the host population and falls within the susceptible-infected (SI) category of models. We propose a mathematical model with nonlinear incidence rates to describe the host-vector interaction between pines and pine sawyers carrying nematode by means of ordinary differential equation.
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