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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 219173, 11 pages
http://dx.doi.org/10.1155/2014/219173
Research Article

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

1Department of Mathematics, Pusan National University, 30 Geumjeong-Gu, Busan 609-735, Republic of Korea
2School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan

Received 29 June 2013; Revised 6 November 2013; Accepted 20 November 2013; Published 6 January 2014

Academic Editor: Elena Braverman

Copyright © 2014 Kwang Sung Lee and Abid Ali Lashari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . 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 , the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. If , 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.