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Abstract and Applied Analysis
Volume 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.

Linked References

  1. F. Takasu, “Individual-based modeling of the spread of pine wilt disease: vector beetle dispersal and the Allee effect,” Population Ecology, vol. 51, no. 3, pp. 399–409, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. M. M. Mota, H. Braasch, M. A. Bravo et al., “First report of Bursaphelenchus xylophilus in Portugal and in Europe,” Nematology, vol. 1, no. 7-8, pp. 727–734, 1999. View at Google Scholar · View at Scopus
  3. B. G. Zhao, K. Futai, J. R. Sutherland, and Y. Takeuchi, Pine Wilt Disease, Springer, New York, NY, USA, 2008.
  4. K. S. Lee and D. Kim, “Global dynamics of a pine wilt disease transmission model with nonlinear incidence rates,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4561–4569, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Shi and G. Song, “Analysis of the mathematical model for the spread of pine wilt disease,” Journal of Applied Mathematics, vol. 2013, Article ID 184054, 10 pages, 2013. View at Publisher · View at Google Scholar
  6. V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1-2, pp. 43–61, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. Cai, S. Guo, X. Li, and M. Ghosh, “Global dynamics of a dengue epidemic mathematical model,” Chaos, Solitons & Fractals, vol. 42, no. 4, pp. 2297–2304, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3531–3541, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–380, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Ozair, A. A. Lashari, I. H. Jung, and K. O. Okosun, “Stability analysis and optimal control of a vector-borne disease with nonlinear incidence,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 595487, 21 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. Esteva and C. Vargas, “A model for dengue disease with variable human population,” Journal of Mathematical Biology, vol. 38, no. 3, pp. 220–240, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Journal of Differential Equations, vol. 188, pp. 135–163, 2003. View at Google Scholar
  13. S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135–163, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. Korobeinikov and P. K. Maini, “Non-linear incidence and stability of infectious disease models,” Mathematical Medicine and Biology, vol. 22, no. 2, pp. 113–128, 2005. View at Publisher · View at Google Scholar · View at Scopus
  15. B. Buonomo and S. Rionero, “On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 4010–4016, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070–1083, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, vol. 125 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989. View at MathSciNet
  18. H. R. Thieme, “Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,” Journal of Mathematical Biology, vol. 30, no. 7, pp. 755–763, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002, John A. Jacquez memorial volume. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. L. J. S. Allen, An Introduction to Mathematical Biology, Pearson Education, Upper Saddle River, NJ, USA, 2007.
  21. J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, Pa, USA, 1976. View at MathSciNet
  22. J. S. Muldowney, “Compound matrices and ordinary differential equations,” The Rocky Mountain Journal of Mathematics, vol. 20, no. 4, pp. 857–872, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Y. Li and J. S. Muldowney, “Global stability for the SEIR model in epidemiology,” Mathematical Biosciences, vol. 125, no. 2, pp. 155–164, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. C. C. McCluskey and P. van den Driessche, “Global analysis of two tuberculosis models,” Journal of Dynamics and Differential Equations, vol. 16, no. 1, pp. 139–166, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Fonda, “Uniformly persistent semidynamical systems,” Proceedings of the American Mathematical Society, vol. 104, no. 1, pp. 111–116, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. B. Buonomo and C. Vargas-De-León, “Stability and bifurcation analysis of a vector-bias model of malaria transmission,” Mathematical Biosciences, vol. 242, no. 1, pp. 59–67, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. Tumwiine, J. Y. T. Mugisha, and L. S. Luboobi, “A host-vector model for malaria with infective immigrants,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 139–149, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet