Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 571469, 15 pages
http://dx.doi.org/10.1155/2012/571469
Research Article

Global Stability of a SLIT TB Model with Staged Progression

Department of Mathematics, North University of China, Shanxi, Taiyuan 030051, China

Received 14 January 2012; Revised 28 July 2012; Accepted 16 August 2012

Academic Editor: Vu Phat

Copyright © 2012 Yakui Xue and Xiaohong Wang. 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. B. R. Bloom, Tuberculosis: Pathogenesis, Protection, and Control, ASM Press, Washington, DC, USA, 1994.
  2. H. Liu and L. Li, “A class age-structured HIV/AIDS model with impulsive drug-treatment strategy,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 758745, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. X. Wang and Y. Xue, “Global stability of a model with fast and slow Tuberculosis,” Submitted.
  4. J. M. Hyman, J. Li, and E. Ann Stanley, “The differential infectivity and staged progression models for the transmission of HIV,” Mathematical Biosciences, vol. 155, no. 2, pp. 77–109, 1999. View at Google Scholar · View at Scopus
  5. H. Guo and M. Y. Li, “Global dynamics of a staged progression model for infectious diseases,” Mathematical Biosciences and Engineering, vol. 3, no. 3, pp. 513–525, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. S. Koopman, J. A. Jacquez, G. W. Welch et al., “The role of early HIV infection in the spread of HIV through populations,” Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology, vol. 14, no. 3, pp. 249–258, 1997. View at Google Scholar · View at Scopus
  7. A. L. Lloyd, “Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods,” Proceedings of the Royal Society B, vol. 268, no. 1470, pp. 985–993, 2001. View at Publisher · View at Google Scholar · View at Scopus
  8. A. L. Lloyd, “Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics,” Theoretical Population Biology, vol. 60, no. 1, pp. 59–71, 2001. View at Publisher · View at Google Scholar · View at Scopus
  9. W. Y. Tan and Z. Ye, “Estimation of HIV infection and incubation via state space models,” Mathematical Biosciences, vol. 167, no. 1, pp. 31–50, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. A. Kaddar, “Stability analysis in a delayed SIR epidemic model with a saturated incidence rate,” Nonlinear Analysis. Modelling and Control, vol. 15, no. 3, pp. 299–306, 2010. View at Google Scholar · View at Zentralblatt MATH
  11. M. De la Sen, “About the properties of a modified generalized Beverton-Holt equation in ecology models,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 592950, p. 23, 2008. View at Google Scholar
  12. M. De la Sen, “The generalized Beverton-Holt equation and the control of populations,” Applied Mathematical Modelling, vol. 32, no. 11, pp. 2312–2328, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. De la Sen and S. Alonso-Quesada, “A control theory point of view on Beverton-Holt equation in population dynamics and some of its generalizations,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 464–481, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. De La Sen and S. Alonso-Quesada, “Model-matching-based control of the Beverton-Holt equation in ecology,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 793512, 21 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. M. De la Sen and S. Alonso-Quesada, “Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: non-adaptive and adaptive cases,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2616–2633, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. De La Sen and S. Alonso-Quesada, “On vaccination control tools for a general SEIRepidemic model,” in 18th IEEE Mediterranean Conference on Control and Automation (MED '10), pp. 1322–1328, June 2010. View at Scopus
  17. M. De la Sen and S. Alonso-Quesada, “Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3888–3904, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. De la Sen, A. Ibeas, S. Alonso-Quesada, and R. Nistal, “On the equilibrium points, boundedness and positivity of a SVEIRS epidemic model under constant regular constrained vaccination,” Informatica, vol. 22, no. 3, pp. 339–370, 2011. View at Google Scholar
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar
  21. A. Korobeinikov and G. C. Wake, “Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,” Applied Mathematics Letters, vol. 15, no. 8, pp. 955–960, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. Korobeinikov, “Lyapunov functions and global properties for SEIR and SEIS epidemic models,” Mathematical Medicine and Biology, vol. 21, no. 2, pp. 75–83, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. Korobeinikov, “Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,” Bulletin of Mathematical Biology, vol. 71, no. 1, pp. 75–83, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. A. Korobeinikov, “Global properties of basic virus dynamics models,” Bulletin of Mathematical Biology, vol. 66, no. 4, pp. 879–883, 2004. View at Publisher · View at Google Scholar
  25. P. Georgescu and Y.-H. Hsieh, “Global stability for a virus dynamics model with nonlinear incidence of infection and removal,” SIAM Journal on Applied Mathematics, vol. 67, no. 2, pp. 337–353, 2006. View at Publisher · View at Google Scholar
  26. 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
  27. A. Korobeinikov, “Global properties of infectious disease models with nonlinear incidence,” Bulletin of Mathematical Biology, vol. 69, no. 6, pp. 1871–1886, 2007. View at Publisher · View at Google Scholar
  28. A. Korobeinikov, “Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,” Bulletin of Mathematical Biology, vol. 68, no. 3, pp. 615–626, 2006. View at Publisher · View at Google Scholar
  29. J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, NY, USA, 1969.
  30. E. A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff, Groningen, The Netherlands, 1970.
  31. J. La Salle and S. Lefschetz, Stability by Liapunovs Direct Method, Academic Press, New York, NY, USA, 1961.