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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 293747, 31 pages
http://dx.doi.org/10.1155/2010/293747
Research Article

Stability of a Two-Strain Tuberculosis Model with General Contact Rate

1Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
2Department of Pediatrics, First Hospital of Lanzhou University, Lanzhou, Gansu 730000, China

Received 1 November 2010; Accepted 31 December 2010

Academic Editor: D. Anderson

Copyright © 2010 Hai-Feng Huo et al. 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. C. Dye, S. Scheele, P. Dolin, V. Pathania, and M. C. Raviglione, “Global burden of tuberculosis: estimated incidence, prevalence, and mortality by country,” Journal of the American Medical Association, vol. 282, no. 7, pp. 677–686, 1999. View at Publisher · View at Google Scholar
  2. T. R. Frieden, T. R. Sterling, S. S. Munsiff, C. J. Watt, and C. Dye, “Tuberculosis,” Lancet, vol. 362, no. 9387, pp. 887–899, 2003. View at Publisher · View at Google Scholar
  3. B. Song, C. Castillo-Chavez, and J. P. Aparicio, “Tuberculosis models with fast and slow dynamics: the role of close and casual contacts,” Mathematical Biosciences, vol. 180, pp. 187–205, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. B. Miller, “Preventive therapy for tuberculosis,” Medical Clinics of North America, vol. 77, no. 6, pp. 1263–1275, 1993. View at Google Scholar
  5. P. Rodrigues, M. G. M. Gomes, and C. Rebelo, “Drug resistance in tuberculosis-a reinfection model,” Theoretical Population Biology, vol. 71, no. 2, pp. 196–212, 2007. View at Publisher · View at Google Scholar
  6. M. A. Espinal et al., “Standard short-course chemotherapy for drug-resistant tuberculosis: treatment outcomes in 6 countries,” Journal of the American Medical Association, vol. 283, no. 19, pp. 2537–2545, 2000. View at Google Scholar
  7. H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. F. Brauer and C. Castillo-Chávez, Mathematical models in population biology and epidemiology, vol. 40 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2001.
  9. B. M. Murphy, B. H. Singer, S. Anderson, and D. Kirschner, “Comparing epidemic tuberculosis in demographically distinct heterogeneous populations,” Mathematical Biosciences, vol. 180, pp. 161–185, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361–404, 2004. View at Google Scholar · View at Zentralblatt MATH
  11. B. M. Murphy, B. H. Singer, and D. Kirschner, “On treatment of tuberculosis in heterogeneous populations,” Journal of Theoretical Biology, vol. 223, no. 4, pp. 391–404, 2003. View at Publisher · View at Google Scholar
  12. C. Castillo-Chavez and Z. Feng, “Global stability of an age-structure model for TB and its applications to optimal vaccination strategies,” Mathematical Biosciences, vol. 151, no. 2, pp. 135–154, 1998. View at Publisher · View at Google Scholar
  13. C. C. McCluskey, “Lyapunov functions for tuberculosis models with fast and slow progression,” Mathematical Biosciences and Engineering, vol. 3, no. 4, pp. 603–614, 2006. View at Google Scholar · View at Zentralblatt MATH
  14. C. Castillo-Chavez and Z. Feng, “To treat or not to treat: the case of tuberculosis,” Journal of Mathematical Biology, vol. 35, no. 6, pp. 629–656, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Bowong, “Optimal control of the transmission dynamics of tuberculosis,” Nonlinear Dynamics, vol. 61, no. 4, pp. 729–748, 2010. View at Publisher · View at Google Scholar
  16. S. Bowong and J. J. Tewa, “Mathematical analysis of a tuberculosis model with differential infectivity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 4010–4021, 2009. View at Publisher · View at Google Scholar
  17. H. McCallum, N. Barlow, and J. Hone, “How should pathogen transmission be modelled?” Trends in Ecology and Evolution, vol. 16, no. 6, pp. 295–300, 2001. View at Publisher · View at Google Scholar
  18. 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
  19. D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419–429, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Z. Yuan and L. Wang, “Global stability of epidemiological models with group mixing and nonlinear incidence rates,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 995–1004, 2010. View at Publisher · View at Google Scholar
  21. Y. Tang, D. Huang, S. Ruan, and W. Zhang, “Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate,” SIAM Journal on Applied Mathematics, vol. 69, no. 2, pp. 621–639, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. 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
  23. H. R. Thieme and C. Castillo-Chavez, “On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic,” in Mathematical and Statistical Approaches to AIDS Epidemiology, vol. 83 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1989. View at Google Scholar
  24. J. Zhang and Z. Ma, “Global dynamics of an SEIR epidemic model with saturating contact rate,” Mathematical Biosciences, vol. 185, no. 1, pp. 15–32, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. A. P. Heesterbeek and J. A. J. Metz, “The saturating contact rate in marriage- and epidemic models,” Journal of Mathematical Biology, vol. 31, no. 5, pp. 529–539, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. S. Bowong and J. J. Tewa, “Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3621–3631, 2010. View at Publisher · View at Google Scholar
  27. 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
  28. C. Castillo-Chavez and H. R. Thieme, “Asymptotically autonomous epidemic models,” in Mathematical Population Dynamics: Analysis of Heterogeneity, O. Arino et al., Ed., vol. 1 of Theory of Epidemics, pp. 33–50, Wuetz, 1995. View at Google Scholar
  29. K. Mischaikow, H. Smith, and H. R. Thieme, “Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1669–1685, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. H. R. Thieme, “Persistence under relaxed point-dissipativity (with application to an endemic model),” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 407–435, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, Pa, USA, 1976, With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics.
  32. J. P. LaSalle, “Stability theory for ordinary differential equations,” Journal of Differential Equations, vol. 4, pp. 57–65, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. A. Korobeinikov and P. K. Maini, “A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,” Mathematical Biosciences and Engineering, vol. 1, no. 1, pp. 57–60, 2004. View at Google Scholar · View at Zentralblatt MATH