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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 131257, 7 pages
http://dx.doi.org/10.1155/2014/131257
Research Article

An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate

Key Laboratory of Eco-Environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 26 December 2013; Accepted 6 January 2014; Published 20 February 2014

Academic Editor: Weiming Wang

Copyright © 2014 Mingming Li and Xianning Liu. 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. M. Gabriela, M. Gomes, L. J. White, and G. F. Medley, “The reinfection threshold,” Journal of Theoretical Biology, vol. 236, no. 1, pp. 111–113, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  2. Y. Zhou and H. Liu, “Stability of periodic solutions for an SIS model with pulse vaccination,” Mathematical and Computer Modelling, vol. 38, no. 3-4, pp. 299–308, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, “A stochastic differential equation SIS epidemic model,” SIAM Journal on Applied Mathematics, vol. 71, no. 3, pp. 876–902, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Song, W. Ma, and Y. Takeuchi, “Permanence of a delayed SIR epidemic model with density dependent birth rate,” Journal of Computational and Applied Mathematics, vol. 201, no. 2, pp. 389–394, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. W. Ma, M. Song, and Y. Takeuchi, “Global stability of an SIR epidemic model with time delay,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1141–1145, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. Zhang, Z. Li, and F. Zhang, “Global stability of an SIR epidemic model with constant infectious period,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 285–291, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. Liu, Y. Takeuchi, and S. Iwami, “SVIR epidemic models with vaccination strategies,” Journal of Theoretical Biology, vol. 253, no. 1, pp. 1–11, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Wang, J. Zhang, and Z. Jin, “Analysis of an SIR model with bilinear incidence rate,” Nonlinear Analysis, vol. 11, no. 4, pp. 2390–2402, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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 · View at MathSciNet
  10. S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, and A. V. Pokrovskii, “Lyapunov functions for SIR and SIRS epidemic models,” Applied Mathematics Letters, vol. 23, no. 4, pp. 446–448, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. Liu, X. Q. Zhao, and Y. Zhou, “A tuberculosis model with seasonality,” Bulletin of Mathematical Biology, vol. 72, no. 4, pp. 931–952, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. N. Yoshida and T. Hara, “Global stability of a delayed SIR epidemic model with density dependent birth and death rates,” Journal of Computational and Applied Mathematics, vol. 201, no. 2, pp. 339–347, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. M. Anderson and R. M. May, “Regulation and stability of host-parasite population interactions. I. Regulatory processes,” Journal of Animal Ecology, vol. 47, no. 1, pp. 219–267, 1978.
  14. C. Wei and L. Chen, “A delayed epidemic model with pulse vaccination,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 746951, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. Zhang, Z. Jin, Q. Liu, and Z. Zhang, “Analysis of a delayed SIR model with nonlinear incidence rate,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 636153, 16 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. Jiang and J. Wei, “Stability and bifurcation analysis in a delayed SIR model,” Chaos, Solitons and Fractals, vol. 35, no. 3, pp. 609–619, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. R. Xu and Z. Ma, “Stability of a delayed SIRS epidemic model with a nonlinear incidence rate,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2319–2325, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. R. Xu, Z. Ma, and Z. Wang, “Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3211–3221, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. Xu and Z. Ma, “Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,” Nonlinear Analysis, vol. 10, no. 5, pp. 3175–3189, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433–443, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. C. C. McCluskey, “Global stability for an SIR epidemic model with delay and nonlinear incidence,” Nonlinear Analysis, vol. 11, no. 4, pp. 3106–3109, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Kaddar, “On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate,” Electronic Journal of Differential Equations, vol. 2009, no. 133, pp. 1–7, 2009. View at Zentralblatt MATH · View at MathSciNet
  23. A. Kaddar, A. Abta, and H. T. Alaoui, “A comparison of delayed SIR and SEIR epidemic models,” Nonlinear Analysis, vol. 16, no. 2, pp. 181–190, 2011. View at Zentralblatt MATH · View at MathSciNet
  24. A. Abta, A. Kaddar, and H. T. Alaoui, “Global stability for delay SIR and SEIR epidemic models with saturated incidence rates,” Electronic Journal of Differential Equations, vol. 2012, no. 23, pp. 1–13, 2012. View at Zentralblatt MATH · View at MathSciNet
  25. Z. Liu, “Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates,” Nonlinear Analysis, vol. 14, no. 3, pp. 1286–1299, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. Korobeinikov and K. Philip Maini, “Nonliear incidence and stability of infectious diseasemodels,” Mathematical Medicine and Biology, vol. 22, pp. 113–128, 2005.
  27. 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 · View at MathSciNet
  28. 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 · View at Zentralblatt MATH · View at MathSciNet
  29. A. Korobeinikov, “Stability of ecosystem: global properties of a general predator-prey model,” Mathematical Medicine and Biology, vol. 26, no. 4, pp. 309–321, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. G. Huang, Y. Takeuchi, W. Ma, and D. Wei, “Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,” Bulletin of Mathematical Biology, vol. 72, no. 5, pp. 1192–1207, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, NY, USA, 1993. View at MathSciNet
  32. S. F. Ellermeyer, “Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth,” SIAM Journal on Applied Mathematics, vol. 54, no. 2, pp. 456–465, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. F. Ellermeyer, J. Hendrix, and N. Ghoochan, “A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria,” Journal of Theoretical Biology, vol. 222, no. 4, pp. 485–494, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  34. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, NY, USA, 2011. View at MathSciNet