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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 705601, 8 pages
http://dx.doi.org/10.1155/2013/705601
Research Article

Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model

Department of Mathematics and Sciences, Hebei Institute of Architecture and Civil Engineering, Zhangjiakou, Hebei 075000, China

Received 5 January 2013; Accepted 8 April 2013

Academic Editor: Qingdu Li

Copyright © 2013 Junhong Li and Ning Cui. 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. L. Cai, X. Li, M. Ghosh, and B. Guo, “Stability analysis of an HIV/AIDS epidemic model with treatment,” Journal of Computational and Applied Mathematics, vol. 229, no. 1, pp. 313–323, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Y.-H. Hsieh, C.-C. King, C. W. S. Chen, M.-S. Ho, S.-B. Hsu, and Y.-C. Wu, “Impact of quarantine on the 2003 SARS outbreak: a retrospective modeling study,” Journal of Theoretical Biology, vol. 244, no. 4, pp. 729–736, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. Iwami, Y. Takeuchi, X. Liu, and S. Nakaoka, “A geographical spread of vaccine-resistance in avian influenza epidemics,” Journal of Theoretical Biology, vol. 259, no. 2, pp. 219–228, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  4. D. Summers, J. G. Cranford, and B. P. Healey, “Chaos in periodically forced discrete-time ecosystem models,” Chaos, Solitons and Fractals, vol. 11, no. 14, pp. 2331–2342, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Vandermeer, L. Stone, and B. Blasius, “Categories of chaos and fractal basin boundaries in forced predator-prey models,” Chaos, Solitons and Fractals, vol. 12, no. 2, pp. 265–276, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. Gray, D. Greenhalgh, X. Mao, and J. Pan, “The SIS epidemic model with Markovian switching,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 496–516, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. 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 MathSciNet
  8. J. Liu and T. Zhang, “Bifurcation analysis of an SIS epidemic model with nonlinear birth rate,” Chaos, Solitons & Fractals, vol. 40, no. 3, pp. 1091–1099, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. Gao and S. Ruan, “An SIS patch model with variable transmission coefficients,” Mathematical Biosciences, vol. 232, no. 2, pp. 110–115, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y. Li and J. Cui, “The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2353–2365, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Q. Wu and X. Fu, “Modelling of discrete-time SIS models with awareness interactions on degree-uncorrelated networks,” Physica A, vol. 390, no. 3, pp. 463–470, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. P. L. Salceanu and H. L. Smith, “Persistence in a discrete-time stage-structured fungal disease model,” Journal of Biological Dynamics, vol. 3, no. 2-3, pp. 271–285, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. E. Franke and A.-A. Yakubu, “Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models,” Journal of Mathematical Biology, vol. 57, no. 6, pp. 755–790, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. Li, Z. Ma, and F. Brauer, “Global analysis of discrete-time SI and SIS epidemic models,” Mathematical Biosciences and Engineering, vol. 4, no. 4, pp. 699–710, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. L. J. S. Allen, Y. Lou, and A. L. Nevai, “Spatial patterns in a discrete-time SIS patch model,” Journal of Mathematical Biology, vol. 58, no. 3, pp. 339–375, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. L. Li, G.-Q. Sun, and Z. Jin, “Bifurcation and chaos in an epidemic model with nonlinear incidence rates,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1226–1234, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer, New York, NY, USA, 1983. View at MathSciNet
  18. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2, Springer, New York, NY, USA, 1990. View at MathSciNet
  19. T. Chen, J. He, and Q. Yin, “Dynamics evolution of credit risk contagion in the CRT market,” Dynamics in Nature and Society, vol. 2013, Article ID 206201, 9 pages, 2013. View at Publisher · View at Google Scholar
  20. Q. Li, L. Zhang, and F. Yang, “An algorithm to automatically detect the Smale horseshoes,” Discrete Dynamics in Nature and Society, Article ID 283179, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Q. Li and X.-S. Yang, “A simple method for finding topological horseshoes,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 2, pp. 467–478, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X.-S. Yang, “Topological horseshoes and computer assisted verification of chaotic dynamics,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 4, pp. 1127–1145, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. L. Qingdu and T. Song, “Algorithm for finding horseshoes in three-dimensinal hyperchaltic maps and its application,” Acta Physica Sinica, vol. 62, no. 2, Article ID 020510, 2013.
  24. M. P. Hassell, H. N. Comins, and R. M. May, “Spatial structure and chaos in insect population dynamics,” Nature, vol. 353, pp. 255–258, 1991.
  25. A. A. Berryman and J. A. Millstein, “Are ecological systems chaotic and if not, why not?” Trends in Ecology & Evolution, vol. 4, no. 1, pp. 26–28, 1989.