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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 592821, 12 pages
http://dx.doi.org/10.1155/2013/592821
Research Article

Survival and Stationary Distribution in a Stochastic SIS Model

1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2Shanghai Medical Instrumentation College, Shanghai 200093, China

Received 10 October 2013; Accepted 12 November 2013

Academic Editor: Zhen Jin

Copyright © 2013 Yanli Zhou 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. 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
  2. W. M. Liu, S. A. Levin, and Y. Iwasa, “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,” Journal of Mathematical Biology, vol. 23, no. 2, pp. 187–204, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. W. Hethcote, “Qualitative analyses of communicable disease models,” Mathematical Biosciences, vol. 28, no. 3-4, pp. 335–356, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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 · View at MathSciNet
  5. 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
  6. 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
  7. Y. Jin, W. Wang, and S. Xiao, “An SIRS model with a nonlinear incidence rate,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1482–1497, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Yang and D. Xiao, “Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models,” Discrete and Continuous Dynamical Systems B, vol. 13, no. 1, pp. 195–211, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. Xu and Z. Ma, “Global stability of a delayed SEIRS epidemic model with saturation incidence rate,” Nonlinear Dynamics, vol. 61, no. 1-2, pp. 229–239, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Xu and Z. Ma, “Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3175–3189, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. W. Hethcote and P. van den Driessche, “Some epidemiological models with nonlinear incidence,” Journal of Mathematical Biology, vol. 29, no. 3, pp. 271–287, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Chen and T. Li, “Stability of stochastic delayed SIR model,” Stochastics and Dynamics, vol. 9, no. 2, pp. 231–252, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. Tornatore, S. M. Buccellato, and P. Vetro, “Stability of a stochastic SIR system,” Physica A, vol. 354, no. 1-4, pp. 111–126, 2005. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Ji, D. Jiang, Q. Yang, and N. Shi, “Dynamics of a multigroup SIR epidemic model with stochastic perturbation,” Automatica, vol. 48, no. 1, pp. 121–131, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. B. Frank and N. Peter, “A general model for stochastic SIR epidemics with two levels of mixing,” Mathematical Biosciences, vol. 180, pp. 73–102, 2002.
  16. J. Yu, D. Jiang, and N. Shi, “Global stability of two-group SIR model with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 235–244, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. I. Nåsell, “Stochastic models of some endemic infections,” Mathematical Biosciences, vol. 179, no. 1, pp. 1–19, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Q. Yang, D. Jiang, N. Shi, and C. Ji, “The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence,” Journal of Mathematical Analysis and Applications, vol. 388, no. 1, pp. 248–271, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Carletti, “Mean-square stability of a stochastic model for bacteriophage infection with time delays,” Mathematical Biosciences, vol. 210, no. 2, pp. 395–414, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Lahrouz and L. Omari, “Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence,” Statistics & Probability Letters, vol. 83, no. 4, pp. 960–968, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics,” Proceedings of the Royal Society A, vol. 115, pp. 700–721, 1927.
  22. D. Jiang, C. Ji, N. Shi, and J. Yu, “The long time behavior of DI SIR epidemic model with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 162–180, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. W. Y. Tan and X. Zhu, “A stochastic model of the HIV epidemic for heterosexual transmission involving married couples and prostitutes—I. The probabilities of HIV transmission and pair formation,” Mathematical and Computer Modelling, vol. 24, no. 11, pp. 47–107, 1996. View at Publisher · View at Google Scholar · View at Scopus
  24. N. Dalal, D. Greenhalgh, and X. Mao, “A stochastic model of AIDS and condom use,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 36–53, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. R. Beddington and R. M. May, “Harvesting natural populations in a randomly fluctuating environment,” Science, vol. 197, no. 4302, pp. 463–465, 1977. View at Scopus
  26. X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
  27. R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, 1980. View at MathSciNet
  28. T. C. Gard, Introduction to Stochastic Differential Equations, vol. 114, Marcel Dekker, New York, NY, USA, 1987. View at MathSciNet
  29. G. Strang, Linear Algebra and Its Applications, Thomson Learning, 1988.
  30. C. Zhu and G. Yin, “Asymptotic properties of hybrid diffusion systems,” SIAM Journal on Control and Optimization, vol. 46, no. 4, pp. 1155–1179, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. C. Ji and D. Jiang, “Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 441–453, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet