About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 792308, 9 pages
http://dx.doi.org/10.1155/2013/792308
Research Article

A SIRS Epidemic Model Incorporating Media Coverage with Random Perturbation

College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China

Received 21 June 2013; Accepted 5 September 2013

Academic Editor: Yong Ren

Copyright © 2013 Wenbin 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. Z. Ma, Y. Zhou, and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, China, 2009.
  2. 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
  3. 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 Scopus
  4. 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, 1985. View at Scopus
  5. 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 Scopus
  6. H. W. Hethcote, “Mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. View at Scopus
  7. 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 Scopus
  8. 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 Scopus
  9. Y. Xiao and S. Tang, “Dynamics of infection with nonlinear incidence in a simple vaccination model,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4154–4163, 2010. View at Publisher · View at Google Scholar · View at Scopus
  10. Y. Xiao, X. Xu, and S. Tang, “Sliding mode control of outbreaks of emerging infectious diseases,” Bulletin of Mathematical Biology, vol. 74, no. 10, pp. 2403–2422, 2012.
  11. J. Zhang, Z. Jin, G.-Q. Sun, T. Zhou, and S. Ruan, “Analysis of rabies in China: transmission dynamics and control,” PloS One, vol. 6, no. 7, article e20891, 2011. View at Scopus
  12. J. Cui, Y. Sun, and H. Zhu, “The impact of media on the control of infectious diseases,” Journal of Dynamics and Differential Equations, vol. 20, no. 1, pp. 31–53, 2008. View at Publisher · View at Google Scholar · View at Scopus
  13. J.-A. Cui, X. Tao, and H. Zhu, “An SIS infection model incorporating media coverage,” Rocky Mountain Journal of Mathematics, vol. 38, no. 5, pp. 1323–1334, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. R. Liu, J. Wu, and H. Zhu, “Media/psychological impact on multiple outbreaks of emerging infectious diseases,” Computational and Mathematical Methods in Medicine, vol. 8, no. 3, pp. 153–164, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. Y. Liu and J.-A. Cui, “The impact of media coverage on the dynamics of infectious disease,” International Journal of Biomathematics, vol. 1, no. 1, pp. 65–74, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. C. Sun, W. Yang, J. Arino, and K. Khan, “Effect of media-induced social distancing on disease transmission in a two patch setting,” Mathematical Biosciences, vol. 230, no. 2, pp. 87–95, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith, and C. T. Bauch, “The impact of media coverage on the transmission dynamics of human influenza,” BMC Public Health, vol. 11, no. 1, article S5, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. Y. Xiao, T. Zhao, and S. Tang, “Dynamics of an infectious diseases with media/psychology induced non-smooth incidence,” Mathematical Biosciences and Engineering, vol. 10, no. 2, pp. 445–461, 2013.
  19. 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
  20. L. J. Allen, “An introduction to stochastic epidemic models,” in Mathematical Epidemiology, F. Brauer, P. van den Driessche, and J. Wu, Eds., pp. 81–130, Springer, Berlin, Germany, 2008.
  21. T. C. Gard, Introduction to Stochastic Differential Equations, Dekker, New York, NY, USA, 1988.
  22. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 1995.
  23. X. Mao, Stochastic Differential Equations and Their Applications, Horwood, 1997.
  24. E. Beretta, V. Kolmanovskii, and L. Shaikhet, “Stability of epidemic model with time delays influenced by stochastic perturbations,” Mathematics and Computers in Simulation, vol. 45, no. 3-4, pp. 269–277, 1998. View at Scopus
  25. R. Z. Khasminskii and F. C. Klebaner, “Long term behavior of solutions of the lotka-volterra system under small random perturbations,” Annals of Applied Probability, vol. 11, no. 3, pp. 952–963, 2001. View at Publisher · View at Google Scholar · View at Scopus
  26. X. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and Their Applications, vol. 97, no. 1, pp. 95–110, 2002. View at Publisher · View at Google Scholar · View at Scopus
  27. 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 Scopus
  28. 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
  29. Q. Luo and X. Mao, “Stochastic population dynamics under regime switching,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 69–84, 2007. View at Publisher · View at Google Scholar · View at Scopus
  30. M. Liu and K. Wang, “Survival analysis of stochastic single-species population models in polluted environments,” Ecological Modelling, vol. 220, no. 9-10, pp. 1347–1357, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. 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 Scopus
  32. T. Britton, “Stochastic epidemic models: a survey,” Mathematical Biosciences, vol. 225, no. 1, pp. 24–35, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. F. Ball, D. Sirl, and P. Trapman, “Analysis of a stochastic SIR epidemic on a random network incorporating household structure,” Mathematical Biosciences, vol. 224, no. 2, pp. 53–73, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. 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 Scopus
  35. D. Jiang, J. Yu, C. Ji, and N. Shi, “Asymptotic behavior of global positive solution to a stochastic SIR model,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 221–232, 2011. View at Publisher · View at Google Scholar · View at Scopus
  36. J. Lv and K. Wang, “Asymptotic properties of a stochastic predator-prey system with Holling II functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4037–4048, 2011. View at Publisher · View at Google Scholar · View at Scopus
  37. 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 Scopus
  38. X. Wang, H. Huang, Y. Cai, and W. Wang, “The complex dynamics of a stochastic predatorprey model,” Abstract and Applied Analysis, vol. 2012, Article ID 401031, 24 pages, 2012. View at Publisher · View at Google Scholar
  39. Y. Cai, X. Wang, W. Wang, and M. Zhao, “Stochastic dynamics of a SIRS epidemic model with ratio-dependent incidence rate,” Abstract and Applied Analysis, vol. 2013, Article ID 172631, 11 pages, 2013. View at Publisher · View at Google Scholar
  40. Z. Liu, “Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates,” Nonlinear Analysis: Real World Applications, vol. 14, no. 3, pp. 1286–1299, 2013.
  41. J. P. LaSalle, “The stability of dynamical systems,” Society for Industrial and Applied Mathematics, vol. 25, 1987.
  42. A. M. Lyapunov, “The general problem of the stability of motion,” International Journal of Control, vol. 55, no. 3, pp. 531–534, 1992.
  43. R. Z. Khas’minskiĭ, Stochastic Stability of Differential Equations, vol. 7, Kluwer Academic Publishers, 1980.
  44. V. N. Afanasiev, V. B. Kolmanovskiĭ, and V. R. Nosov, Mathematical Theory of Control Systems Design, Kluwer Academic Publishers, 1996.
  45. 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 Scopus