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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 172631, 11 pages
http://dx.doi.org/10.1155/2013/172631
Research Article

Stochastic Dynamics of an SIRS Epidemic Model with Ratio-Dependent Incidence Rate

1College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
2School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China
3College of Life and Environmental Science, Wenzhou University, Wenzhou 325035, China

Received 21 April 2013; Revised 23 May 2013; Accepted 23 May 2013

Academic Editor: Mark A. McKibben

Copyright © 2013 Yongli Cai 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. W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics—I,” Bulletin of Mathematical Biology, vol. 53, no. 1-2, pp. 33–55, 1991. View at Publisher · View at Google Scholar · View at Scopus
  2. Z. Ma, Y. Zhou, and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, China, 2009.
  3. C. Bain, “Applied mathematical ecology,” Journal of Epidemiology and Community Health, vol. 44, no. 3, p. 254, 1990.
  4. 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
  5. J. Ma and Z. Ma, “Epidemic threshold conditions for seasonally forced SEIR models,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 161–172, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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
  7. V. Capasso, Mathematical Structures of Epidemic Systems, vol. 97, Springer, Berlin, Germany, 2008. View at MathSciNet
  8. 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
  9. 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
  10. J. A. Jacquez and P. O'Neill, “Reproduction numbers and thresholds in stochastic epidemic models I. Homogeneous populations,” Mathematical Biosciences, vol. 107, no. 2, pp. 161–186, 1991. View at Publisher · View at Google Scholar · View at Scopus
  11. W. R. Derrick and P. van den Driessche, “A disease transmission model in a nonconstant population,” Journal of Mathematical Biology, vol. 31, no. 5, pp. 495–512, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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 · View at MathSciNet
  13. H. W. Hethcote and S. A. Levin, “Periodicity in epidemiological models,” Applied Mathematical Ecology, vol. 18, pp. 193–211, 1989.
  14. M. E. Alexander and S. M. Moghadas, “Periodicity in an epidemic model with a generalized non-linear incidence,” Mathematical Biosciences, vol. 189, no. 1, pp. 75–96, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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
  16. 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
  17. X.-Z. Li, W.-S. Li, and M. Ghosh, “Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 141–150, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Yuan and B. Li, “Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 609306, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Li, S. Yuan, and W. Zhang, “Analysis on an epidemic model with a ratio-dependent nonlinear incidence rate,” International Journal of Biomathematics, vol. 4, no. 2, pp. 227–239, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. 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
  21. L. J. S. Allen, “An introduction to stochastic epidemic models,” in Mathematical Epidemiology, F. Brauer, P. van den Driessche, and J. Wu, Eds., vol. 1945, pp. 81–130, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. T. C. Gard, Introduction to Stochastic Differential Equations, vol. 114, Marcel Dekker, New York, NY, USA, 1988. View at MathSciNet
  23. B. Øksendal, Stochastic Differential Equations, Springer, Berlin, Germany, 4th edition, 1995. View at MathSciNet
  24. X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited,, Chichester, UK, 1997. View at MathSciNet
  25. R. Z. Khasminskii and F. C. Klebaner, “Long term behavior of solutions of the Lotka-Volterra system under small random perturbations,” The Annals of Applied Probability, vol. 11, no. 3, pp. 952–963, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 Zentralblatt MATH · View at MathSciNet
  27. M. Bandyopadhyay and J. Chattopadhyay, “Ratio-dependent predator-prey model: effect of environmental fluctuation and stability,” Nonlinearity, vol. 18, no. 2, pp. 913–936, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. D. Jiang and N. Shi, “A note on nonautonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 164–172, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 Zentralblatt MATH · View at MathSciNet
  30. S. Chatterjee, M. Isaia, F. Bona, G. Badino, and E. Venturino, “Modelling environmental influences on wanderer spiders in the Langhe region (Piemonte-NW Italy),” Journal of Numerical Analysis, Industrial and Applied Mathematics, vol. 3, no. 3-4, pp. 193–209, 2008. View at Zentralblatt MATH · View at MathSciNet
  31. X. Li and X. Mao, “Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation,” Discrete and Continuous Dynamical Systems Series A, vol. 24, no. 2, pp. 523–545, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. 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
  33. 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 Zentralblatt MATH · View at MathSciNet
  34. X. Wang, H. Huang, Y. Cai, and W. Wang, “The complex dynamics of a stochastic predator-prey model,” Abstract and Applied Analysis, vol. 2012, Article ID 401031, 24 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. M. Carletti, “On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment,” Mathematical Biosciences, vol. 175, no. 2, pp. 117–131, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. 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
  38. 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
  39. 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
  40. N. Dalal, D. Greenhalgh, and X. Mao, “A stochastic model for internal HIV dynamics,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1084–1101, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. 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
  42. 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 Zentralblatt MATH · View at MathSciNet
  43. 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 Zentralblatt MATH · View at MathSciNet
  44. 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
  45. 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 Zentralblatt MATH · View at MathSciNet
  46. 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
  47. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. L. Imhof and S. Walcher, “Exclusion and persistence in deterministic and stochastic chemostat models,” Journal of Differential Equations, vol. 217, no. 1, pp. 26–53, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. 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
  50. V. N. Afanasiev, V. B. Kolmanovskii, and V. R. Nosov, Mathematical Theory of Control Systems Design, vol. 341, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1996. View at MathSciNet