- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 592821, 12 pages
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.
- 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.
- 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.
- H. W. Hethcote, “Qualitative analyses of communicable disease models,” Mathematical Biosciences, vol. 28, no. 3-4, pp. 335–356, 1976.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- G. Chen and T. Li, “Stability of stochastic delayed SIR model,” Stochastics and Dynamics, vol. 9, no. 2, pp. 231–252, 2009.
- 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.
- 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.
- 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.
- 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.
- I. Nåsell, “Stochastic models of some endemic infections,” Mathematical Biosciences, vol. 179, no. 1, pp. 1–19, 2002.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J. R. Beddington and R. M. May, “Harvesting natural populations in a randomly fluctuating environment,” Science, vol. 197, no. 4302, pp. 463–465, 1977.
- X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
- R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, 1980.
- T. C. Gard, Introduction to Stochastic Differential Equations, vol. 114, Marcel Dekker, New York, NY, USA, 1987.
- G. Strang, Linear Algebra and Its Applications, Thomson Learning, 1988.
- 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.
- 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.
- D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.