Abstract and Applied Analysis

The Stability of SI Epidemic Model in Complex Networks with Stochastic Perturbation

Jinqing Zhao, Maoxing Liu, Wanwan Wang, and Panzu Yang

Full-text: Open access

Abstract

We investigate a stochastic SI epidemic model in the complex networks. We show that this model has a unique global positive solution. Then we consider the asymptotic behavior of the model around the disease-free equilibrium and show that the solution will oscillate around the disease-free equilibrium of deterministic system when R 0 1 . Furthermore, we derive that the disease will be persistent when R 0 > 1 . Finally, a series of numerical simulations are presented to illustrate our mathematical findings. A new result is given such that, when R 0 1 , with the increase of noise intensity the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 610959, 14 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607553

Digital Object Identifier
doi:10.1155/2014/610959

Mathematical Reviews number (MathSciNet)
MR3182294

Citation

Zhao, Jinqing; Liu, Maoxing; Wang, Wanwan; Yang, Panzu. The Stability of SI Epidemic Model in Complex Networks with Stochastic Perturbation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 610959, 14 pages. doi:10.1155/2014/610959. https://projecteuclid.org/euclid.aaa/1412607553


Export citation

References

  • N. T. Bailey, The Mathematical Theory of Infectious Disease, Hafner Press, New York, NY, USA, 2nd edition, 1975.
  • R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, Oxford, UK, 1992.
  • O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio \emphR$_{0}$ in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990.
  • O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, John Wiley & Sons, Chichester, UK, 2000, Model building, analysis and interpretation.
  • H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000.
  • J. Wang, M. Liu, and Y. Li, “Analysis of epidemic models with demographics in metapopulation networks,” Physica A, vol. 392, no. 7, pp. 1621–1630, 2013.
  • S. Eubank, H. Guclu, V. S. A. Kumar et al., “Modelling disease outbreaks in realistic urban social networks,” Nature, vol. 429, no. 6988, pp. 180–184, 2004.
  • M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, “Dynamical patterns of epidemic outbreaks in complex heterogeneous networks,” Journal of Theoretical Biology, vol. 235, no. 2, pp. 275–288, 2005.
  • L. Wang and G.-Z. Dai, “Global stability of virus spreading in complex heterogeneous networks,” SIAM Journal on Applied Mathematics, vol. 68, no. 5, pp. 1495–1502, 2008.
  • L. Mao-Xing and R. Jiong, “Modelling the spread of sexually transmitted diseases on scale-free networks,” Chinese Physics B, vol. 18, no. 6, pp. 2115–2120, 2009.
  • R. Pastor-Satorras and A. Vespignani, “Epidemic spreading in scale-free networks,” Physical Review Letters, vol. 86, no. 14, pp. 3200–3203, 2001.
  • R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics in finite size scale-free networks,” Physical Review E, vol. 65, no. 3, Article ID 035108, pp. 035108/1–035108/4, 2002.
  • J.-p. Zhang and Z. Jin, “The analysis of an epidemic model on networks,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7053–7064, 2011.
  • J. Z. Liu, Y. F. Tang, and Z. R. Yang, “The spread of disease with birth and death on networks,” Journal of Statistical Mechanics, vol. 2004, Article ID P08008, 2004.
  • X. Fu, M. Small, D. M. Walker, and H. Zhang, “Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization,” Physical Review E, vol. 77, no. 3, pp. 036113/1–036113/8, 2008.
  • T. Zhou, J.-G. Liu, W.-J. Bai, G. Chen, and B.-H. Wang, “Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity,” Physical Review E, vol. 74, no. 5, pp. 056109/1–056109/6, 2006.
  • M.-X. Liu and J. Ruan, “A stochastic epidemic model on homogeneous networks,” Chinese Physics B, vol. 18, no. 12, pp. 5111–5116, 2009.
  • C. Q. Xu, S. L. Yuan, and T. H. Zhang, “Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate,” Abstract and Applied Analysis, vol. 2013, Article ID 423154, 11 pages, 2013.
  • 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.
  • C. Ji, D. Jiang, and N. Shi, “Multigroup SIR epidemic model with stochastic perturbation,” Physica A, vol. 390, no. 10, pp. 1747–1762, 2011.
  • C. Yuan, D. Jiang, D. O'Regan, and R. P. Agarwal, “Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2501–2516, 2012.
  • 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.
  • P. J. Witbooi, “Stability of an SEIR epidemic model with independent stochastic perturbations,” Physica A, vol. 392, no. 20, pp. 4928–4936, 2013.
  • Y. Zhao, D. Jiang, and D. O'Regan, “The extinction and persistence of the stochastic SIS epidemic model with vaccination,” Physica A, vol. 392, no. 20, pp. 4916–4927, 2013.
  • G. Hu, M. Liu, and K. Wang, “The asymptotic behaviours of an epidemic model with two correlated stochastic perturbations,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10520–10532, 2012.
  • 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.
  • L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley Press, New York, NY, USA, 1974.
  • X. Mao, Stochastic Differential Equations and Applications, Horwood Press, Chichester, UK, 1997.
  • 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.
  • 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.
  • D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001. \endinput