## Abstract and Applied Analysis

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

#### 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 ${R}_{0}\le 1$. 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 ${R}_{0}\le 1$, 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

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

#### 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