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Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

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  • Can Chen
  • Yanmei Kang

Abstract

We introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and its pth moment exponential stability, which indicates that the pth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.

Suggested Citation

  • Can Chen & Yanmei Kang, 2014. "Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:723825
    DOI: 10.1155/2014/723825
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    References listed on IDEAS

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    1. Yanan Zhao & Daqing Jiang, 2013. "Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, September.
    2. Yanan Zhao & Daqing Jiang, 2013. "Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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    4. Lin, Yuguo & Jiang, Daqing & Wang, Shuai, 2014. "Stationary distribution of a stochastic SIS epidemic model with vaccination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 187-197.
    5. Tornatore, Elisabetta & Maria Buccellato, Stefania & Vetro, Pasquale, 2005. "Stability of a stochastic SIR system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 111-126.
    6. Ji, Chunyan & Jiang, Daqing & Shi, Ningzhong, 2011. "Multigroup SIR epidemic model with stochastic perturbation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(10), pages 1747-1762.
    7. Zhao, Yanan & Jiang, Daqing & O’Regan, Donal, 2013. "The extinction and persistence of the stochastic SIS epidemic model with vaccination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(20), pages 4916-4927.
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