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Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching

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  • Greenhalgh, D.
  • Liang, Y.
  • Mao, X.

Abstract

We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible–infectious–recovered–susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence. We then explain how the results can be generalised to a finite-state MC. The results for the SIR (Susceptible–Infectious–Removed) model with Markovian Switching (MS) are a special case. Numerical simulations are produced to confirm our theoretical results.

Suggested Citation

  • Greenhalgh, D. & Liang, Y. & Mao, X., 2016. "Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 684-704.
    • Handle: RePEc:eee:phsmap:v:462:y:2016:i:c:p:684-704
    DOI: 10.1016/j.physa.2016.06.125
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    References listed on IDEAS

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    1. Vargas-De-León, Cruz, 2011. "On the global stability of SIS, SIR and SIRS epidemic models with standard incidence," Chaos, Solitons & Fractals, Elsevier, vol. 44(12), pages 1106-1110.
    2. Lu, Qiuying, 2009. "Stability of SIRS system with random perturbations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(18), pages 3677-3686.
    3. 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.
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    Cited by:

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    8. Guo, Xiaoxia & Luo, Jiaowan, 2018. "Stationary distribution and extinction of SIR model with nonlinear incident rate under Markovian switching," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 471-481.

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