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SIS Epidemic Model Birth-and-Death Markov Chain Approach

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  • A.H. Nzokem

Abstract

We are interested in describing the dynamics of the infected size of the SIS Epidemic model using the Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size M; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter α; the disease spreads within the population is modelled by a Poisson process with a rate λ_I. λ_I=βI(1-I/M) is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number (R=β/α) is greater than one. As methodology, we use both numerical and analytical approaches. The numerical approach shows that the infected size dynamics converge to a stationary stochastic process. And the analytical results determine the distribution of the stationary stochastic process as a normal distribution with mean (1-1/R)M and Variance M/R when M becomes larger.

Suggested Citation

  • A.H. Nzokem, 2021. "SIS Epidemic Model Birth-and-Death Markov Chain Approach," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(4), pages 1-10, July.
    • Handle: RePEc:ibn:ijspjl:v:10:y:2021:i:4:p:10
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    References listed on IDEAS

    as
    1. Damian Clancy & Sang Taphou Mendy, 2011. "Approximating the Quasi-stationary Distribution of the SIS Model for Endemic Infection," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 603-618, September.
    2. Wierman, John C. & Marchette, David J., 2004. "Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction," Computational Statistics & Data Analysis, Elsevier, vol. 45(1), pages 3-23, February.
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    Citations

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    Cited by:

    1. A. H. Nzokem, 2022. "Pricing European Options under Stochastic Volatility Models: Case of five-Parameter Variance-Gamma Process," Papers 2201.03378, arXiv.org, revised Jan 2023.
    2. A. H. Nzokem, 2023. "Bitcoin versus S&P 500 Index: Return and Risk Analysis," Papers 2310.02436, arXiv.org.
    3. A. H. Nzokem, 2023. "European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis," Papers 2304.06060, arXiv.org, revised Aug 2023.
    4. Xie, Meiling & Li, Yuhan & Feng, Minyu & Kurths, Jürgen, 2023. "Contact-dependent infection and mobility in the metapopulation SIR model from a birth–death process perspective," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    5. Javier Cifuentes-Faura & Ursula Faura-Martínez & Matilde Lafuente-Lechuga, 2022. "Mathematical Modeling and the Use of Network Models as Epidemiological Tools," Mathematics, MDPI, vol. 10(18), pages 1-14, September.

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    JEL classification:

    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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