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SIR epidemics with stochastic infectious periods

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  • Simon, Matthieu

Abstract

We consider an SIR model for the spread of an epidemic in a closed and homogeneously mixing population, where the infectious periods are represented by an arbitrary absorbing Markov process. A version of this process starts whenever an infection occurs, and the contamination rate of the newly infected individual is a function of its state. When his process is absorbed, the individual becomes a removed case. We use a martingale approach to derive the distribution of the final epidemic size and severity for this class of models. Next, we examine some special cases. In particular, we focus on situations where the infection processes are Brownian motions and where they are Markov-modulated fluid flows. In the latter case, we use matrix-analytic methods to provide more explicit results. We conclude with some numerical illustrations.

Suggested Citation

  • Simon, Matthieu, 2020. "SIR epidemics with stochastic infectious periods," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4252-4274.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:7:p:4252-4274
    DOI: 10.1016/j.spa.2019.12.003
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    References listed on IDEAS

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    1. Zhengjun Jiang & Martijn Pistorius, 2008. "On perpetual American put valuation and first-passage in a regime-switching model with jumps," Finance and Stochastics, Springer, vol. 12(3), pages 331-355, July.
    2. Z. Jiang & M. R. Pistorius, 2008. "On perpetual American put valuation and first-passage in a regime-switching model with jumps," Papers 0803.2302, arXiv.org.
    3. Ball, Frank & Donnelly, Peter, 1995. "Strong approximations for epidemic models," Stochastic Processes and their Applications, Elsevier, vol. 55(1), pages 1-21, January.
    4. Clancy, Damian, 2014. "SIR epidemic models with general infectious period distribution," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 1-5.
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    1. Barraza, Néstor Ruben & Pena, Gabriel & Moreno, Verónica, 2020. "A non-homogeneous Markov early epidemic growth dynamics model. Application to the SARS-CoV-2 pandemic," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).

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