IDEAS home Printed from https://ideas.repec.org/a/plo/pone00/0036160.html
   My bibliography  Save this article

Numerical Integration of the Master Equation in Some Models of Stochastic Epidemiology

Author

Listed:
  • Garrett Jenkinson
  • John Goutsias

Abstract

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation – up to a desired precision – in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.

Suggested Citation

  • Garrett Jenkinson & John Goutsias, 2012. "Numerical Integration of the Master Equation in Some Models of Stochastic Epidemiology," PLOS ONE, Public Library of Science, vol. 7(5), pages 1-9, May.
    • Handle: RePEc:plo:pone00:0036160
    DOI: 10.1371/journal.pone.0036160
    as

    Download full text from publisher

    File URL: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0036160
    Download Restriction: no
    File URL: https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0036160&type=printable
    Download Restriction: no
    File URL: https://libkey.io/10.1371/journal.pone.0036160?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Sidje, Roger B. & Stewart, William J., 1999. "A numerical study of large sparse matrix exponentials arising in Markov chains," Computational Statistics & Data Analysis, Elsevier, vol. 29(3), pages 345-368, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ankit Gupta & Corentin Briat & Mustafa Khammash, 2014. "A Scalable Computational Framework for Establishing Long-Term Behavior of Stochastic Reaction Networks," PLOS Computational Biology, Public Library of Science, vol. 10(6), pages 1-16, June.
    2. Chris Sherlock, 2021. "Direct statistical inference for finite Markov jump processes via the matrix exponential," Computational Statistics, Springer, vol. 36(4), pages 2863-2887, December.
    3. Bashkirtseva, Irina & Perevalova, Tatyana & Ryashko, Lev, 2022. "Regular and chaotic variability caused by random disturbances in a predator–prey system with disease in predator," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    4. Timothy Kinyanjui & Jo Middleton & Stefan Güttel & Jackie Cassell & Joshua Ross & Thomas House, 2018. "Scabies in residential care homes: Modelling, inference and interventions for well-connected population sub-units," PLOS Computational Biology, Public Library of Science, vol. 14(3), pages 1-24, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Igor Halperin & Andrey Itkin, 2013. "USLV: Unspanned Stochastic Local Volatility Model," Papers 1301.4442, arXiv.org, revised Mar 2013.
    2. Mercier, Sophie, 2008. "Bounds and approximations for continuous-time Markovian transition probabilities and large systems," European Journal of Operational Research, Elsevier, vol. 185(1), pages 216-234, February.
    3. Alexander Herbertsson, 2011. "Modelling default contagion using multivariate phase-type distributions," Review of Derivatives Research, Springer, vol. 14(1), pages 1-36, April.
    4. Vo, H.D. & Sidje, R.B., 2017. "Implementation of variable parameters in the Krylov-based finite state projection for solving the chemical master equation," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 334-344.
    5. Herbertsson, Alexander, 2007. "Modelling Default Contagion Using Multivariate Phase-Type Distributions," Working Papers in Economics 271, University of Gothenburg, Department of Economics.
    6. Suñé, Víctor & Carrasco, Juan Antonio, 2017. "Implicit ODE solvers with good local error control for the transient analysis of Markov models," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 96-111.
    7. Chris Sherlock, 2021. "Direct statistical inference for finite Markov jump processes via the matrix exponential," Computational Statistics, Springer, vol. 36(4), pages 2863-2887, December.
    8. Wu, Xiaoxia & Zhang, Lianzhu, 2019. "On structural properties of ABC-minimal chemical trees," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    9. Herbertsson, Alexander & Rootzén, Holger, 2007. "Pricing k-th-to-default Swaps under Default Contagion: The Matrix-Analytic Approach," Working Papers in Economics 269, University of Gothenburg, Department of Economics.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:plo:pone00:0036160. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: plosone (email available below). General contact details of provider: https://journals.plos.org/plosone/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.