The relationships between message passing, pairwise, Kermack–McKendrick and stochastic SIR epidemic models View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2017-12

AUTHORS

Robert R. Wilkinson, Frank G. Ball, Kieran J. Sharkey

ABSTRACT

We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time [Formula: see text]. We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack-McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time [Formula: see text] than the Kermack-McKendrick model. More... »

PAGES

1563-1590

References to SciGraph publications

  • 2017-02. Dangerous connections: on binding site models of infectious disease dynamics in JOURNAL OF MATHEMATICAL BIOLOGY
  • 2001. Algebraic Graph Theory in NONE
  • 2012-09. A Note on the Derivation of Epidemic Final Sizes in BULLETIN OF MATHEMATICAL BIOLOGY
  • 2008-09. Deterministic epidemiological models at the individual level in JOURNAL OF MATHEMATICAL BIOLOGY
  • 2011-03. A note on a paper by Erik Volz: SIR dynamics in random networks in JOURNAL OF MATHEMATICAL BIOLOGY
  • 2018-02. Mean-field models for non-Markovian epidemics on networks in JOURNAL OF MATHEMATICAL BIOLOGY
  • 2011-04. Exact epidemic models on graphs using graph-automorphism driven lumping in JOURNAL OF MATHEMATICAL BIOLOGY
  • 2008-03. SIR dynamics in random networks with heterogeneous connectivity in JOURNAL OF MATHEMATICAL BIOLOGY
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00285-017-1123-8

    DOI

    http://dx.doi.org/10.1007/s00285-017-1123-8

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1084804574

    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/28409223


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    44 schema:description We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time [Formula: see text]. We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack-McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time [Formula: see text] than the Kermack-McKendrick model.
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