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

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


Robert R. Wilkinson

Professor of Applied Probability

Kieran J. Sharkey


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 t > 0. 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 t > 0 than the Kermack–McKendrick model.


Wilkinson, R. R., Ball, F. G., & Sharkey, K. J. (2017). The relationships between message passing, pairwise, Kermack–McKendrick and stochastic SIR epidemic models. Journal of Mathematical Biology, 75(6-7), 1563-1590.

Journal Article Type Article
Acceptance Date Mar 23, 2017
Online Publication Date Apr 13, 2017
Publication Date 2017-12
Deposit Date Apr 21, 2017
Publicly Available Date Apr 21, 2017
Journal Journal of Mathematical Biology
Print ISSN 0303-6812
Electronic ISSN 1432-1416
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 75
Issue 6-7
Pages 1563-1590
Keywords Stochastic SIR epidemic, Kermack–McKendrick model, Non-Markovian, Message passing, Pairwise, Network
Public URL
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art%3A10.1007%2Fs00285-017-1123-8.pdf (560 Kb)

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