Fast Variables Determine the Epidemic Threshold in the Pairwise Model with an Improved Closure

  • István Z. KissEmail author
  • Joel C. Miller
  • Péter L. Simon
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)


Pairwise models are widely used to model epidemic spread on networks. This includes the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact tracing on more exotic networks exhibiting degree heterogeneity, directed and/or weighted links and clustering. However, extra features of the disease dynamics or of the network lead to an increase in system size and analytical tractability becomes problematic. Various “closures” can keep the system tractable. Focusing on SIR epidemics on regular but clustered networks, we show that even for the most complex closure we can determine the epidemic threshold as an asymptotic expansion in terms of the clustering coefficient. We do this by exploiting the presence of a system of fast variables, specified by the correlation structure of the epidemic, whose steady state determines the epidemic threshold. While we do not find the steady state analytically, we create an elegant asymptotic expansion of it. We validate this new threshold by comparing it to the numerical solution of the full system and find excellent agreement over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. The technique carries over to pairwise models with other closures [1], and we note that the epidemic threshold will be model dependent. This emphasises the importance of model choice when dealing with realistic outbreaks.


Epidemic Network Clustering Threshold Pairwise model 



István Z. Kiss acknowledges support from the Leverhulme Trust Research Project Grant (RPG-2017-370). Péter L. Simon acknowledges support from Hungarian Scientific Research Fund, OTKA, (grant no. 115926). Joel C. Miller acknowledges support from Global Good.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • István Z. Kiss
    • 1
    Email author
  • Joel C. Miller
    • 2
  • Péter L. Simon
    • 3
    • 4
  1. 1.Department of Mathematics, School of Mathematical and Physical SciencesUniversity of SussexFalmer, BrightonUK
  2. 2.Institute for Disease ModelingBellevueUnited States of America
  3. 3.Institute of MathematicsEötvös Loránd UniversityBudapestHungary
  4. 4.Numerical Analysis and Large Networks Research GroupHungarian Academy of SciencesBudapestHungary

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