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Fast variables determine the epidemic threshold in the pairwise model with an improved closure

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conference contribution
posted on 2023-06-09, 15:53 authored by Istvan Kiss, Joel C Miller, Péter L Simon
Pairwise models are used widely to model epidemic spread on networks. These include 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 be used to 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.


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Proceedings of Complex Networks 2018 (The Seventh International Conference on Complex Networks and their Applications


Springer Verlag

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Event name

Complex Networks 2018: The 7th International Conference on Complex Networks and Their Applications

Event location

Cambridge, United Kingdom

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December 11-13, 2018



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  • Mathematics Publications

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  • Mathematics Applied to Biology Research Group Publications

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