Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Barnard, R.C., Berthouze, L., Simon, P.L., Kiss, I.Z.: Epidemic threshold in pairwise models for clustered networks: closures and fast correlations. arXiv preprint arXiv:1806.06135 (2018)
Eames, K.T.: Modelling disease spread through random and regular contacts in clustered populations. Theor. Popul. Biol. 73(1), 104–111 (2008)
Eames, K.T., Keeling, M.J.: Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Natl. Acad. Sci. 99(20), 13330–13335 (2002)
House, T., Davies, G., Danon, L., Keeling, M.J.: A motif-based approach to network epidemics. Bull. Math. Biol. 71(7), 1693–1706 (2009)
House, T., Keeling, M.J.: The impact of contact tracing in clustered populations. PLoS Comput. Biol. 6(3), e1000,721 (2010)
Karrer, B., Newman, M.E.: Message passing approach for general epidemic models. Phys. Rev. E 82(1), 016,101 (2010)
Karrer, B., Newman, M.E.: Random graphs containing arbitrary distributions of subgraphs. Phys. Rev. E 82(6), 066,118 (2010)
Keeling, M.J.: The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B Biol. Sci. 266(1421), 859–867 (1999)
Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks. Springer, Berlin (2017)
Li, J., Li, W., Jin, Z.: The epidemic model based on the approximation for third-order motifs on networks. Math. Biosci. 297, 12–26 (2018)
Lindquist, J., Ma, J., Van den Driessche, P., Willeboordse, F.H.: Effective degree network disease models. J. Math. Biol. 62(2), 143–164 (2011)
Miller, J.C.: Percolation and epidemics in random clustered networks. Phys. Rev. E 80(2), 020,901 (2009)
Miller, J.C.: Spread of infectious disease through clustered populations. J. R. Soc. Interface 6, rsif–2008 (2009)
Miller, J.C., Slim, A.C., Volz, E.M.: Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface 9(70), 890–906 (2012)
Miller, J.C., Volz, E.M.: Model hierarchies in edge-based compartmental modeling for infectious disease spread. J. Math. Biol. 67(4), 869–899 (2013)
Newman, M.E.: Random graphs with clustering. Phys. Rev. Lett. 103(5), 058,701 (2009)
Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925 (2015)
Pastor-Satorras, R., Vespignani, A.: Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63(6), 066,117 (2001)
Rand, D.: Correlation equations and pair approximations for spatial ecologies. Advanced Ecological Theory: Principles and Applications, vol. 100. Blackwell Science, London (1999)
Rattana, P., Blyuss, K.B., Eames, K.T., Kiss, I.Z.: A class of pairwise models for epidemic dynamics on weighted networks. Bull. Math. Biol. 75(3), 466–490 (2013)
Ritchie, M., Berthouze, L., Kiss, I.Z.: Beyond clustering: mean-field dynamics on networks with arbitrary subgraph composition. J. Math. Biol. 72(1–2), 255–281 (2016)
Sharkey, K.J., et al.: Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks. J. Math. Biol. 53(1), 61–85 (2006)
Sherborne, N., Miller, J.C., Blyuss, K.B., Kiss, I.Z.: Mean-field models for non-markovian epidemics on networks. J. Math. Biol. 76(3), 755–778 (2018)
Trapman, P.: On analytical approaches to epidemics on networks. Theor. Popul. Biol. 71(2), 160–173 (2007)
Volz, E.M., Miller, J.C., Galvani, A., Meyers, L.A.: Effects of heterogeneous and clustered contact patterns on infectious disease dynamics. PLoS Comput. Biol. 7(6), e1002,042 (2011)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kiss, I.Z., Miller, J.C., Simon, P.L. (2019). Fast Variables Determine the Epidemic Threshold in the Pairwise Model with an Improved Closure. In: Aiello, L., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L. (eds) Complex Networks and Their Applications VII. COMPLEX NETWORKS 2018. Studies in Computational Intelligence, vol 812. Springer, Cham. https://doi.org/10.1007/978-3-030-05411-3_30
Download citation
DOI: https://doi.org/10.1007/978-3-030-05411-3_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05410-6
Online ISBN: 978-3-030-05411-3
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)