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An SIR epidemic on a weighted network

Published online by Cambridge University Press:  26 December 2019

Kristoffer Spricer Open the ORCID record for Kristoffer Spricer [Opens in a new window]  and
Tom Britton
Kristoffer Spricer*
Affiliation:
Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
Tom Britton
Affiliation:
Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
*
*Corresponding author. Email: [email protected]
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Abstract

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We introduce a weighted configuration model graph, where edge weights correspond to the probability of infection in an epidemic on the graph. On these graphs, we study the development of a Susceptible–Infectious–Recovered epidemic using both Reed–Frost and Markovian settings. For the special case of having two different edge types, we determine the basic reproduction numberR0, the probability of a major outbreak, and the relative final size of a major outbreak. Results are compared with those for a calibrated unweighted graph. The degree distributions are based on both theoretical constructs and empirical network data. In addition, bivariate standard normal copulas are used to model the dependence between the degrees of the two edge types, allowing for modeling the correlation between edge types over a wide range. Among the results are that the weighted graph produces much richer results than the unweighted graph. Also, while R0 always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of a major outbreak nor for the relative final size of a major outbreak. When using copulas we see that these can produce results that are similar to those of the empirical degree distributions, indicating that in some cases a copula is a viable alternative to using the full empirical data.

Keywords

epidemicsbasic reproduction numberweighted graphconfiguration modelfinal sizemajor outbreakcopula
Type
Original Article
Information
Network Science , Volume 7 , Issue 4 , December 2019 , pp. 556 - 580
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge University Press 2019

References

Adamic, L. A., & Huberman, B. A. (2000). Power-law distribution of the world wide web. Science, 287(5461), 2115.CrossRefGoogle Scholar
Bailey, N. T. J. (1975). The mathematical theory of infectious diseases and its applications. Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE.Google Scholar
Ball, F., & Neal, P. (2002). A general model for stochastic sir epidemics with two levels of mixing. Mathematical Biosciences, 180(1–2), 73–102.CrossRefGoogle Scholar
Ball, F., & Neal, P. (2008). Network epidemic models with two levels of mixing. Mathematical Biosciences, 212(1), 69–87.CrossRefGoogle Scholar
Ball, F., & Sirl, D. (2012). An sir epidemic model on a population with random network and household structure, and several types of individuals. Advances in Applied Probability, 44(1), 63–86.CrossRefGoogle Scholar
Barabási, A.-L., & Bonabeau, E. (2003). Scale-free networks. Scientific American, 288(5), 60–69.CrossRefGoogle ScholarPubMed
Biswas, A., & Hwang, J.-S. (2002). A new bivariate binomial distribution. Statistics & Probability Letters, 60(2), 231–240.CrossRefGoogle Scholar
Bollobás, B. (2001). Random graphs (2nd ed.), Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Britton, T. (2010). Stochastic epidemic models: A survey. Mathematical Biosciences, 225(1), 24–35.CrossRefGoogle ScholarPubMed
Britton, T., Deijfen, M., & Liljeros, F. (2011). A weighted configuration model and inhomogeneous epidemics. Journal of Statistical Physics, 145(5), 1368–1384.CrossRefGoogle Scholar
Britton, T., Deijfen, M., & Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. Journal of Statistical Physics, 124(6), 1377–1397.CrossRefGoogle Scholar
Britton, T., Janson, S., & Martin-Löf, A. (2007). Graphs with specified degree distributions, simple epidemics, and local vaccination strategies. Advances in Applied Probability, 39(4), 922–948.CrossRefGoogle Scholar
Deijfen, M., & Fitzner, R. (2017). Birds of a feather or opposites attract - effects in network modelling. Internet Mathematics, 1(1).Google Scholar
Diekmann, O., Heesterbeek, H., & Britton, T. (2012). Mathematical tools for understanding infectious disease dynamics. Princeton, NJ: Princeton University Press.Google Scholar
Grassberger, P. (1983). On the critical behavior of the general epidemic process and dynamical percolation. Mathematical Biosciences, 63(2), 157–172.CrossRefGoogle Scholar
Hansson, D., Fridlund, V., Stenqvist, K., Britton, T., & Liljeros, F. (2018). Inferring individual sexual action dispositions from egocentric network data on dyadic sexual outcomes. Plos One, 13(11), e0207116.CrossRefGoogle ScholarPubMed
Harris, T. E. (2002). The theory of branching processes. Dover Phoenix Editions. Mineola, NY: Dover Publications Inc. Corrected reprint of the 1963 original [Springer, Berlin; MR0163361 (29 #664)].Google Scholar
Holm, E., Lindgren, U., Lundevaller, E., & Strömgren, M. (2006). The SVERIGE spatial microsimulation model. Paper presented at the 8th Nordic Seminar on Microsimulation Models, Oslo (pp. 8–9).Google Scholar
Kamp, C., Moslonka-Lefebvre, M., & Alizon, S. (2013). Epidemic spread on weighted networks. Plos Computational Biology, 9(12), e1003352.CrossRefGoogle ScholarPubMed
Kenah, E., & Robins, J. M. (2007). Second look at the spread of epidemics on networks. Physical Review E, 76(3), 036113.CrossRefGoogle ScholarPubMed
Larson, J. M. (2017). The weakness of weak ties for novel information diffusion. Applied Network Science, 2(1), 14.CrossRefGoogle ScholarPubMed
Lefèvre, C. (1990). Stochastic epidemic models for sir infectious diseases: A brief survey of the recent general theory. In Gabriel, J.-P., Lefevre, C., & Picard, P. (Eds.), Stochastic processes in epidemic theory (pp. 1–12). Springer-Verlag Berlin Heidelberg.Google Scholar
Liljeros, F., Edling, C. R., Amaral, L. A. N., Stanley, H. E., & Åberg, Y. (2001). The web of human sexual contacts. Nature, 411(6840), 907.CrossRefGoogle ScholarPubMed
Miller, J. C., & Volz, E. M. (2013). Incorporating disease and population structure into models of sir disease in contact networks. Plos One, 8(8), e69162.CrossRefGoogle ScholarPubMed
Molloy, M., & Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6(2–3), 161–180.CrossRefGoogle Scholar
Nelsen, R. B. (2007). An introduction to copulas. New York: Springer Science+Business Media, Inc.Google Scholar
Newman, M. E. J. (2002). Spread of epidemic disease on networks. Physical Review E, 66(1), 016128.CrossRefGoogle ScholarPubMed
Shu, P., Tang, M., Gong, K., & Liu, Y. (2012). Effects of weak ties on epidemic predictability on community networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4), 043124.CrossRefGoogle ScholarPubMed