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Epidemic Spread in Networks: Existing Methods and CurrentChallenges

Published online by Cambridge University Press:  24 April 2014

J. C. Miller  and
I. Z. Kiss
J. C. Miller*
Affiliation:
School of Mathematical Sciences, School of Biological Sciences, and Monash Academy for Cross & Interdisciplinary Mathematics, Monash University, VIC 3800, Australia
I. Z. Kiss
Affiliation:
School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK
*
⋆JCM dedicates this work to the memory of Bob Borrelli, who taught him how to useintegrating factors and much more.
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Abstract

We consider the spread of infectious disease through contact networks of ConfigurationModel type. We assume that the disease spreads through contacts and infected individualsrecover into an immune state. We discuss a number of existing mathematical models used toinvestigate this system, and show relations between the underlying assumptions of themodels. In the process we offer simplifications of some of the existing models. Thedistinctions between the underlying assumptions are subtle, and in many if not most casesthis subtlety is irrelevant. Indeed, under appropriate conditions the models areequivalent. We compare the benefits and disadvantages of the different models, and discusstheir application to other populations (e.g., clustered networks).Finally we discuss ongoing challenges for network-based epidemic modeling.

Keywords

epidemicSIR diseasenetworkpairwiseeffective degreeedge-based compartmental modelsconfiguration model networks
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 2: Epidemics models on networks , 2014 , pp. 4 - 42
Copyright
© EDP Sciences, 2014

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References

R. M. Anderson, R. M. May. Infectious Diseases of Humans. Oxford University Press, Oxford (1991).
Ball, F., Neal, P.. Network epidemic models with two levels of mixing. Mathematical Biosciences, 212 (2008), no. 1, 69–87. CrossRefGoogle Scholar
Boguñá, M., Castellano, C., Pastor-Satorras, R.. Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in networks. Physical Review Letters, 111 (2013), no. 6, 068701, CrossRefGoogle ScholarPubMed
Britton, T., Deijfen, M., Lageras, A., Lindholm, M.. Epidemics on random graphs with tunable clustering. Journal of Applied Probability, 45 (2008), no. 3, 743–756. CrossRefGoogle Scholar
Chatterjee, S., Durrett, R.. Contact processes on random graphs with power law degree distributions have critical value 0. The Annals of Probability, 37 (2009), no. 6, 2332–2356. CrossRefGoogle Scholar
Diekmann, O., De Jong, M. C. M., Metz, J. A. J.. A deterministic epidemic model taking account of repeated contacts between the same individuals. Journal of Applied Probability, 35 (1998), no. 2, 448–462. CrossRefGoogle Scholar
Eames, K., Keeling, M.. Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proceedings of the National Academy of Sciences, 99 (2002), no. 20, 13330–13335. CrossRefGoogle ScholarPubMed
Eames, K. T. D.. Modelling disease spread through random and regular contacts in clustered populations. Theoretical Population Biology, 73 (2008), no. 1, 104–111. CrossRefGoogle Scholar
Gleeson, J., Melnik, S., Hackett, A.. How clustering affects the bond percolation threshold in complex networks. Physical Review E, 81 (2010), no. 6, 066114. CrossRefGoogle ScholarPubMed
Green, D. M., Kiss, I. Z.. Large-scale properties of clustered networks: Implications for disease dynamics. Journal of Biological Dynamics, 4 (2010), no. 5, 431–445. CrossRefGoogle ScholarPubMed
Hébert-Dufresne, L., Patterson-Lomba, O., Goerg, G. M., Althouse, B. M.. Pathogen mutation modeled by competition between site and bond percolation. Physical Review Letters, 110 (2013), no. 10, 108103. CrossRefGoogle ScholarPubMed
van der Hoef, M. A., van Sint Annaland, M., Deen, N., Kuipers, J.. Numerical simulation of dense gas-solid fluidized beds: A multiscale modeling strategy. Annual Review of Fluid Mechanics, 40 (2008) 47–70. CrossRefGoogle Scholar
House, T., Davies, G., Danon, L., Keeling, M. J.. A motif-based approach to network epidemics. Bulletin of Mathematical Biology, 71 (2009), no. 7, 1693–1706. CrossRefGoogle Scholar
House, T., Keeling, M.. Insights from unifying modern approximations to infections on networks. Journal of The Royal Society Interface, 8 (2011), no. 54, 67–73, ISSN 1742-5689. CrossRefGoogle ScholarPubMed
Jaynes, E. T.. Information theory and statistical mechanics. Physical review, 106 (1957), no. 4, 620. CrossRefGoogle Scholar
Karrer, B., Newman, M. E. J.. Random graphs containing arbitrary distributions of subgraphs. Physical Review E, 82 (2010), no. 6, 066118. CrossRefGoogle ScholarPubMed
Keeling, M. J.. The implications of network structure for epidemic dynamics. Theoretical Population Biolology, 67 (2005), no. 1, 1–8. CrossRefGoogle ScholarPubMed
Kenah, E., Robins, J. M.. Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing. Journal of Theoretical Biology, 249 (2007), no. 4, 706–722. CrossRefGoogle ScholarPubMed
A. N. Kolmogorov. Dissipation of energy in locally isotropic turbulence. In Dokl. Akad. Nauk SSSR, volume 32, pages 16–18.
A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. In Dokl. Akad. Nauk SSSR, volume 30, pages 299–303.
Kretzschmar, M., White, R., Caraël, M.. Concurrency is more complex than it seems. AIDS (London, England), 24 (2010), no. 2, 313. CrossRefGoogle ScholarPubMed
Lindquist, J., Ma, J., van den Driessche, P., Willeboordse, F.. Effective degree network disease models. Journal of Mathematical Biology, 62 (2011), no. 2, 143–164, ISSN 0303-6812. CrossRefGoogle ScholarPubMed
May, R. M., Anderson, R. M.. The transmission dynamics of human immunodeficiency virus (HIV). Philosophical Transactions of the Royal Society London B, 321 (1988), no. 1207, 565–607. CrossRefGoogle Scholar
May, R. M., Lloyd, A. L.. Infection dynamics on scale-free networks. Physical Review E, 64 (2001), no. 6, 066112. CrossRefGoogle ScholarPubMed
McBryde, E. S.. Network structure can play a role in vaccination thresholds and herd immunity: a simulation using a network mathematical model. Clinical infectious diseases, 48 (2009), no. 5, 685–686. CrossRefGoogle Scholar
Melnik, S., Hackett, A., Porter, M., Mucha, P., Gleeson, J.. The unreasonable effectiveness of tree-based theory for networks with clustering. Physical Review E, 83 (2011), no. 3, 036112. CrossRefGoogle ScholarPubMed
S. Melnick, M. A. Porter, P. J. Mucha, J. P. Gleeson. Dynamics on modular networks with heterogeneous correlations. Chaos, (In Press), available at http://arxiv.org/abs/1207.1809.
Meyers, L. A.. Contact network epidemiology: Bond percolation applied to infectious disease prediction and control. Bulletin of the American Mathematical Society, 44 (2007), no. 1, 63–86. CrossRefGoogle Scholar
Meyers, L. A., Pourbohloul, B., Newman, M. E. J., Skowronski, D. M., Brunham, R. C., Network theory and SARS: predicting outbreak diversity. Journal of Theoretical Biology, 232 (2005), no. 1, 71–81. CrossRefGoogle ScholarPubMed
Miller, J. C.. Percolation and epidemics in random clustered networks. Physical Review E, 80 (2009), no. 2, 020901(R). CrossRefGoogle Scholar
Miller, J. C.. Spread of infectious disease through clustered populations. Journal of The Royal Society Interface, 6 (2009), no. 41, 1121. CrossRefGoogle ScholarPubMed
Miller, J. C.. A note on a paper by Erik Volz: SIR dynamics in random networks. Journal of Mathematical Biology, 62 (2011), no. 3, 349–358, ISSN 0303-6812. CrossRefGoogle Scholar
Miller, J. C.. A note on the derivation of epidemic final sizes. Bulletin of Mathematical Biology, 74 (2012), no. 9, 2125–2141. CrossRefGoogle ScholarPubMed
Miller, J. C., Slim, A. C., Volz, E. M.. Edge-based compartmental modelling for infectious disease spread. Journal of the Royal Society Interface, 9 (2012), no. 70, 890–906. CrossRefGoogle ScholarPubMed
Miller, J. C., Volz, E. M.. Incorporating disease and population structure into models of SIR disease in contact networks. PloS One, 8 (2013), no. 8, e69162. CrossRefGoogle ScholarPubMed
Miller, J. C., Volz, E. M.. Model hierarchies in edge-based compartmental modeling for infectious disease spread. Journal of Mathematical Biology, 67 (2013), no. 4, 869–899. CrossRefGoogle ScholarPubMed
Molloy, M., Reed, B.. A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6 (1995), no. 2, 161–179. CrossRefGoogle Scholar
Moreno, Y., Pastor-Satorras, R., Vespignani, A.. Epidemic outbreaks in complex heterogeneous networks. The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002), no. 4, 521–529. CrossRefGoogle Scholar
Newman, M. E. J.. Spread of epidemic disease on networks. Physical Review E, 66 (2002), no. 1, 016128. CrossRefGoogle ScholarPubMed
Newman, M. E. J.. The structure and function of complex networks. SIAM Review, 45 (2003), no. 2, 167–256. CrossRefGoogle Scholar
Newman, M. E. J.. Random graphs with clustering. Physical Review Letters, 103 (2009), no. 5, 58701. CrossRefGoogle ScholarPubMed
Noël, P.-A., Davoudi, B., Brunham, R. C., Dubé, L. J., Pourbohloul, B., Time evolution of disease spread on finite and infinite networks. Physical Review E, 79 (2009), no. 2, 026101. CrossRefGoogle ScholarPubMed
Pastor-Satorras, R., Vespignani, A.. Epidemic spreading in scale-free networks. Physical Review Letters, 86 (2001), no. 14, 3200–3203. CrossRefGoogle ScholarPubMed
L. F. Richardson, S. Chapman. Weather prediction by numerical process. Dover publications New York (1965).
Rogers, T.. Maximum-entropy moment-closure for stochastic systems on networks. Journal of Statistical Mechanics: Theory and Experiment, 2011 (2011), no. 05, P05007. CrossRefGoogle Scholar
P. Sagaut. Large eddy simulation for incompressible flows, volume 3. Springer Berlin (2000).
Serrano, M., Boguñá, M.. Percolation and epidemic thresholds in clustered networks. Physical Review Letters, 97 (2006), no. 8, 088701. CrossRefGoogle ScholarPubMed
T. J. Taylor, I. Z. Kiss. Interdependency and hierarchy of exact and approximate epidemic models on networks. Journal of Mathematical Biology, (In Press), available at http://arxiv.org/abs/1212.3124.
Volz, E. M.. SIR dynamics in random networks with heterogeneous connectivity. Journal of Mathematical Biology, 56 (2008), no. 3, 293–310. CrossRefGoogle ScholarPubMed
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 (2011), no. 6, e1002042. CrossRefGoogle ScholarPubMed