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Non-Markovian Stochastic Epidemics in Extremely HeterogeneousPopulations

Published online by Cambridge University Press:  24 April 2014

T. House
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Abstract

A feature often observed in epidemiological networks is significant heterogeneity indegree. A popular modelling approach to this has been to consider large populations withhighly heterogeneous discrete contact rates. This paper defines an individual-levelnon-Markovian stochastic process that converges on standard ODE models of such populationsin the appropriate asymptotic limit. A generalised Sellke construction is derived for thismodel, and this is then used to consider final outcomes in the case where heterogeneityfollows a truncated Zipf distribution.

Keywords

ZipfSellkeSIR

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Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 2: Epidemics models on networks , 2014 , pp. 153 - 160
Copyright
© EDP Sciences, 2014

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References

R. M. Anderson, R. M. May. Infectious Diseases of Humans. Oxford University Press, Oxford, 1991.
H. Andersson, T. Britton. Stochastic Epidemic Models and Their Statistical Analysis, volume 151 of Springer Lectures Notes in Statistics. Springer, Berlin, 2000.
Bansal, S., Grenfell, B. T., Meyers, L. A.. When individual behaviour matters: homogeneous and network models in epidemiology. Journal of the Royal Society Interface, 4 (2007), no. 16, 879–91. CrossRefGoogle ScholarPubMed
Barabási, A. L., Albert, R.. Emergence of scaling in random networks. Science, 286 (1999), no. 5439, 509–512. Google ScholarPubMed
N. Berger, C. Borgs, J. T. Chayes, A. Saberi. On the spread of viruses on the internet. In Proceedings of the 16th Symposium on Discrete Algorithms, 2005.
C. Castellano,Pastor-Satorras, R.. Thresholds for epidemic spreading in networks. Physical Review Letters, 105 (2010), no. 21, 218701. Google Scholar
Clauset, A., Shalizi, C. R., Newman, M. E. J.. Power-law distributions in empirical data. SIAM Review, 51 (2009), no. 4, 661–703. CrossRefGoogle Scholar
L. Danon, A. P. Ford, T. House, C. P. Jewell, M. J. Keeling, G. O. Roberts, J. V. Ross, and M. C. Vernon. Networks and the epidemiology of infectious disease. Interdisciplinary Perspectives on Infectious Diseases, (2011), 1–28.
Danon, L., House, T., Keeling, M. J., Read, J. M.. Social encounter networks: collective properties and disease transmission. Journal of the Royal Society Interface, 9 (2012), no. 76, 2826–2833. CrossRefGoogle ScholarPubMed
O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. John Wiley & Sons Ltd., 2000.
Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J.. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28 (1990), no. 4, 365–382. CrossRefGoogle ScholarPubMed
R. Durrett. Random Graph Dynamics. Cambridge University Press, 2007.
Eubank, S., Guclu, H., Kumar, V. S. A., Marathe, M. V., Srinivasan, A., Toroczkai, Z., Wang, N.. Modelling disease outbreaks in realistic urban social networks. Nature, 429 (2004), no. 6988, 180–184. CrossRefGoogle Scholar
G. R. Grimmett, D. R. Stirzaker. Probability and Random Processes. Oxford University Press, 3rd edition, 2001.
House, T., Ross, J. V., Sirl, D.. How big is an outbreak likely to be? Methods for epidemic final-size calculation. Proceedings of the Royal Society A, 469 (2013), no. 2150, 20120436. CrossRefGoogle Scholar
M. J. Keeling, P. Rohani. Modeling Infectious Diseases in Humans and Animals. Princeton University Press, New Jersey, 2007.
Kiss, I. Z., Green, D. M., Kao, R. R.. The effect of contact heterogeneity and multiple routes of transmission on final epidemic size. Mathematical Biosciences, 203 (2006), no. 1, 124–36. CrossRefGoogle Scholar
Kurtz, T. G.. Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability, 7 (1970), no. 1, 49–58. CrossRefGoogle Scholar
Kurtz, T. G.. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8 (1971), no. 2, 344–356. CrossRefGoogle Scholar
May, R. M., Anderson, R. M.. The transmission dynamics of human immunodeficiency virus (HIV). Philosophical Transactions of the Royal Society of London Series 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), 066112. CrossRefGoogle ScholarPubMed
Pastor-Satorras, R., Vespignani, A.. Epidemic dynamics and endemic states in complex networks. Physical Review E, 63 (2001), 066117. CrossRefGoogle ScholarPubMed
Pastor-Satorras, R., Vespignani, A.. Epidemic dynamics in finite size scale-free networks. Physical Review E, 65 (2002), no. 3, 035108. CrossRefGoogle ScholarPubMed
Schneeberger, A., Mercer, C. H., Gregson, S. A. J., Ferguson, N. M., Nyamukapa, C. A., Anderson, R. M., Johnson, A. M., Garnett, G. P.. Scale-free networks and sexually transmitted diseases: a description of observed patterns of sexual contacts in Britain and Zimbabwe. Sexually Transmitted Diseases, 31 (2004), no. 6, 380–7. CrossRefGoogle ScholarPubMed