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Non-Markovian Stochastic Epidemics in Extremely Heterogeneous Populations

  • T. House

Abstract

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

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Corresponding author

Corresponding author. E-mail: T.A.House@warwick.ac.uk

References

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[1] R. M. Anderson, R. M. May. Infectious Diseases of Humans. Oxford University Press, Oxford, 1991.
[2] H. Andersson, T. Britton. Stochastic Epidemic Models and Their Statistical Analysis, volume 151 of Springer Lectures Notes in Statistics. Springer, Berlin, 2000.
[3] 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, 87991.
[4] Barabási, A. L., Albert, R.. Emergence of scaling in random networks. Science, 286 (1999), no. 5439, 509512.
[5] 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.
[6] C. Castellano,Pastor-Satorras, R.. Thresholds for epidemic spreading in networks. Physical Review Letters, 105 (2010), no. 21, 218701.
[7] Clauset, A., Shalizi, C. R., Newman, M. E. J.. Power-law distributions in empirical data. SIAM Review, 51 (2009), no. 4, 661703.
[8] 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.
[9] 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, 28262833.
[10] O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. John Wiley & Sons Ltd., 2000.
[11] 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, 365382.
[12] R. Durrett. Random Graph Dynamics. Cambridge University Press, 2007.
[13] 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, 180184.
[14] G. R. Grimmett, D. R. Stirzaker. Probability and Random Processes. Oxford University Press, 3rd edition, 2001.
[15] 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.
[16] M. J. Keeling, P. Rohani. Modeling Infectious Diseases in Humans and Animals. Princeton University Press, New Jersey, 2007.
[17] 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, 12436.
[18] Kurtz, T. G.. Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability, 7 (1970), no. 1, 4958.
[19] Kurtz, T. G.. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8 (1971), no. 2, 344356.
[20] 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, 565607.
[21] May, R. M., Lloyd, A. L.. Infection dynamics on scale-free networks. Physical Review E, 64 (2001), 066112.
[22] Pastor-Satorras, R., Vespignani, A.. Epidemic dynamics and endemic states in complex networks. Physical Review E, 63 (2001), 066117.
[23] Pastor-Satorras, R., Vespignani, A.. Epidemic dynamics in finite size scale-free networks. Physical Review E, 65 (2002), no. 3, 035108.
[24] 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, 3807.

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Non-Markovian Stochastic Epidemics in Extremely Heterogeneous Populations

  • T. House

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