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A law of large numbers approximation for Markov population processes with countably many types
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  • Published: 07 April 2011

A law of large numbers approximation for Markov population processes with countably many types

  • A. D. Barbour1 &
  • M. J. Luczak2 

Probability Theory and Related Fields volume 153, pages 727–757 (2012)Cite this article

Abstract

When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for quite general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted ℓ 1 norm.

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Author information

Authors and Affiliations

  1. Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    A. D. Barbour

  2. Department of Mathematics, London School of Economics, Houghton St., London, WC2A 2AE, UK

    M. J. Luczak

Authors
  1. A. D. Barbour
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  2. M. J. Luczak
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Corresponding author

Correspondence to A. D. Barbour.

Additional information

A.D. Barbour was supported in part by Schweizerischer Nationalfonds Projekt Nr. 20–107935/1.

M.J. Luczak was supported in part by a STICERD grant.

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Cite this article

Barbour, A.D., Luczak, M.J. A law of large numbers approximation for Markov population processes with countably many types. Probab. Theory Relat. Fields 153, 727–757 (2012). https://doi.org/10.1007/s00440-011-0359-2

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  • Received: 02 September 2010

  • Revised: 24 March 2011

  • Published: 07 April 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0359-2

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Keywords

  • Epidemic models
  • Metapopulation processes
  • Countably many types
  • Quantitative law of large numbers
  • Markov population processes

Mathematics Subject Classification (2000)

  • 92D30
  • 60J27
  • 60B12
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