Coupling a branching process to an infinite dimensional epidemic process***

Andrew D. Barbour

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 53-64
  • ISSN: 1292-8100

Abstract

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Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence can be achieved with asymptotically high probability until MN infections have occurred, as long as MN = o(N2/3), where N denotes the total number of hosts.

How to cite

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Barbour, Andrew D.. "Coupling a branching process to an infinite dimensional epidemic process***." ESAIM: Probability and Statistics 14 (2010): 53-64. <http://eudml.org/doc/252259>.

@article{Barbour2010,
abstract = { Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence can be achieved with asymptotically high probability until MN infections have occurred, as long as MN = o(N2/3), where N denotes the total number of hosts. },
author = {Barbour, Andrew D.},
journal = {ESAIM: Probability and Statistics},
keywords = {Likelihood ratio coupling; branching process approximation; epidemic process.; likelihood ratio coupling},
language = {eng},
month = {3},
pages = {53-64},
publisher = {EDP Sciences},
title = {Coupling a branching process to an infinite dimensional epidemic process***},
url = {http://eudml.org/doc/252259},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Barbour, Andrew D.
TI - Coupling a branching process to an infinite dimensional epidemic process***
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 14
SP - 53
EP - 64
AB - Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence can be achieved with asymptotically high probability until MN infections have occurred, as long as MN = o(N2/3), where N denotes the total number of hosts.
LA - eng
KW - Likelihood ratio coupling; branching process approximation; epidemic process.; likelihood ratio coupling
UR - http://eudml.org/doc/252259
ER -

References

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  1. F.G. Ball, The threshold behaviour of epidemic models. J. Appl. Probab. 20 (1983) 227–241.  Zbl0519.92023
  2. F.G. Ball and P. Donnelly, Strong approximations for epidemic models. Stoch. Proc. Appl.55 (1995) 1–21.  Zbl0823.92024
  3. A.D. Barbour and M. Kafetzaki, A host–parasite model yielding heterogeneous parasite loads. J. Math. Biol.31 (1993) 157–176.  Zbl0853.92020
  4. A.D. Barbour and S. Utev, Approximating the Reed-Frost epidemic process. Stoch. Proc. Appl.113 (2004) 173–197.  Zbl1113.92054
  5. M.S. Bartlett, An introduction to stochastic processes. Cambridge University Press (1956).  
  6. O. Diekmann and J.A.P. Heesterbeek, Mathematical epidemiology of infectious diseases. Wiley, New York (2000).  Zbl0997.92505
  7. J.A.P. Heesterbeek, R0. CWI, Amsterdam (1992).  
  8. D.G. Kendall, Deterministic and stochastic epidemics in closed populations. Proc. Third Berk. Symp. Math. Stat. Probab.4 (1956) 149–165.  Zbl0070.15101
  9. T.G. Kurtz, Limit theorems and diffusion approximations for density dependent Markov chains. Math. Prog. Study5 (1976) 67–78.  Zbl0373.60081
  10. T.G. Kurtz, Approximation of population processes, volume 36 of CBMS-NSF Regional Conf. Series in Appl. Math. SIAM, Philadelphia (1981).  Zbl0465.60078
  11. C.J. Luchsinger, Stochastic models of a parasitic infection, exhibiting three basic reproduction ratios. J. Math. Biol.42 (2002) 532–554.  Zbl0984.92030
  12. C.J. Luchsinger, Approximating the long-term behaviour of a model for parasitic infection. J. Math. Biol.42 (2002) 555–581.  Zbl0989.92021
  13. P. Whittle, The outcome of a stochastic epidemic – a note on Bailey's paper. Biometrika42 (1955) 116–122.  Zbl0064.39103

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