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

ESAIM: Probability and Statistics (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topBarbour, 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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.