Coupling a branching process to an infinite dimensional epidemic process
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 53-64.

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.

DOI : 10.1051/ps:2008023
Classification : 92D30, 62E17
Mots clés : likelihood ratio coupling, branching process approximation, epidemic process
@article{PS_2010__14__53_0,
     author = {Barbour, Andrew D.},
     title = {Coupling a branching process to an infinite dimensional epidemic process},
     journal = {ESAIM: Probability and Statistics},
     pages = {53--64},
     publisher = {EDP-Sciences},
     volume = {14},
     year = {2010},
     doi = {10.1051/ps:2008023},
     mrnumber = {2654547},
     zbl = {1208.92061},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2008023/}
}
TY  - JOUR
AU  - Barbour, Andrew D.
TI  - Coupling a branching process to an infinite dimensional epidemic process
JO  - ESAIM: Probability and Statistics
PY  - 2010
SP  - 53
EP  - 64
VL  - 14
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2008023/
DO  - 10.1051/ps:2008023
LA  - en
ID  - PS_2010__14__53_0
ER  - 
%0 Journal Article
%A Barbour, Andrew D.
%T Coupling a branching process to an infinite dimensional epidemic process
%J ESAIM: Probability and Statistics
%D 2010
%P 53-64
%V 14
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2008023/
%R 10.1051/ps:2008023
%G en
%F PS_2010__14__53_0
Barbour, Andrew D. Coupling a branching process to an infinite dimensional epidemic process. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 53-64. doi : 10.1051/ps:2008023. http://www.numdam.org/articles/10.1051/ps:2008023/

[1] F.G. Ball, The threshold behaviour of epidemic models. J. Appl. Probab. 20 (1983) 227-241. | Zbl

[2] F.G. Ball and P. Donnelly, Strong approximations for epidemic models. Stoch. Proc. Appl. 55 (1995) 1-21. | Zbl

[3] A.D. Barbour and M. Kafetzaki, A host-parasite model yielding heterogeneous parasite loads. J. Math. Biol. 31 (1993) 157-176. | Zbl

[4] A.D. Barbour and S. Utev, Approximating the Reed-Frost epidemic process. Stoch. Proc. Appl. 113 (2004) 173-197. | Zbl

[5] M.S. Bartlett, An introduction to stochastic processes. Cambridge University Press (1956). | Zbl

[6] O. Diekmann and J.A.P. Heesterbeek, Mathematical epidemiology of infectious diseases. Wiley, New York (2000). | Zbl

[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. | Zbl

[9] T.G. Kurtz, Limit theorems and diffusion approximations for density dependent Markov chains. Math. Prog. Study 5 (1976) 67-78. | Zbl

[10] T.G. Kurtz, Approximation of population processes, volume 36 of CBMS-NSF Regional Conf. Series in Appl. Math. SIAM, Philadelphia (1981). | Zbl

[11] C.J. Luchsinger, Stochastic models of a parasitic infection, exhibiting three basic reproduction ratios. J. Math. Biol. 42 (2002) 532-554. | Zbl

[12] C.J. Luchsinger, Approximating the long-term behaviour of a model for parasitic infection. J. Math. Biol. 42 (2002) 555-581. | Zbl

[13] P. Whittle, The outcome of a stochastic epidemic - a note on Bailey's paper. Biometrika 42 (1955) 116-122. | Zbl

Cité par Sources :