The Annals of Probability

Poisson approximations for epidemics with two levels of mixing

Frank Ball and Peter Neal

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This paper is concerned with a stochastic model for the spread of an epidemic among a population of n individuals, labeled $1,2,\ldots,n$, in which a typical infected individual, i say, makes global contacts, with individuals chosen independently and uniformly from the whole population, and local contacts, with individuals chosen independently according to the contact distribution ${V_{i}^{n} = \{ v_{i,j}^{n} ; j=1,2, \ldots, n \}}$, at the points of independent Poisson processes with rates $\lambda_G^{n}$ and $\lambda_L^{n}$, respectively, throughout an infectious period that follows an arbitrary but specified distribution. The population initially comprises $m_n$ infectives and $n-m_n$ susceptibles. A sufficient condition is derived for the number of individuals who survive the epidemic to converge weakly to a Poisson distribution as $n \to \infty$. The result is specialized to the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; the overlapping groups model, in which the population is partitioned in several ways and local mixing is uniform within the elements of the partitions; and the great circle model, in which $v_{i,j}^{n} = v_{(i-j)_{\mod n}}^{n}$.

Article information

Ann. Probab., Volume 32, Number 1B (2004), 1168-1200.

First available in Project Euclid: 11 March 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 92D30: Epidemiology
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Epidemic models local and global mixing Poisson convergence random graph positively related coupling


Ball, Frank; Neal, Peter. Poisson approximations for epidemics with two levels of mixing. Ann. Probab. 32 (2004), no. 1B, 1168--1200. doi:10.1214/aop/1079021475.

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  • Aldous, D. J. (1985). Exchangability and related topics. École d'ete de Probabilités de Saint-Flour XIII. Lecture Notes in Math. 1117 1--198. Springer, Berlin.
  • Anderson, R. M. and May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford Univ. Press.
  • Andersson, H. (1999). Epidemic models and social networks. Math. Sci. 24 128--147.
  • Ball, F. G. (1996). Threshold behavior in stochastic epidemics among households. Athens Conference on Applied Probability and Time Series Analysis. Vol. I. Lecture Notes in Statist. 114 253--266. Springer, Berlin.
  • Ball, F. G. and Barbour, A. D. (1990). Poisson approximation for some epidemic models. J. Appl. Probab. 27 479--490.
  • Ball, F. G. and Donnelly, P. J. (1995). Strong approximations for epidemic models. Stochastic Process. Appl. 55 1--21.
  • Ball, F. G. and Lyne, O. D. (2001). Stochastic multitype SIR epidemics among a population partitioned into households. Adv. in Appl. Probab. 33 99--123.
  • Ball, F. G., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Probab. 7 46--89.
  • Ball, F. G. and Neal, P. J. (2002). A general model for stochastic SIR epidemics with two levels of mixing. Math. Biosci. 180 73--102.
  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press.
  • Barbour, A. D. and Reinert, G. (2001). Small worlds. Random Struct. Algorithms 19 54--74.
  • Becker, N. G. and Dietz, K. (1995). The effect of the household distribution on transmission and control of highly infectious diseases. Math. Biosci. 127 207--219.
  • Becker, N. G. and Hasofer, A. M. (1998). Estimating the transmission rate for a highly infectious disease. Biometrics 54 730--738.
  • Daniels, H. E. (1967). The distribution of the total size of an epidemic. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 4 281--293. Univ. California Press, Berkeley.
  • Halloran, M. E., Longini, I. M., Nizam, A. and Yang, Y. (2002). Containing bioterrorist smallpox. Science 298 1428--1432.
  • Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934). Inequalities. Cambridge Univ. Press.
  • Kaplan, E. H., Craft, D. L. and Wein, L. M. (2002). Emergency response to a smallpox attack: The case for mass vaccination. Proc. Natl. Acad. Sci. U.S.A. 99 10935--10940.
  • Kendall, D. G. (1956). Deterministic and stochastic epidemics in closed populations. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 4 149--165. Univ. California Press, Berkeley.
  • Kendall, W. S. (1994). Contribution to the discussion of ``Epidemics: Models and data'' by D. Mollison, V. Isham and B. T. Grenfell. J. Roy. Statist. Soc. Ser. A 157 143.
  • Lefèvre, C. and Utev, S. (1995). Poisson approximation for the final state of a generalised epidemic process. Ann. Probab. 23 1139--1162.
  • Lefèvre, C. and Utev, S. (1997). Mixed Poisson approximation in the collective epidemic model. Stochastic Process. Appl. 69 217--246.
  • Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Probab. 23 265--282.
  • Picard, P. and Lefèvre, C. (1990). A unified analysis of the final size and severity distribution in collective Reed--Frost epidemic processes. Adv. in Appl. Probab. 22 269--294.
  • Scalia-Tomba, G. (1985). Asymptotic final size distribution for some chain-binomial processes. Adv. in Appl. Prob. 17 477--495.
  • Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Probab. 20 390--394.
  • von Bahr, B. and Martin-Löf, A. (1980). Threshold limit theorems for some epidemic processes. Adv. in Appl. Probab. 12 319--349.
  • Watts, D. J. (1999). Small Worlds. Princeton Univ. Press.
  • Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of ``small-world'' networks. Nature 393 440--442.
  • Whittle, P. (1955). The outcome of a stochastic epidemic---a note on Bailey's paper. Biometrika 42 116--122.