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Random contact processes, snowball sampling and factorial series distributions

Published online by Cambridge University Press:  14 July 2016

Sven Berg
Sven Berg*
Affiliation:
University of Lund
*
∗ Postal address: Department of Statistics, University of Lund, Box 7008, S-220 07 Lund, Sweden.
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Abstract

A contact process first posed as a random mapping problem is discussed together with a model for the spread of a rumour through an archaic telephone system. Snowball sampling and the well-known Reed–Frost chain binomial model are also commented on. Our main interest and emphasis is on distributional properties of discrete-time models. Simple combinatorial arguments are used along with a class of discrete distributions originally proposed for catch–recapture sampling.

Keywords

RANDOM MAPPINGSSPREAD OF RUMOURSCHAIN BINOMIAL MODELSSEQUENTIAL CATCH–RECAPTURESNOWBALL SAMPLINGEPIDEMIC CURVESIZE OF EPIDEMICSTIRLING NUMBERS OF THE SECOND KINDFACTORIAL SERIES DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 31 - 46
Copyright
Copyright © Applied Probability Trust 1983 

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References

Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.CrossRefGoogle Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases, 2nd edn. Griffin, London.Google Scholar
Berg, S. (1974) Factorial series distributions with applications to catch–recapture problems. Scand. J. Statist. 1, 145–152.Google Scholar
Berg, S. (1981) On snowball sampling, random mappings and related problems. J. Appl. Prob. 18, 283–290.CrossRefGoogle Scholar
Cane, V. R. (1966) A note on the size of epidemics and the number of people hearing a rumour. J. R. Statist. Soc. B 28, 487–490.Google Scholar
Daley, D. J. and Kendall, D. G. (1965) Stochastic rumours. J. Inst. Math. Appl. 1, 42–55.CrossRefGoogle Scholar
Frank, O. (1979) Estimation of population totals by use of snowball samples. In Perspectives on Social Network Research, ed. Holland, P. P., and Leinhardt, S.. Academic Press, New York, 319–347.CrossRefGoogle Scholar
Frauenthal, J. C. (1980) Mathematical Modeling in Epidemiology. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco.Google Scholar
Gani, J. and Jerwood, D. (1971) Markov chain methods in chain binomial epidemic models. Biometrics 57, 591–603.CrossRefGoogle Scholar
Gertsbakh, I. B. (1977) Epidemic process on a random graph: some preliminary results. J. Appl. Prob. 14, 427–438.CrossRefGoogle Scholar
Goodman, L. A. (1953) Sequential sampling tagging for population size problems. Ann. Math. Statist. 24, 56–69.CrossRefGoogle Scholar
Goodman, L. A. (1961) Snowball sampling. Ann. Math. Statist. 32, 148–170.CrossRefGoogle Scholar
Harris, B. (1960) Probability distributions related to random mappings. Ann. Math. Statist. 31, 1045–1062.CrossRefGoogle Scholar
Holst, L. (1980) On numbers related to partitions of unlike objects and occupancy problems. MS, Upsala University.Google Scholar
Johnson, N. L. and Kotz, S. (1977) Urn Models and Their Applications. Wiley, New York.Google Scholar
Landau, H. G. (1952) On some problems of random nets. Bull. Math. Biophys. 14, 203–212.CrossRefGoogle Scholar
Maki, D. P. and Thompson, M. (1973) Mathematical Models and Applications. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Rapoport, A. (1979) A probabilistic approach to networks. Social Networks 2, 1–18.CrossRefGoogle Scholar
Samuel, E. (1968) Sequential maximum likelihood estimation of the size of a population. Ann. Math. Statist. 39, 1057–1068.CrossRefGoogle Scholar