Skip to main content Accessibility help

An unusual stochastic order relation with some applications in sampling and epidemic theory

  • Claude Lefevre (a1) and Philippe Picard (a2)


One expects, intuitively, that the total damage caused by an epidemic increases, in a certain sense, with the infection intensity exerted by the infectives during their lifelength. The original object of the present work is to make precise in which probabilistic terms such a statement does indeed hold true, when the spread of the disease is described by a collective Reed–Frost model and the global cost is represented by the final size and severity. Surprisingly, this problem leads us to introduce an order relation for -valued random variables, unusual in the literature, based on the descending factorial moments. Further applications of the ordering occur when comparing certain sampling procedures through the number of un-sampled individuals. In particular, it is used to reinforce slightly comparison results obtained earlier for two such samplings.


Corresponding author

Postal address: Institut de Statistique, C.P. 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
∗∗Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France.


Hide All
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.
Ball, F. and Donnelly, P. (1987) Interparticle correlation in death processes with application to variability in compartmental models. Adv. Appl. Prob. 19, 755766.
Bernard, S. R. (1977) An urn model study of variability within a compartment. Bull. Math. Biol. 39, 463470.
Daley, D. J. (1990) The size of epidemics with variable infectious periods. Submitted for publication.
Donnelly, P. and Whitt, W. (1989) On reinforcement–depletion compartmental urn models. J. Appl. Prob. 26, 477489.
Klefsjö, B. (1983) A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615626.
Lefevre, Cl. and Picard, Ph. (1989) On the formulation of discrete-time epidemic models. Math. Biosci. 95, 2735.
Lefevre, Cl. (1990) Stochastic epidemic models for S.I.R. infectious diseases: a brief survey of the recent general theory. In Stochastic Processes in Epidemic Theory, ed. Gabriel, J. P., Lefèvre, CL. and Picard, Ph., Lecture Notes in Biomathematics 86, pp. 112, Springer-Verlag, New York.
Lefevre, Cl. and Michaletzky, G. (1990) Interparticle dependence in a linear death process subjected to a random environment. J. Appl. Prob. 27, 491498.
Lefevre, Cl. and Picard, Ph. (1990) A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 2548.
Lefevre, Cl. and Picard, Ph. (1992) A multivariate stochastic ordering by the mixed descending factorial moments with applications. In Stochastic Inequalities, ed. Shaked, M. and Tong, Y. L., pp. 235252, IMS Lecture Notes Series.
Malice, M. P. and Kryscio, R. J. (1989) On the role of variable incubation periods in simple epidemic models. I.M.A. J. Math. Appl. Med. Biol. 6, 233242.
Malice, M. P. and Lefevre, Cl. (1988) On some effects of variability in the Weiss epidemic model. Commun. Stat.-Theory Methods 17, 37233731.
Martin-Löf, A. (1986) Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265282.
Picard, Ph. and Lefevre, Cl. (1990) A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 269294.
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.
Shenton, L. R. (1981) A reinforcement–depletion urn problem. I. Basic theory. Bull. Math. Biol. 43, 327340.
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.


MSC classification

An unusual stochastic order relation with some applications in sampling and epidemic theory

  • Claude Lefevre (a1) and Philippe Picard (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed