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We determine the distributions of some random variables related to a simplemodel of an epidemic with contact tracing and cluster isolation. This enablesus to apply general limit theorems for super-critical Crump-Mode-Jagersbranching processes. Notably, we compute explicitly the asymptotic proportionof isolated clusters with a given size amongst all isolated clusters,conditionally on survival of the epidemic. Somewhat surprisingly, the latterdiffers from the distribution of the size of a typical cluster at the time ofits detection; and we explain the reasons behind this seeming paradox.

A model for an epidemic with contact tracing and cluster isolation, and a detection paradox