  We determine the distributions of some random variables related to a simple
model of an epidemic with contact tracing and cluster isolation. This enables
us to apply general limit theorems for super-critical Crump-Mode-Jagers
branching processes. Notably, we compute explicitly the asymptotic proportion
of isolated clusters with a given size amongst all isolated clusters,
conditionally on survival of the epidemic. Somewhat surprisingly, the latter
differs from the distribution of the size of a typical cluster at the time of
its detection; and we explain the reasons behind this seeming paradox.
