  We consider a threshold epidemic model on a clustered random graph with
overlapping communities. In other words, our epidemic model is such that an
individual becomes infected as soon as the proportion of her infected neighbors
exceeds the threshold q of the epidemic. In our random graph model, each
individual can belong to several communities. The distributions for the
community sizes and the number of communities an individual belongs to are
arbitrary.
  We consider the case where the epidemic starts from a single individual, and
we prove a phase transition (when the parameter q of the model varies) for the
appearance of a cascade, i.e. when the epidemic can be propagated to an
infinite part of the population. More precisely, we show that our epidemic is
entirely described by a multi-type (and alternating) branching process, and
then we apply Sevastyanov's theorem about the phase transition of multi-type
Galton-Watson branching processes. In addition, we compute the entries of the
matrix whose largest eigenvalue gives the phase transition.
