  In this paper we consider a model for the spread of a stochastic SIR
(Susceptible $\to$ Infectious $\to$ Recovered) epidemic on a network of
individuals described by a random intersection graph. Individuals belong to a
random number of cliques, each of random size, and infection can be transmitted
between two individuals if and only if there is a clique they both belong to.
Both the clique sizes and the number of cliques an individual belongs to follow
mixed Poisson distributions. An infinite-type branching process approximation
(with type being given by the length of an individual's infectious period) for
the early stages of an epidemic is developed and made fully rigorous by proving
an associated limit theorem as the population size tends to infinity. This
leads to a threshold parameter $R_*$, so that in a large population an epidemic
with few initial infectives can give rise to a large outbreak if and only if
$R_*>1$. A functional equation for the survival probability of the
approximating infinite-type branching process is determined; if $R_*\le1$, this
equation has no nonzero solution, while if $R_*>1$, it is shown to have
precisely one nonzero solution. A law of large numbers for the size of such a
large outbreak is proved by exploiting a single-type branching process that
approximates the size of the susceptibility set of a typical individual.
