  This paper is concerned with the growth rate of SIR
(Susceptible-Infectious-Recovered) epidemics with general infectious period
distribution on random intersection graphs. This type of graph is characterized
by the presence of cliques (fully connected subgraphs). We study epidemics on
random intersection graphs with a mixed Poisson degree distribution and show
that in the limit of large population sizes the number of infected individuals
grows exponentially during the early phase of the epidemic, as is generally the
case for epidemics on asymptotically unclustered networks. The Malthusian
parameter is shown to satisfy a variant of the classical Euler-Lotka equation.
To obtain these results we construct a coupling of the epidemic process and a
continuous-time multitype branching process, where the type of an individual is
(essentially) given by the length of its infectious period. Asymptotic results
are then obtained via an embedded single-type Crump-Mode-Jagers branching
process.
