  In this paper we consider SIR epidemics on random graphs with clustering. To
incorporate group structure of the underlying social network, we use a
generalized version of the configuration model in which each node is a member
of a specified number of triangles. SIR epidemics on this type of graph have
earlier been investigated under the assumption of homogeneous infectivity and
also under the assumption of Poisson transmission and recovery rates. We extend
known results from literature by relaxing the assumption of homogeneous
infectivity. An important special case of the epidemic model analyzed in this
paper is epidemics in continuous time with arbitrary infectious period
distribution. We use branching process approximations of the spread of the
disease to provide expressions for the basic reproduction number R0, the
probability of a major outbreak and the expected final size. In addition, the
impact of random vaccination with a perfect vaccine on the final outcome of the
epidemic is investigated. We find that, for this particular model, R0 equals
the perfect vaccine-associated reproduction number. Generalizations to groups
larger than three are discussed briefly.
