  We consider an infectious disease spreading along the edges of a network
which may have significant clustering. The individuals in the population have
heterogeneous infectiousness and/or susceptibility. We define the
out-transmissibility of a node to be the marginal probability that it would
infect a randomly chosen neighbor given its infectiousness and the distribution
of susceptibility. For a given distribution of out-transmissibility, we find
the conditions which give the upper [or lower] bounds on size and probability
of an epidemic, under weak assumptions on the transmission properties, but very
general assumptions on the network. We find similar bounds for a given
distribution of in-transmissibility (the marginal probability of being infected
by a neighbor). We also find conditions giving global upper bounds on size and
probability. The distributions leading to these bounds are network-independent.
In the special case of networks with high girth (locally tree-like), we are
able to prove stronger results. In general the probability and size of
epidemics are maximal when the population is homogeneous and minimal when the
variance of in- or out-transmissibility is maximal.
