  Infectious disease remains, despite centuries of work to control and mitigate
its effects, a major problem facing humanity. This paper reviews the
mathematical modelling of infectious disease epidemics on networks, starting
from the simplest Erdos-Renyi random graphs, and building up structure in the
form of correlations, heterogeneity and preference, paying particular attention
to the links between random graph theory, percolation and dynamical systems
representing transmission. Finally, the problems posed by networks with a large
number of short closed looks are discussed.
