  A random network model which allows for tunable, quite general forms of
clustering, degree correlation and degree distribution is defined. The model is
an extension of the configuration model, in which stubs (half-edges) are paired
to form a network. Clustering is obtained by forming small completely connected
subgroups, and positive (negative) degree correlation is obtained by connecting
a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR
(Susceptible -> Infective -> Recovered) epidemic model is defined on this
network. Asymptotic properties of both the network and the epidemic, as the
population size tends to infinity, are derived: the degree distribution, degree
correlation and clustering coefficient, as well as a reproduction number $R_*$,
the probability of a major outbreak and the relative size of such an outbreak.
The theory is illustrated by Monte Carlo simulations and numerical examples.
The main findings are that clustering tends to decrease the spread of disease,
the effect of degree correlation is appreciably greater when the disease is
close to threshold than when it is well above threshold and disease spread
broadly increases with degree correlation $\rho$ when $R_*$ is just above its
threshold value of one and decreases with $\rho$ when $R_*$ is well above one.
