  Random networks with specified degree distributions have been proposed as
realistic models of population structure, yet the problem of dynamically
modeling SIR-type epidemics in random networks remains complex. I resolve this
dilemma by showing how the SIR dynamics can be modeled with a system of three
nonlinear ODE's. The method makes use of the probability generating function
(PGF) formalism for representing the degree distribution of a random network
and makes use of network-centric quantities such as the number of edges in a
well-defined category rather than node-centric quantities such as the number of
infecteds or susceptibles. The PGF provides a simple means of translating
between network and node-centric variables and determining the epidemic
incidence at any time. The theory also provides a simple means of tracking the
evolution of the degree distribution among susceptibles or infecteds. The
equations are used to demonstrate the dramatic effects that the degree
distribution plays on the final size of an epidemic as well as the speed with
which it spreads through the population. Power law degree distributions are
observed to generate an almost immediate expansion phase yet have a smaller
final size compared to homogeneous degree distributions such as the Poisson.
The equations are compared to stochastic simulations, which show good agreement
with the theory. Finally, the dynamic equations provide an alternative way of
determining the epidemic threshold where large-scale epidemics are expected to
occur, and below which epidemic behavior is limited to finite-sized outbreaks.
