We extrapolate from the exact master equations of epidemic dynamics on fully connected
graphs to non-fully connected by keeping the size of the state space N + 1, where
N is the
number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary
differential equations) where the challenge is to compute/approximate analytically the
transmission rates. We show that this is possible for graphs with arbitrary degree
distributions built according to the configuration model. Numerical tests confirm that:
(a) the agreement of the approximate ODEs system with simulation is excellent and (b) that
the approach remains valid for clustered graphs with the analytical calculations of the
transmission rates still pending. The marked reduction in state space gives good results,
and where the transmission rates can be analytically approximated, the model provides a
strong alternative approximate model that agrees well with simulation. Given that the
transmission rates encompass information both about the dynamics and graph properties, the
specific shape of the curve, defined by the transmission rate versus the number of
infected nodes, can provide a new and different measure of network structure, and the
model could serve as a link between inferring network structure from prevalence or
incidence data.