  We predict the future course of ongoing susceptible-infected-susceptible
(SIS) epidemics on regular, Erd\H{o}s-R\'{e}nyi and Barab\'asi-Albert networks.
It is known that the contact network influences the spread of an epidemic
within a population. Therefore, observations of an epidemic, in this case at
the population-level, contain information about the underlying network. This
information, in turn, is useful for predicting the future course of an ongoing
epidemic. To exploit this in a prediction framework, the exact high-dimensional
stochastic model of an SIS epidemic on a network is approximated by a
lower-dimensional surrogate model. The surrogate model is based on a
birth-and-death process; the effect of the underlying network is described by a
parametric model for the birth rates. We demonstrate empirically that the
surrogate model captures the intrinsic stochasticity of the epidemic once it
reaches a point from which it will not die out. Bayesian parameter inference
allows for uncertainty about the model parameters and the class of the
underlying network to be incorporated directly into probabilistic predictions.
An evaluation of a number of scenarios shows that in most cases the resulting
prediction intervals adequately quantify the prediction uncertainty. As long as
the population-level data is available over a long-enough period, even if not
sampled frequently, the model leads to excellent predictions where the
underlying network is correctly identified and prediction uncertainty mainly
reflects the intrinsic stochasticity of the spreading epidemic. For predictions
inferred from shorter observational periods, uncertainty about parameters and
network class dominate prediction uncertainty. The proposed method relies on
minimal data and is numerically efficient, which makes it attractive either as
a standalone inference and prediction scheme or in conjunction with other
methods.
