  Using the continuous-time susceptible-infected-susceptible (SIS) model on
networks, we investigate the problem of inferring the class of the underlying
network when epidemic data is only available at population-level (i.e. the
number of infected individuals at a finite set of discrete times of a single
realisation of the epidemic), the only information likely to be available in
real world settings. To tackle this, epidemics on networks are approximated by
a Birth-and-Death process which keeps track of the number of infected nodes at
population level. The rates of this surrogate model encode both the structure
of the underlying network and disease dynamics. We use extensive simulations
over Regular, Erd\H{o}s-R\'enyi and Barab\'asi-Albert networks to build network
class-specific priors for these rates. % show that different well-known network
classes map onto distinct regions of the parameter space of this model. We then
use Bayesian model selection to recover the most likely underlying network
class, based only on a single realisation of the epidemic. We show that the
proposed methodology yields good results on both synthetic and real-world
networks.
