  Infectious diseases typically spread over a contact network with millions of
individuals, whose sheer size is a tremendous challenge to analysing and
controlling an epidemic outbreak. For some contact networks, it is possible to
group individuals into clusters. A high-level description of the epidemic
between a few clusters is considerably simpler than on an individual level.
However, to cluster individuals, most studies rely on equitable partitions, a
rather restrictive structural property of the contact network. In this work, we
focus on Susceptible-Infected-Susceptible (SIS) epidemics, and our contribution
is threefold. First, we propose a geometric approach to specify all networks
for which an epidemic outbreak simplifies to the interaction of only a few
clusters. Second, for the complete graph and any initial viral state vectors,
we derive the closed-form solution of the nonlinear differential equations of
the N-Intertwined Mean-Field Approximation (NIMFA) of the SIS process. Third,
by relaxing the notion of equitable partitions, we derive low-complexity
approximations and bounds for epidemics on arbitrary contact networks. Our
results are an important step towards understanding and controlling epidemics
on large networks.
