  We study the diffusion of epidemics on networks that are partitioned into
local communities. The gross structure of hierarchical networks of this kind
can be described by a quotient graph. The rationale of this approach is that
individuals infect those belonging to the same community with higher
probability than individuals in other communities. In community models the
nodal infection probability is thus expected to depend mainly on the
interaction of a few, large interconnected clusters. In this work, we describe
the epidemic process as a continuous-time individual-based
susceptible-infected-susceptible (SIS) model using a first-order mean-field
approximation. A key feature of our model is that the spectral radius of this
smaller quotient graph (which only captures the macroscopic structure of the
community network) is all we need to know in order to decide whether the
overall healthy-state defines a globally asymptotically stable or an unstable
equilibrium. Indeed, the spectral radius is related to the epidemic threshold
of the system. Moreover we prove that, above the threshold, another
steady-state exists that can be computed using a lower-dimensional dynamical
system associated with the evolution of the process on the quotient graph. Our
investigations are based on the graph-theoretical notion of equitable partition
and of its recent and rather flexible generalization, that of almost equitable
partition.
