We study the diffusion of epidemics on networks that are partitioned intolocal communities. The gross structure of hierarchical networks of this kindcan be described by a quotient graph. The rationale of this approach is thatindividuals infect those belonging to the same community with higherprobability than individuals in other communities. In community models thenodal infection probability is thus expected to depend mainly on theinteraction of a few, large interconnected clusters. In this work, we describethe epidemic process as a continuous-time individual-basedsusceptible-infected-susceptible (SIS) model using a first-order mean-fieldapproximation. A key feature of our model is that the spectral radius of thissmaller quotient graph (which only captures the macroscopic structure of thecommunity network) is all we need to know in order to decide whether theoverall healthy-state defines a globally asymptotically stable or an unstableequilibrium. Indeed, the spectral radius is related to the epidemic thresholdof the system. Moreover we prove that, above the threshold, anothersteady-state exists that can be computed using a lower-dimensional dynamicalsystem associated with the evolution of the process on the quotient graph. Ourinvestigations are based on the graph-theoretical notion of equitable partitionand of its recent and rather flexible generalization, that of almost equitablepartition.
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