  We consider a multi-population epidemic model with one or more (almost)
isolated communities and one mobile community. Each of the isolated communities
has contact within itself and, in addition, contact with the outside world but
only through the mobile community. The contact rate between the mobile
community and the other communities is assumed to be controlled. We first
derive a multidimensional ordinary differential equation (ODE) as a mean-field
fluid approximation to the process of the number of infected nodes, after
appropriate scaling. We show that the approximation becomes tight as the sizes
of the communities grow. We then use a singular perturbation approach to reduce
the dimension of the ODE and identify an optimal control policy on this system
over a fixed time horizon via Pontryagin's minimum principle. We then show that
this policy is close to optimal, within a certain class, on the original
problem for large enough communities. From a phenomenological perspective, we
show that the epidemic may sustain in time in all communities (and thus the
system has a nontrivial metastable regime) even though in the absence of the
mobile nodes the epidemic would die out quickly within each of the isolated
communities.
