  We formalize and study the problem of optimal allocation strategies for a
(perfect) vaccine in the infinite-dimensional SIS model. The question may be
viewed as a bi-objective minimization problem, where one tries to minimize
simultaneously the cost of the vaccination, and a loss that may be either the
effective reproduction number, or the proportion of the infected population in
the endemic state. We prove the existence of Pareto optimal strategies,
describe the corresponding Pareto frontier in both cases, and study its
convexity and stability properties. We also show that vaccinating according to
the profile of the endemic state is a critical allocation, in the sense that,
if the initial reproduction number is larger than 1, then this vaccination
strategy yields an effective reproduction number equal to 1.
