  We consider the bi-objective problem of allocating doses of a (perfect)
vaccine to an infinite-dimensional metapopulation in order to minimize
simultaneously the vaccination cost and the effective reproduction number
$R_e$, which is defined as the spectral radius of the effective next-generation
operator.
  In this general framework, we prove that a cordon sanitaire, that is, a
strategy that effectively disconnects the non-vaccinated population, might not
be optimal, but it is still better than the "worst" vaccination strategies.
Inspired by graph theory, we also compute the minimal cost which ensures that
no infection occurs using independent sets. Using Frobenius decomposition of
the whole population into irreducible sub-populations, we give some explicit
formulae for optimal ("best" and "worst") vaccinations strategies. Eventually,
we provide some sufficient conditions for a scaling of an optimal strategy to
still be optimal.
