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We consider the problem of optimal allocation strategies for a (perfect)vaccine in an infinite-metapopulation model (including SIS, SIR, SEIR,. . .),when the loss function is given by the effective reproduction number Re, whichis defined as the spectral radius of the effective next generation matrix (infinite dimension) or more generally of the effective next generation operator(in infinite dimension). We give sufficient conditions for Re to be a convex ora concave function of the vaccination strategy. Then, following a previouswork, we consider the bi-objective problem of minimizing simultaneously thecost and the loss of the vaccination strategies. In particular, we prove that acordon sanitaire might not be optimal, but it is still better than the "worst"vaccination strategies. Inspired by the graph theory, we compute the minimalcost which ensures that no infection occurs using independent sets. UsingFrobenius decomposition of the whole population into irreduciblesub-populations, we give some explicit formulae for optimal ("best" and"worst") vaccinations strategies. Eventually, we provide equivalence propertieson models which ensure that the function Re is unchanged.

Effective reproduction number: Convexity, invariance and cordons sanitaires