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We consider the simple epidemiological SIS model for a general heterogeneouspopulation introduced by Lajmanovich and Yorke (1976) in finite dimension, andits infinite dimensional generalization we introduced in previous works. Inthis model the basic reproducing number $R_0$ is given by the spectral radiusof an integral operator. If $R_0>1$, then there exists a maximal endemicequilibrium. In this very general heterogeneous SIS model, we prove thatvaccinating according to the profile of this maximal endemic equilibriumensures herd immunity. Moreover, this vaccination strategy is critical: theresulting effective reproduction number is exactly equal to one. As anapplication, we estimate that if $R_0 = 2$ in an age-structured community withmixing rates fitted to social activity, applying this strategy would requireapproximately 29% less vaccine doses than the strategy which consists invaccinating uniformly a proportion $1 - 1/R_0$ of the population.  From a dynamical systems point of view, we prove that the non-maximality ofan equilibrium $g$ is equivalent to its linear instability in the originaldynamics, and to the linear instability of the disease-free state in themodified dynamics where we vaccinate according to $g$.

Vaccinating according to the maximal endemic equilibrium achieves herd immunity