VACCINATING ACCORDING TO THE MAXIMAL ENDEMIC EQUILIBRIUM ACHIEVES HERD IMMUNITY

We consider the simple epidemiological SIS model for a general heterogeneous population introduced by Lajmanovich and Yorke (1976) in finite dimensions, and its infinite-dimensional generalization we introduced in previous works. In this model the basic reproducing number R₀ is given by the spectral radius of an integral operator. If the basic reproducing number R₀ is greater than 1 (R₀ > 1), then there exists a maximal endemic equilibrium. In this very general heterogeneous SIS model, we prove that vaccinating according to the profile of this maximal endemic equilibrium ensures herd immunity. Moreover, this vaccination strategy is critical: the resulting effective reproduction number is exactly equal to one. As an application, we estimate in an example from Britton, Ball, and Trapman (2020) that if R₀ = 2 in an age-structured community with mixing rates fitted to social activity, applying this strategy would require approximately 29% fewer vaccine doses than the strategy which consists in vaccinating uniformly a proportion 1 − 1/R₀ of the population. From a dynamical systems point of view, we prove that the nonmaximality of an equilibrium g is equivalent to its linear instability in the original dynamics, and to the linear instability of the disease-free state in the modified dynamics where we vaccinate according to g.