Vaccinating according to the maximal endemic equilibrium achieves herd immunity

TL;DR

This paper demonstrates that vaccinating according to the maximal endemic equilibrium in a heterogeneous SIS model guarantees herd immunity and is more efficient than uniform vaccination, reducing vaccine needs by about 29% in certain cases.

Contribution

It proves that vaccination based on the maximal endemic equilibrium profile ensures herd immunity and establishes its critical nature in heterogeneous populations.

Findings

Vaccination according to the maximal endemic equilibrium achieves herd immunity.

This strategy reduces vaccine doses by approximately 29% compared to uniform vaccination.

The non-maximality of an equilibrium correlates with its linear instability.

Abstract

We consider the simple epidemiological SIS model for a general heterogeneous population introduced by Lajmanovich and Yorke (1976) in finite dimension, and its infinite dimensional generalization we introduced in previous works. In this model the basic reproducing number $R_0$ is given by the spectral radius of an integral operator. If $R_0>1$, 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 that if $R_0 = 2$ in an age-structured community with mixing rates fitted to social activity, applying this strategy would require approximately 29% less vaccine doses than the strategy which consists in vaccinating uniformly a proportion $1 - 1/R_0$ of the population. From a dynamical systems point of view, we prove that the non-maximality 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 $g$.