The effective reproduction number: Convexity, concavity and invariance

Abstract

Motivated by the question of optimal vaccine allocation strategies in heterogeneous population for epidemic models, we study various properties of the effective reproduction number. In the simplest case, given a fixed non-negative matrix $K$, this corresponds mathematically to the study of the spectral radius $R_e(\eta )$ of the matrix product $\mathrm{Diag}(\eta )K$, as a function of $\eta \in {\mathbb{R}}_{+}^n$. The matrix $K$ and the vector $\eta$ can be interpreted as a next-generation operator and a vaccination strategy. This can be generalized in an infinite-dimensional case where the matrix $K$ is replaced by a positive integral compact operator, which is composed with a multiplication by a non-negative function $\eta$. We give sufficient conditions for the function $R_e$ to be convex or a concave. Eventually, we provide equivalence properties on models which ensure that the function $R_e$ is unchanged.