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 \emph{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.

20 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:2110.12693