The effective reproduction number: convexity, concavity and invariance

TL;DR

This paper analyzes the mathematical properties of the effective reproduction number in epidemic models, focusing on convexity, concavity, and invariance, to inform optimal vaccination strategies in heterogeneous populations.

Contribution

It provides new conditions under which the effective reproduction number function is convex, concave, or invariant, extending to infinite-dimensional models with integral operators.

Findings

Identifies conditions for convexity and concavity of R_e.

Establishes invariance properties of R_e under certain model transformations.

Extends analysis to infinite-dimensional integral operator models.

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.