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Article Dans Une Revue Journal of the European Mathematical Society Année : 2024

The effective reproduction number: convexity, concavity and invariance

Résumé

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.

Dates et versions

hal-04231412 , version 1 (06-10-2023)

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Citer

Jean-François Delmas, Dylan Dronnier, Pierre-André Zitt. The effective reproduction number: convexity, concavity and invariance. Journal of the European Mathematical Society, 2024, ⟨10.4171/jems/1431⟩. ⟨hal-04231412⟩
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