Convex Optimization of the Basic Reproduction Number

TL;DR

This paper introduces convex optimization methods for the basic reproduction number $R_0$, enabling optimal resource allocation in epidemiology through novel geometric programming techniques.

Contribution

It provides new stability and geometric program characterizations of $R_0$, facilitating convex optimization approaches for epidemic control strategies.

Findings

Targeting $R_0$ yields different optimal solutions compared to spectral abscissa methods.

Geometric program formulation simplifies $R_0$-constrained optimization problems.

Numerical examples demonstrate effective vaccine and antidote allocation strategies.

Abstract

The basic reproduction number $R_0$ is a fundamental quantity in epidemiological modeling, reflecting the typical number of secondary infections that arise from a single infected individual. While $R_0$ is widely known to scientists, policymakers, and the general public, it has received comparatively little attention in the controls community. This note provides two novel characterizations of $R_0$: a stability characterization and a geometric program characterization. The geometric program characterization allows us to write $R_0$-constrained and budget-constrained optimal resource allocation problems as geometric programs, which are easily transformed into convex optimization problems. We apply these programs to allocating vaccines and antidotes in numerical examples, finding that targeting $R_0$ instead of the spectral abscissa of the Jacobian matrix (a common target in the controls literature) leads to qualitatively different solutions.