Convex Optimization of the Basic Reproduction Number

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 a case study of allocating vaccines and antidotes, 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.