Convex Optimization of the Basic Reproduction Number

The basic reproduction number R0 is a fundamental quantity in epidemiological modeling, reflecting the typical number of secondary infections that arise from a single infected individual. While R0 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 R0: a stability characterization and a geometric program characterization. The geometric program characterization allows us to write R0-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 R0 instead of the spectral abscissa of the Jacobian matrix (a common target in the controls literature) leads to qualitatively different solutions.