Convex Optimization of the Basic Reproduction Number

• Published in: IEEE transactions on automatic control Vol. 68; no. 7; pp. 4398 - 4404
• Main Authors: Smith, Kevin D., Bullo, Francesco
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Compartmental models; convex optimization; Convexity; Diseases; Epidemics; geometric programming; Jacobi matrix method; Jacobian matrices; Jacobian matrix; optimal resource allocation; Optimization; Resource allocation; Resource management; Sociology; Statistics
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3212012

The basic reproduction number <inline-formula><tex-math notation="LaTeX">R_{0}</tex-math></inline-formula> is a fundamental quantity in epidemiological modeling, reflecting the typical number of secondary infections that arise from a single infected individual. While <inline-formula><tex-math notation="LaTeX">R_{0}</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">R_{0}</tex-math></inline-formula>: a stability characterization and a geometric program characterization. The geometric program characterization allows us to write <inline-formula><tex-math notation="LaTeX">R_{0}</tex-math></inline-formula>-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 <inline-formula><tex-math notation="LaTeX">R_{0}</tex-math></inline-formula> instead of the spectral abscissa of the Jacobian matrix (a common target in the controls literature) leads to qualitatively different solutions.