The Sooner, the Better? Optimal Vaccination Policy with Limited Vaccine Supply

We study the optimal single-dose vaccination policy during an infectious disease outbreak, considering both limited vaccine supply and imperfect efficacy, which provides partial immunity to each vaccinated individual. The inclusion of imperfect efficacy introduces an additional compartment to the celebrated Susceptible-Infectious-Recovered (SIR) model, giving rise to an infinite horizon nonlinear optimal control problem. We derive a closed-form expression for the optimal vaccination policy under infinite administrative capacity and establish theoretical structures of the optimal policy under finite capacity. Our results suggest that delaying the start of the vaccination process may be optimal, especially when the vaccine is less effective, the vaccine supply is more limited, and the disease is more infectious. We identify key terms as an integral part of the analysis that characterize the individual-level benefit of vaccination and, importantly, provide insights into the intuition behind vaccination delay. Building on these theoretical findings, our numerical study verifies these insights based on sensitivity results. We demonstrate the significant benefit of delay in reducing the total number of infections compared to policies without delay. Furthermore, we extend our analysis to alternative model dynamics and objective functions, confirming that our key findings remain robust and our methodology applicable across different settings. Additionally, we develop a general framework and provide guidance for applying the methodology to broader epidemic settings. Our study contributes to the methodological framework to analyze optimal vaccination policies for infectious disease control, and uncovers an important policy structure — vaccination delay — that has been overlooked in the literature but has important implications for practice.