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

Abstract

We study the optimal single-dose vaccination policy in an infectious disease outbreak, considering both the limited total supply of the vaccine and its 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. To facilitate theoretical analysis, we propose a novel variable transformation that converts the problem into an equivalent form with linear dynamics. Leveraging this transformation, we derive a closed-form expression for the optimal vaccination policy under infinite administrative capacity and establish theoretical structures for 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. The optimality of delay occurs because the individual-level benefit of vaccination with imperfect vaccines, in terms of the reduction of infection risk over the course of a disease outbreak, is non-monotonic in time; hence maximizing the overall benefit of vaccination requires a strategic allocation of limited vaccine supply over time. 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. Our study contributes to the methodology of solving optimal control problems in the context of infectious disease outbreaks. Moreover, it highlights scenarios in which delaying the start of the vaccination process can be beneficial -- an aspect of vaccination policy design that has been overlooked in the literature but has important implications for practice.

Note:

Funding Information: We have no funding.

Conflict of Interests: We have no conflict of interest to declare.

Keywords: Vaccination, SIR model, Resource allocation, Optimal control

JEL Classification: C61, I18