EXPRESS: Partial or Full Doses First? Vaccine Allocation Under Limited Supply

Vaccines are proven highly effective in curbing disease spread in the face of sudden large-scale infectious diseases. However, the challenge of limited vaccine supplies during initial mass production or in certain regions such as Africa poses a significant hurdle. To address this issue, one potential solution is the implementation of a fractional dosing approach, whereby individuals receive partial doses of the vaccine to achieve partial immunity. In this paper, we investigate the optimal vaccine allocation strategy in scenarios where the vaccine supply is scarce. By exploring the trade-offs between administering fractional and full dosages, we formulate the problem as an optimal control problem under disease transmission dynamics to minimize the total number of infections over a planning horizon. Specifically, we extend the SIR model by introducing two new compartments for partially and fully vaccinated individuals. Under mild conditions, we demonstrate that an attainable optimal vaccine allocation strategy exhibits a "bang-bang" structure, suggesting that either the "Partial Doses First'' (PDF) or the "Full Doses First" (FDF) allocation policy achieves optimality at any time. Furthermore, we compare the committed PDF and FDF policies for the entire planning horizon and identify a critical threshold on the fractional-dose vaccine efficacy, such that when the efficacy is above (resp., below) this threshold, choosing the committed PDF (resp., FDF) policy is better. Using COVID-19 vaccination data, we calculate thresholds for different countries in January 2021 to recommend to governments whether the priority should be given to allocating the first dose or to prioritizing the allocation of both first and second doses to the population. We also illustrate the versatility of our proposed scarce capacity allocation model in addressing a broad spectrum of resource allocation problems under diffusion dynamics, for which partial resource allocation can still yield some, though not full, benefits, and the optimal allocation decision involves balancing the coverage and protection level.