Dose-optimal vaccine allocation over multiple populations

For a large number of infectious diseases, vaccination is the most effective way to prevent an epidemic. However, the vaccine stockpile is hardly ever sufficient to treat the entire population, which brings about the challenge of vaccine allocation. To aid decision makers facing this challenge, we provide insights into the structure of this problem. We first investigate the dependence of health benefit on the fraction of people that receive vaccination, where we define health benefit as the total number of people that escape infection. We start with the seminal SIR compartmental model. Using implicit function analysis, we prove the existence of a unique vaccination fraction that maxi- mizes the health benefit per dose of vaccine, and that the health benefit per dose of vaccine decreases monotonically when moving away from this fraction in either direc- tion. Surprisingly, this fraction does not coincide with the so-called critical vaccination coverage that has been advocated in literature. We extend these insights to other compartmental models such as the SEIR model. These results allow us to provide new insights into vaccine allocation to multiple non-interacting or weakly interacting populations. We explain the counter-intuitive switching behavior of optimal allocation. We show that allocations that maximize health benefits are rarely equitable, while equitable allocations may be significantly non-optimal.