DOSE-OPTIMAL VACCINE ALLOCATION OVER MULTIPLE POPULATIONS
Method(s): A sensible objective for vaccine allocation is maximizing the health benefit, defined in this paper as the number of people that escape infection. In literature this objective is often achieved by evaluating the eventual outcome of alternative allocations using numerical methods or simulation. This approach does not give a high-level explanation why certain allocations yield a higher health benefit. This is especially problematic because the resulting allocations are often inequitable and behave counter-intuitively. We propose to apply analytical methods to vaccine allocation to obtain a high-level understanding of these inequitable and seemingly counter-intuitive outcomes. Thereto, we first investigate the dependence of health benefit on the fraction of people that receive vaccination. We study the seminal SIR compartmental model to model the total health benefit as a function of the vaccination fraction that is used.
Result(s): Using implicit function analysis, we prove that the health benefits as a function of the vaccination fraction have increasing returns to scale for small vaccination fractions and decreasing returns to scale for vaccination fractions larger than a certain threshold. This implies the existence of a unique vaccination fraction that maximizes the health benefit per dose of vaccine, which we refer to as the `dose-optimal vaccination fraction'. We show that the health benefit per dose of vaccine decreases monotonically when moving away from this fraction in either direction. Surprisingly, this fraction does not coincide with the so-called critical vaccination coverage that has been advocated in literature. We show that the optimal allocation is governed by the dose-optimal vaccination fraction, as vaccinating with this fraction is the most effective way of using the available resources.
Conclusion(s): These results allow us to provide new insights into vaccine allocation to multiple non-interacting and weakly interacting populations. We explain the counter-intuitive switching type behavior of optimal allocations. We also show that allocations that maximize health benefits are rarely equitable, while equitable allocations may be significantly non-optimal.