Vaccine Allocation under Scarcity
Multi-objective allocation · lives saved · equity
In the early months of a pandemic, vaccines are scarce: millions of doses, billions of people, no way to vaccinate everyone at once. Who goes first? The decision is an optimisation under scarcity problem with multiple competing objectives: maximise lives saved (target high-mortality groups), minimise transmission (target high-contact groups), protect healthcare capacity (target healthcare workers), achieve demographic equity (balance across race / income / geography). These objectives systematically conflict. Bertsimas et al. (2022) formalised the COVID-era allocation problem as multi-objective LP; the ethical framework literature (Emanuel et al. 2020) gave the normative vocabulary.
Problem context
Supply-constrained allocation, multi-objective ethics
In March 2020, as COVID-19 spread, every country faced the same decision: with supply arriving gradually and demand exceeding supply by orders of magnitude, who gets vaccinated first? The US CDC Advisory Committee on Immunization Practices (ACIP) and the WHO SAGE Working Group both worked through the problem with ethics-first frameworks. The pivotal Emanuel, Persad, Upshur, et al. (2020) New England Journal of Medicine paper proposed four ethical values: maximising benefits, treating people equally, promoting instrumental value (those whose work has downstream benefits), and priority for the worst-off.
Bertsimas and colleagues (2022) formalised the operational problem as a large-scale MILP allocating doses across regions and priority groups over time, taking SIR / SEIR dynamics, age-stratified mortality risk, and vaccine efficacy as inputs. Their DELPHI-V-OPT framework combined epidemic modelling with optimisation and was used in the early US pandemic response. Related work by Matrajt et al. (2021) compared age-prioritised vs age-and-contact-prioritised strategies, finding that mortality minimisation generally favours age prioritisation while transmission minimisation favours young adult / high-contact targeting.
Globally, the COVAX facility attempted equity-across-countries allocation but was badly under-funded; Duch et al. (2021)'s Nature study on public preferences across 13 countries showed strong support for healthcare-worker priority universally but diverging views on other groups. Global vaccine inequity remains a first-order failure — not of optimisation, but of the political and financial structures around it.
Multi-stakeholder framing
Mathematical formulation
Multi-objective allocation over regions × priority groups × time
| Symbol | Definition | Domain |
|---|---|---|
| R | Set of regions (states / countries) | |
| G | Set of priority groups (age × risk) | |
| T | Time horizon (weeks) | |
| Nrg | Population of group g in region r | ℕ |
| μrg | Expected mortality rate if infected (age × comorbidity) | [0, 1] |
| βrg | Average contacts per day (transmission weight) | ℝ⁺ |
| St | Vaccine supply arriving at time t | ℕ |
| η | Vaccine efficacy (reduction in mortality / transmission) | [0, 1] |
| xrgt | Doses allocated to group g in region r at time t | ℝ⁺₀ |
Σrg μrg · Irg // expected deaths (SIR-driven)
Σr max(0, incidencer − ICU_capr) // healthcare overload
Gini(coverage by demographic) // equity term
s.t. Σrg xrgt ≤ St ∀ t // weekly supply
Σt xrgt ≤ Nrg ∀ r, g // cannot over-vaccinate
capacityrt ≤ logistical delivery capacity // admin throughput
priority-set precedence: lower-priority groups wait until higher have threshold coverage
Irg(t+1) = SEIR-state-transition(Irg(t), xrgt, η, βrg)
Complexity Coupled with SEIR dynamics this is a large nonlinear MILP. Bertsimas et al. (2022) use time-discretised LP relaxation plus rollout for tractability. Exact solutions infeasible at national scale; approximation and scenario-based methods dominate.
The objectives conflict. Mortality-minimising allocation favours elderly and immunocompromised. Transmission-minimising allocation favours high-contact young adults (students, service workers). Equity-maximising allocation spreads doses across demographic groups and geographies. Healthcare-protection allocation fronts healthcare workers. These are distinct solutions; no single weighting is objectively correct.
Interactive solver
Allocate supply across regions and priority groups under competing objectives
Vaccine Allocation Solver
Solution interpretation
Three objectives, three different answers
Mortality-minimisation concentrates early doses in high-mortality (elderly, immunocompromised) groups. This was the dominant approach in most Western countries during COVID-19 rollouts in 2020-2021.
Transmission-minimisation targets high-contact demographics (young adults, service workers, students). When the vaccine blocks transmission well, this can save more lives in indirect protection than the mortality-minimising approach saves directly — but timing and efficacy have to be right.
Equity-maximising spreads doses evenly across demographic groups. It accepts higher aggregate mortality in exchange for no systematic under-coverage. Many countries used a version of this as a hard constraint (minimum coverage per region) rather than a sole objective.
Weighted combinations are the operational default. The weights are a political choice, not a technical one. Showing the frontier of tradeoffs is more useful to decision-makers than recommending a single point.
Limitations & Critique
What the model does not see
Global inequity is the binding failure
Optimisation within a wealthy country's supply cannot solve the fact that lower-income countries received doses far later. COVAX fell dramatically short of its targets. No allocation algorithm compensates for global supply inequity — that's a political and financing problem.
Hesitancy is endogenous
Vaccine hesitancy varies by demographic and responds to allocation strategy. Groups that feel de-prioritised or coerced develop lasting mistrust. Communication and community engagement are not separable from the allocation decision.
Epidemiological parameters are uncertain
μ, β, η are estimates with substantial error bands, especially early in a pandemic. Optimal allocation under one estimate can be far from optimal under another. Robust and scenario-based formulations exist; static deterministic models overstate confidence.
Vaccines ≠ sterilising immunity
Models that assume vaccines prevent all onward transmission over-predict benefit of transmission- focused allocation. COVID vaccines reduced but did not eliminate transmission; strategies calibrated on sterilising-immunity assumptions performed worse than expected.
DALY / QALY weighting is contested
Quality-adjusted life-year calculations systematically value young lives more than old; disability- adjusted calculations embed further contested choices. Using μrg without acknowledging the weighting it implies is a silent ethical commitment.
Delivery logistics are often the binding constraint
Allocation planning assumes vaccines reach eligible recipients. In practice, cold-chain failures, rural-access gaps, and appointment-booking friction shifted real coverage from the optimised plan — often in ways that reduced equity.
Extensions & variants
Where vaccine-allocation research is going
Coupled epidemic-allocation MILP
Bertsimas et al. DELPHI-V-OPT; full SEIR + optimisation solved jointly.
Robust & scenario-based
Multiple scenarios for μ, β, η; solve for robust allocation across scenarios.
Dynamic reallocation
Re-optimise as new data arrives; rolling-horizon formulations.
Multi-dose schedules
Two-dose vs one-dose decisions under supply constraint; Matrajt et al. (2021).
Variant response
New variant emergence shifts vaccine efficacy; booster-allocation policy.
Ventilator allocation (related)
Even-more-scarce resources; triage ethics; White & Lo 2020 for COVID hospital triage frameworks.
Global fairness allocation
COVAX-style cross-country equity; Emanuel et al. 2020 fair allocation framework.
Hesitancy-aware allocation
Incorporate acceptance rates per group; redirect doses when demand low.
Key references
Cited above
Related applications
Allocation, matching, and scarcity siblings