# Analyzing greedy vaccine allocation algorithms for metapopulation disease models

## TL;DR

This paper studies how to best allocate limited vaccines across different regions to reduce the spread of disease, showing that simple greedy algorithms work well and can be scaled to large populations.

## Contribution

The paper introduces a novel analysis of greedy vaccine allocation algorithms for metapopulation disease models, showing their effectiveness and scalability.

## Key findings

- Greedy algorithms for vaccine allocation are scalable and effective across different population sizes.
- These algorithms outperform existing methods in most tested scenarios.
- The performance of greedy algorithms is linked to the submodularity ratio of the objective function.

## Abstract

As observed in the case of COVID-19, effective vaccines for an emerging pandemic tend to be in limited supply initially and must be allocated strategically. The allocation of vaccines can be modeled as a discrete optimization problem that prior research has shown to be computationally difficult (i.e., NP-hard) to solve even approximately. Using a combination of theoretical and experimental results, we show that this hardness result may be circumvented. We present our results in the context of a metapopulation model, which views a population as composed of geographically dispersed heterogeneous subpopulations, with arbitrary travel patterns between them. In this setting, vaccine bundles are allocated at a subpopulation level, and so the vaccine allocation problem can be formulated as a problem of maximizing an integer lattice function g:ℤ+K→ℝ subject to a budget constraint ‖x‖1≤D. We consider a variety of simple, well-known greedy algorithms for this problem and show the effectiveness of these algorithms for three problem instances at different scales: New Hampshire (10 counties, population 1.4 million), Iowa (99 counties, population 3.2 million), and Texas (254 counties, population 30.03 million). We provide a theoretical explanation for this effectiveness by showing that the approximation factor (a measure of how well the algorithmic output for a problem instance compares to its theoretical optimum) of these algorithms depends on the submodularity ratio of the objective function g. The submodularity ratio of a function is a measure of how distant g is from being submodular; here submodularity refers to the very useful “diminishing returns” property of set and lattice functions, i.e., the property that as the function inputs are increased the function value increases, but not by as much.

Strategic and timely allocation of vaccines is crucial in combating epidemic outbreaks. Developing strategies to allocate vaccines over subpopulations rather than to individuals leads to policy recommendations that are more feasible in practice. Despite this, vaccine allocation over subpopulations has only received limited research interest, and the associated computational challenges are relatively unknown. To address this gap, we study vaccine allocation problems over geographically distinct subpopulations in this paper. We formulate our problems to reduce either i) the total infections or ii) the sum of peak infections over meta-population disease models. We first demonstrate that these problems are computationally challenging even to approximate and then show that a family of simple, well-known greedy algorithms exhibit provable guarantees. We conduct realistic experiments on state-level mobility graphs derived from real-world data in three states of distinct population levels: New Hampshire, Iowa, and Texas. Our results show that the greedy algorithms we consider are i) scalable and ii) outperform both state-of-the-art and natural baselines in a majority of settings.