Analyzing greedy vaccine allocation algorithms for metapopulation disease models

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 [Formula: see text] subject to a budget constraint [Formula: see text]. 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.