Fair and efficient vaccine allocation: A generalized Gini index approach

The paper proposes an optimization model for the allocation of vaccines to a heterogeneous population composed of several subpopulations with different sizes and epidemiological disease transmission parameters. As the objective, an aggregated function combining a standard utilitarian efficiency criterion with a Gini index–related penalty term is considered. Contrary to previous work, we adopt an outcome equity view: The inequity measure is not based on vaccination fractions or other input factors, but on the fractions of individuals escaping infection, as predicted by an susceptible‐infectious‐removed (SIR) model. An adjusted pro rata (APR) policy of vaccine allocation minimizing inequity in this outcome view is introduced, and a numerical procedure for its determination is presented. The concepts are developed both for the case of segregated subpopulations and for that of interactions between the subpopulations. Interestingly, in a large number of instances, the optimal solution under the aggregated objective function turns out to be identical to APR. Whether APR is locally or even globally optimal in a concrete case depends on the relation of an inequity aversion parameter to certain threshold values. While the local optimality threshold can be determined by linear programming, the determination of the global optimality threshold, as the vaccine allocation problem itself, is a problem of nonconvex optimization. We suggest an exact optimization approach for smaller instances, and propose algorithms building on particle swarm optimization for threshold determination and allocation optimization at larger instances. Extensions to alternative outcome measures such as the number of fatalities are presented as well. In addition to the investigation of randomly generated instances, two test cases from the literature are revisited in the context of the present work. Moreover, a new case study based on data from the COVID‐19 outbreak in Austria in 2020 is introduced and analyzed.