Balancing the benefits of vaccination: An envy-free strategy

Abstract

The Covid-19 pandemic revealed the difficulties of vaccinating a population under the circumstances marked by urgency and limited availability of doses while balancing benefits associated with distinct guidelines satisfying specific ethical criteria. We offer a vaccination strategy that may be useful in this regard. It relies on the mathematical concept of envy-freeness. We consider finding balance by allocating the resource among individuals that seem heterogeneous concerning the direct and indirect benefits of vaccination, depending on age. The proposed strategy adapts a constructive approach in the literature based on Sperner's Lemma to point out an approximate division of doses guaranteeing that both benefits are optimized each time a batch becomes available. Applications using data about population age distributions from diverse countries suggest that, among other features, this strategy maintains the desired balance, throughout the entire vaccination period. We discuss complementary aspects of the method in the context of epidemiological models of age-stratified Susceptible - Infected - Recovered (SIR) type.

Keywords: balanced vaccine allocation; decision making process; direct and indirect benefits; envy-free division; pandemic preparedness.

Fig. 1.

a) A defined partitioned simplex with $d=7$. The scheme is adapted from Ref. (10). Counselor $C_{\mathrm{A}}$ chooses a side at all points A and counselor $C_{\mathrm{B}}$ chooses a side at all points B. b) The traces of utility functions ( $u_{\mathrm{A}}$ purple, $u_{\mathrm{B}}$ green) are shown to illustrate the choice of $C_{\mathrm{A}}$, at an arbitrary point $p_i=2/7$. c) Benefit evaluation on each side as the areas below the curves. d) Comparing benefits. e) Labeled simplex after both counselors, at each corresponding point, experience (b)–(d) and express their preferences for vaccinating on side I or on side II. In this example, the region shown defined by endpoints $(p_{\mathrm{L}},p_{\mathrm{R}})$ encloses a point $p^{*}$ that sets an envy-free division of the simplex.

Fig. 2.

Illustrative example. Utility density functions $u_{\eta }(y)$, $\eta =A,B$ evaluated from Eq. 1 for counselors $C_{\mathrm{A}}$ (purple) and $C_{\mathrm{B}}$ (green), respectively, with the continuous functions ${\rho }_{\eta }(y)$ (supplementary material) built up using the four combinations of utilities in Table 1, for the population age groups of the United States.

Fig. 3.

Simulated time series for the benefits [2] acquired by each counselor, $C_{\mathrm{A}}$ (orange) and $C_{\mathrm{B}}$ (blue) through a) a random procedure and strategies, b) maximize-benefit, c) oldest-first, d) envy-free. The utility density functions employed correspond to the combination defined as Default in Fig. 2 and the results shown are for the population age distribution of the United States.

Fig. 4.

Time average of the differences (absolute values) [15] between the contributions of the two counselors to the benefits obtained through each strategy, extended to the population age groups of the selected countries, as specified. The utility density functions employed correspond to the combination defined as Default.

Fig. 5.

Simulated time series for the cumulative benefits ${\mathrm{\Phi }}_{\eta }(t)$ [16] acquired by each counselor, $C_{\mathrm{A}}$ (orange) and $C_{\mathrm{B}}$ (blue) through a) a random procedure and strategies, b) maximize-benefit, c) oldest-first, d) envy-free. The utility density functions employed correspond to the combination defined as Default in Fig. 2. The results shown are for the population age distribution of the United States.

Fig. 6.

Time average of the differences (absolute values) $\overline{\mathrm{\Delta }\mathrm{\Phi }}$ [18] between the contributions of the two counselors to the cumulative benefits obtained through each strategy, extended to the population age distributions of the selected countries, as specified. The utility density functions employed correspond to the combination defined as Default.

Fig. 7.

Time average of the mean cumulative benefit $\overline{\mathrm{\Phi }}$ [19] between the contributions of the two counselors, for each strategy and selected countries. The utility density functions employed correspond to the combination defined as Default.

Fig. 8.

Time averages of a) the cumulative differences (absolute values) $\overline{\mathrm{\Delta }\mathrm{\Phi }}$ [18], and b) the instantaneous differences (absolute values) $\overline{\mathrm{\Delta }\mathcal{U}}$ [15] against the mean $\overline{\mathrm{\Phi }}$ [19], evaluated for the population age distributions of 236 selected countries for strategies and utility combinations, as indicated by colors. The distributions of the points are indicated by the lateral diagrams. Maximize-benefit (blue) minimize-benefit (green), oldest-first (brown), envy-free (red), and random procedure (orange).

Fig. 9.

The fraction of the doses to be allocated to each age group, at each time, to follow the envy-free strategy considering the different combinations of utilities, as indicated in Table 1. The results shown are for the population age distribution of the United States.