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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 (J.W. Wu, S.D. John, E.Y. Adashi, Allocating Vaccines in the Pandemic: The Ethical Dimension, The Am. J. of Medicine V.33(11): 1241 - 1242 (2020)). 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 to be 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., Comment: 28 pages, 11 figures

Evolutionary dynamics experienced by mixed microbial populations of cooperators and cheaters has been examined in experiments in the literature using a protocol of periodic dilution to investigate the properties of resilience and adaptability to environmental changes. Data depicted on an appropriate phase diagram indicate, among other features, a stable equilibrium point at which cooperators and cheaters coexist [A. Sanchez, J. Gore, PLOs Biology, 11 (4), e1001547 (2013)]. We present here a phenomenological analysis of these data focusing on an eco-evolutionary-game perspective. To that end, we work on an extension of the model proposed by Tao and Cressman Y. Tao, R. Cressman, Bull. Math. Biol. 69, 1377 - 1399 (2007). It's original version takes into account changes of the total population density while the individuals experience pairwise Prisoner's Dilemma game. The extension devised here contains a dilution factor to be conform with the experimental procedure, in addition of a term accounting for Allee effects. Differently from other descriptions proposed in similar contexts, however, the model here does not account for assortative encounters, group or kin selection. Nonetheless, it describes surprisingly well both qualitatively and quantitatively the features of the observed phase diagram. We discuss these results in terms of the behavior of an effective payoff matrix defined accordingly.