**5. Conclusions**

The widespread vaccination campaign undertaken in Western countries to counteract the evolution of the COVID-19 epidemic and its economic effects depends in large part on the efficacy of vaccines. Mathematical models capable of predicting the evolution of the economy in relation to the effectiveness of the vaccination campaign can play a fundamental role in configuring possible scenarios and suggesting further measures to be taken by governments. In this paper we analyzed, at the level of wealth distribution, the economic improvements induced by the vaccination campaign in terms of its percentage of effectiveness. Following the ideas developed in [1,8], the interplay between the economic trend and the pandemic has been evaluated, resorting to a mathematical model combining a kinetic model for wealth exchanges based on binary interactions with a classical SIR compartmental epidemic model, including the compartment of vaccinated individuals. Extensions of the presented methodology are possible to include disease-related mortality and redistribution operators. Moreover, since a direct comparison of the results of similar compartmental kinetic models—in the case of social aspects related to the transience of the epidemic—outlined a good agreemen<sup>t</sup> with the actual data [8,38,39], we can assume that the present approach is able to follow the real evolution of the economic parameters of a country over a sufficiently long period of time. Indeed, even though the model introduced here necessarily represents a strong simplification of an extremely complex phenomenon, its qualitative behavior is capable of describing the essential features of the pandemic's impact on individuals' wealth. A key aspect of the model is, in fact, the possibility of obtaining explicit configurations of the stationary wealth distributions in the form of inverse Gamma densities, with the essential parameters depending on the percentage of vaccinated and recovered individuals, thus relating the effectiveness of the vaccination campaign to the formation of wealth inequalities. Several numerical experiments have also been conducted

to quantify how a highly effective vaccination campaign has a direct effect on the decrease over time of the Gini coefficient, a classic measure of inequality in the distribution of wealth in Western societies.

**Author Contributions:** Conceptualization, L.P. and M.Z.; Data curation, E.B.; Investigation, E.B.; Methodology, L.P.; Project administration, G.T. and M.Z.; Supervision, L.P. and G.T.; Writing—original draft, E.B.; Writing—review & editing, M.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Ministero dell'Università e della Ricerca, gran<sup>t</sup> number 2020JLWP23 PRIN2020 and Università di Ferrara, gran<sup>t</sup> number FIR2021.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work was conducted within the activities of the GNFM and GNCS groups of INdAM (National Institute of High Mathematics). M.Z. acknowledges the partial support of MUR-PRIN2020 Project (No. 2020JLWP23) "Integrated mathematical approaches to socio-epidemiological dynamics". The research of M.Z. was partially supported by MIUR, Dipartimenti di Eccellenza Program (2018–2022), and Department of Mathematics "F. Casorati", University of Pavia. The research of L.P. was partially supported by FIR project "No hesitation. For effective communication of COVID-19 vaccination", University of Ferrara.

**Conflicts of Interest:** The authors declare no conflict of interest.
