Abstract Compartmental models have gained significant attention not only in public health studies but also in fields such as Operations Research (OR), social sciences, and logistics, particularly following the COVID-19 pandemic. Their broad applicability in epidemiology and their utility in understanding, predicting, and controlling the global spread of infectious diseases have made them indispensable across various disciplines. The appeal of these models lies in their simplicity yet effectiveness in capturing the essential dynamics of disease transmission. This paper provides a comprehensive review of compartmental models, focusing on the Susceptible-Infectious-Recovered (SIR) models and the key aspects of their structure. The primary objective of this review is to enhance the ability of researchers and practitioners to understand and manage infectious disease outbreaks through a twofold approach: (1) an evaluation of the assumptions, equations, and methodologies used for estimating critical parameters in SIR models, and (2) an exploration of the relationship between SIR models and optimization models. Additionally, a systematic micro-level review has identified the most significant research gaps in the literature on compartmental models, leading to recommendations for future research. A key finding emphasizes the need to revisit various assumptions to clarify the connection between SIR models and optimization approaches, which is expected to offer valuable insights for epidemic disease modeling.

Abstract This paper studies the relationship between the student abilities in the second year of high school and the infrastructural endowment in all Italian municipalities, using spatial Bayesian modelling. Municipal student scores are obtained by averaging standardised and spatially homogeneous indicators of student outcomes provided by the INVALSI Institute for two subjects: Italian and mathematics. Given the nature of the data, we employ a multilevel regression model assuming a bivariate intrinsic conditionally autoregressive (ICAR) latent effect to explain the spatial variability and account for the correlation between the two subjects. Bayesian model estimation is obtained using the integrated nested Laplace approximation (INLA), implemented in the package. We find that along with a significant association with the current state of school infrastructure and facilities, spatially structured latent effects are still necessary to explain the different student outcomes across municipalities.

Abstract Environmental pollution represents a significant negative externality for society by imposing substantial costs on it. By incentivizing eco-friendly production, one can mitigate these costs and promote sustainable development. Given this framework, the present article formulates and solves a multi-stage game-theoretical pricing model involving two firms: one commercializes a non-green product; the other one is involved in the production of a green product. The model also involves the State, which can influence the two firms through a dual mechanism based on imposing an excise duty and promoting an advertising campaign. Firms’ profits and social welfare at the subgame perfect equilibrium are obtained in closed form, and their dependence on exogenous parameters is analyzed both theoretically and numerically. An interesting outcome of our analysis is that, in certain cases, the excise duty is found to be negative at the equilibrium and therefore corresponds to a subsidy rather than a green tax. Finally, possible future developments of the proposed model are discussed.