Mean-field-game approach to nonpharmaceutical interventions in a social-structure model of epidemics

The design of coherent and efficient policies to address infectious diseases and their consequences requires modeling not only epidemics dynamics but also individual behaviors, as the latter has a strong influence on the former. In our work, we provide a theoretical model for this problem, taking into account the social structure of a population. This model is based on a mean-field-game version of a SIR compartmental model, in which individuals are grouped by their age class and interact together in different settings. This social heterogeneity allows us to reproduce realistic situations while remaining usable in practice. In our game theoretical approach, individuals can choose to limit their contacts by making a trade-off between the risks incurred by infection and the cost of being confined. The aggregation of all these individual choices and optimizations forms a Nash equilibrium through a system of coupled equations that we derive and solve numerically. The global cost born by the population within this scenario is then compared to its societal optimum counterpart (i.e., the cost associated with the optimal set of strategies from the point of view of the society as a whole), and we investigate how the gap between these two costs can be partially bridged within a constrained Nash equilibrium for which a governmental institution would, under specific conditions, impose "partial lockdowns" such as the ones that were imposed during the COVID-19 pandemic. Finally, we consider the consequences of the finiteness of the population size N_{tot}, or of a time T at which an external event (e.g., a vaccine) would end the epidemic, and show that the variation of these parameters could lead to first-order phase transitions in the choice of optimal strategies. In this paper, all the strategies considered to mitigate epidemics correspond to nonpharmaceutical interventions, and we provide here a theoretical framework within which guidelines for public policies depending on the characteristics of an epidemic and on the cost of restrictions on the society could be assessed.