A multi-agent approach to the impact of epidemic spreading on commercial activities

Abstract

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of the amount of consumer spending transactions of a multi-agent society, that, in the presence of social distancing laws consequent to an epidemic spreading, can reduce the volume of a consistent part of economic commercial activities requiring the presence. At the Boltzmann level, the microscopic variation of the number of transactions around a desired commercial target, is built up by joining a random risk with a deterministic change. Both the risk and target value parameters are here dependent of the number of social contacts of the multi-agent system, which can vary in time in reason of the governments decisions to control the epidemic spreading by introducing restrictions on the social activities. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck-type equation with variable coefficients of diffusion and drift, characterized by the details of the elementary interaction. The economic description is then coupled with the evolution equations of a new SIR-type compartmental epidemic system suitable to describe both the classical epidemic spreading and the social contacts evolution in dependence of the multi-agent social heterogeneity. The (local in time) steady states of the statistical distribution of the amount of economic transactions predicted by the Fokker–Planck equation are shown to be inverse Gamma densities, with time-dependent polynomial tails at infinity which characterize the consequences of the control policies on the time evolution of the Pareto index.