From individual-based models to macroscopic dynamics of antimicrobial resistance

We introduce and discuss a kinetic framework describing the time evolution of the statistical distributions of a population divided into the compartments of susceptible, infectious, recovered, and resistant in the presence of a microbial infection driven by susceptible-infectious interactions. Our main objective is to quantify the impact of excessive and inappropriate antimicrobial use, which accelerates the spread of resistance by enabling a fraction of infectious individuals to transition into the resistant compartment. The model consists of a system of Boltzmann-type equations capturing binary interactions between susceptible and infectious individuals, complemented by linear redistribution operators that represent recovery, the development of resistance, and reinfection processes. In the grazing-collision limit, we show that this Boltzmann system is well approximated by a system of coupled Fokker-Planck equations. This limiting description allows for a more tractable analysis of the dynamics, including the characterization of the long-time behavior of the population densities. Our analysis highlights how interaction terms drive the system toward a stable equilibrium and quantifies the effects of inappropriate antimicrobial use on the distribution of resistant individuals. Overall, the results offer a multi-scale perspective that bridges kinetic theory with classical epidemic modeling.