A Mass Preserving Numerical Scheme for Kinetic Equations that Model Social Phenomena

Abstract

In recent years, kinetic equations have been used to model many social phenomena. A key feature of these models is that transition rate kernels involve Dirac delta functions, which capture sudden, discontinuous state changes. Here, we study kinetic equations with transition rates of the form $$ T(x,y,u) = \delta_{\phi(x,y) - u}. $$ We establish the global existence and uniqueness of solutions for these systems and introduce a fully deterministic scheme, the \emph{Mass Preserving Collocation Method}, which enables efficient, high fidelity simulation of models with multiple subsystems. We validate the accuracy, efficiency, and consistency of the solver on models with up to five subsystems, and compare its performance against two state-of-the-art agent-based methods: Tau-leaping and hybrid methods. Our scheme resolves subsystem distributions captured by these stochastic approaches while preserving mass numerically, requiring significantly less computational time and resources, and avoiding variability and hyperparameter tuning characteristic of these methods.