Kinetic modeling of coupled epidemic and behavior dynamics: The social impact of public policies

We study the propagation of a disease in a population where agents are characterized by their awareness level, representing the measures they take to avoid the infection. We introduce another agent, the government, which is constantly sending a message to the population trying to steer the mean awareness to a value which should ensure the extinction of the disease. We propose three levels to analyze this model. First, an agent-based model, which we use later to derive a mean-field system of ordinary differential equations; and finally, we propose a kinetic approach to model the evolution of the distribution of agents on the awareness levels. We obtain nonlinear systems of different dimension, first an ODE-Boltzmann system and later an ODEs-PDE system, where a Boltzmann or a first order, non-local partial differential equation are coupled with two ordinary differential equations that describe the evolution of the epidemic and the response of the government. We prove the existence and uniqueness of solutions in an abstract setting. Finally, we consider stubborn agents that are not willing to apply protection measures.