Kinetic formulation of compartmental epidemic models

Abstract

We introduce a kinetic model that couples the movement of a population of individuals with the dynamics of a pathogen in the same population. We consider that transmission occurs when a susceptible and an infectious individual are sufficiently close for a sufficiently long time. We show that the model is formally compatible with the well-known SIRS model in mathematical epidemiology. Namely, after identifying an appropriate dimensionless variable and considering the limit when that variable is small, we introduce a partial differential equation model of advection-drift-diffusion type (mesoscopic model), which for spatially homogeneous solutions reduces to the SIRS model. We prove the existence and uniqueness of solutions in appropriate spaces for particular instances of the model. We finish with some examples and discuss possible applications and generalisation of this modelling approach, linking kinetic models, evolutionary game theory, and mathematical epidemiology.