Kinetic formulation of compartmental epidemic models

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.

💡 Research Summary

The paper presents a novel kinetic framework that couples the spatial movement of individuals with the epidemiological dynamics of a pathogen, thereby extending the classic SIRS compartmental model to a mesoscopic level that retains information about position and velocity. The authors begin by defining density functions f S, f I, and f R over time, space, and velocity, and they write a Boltzmann‑type equation for each compartment: ∂ₜf j + v·∇ₓf j = Q j