An interacting particle system for the front of an epidemic advancing through a susceptible population

Abstract

We introduce an interacting particle system that models the spread of an epidemic in terms of heterogeneous diffusive dynamics, rather than exogenous contact and transmission rates at the population level as in classical compartmental models. Each individual has a one-dimensional level of shielding that evolves according to a stochastic differential equation reflected at the advancing front of the epidemic. The front is driven by cumulative infections, and collisions with it represent at-risk situations which may lead to infection depending on a non-Markovian mechanism that involves the local time, the intrinsic transmissibility, and the current contagiousness within the population. We give a rigorous construction of the system and develop two key technical tools: a compensated martingale property for the infected proportion and a general result on how local time transforms under a random time-dependent bijection of the state space. The former yields a decomposition of the expected number of new infections that parallels a corresponding decomposition in the SIR model. The latter allows us to represent the law of each particle, after suitable conditioning, as a generalised elastic Brownian motion with drift.