An exploration of finite-size effects in the dynamics of epidemic compartmental modelling

Abstract

Finite-size effects arise when the dynamical behaviour of human-interacting or particle-interacting systems with inherent randomness deviates from the predictions of the corresponding mean-field equations. Despite their frequent observation in practice, the mechanisms by which finite-size effects shape and influence real-world epidemic dynamics remain largely unexplored. To this end, in this article, we study these effects in the context of both agent-based and mean-field epidemic modelling frameworks, with particular attention to dynamical regimes characterized by repeated outbreaks. Our ultimate objective is to inform and improve public strategies for epidemic prediction and prevention. As an initial step towards our broader goal, we focus on a minimal epidemic model that naturally captures recurrent epidemic waves, namely the SIHRS compartmental model, which assumes waning immunity in recovered individual. This model may be viewed as a natural extension of the classical SIHR and SIRS compartmental models. Rather than obscuring the underlying mechanisms with an overly intricate model structure, the SIHRS framework allows us to isolate, identify and track finite-size effects over time. We carry out our analysis using a martingale-based early-time-step method, in which martingale formulations of the underlying Markov processes are derived and exploited. In particular, the martingale formulations are extended beyond compartment variables to higher-order quantities, which constitutes a key step towards advancing the analysis beyond short-time approximations. Our analysis delivers a quantitative and systematic understanding of finite-size effects in epidemic dynamics, demonstrating that their impact is shaped by population size, compartment scale and temporal structure. These findings are supported by extensive numerical simulations calibrated with both national-scale and county-level COVID-19 data.