Open AccessA martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies
Author(s) -
Xia Li,
Chuntian Wang,
Hao Li,
Andrea L. Bertozzi
Publication year - 2022
Publication title -
networks and heterogeneous media
Language(s) - English
Resource type - Journals
eISSN - 1556-181X
pISSN - 1556-1801
DOI - 10.3934/nhm.2022009
Subject(s) - mathematics , martingale (probability theory) , population , stochastic modelling , infinitesimal , statistical physics , statistics , mathematical analysis , physics , demography , sociology
Deterministic compartmental models for infectious diseases give the mean behaviour of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. In this article, we consider the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derive a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects . Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data.