Exact description of limiting SIR and SEIR dynamics on locally tree-like graphs

TL;DR

This paper derives exact ODE systems for SIR and SEIR epidemic models on locally tree-like networks, providing precise outbreak size predictions and comparing constant versus time-varying rate scenarios.

Contribution

It presents a novel exact characterization of epidemic dynamics on certain networks, including explicit outbreak size formulas and analysis of mean-field overestimations.

Findings

Exact ODE systems describe epidemic dynamics on locally tree-like graphs.

Outbreak size for constant rates is given by a specific functional's zero.

Mean-field predictions overestimate outbreak sizes for constant rates.

Abstract

We study the Susceptible-Infected-Recovered (SIR) and the Susceptible-Exposed-Infected-Recovered (SEIR) models of epidemics, with possibly time-varying rates, on a class of networks that are locally tree-like, which includes sparse Erd\H{o}s-R\`enyi random graphs, random regular graphs, and other configuration models. We identify tractable systems of ODEs that exactly describe the dynamics of the SIR and SEIR processes in a suitable asymptotic regime in which the population size goes to infinity. Moreover, in the case of constant recovery and infection rates, we characterize the outbreak size as the unique zero of an explicit functional. We use this to show that a (suitably defined) mean-field prediction always overestimates the outbreak size, and that the outbreak sizes for SIR and SEIR processes with the same initial condition and constant infection and recovery rates coincide. In contrast, we show that the outbreak sizes for SIR and SEIR processes with the same time-varying infection and recovery rates can in general be quite different. We also demonstrate via simulations the efficacy of our approximations for populations of moderate size.