Equivalence of Mass Action and Poisson Network SIR Epidemic Models

TL;DR

This paper shows that the classical mass-action SIR model closely approximates the Poisson network SIR model in many cases, especially for susceptible decay, simplifying epidemic modeling without significant loss of accuracy.

Contribution

It reveals the equivalence between the classical and network-based SIR models, highlighting practical implications for epidemic forecasting.

Findings

Susceptible decay curves are identical in both models.

Infection curves differ but converge in high-degree networks.

Classical SIR model can approximate Poisson network SIR models effectively.

Abstract

This brief note highlights a largely overlooked similarity between the SIR ordinary differential equations used for epidemics on the configuration model of a Poisson network and the classical mass-action SIR equations introduced nearly a century ago by Kermack and McKendrick. We demonstrate that the decline pattern in susceptibles is identical for both models. This equivalence carries practical implications: the susceptibles decay curve, often referred to as the epidemic or incidence curve, is frequently used in empirical studies to forecast epidemic dynamics. Although the curves for susceptibles align perfectly, those for infections do differ. Yet, the infection curves tend to converge and become almost indistinguishable in high-degree networks. In summary, our analysis suggests that under many practical scenarios, it is acceptable to use the classical SIR model as a close approximation to the Poisson SIR network model.