Relatedness of the Incidence Decay with Exponential Adjustment (IDEA) Model, "Farr's Law" and Compartmental Difference Equation SIR Models

TL;DR

This paper demonstrates that Farr's law, the IDEA model, and SIR models are mathematically equivalent, highlighting the fundamental role of control measures and the reproduction number in epidemic dynamics.

Contribution

It shows the equivalence of Farr's law, the IDEA model, and SIR models, revealing the implicit presence of the reproduction number and control in epidemic modeling.

Findings

Farr's law and the IDEA model are mathematically identical.

The models can be expressed in terms of each other and SIR models.

Control measures are integral to epidemic dynamics, as shown by the models.

Abstract

Mathematical models are often regarded as recent innovations in the description and analysis of infectious disease outbreaks and epidemics, but simple models have been in use for projection of epidemic trajectories for more than a century. We recently described a single equation model (the incidence decay with exponential adjustment, or IDEA, model) that can be used for short term forecasting. In the mid-19th century, Dr. William Farr developed a single equation approach (Farr's law) for epidemic forecasting. We show here that the two models are in fact identical, and can be expressed in terms of one another, and also in terms of a susceptible-infectious-removed (SIR) compartmental model with improving control. This demonstrates that the concept of the reproduction number, R0, is implicit to Farr's (pre-microbial era) work, and also suggests that control of epidemics, whether via behavior change or intervention, is as integral to the natural history of epidemics as is the dynamics of disease transmission.