Relatedness of the incidence decay with exponential adjustment (IDEA) model, “Farr's law” and SIR compartmental difference equation models

Mathematical models are often regarded as recent innovations in the description and analysis of infectious disease outbreaks and epidemics, but simple mathematical expressions have been in use for projection of epidemic trajectories for more than a century. We recently introduced a single equation model (the incidence decay with exponential adjustment, or IDEA model) that can be used for short-term epidemiological forecasting. In the mid-19th century, Dr. William Farr made the observation that epidemic events rise and fall in a roughly symmetrical pattern that can be approximated by a bell-shaped curve. He noticed that this time-evolution behavior could be captured by a single mathematical formula ("Farr's law") that could be used for epidemic forecasting. We show here that the IDEA model follows Farr's law, and show that for intuitive assumptions, Farr's Law can be derived from the IDEA model. Moreover, we show that both mathematical approaches, Farr's Law and the IDEA model, resemble solutions of a susceptible-infectious-removed (SIR) compartmental differential-equation model in an asymptotic limit, where the changes of disease transmission respond to control measures, and not only to the depletion of susceptible individuals. This suggests that the concept of the reproduction number $\left(R_0\right)$ was implicitly captured in 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.