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

Santillana, Mauricio; Tuite, Ashleigh; Nasserie, Tahmina; Fine, Paul; Champredon, David; Chindelevitch, Leonid; Dushoff, Jonathan; Fisman, David

When novel infectious diseases emerge or familiar diseases resurge, mathematical models can serve as useful tools for the synthesis of available data, management of uncertainty, and projection of likely epidemic trajectories (Fisman, 2009) . While it may be challenging to parameterize detailed mechanistic mathematical models when there is little information on mechanisms of transmission, baseline immunity in a community, or the nature of the infecting pathogen, a number of descriptive approaches exist which may permit fitting, and forecasting, of an epidemic curve. One single equation approach that has been applied to emerging infections is the Richards model, which treats cumulative infections as a logistic growth process (Hsieh & Chen, 2009; Wang, Wu, & Yang, 2012) . However, the concept of modeling an epidemic curve as a simple function, without reference to mechanisms of transmission, is in fact much older, and may originate in the work of the English polymath Dr. William Farr (1807e1883) , who rose from humble beginnings to become a physician, mathematician, hygienist and protege of Lancet founder Dr. Thomas Wakley (Brownlee, 1915a; Fine, 1979; Greenwood, 1933) . Dr. Farr spent almost 40 years at the General Register Office of the United Kingdom, and the esteem in which he was held is apparent in the "letters" he published annually as appendices to the reports of the Registrar General, in which he supplemented the dry statistical reports with thoughtful and creative musings on topics as wide-ranging as the relationships between occupation and disease, suicide and mortality in the mentally ill, population density and mortality, and as above the "laws" governing epidemics (Farr, 1840) .