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

William Farr's analysis is a classic in the epidemiology literature. Farr examined the course of mortality attributable to smallpox between mid 1837 (when death registration was introduced into England and Wales) and 1839, and noted that the numbers peaked in the spring quarter of 1838 and then declined until summer 1839 ( Fig. 1) (Farr, 1840) . He noted that the pattern of decline was very close to what would be predicted if the ratios of cases in successive quarters declined at a constant rate. He provided numbers demonstrating this in his annual report to the Registrar General in 1840 ( Fig. 1 ), but did not develop the idea at length. Looking back on this, we may note that this approach is analogous to assuming that the number of transmissions per case (or the "reproduction number" in modern terminology), were to decline at a constant rate during the course of an epidemic. The key difference is that Farr worked before the germ theory, and analysed data in terms of successive calendar time periods rather than successive generations of cases. There is a further irony to the story, in that he never returned to this idea until 1866, at which time there was a major epidemic of rinderpest, which some feared would destroy the British cattle population (Brownlee, 1915a) . Farr applied a similar analysis, but this time based upon assuming that the third ratio of cases per month was a constant (in effect assuming that the reproduction number declined at a constantly accelerating pace). He used this approach to predict that the epidemic would decline rapidly over the subsequent six months, and published this, including predicted monthly incidence numbers, in the Daily News of London in February 1866. His predictions were close to what subsequently happened (Brownlee, 1915a) .