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

In that work, we demonstrated concordance with projections derived using a 3-compartment difference equation model (damped SIR model). We have subsequently realized, and demonstrate above, that our approach simply represented a restatement of Farr's work, albeit in a manner that is tied to the concept of R 0 . According to Brownlee, Farr promised to describe the derivation of his model in greater detail in future reports, but never did so (Brownlee, 1915a) , and much of the mathematical elaboration of Farr's work was in fact done by Brownlee after Farr's death (Brownlee, 1915c) . Nonetheless, Brownlee notes that to Farr, the predictive accuracy of his approach reflected three characteristics of epidemics, according to Farr's (pre-microbial) understanding: (i) diminution in the number of susceptibles over time due to recovery from infection ("immunity", though to use this term in application to Farr is an anachronism); (ii) diminished population density due to death from infection; and (iii) diminishing pathogenicity of the disease with each passing generation of infection as a result of (to quote Farr) "[loss of] part of the force of infection in every body through which they pass…the matter…diminishes in strength at every transmission by innoculation" (Brownlee, 1915a) . The first two characteristics are not incompatible with the current understanding of epidemic dynamics, whereas the third is not (though it does anticipate more modern ideas around evolution of virulence and disease ecology (Ewald, 2004; Lipsitch & Moxon, 1997) ).