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 we applied IDEA to current day outbreaks and epidemics, we have remained agnostic about the factors that cause second order deceleration of epidemic growth. Referring to first principles, the components of a reproduction number are duration of infectiousness, contact rate, and probability of transmission conditional on contact, as well as susceptibility in a population (Vynnycky & White, 2010) . We presume that public health interventions, population behavior change (as a result of education or rumours, prudence or fear), and the occurrence of silent infections with immunity could all contribute to deceleration of epidemic spread, even when the effective reproduction number is expected to be greater than 1 due to widespread susceptibility in the population. Furthermore, we note that Farr's original "time step" appears to have been arbitrary (reflecting the form of data available to him: weekly for rinderpest, quarterly for smallpox), whereas we have used generations as time steps in our more recent applications of Farr's "law", and in IDEA and SIR models. This potential discrepancy between Farr's initial efforts and our more recent efforts can be addressed by a simple rescaling of the time parameter, and is thus immaterial to our conclusions.