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 demonstrating that the IDEA model and Farr's model are mathematically equivalent (and can be virtually identical to an SIR model with abundant susceptibility in the population) we demonstrate that recognizing the underlying mechanism of epidemic control may be unimportant for generating reasonable forecasts of epidemics with control, or recognizing when their fundamental dynamics have changed. Our contribution in the current work is to show that Farr's law, while derived in the pre-microbial era, can be reformulated in terms of the concept of the basic reproduction number, combined with exponential increase in control via an unspecified mechanism. We observe, unexpectedly, that Farr's K can be expressed as a function of the IDEA d parameter alone, independent of R 0 , implying that epidemic trajectory is (and has historically been) more a function of control efforts and changing behavior than of the fundamental characteristics of a given infectious disease. Whether or not the ratio K can have stand-alone value as a tool to identify unexpected shifts in epidemic trajectory (e.g., the two wave epidemic of Chikungunya referred to in Fig. 7 above) will be the subject of future work.