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

Although the real-time application of mathematical modeling to understanding and control of outbreaks is often perceived as representing a recent development in infectious disease epidemiology (Heesterbeek et al., 2015) , disease modeling has deeper historical roots, including work by Bernoulli on smallpox in the 18th century (Greenwood, 1941) ; work by Ross on malaria transmission (Smith et al., 2012) , and as mentioned above, Farr's work on the growth and cessation of Fig. 3 . The right sided panel uses a combination of values (high R 0;SIR and/or low r) where susceptible depletion cannot be ignored (i.e., corresponding to the white area in Fig. 3 3) . It can be seen that IDEA and the damped SIR models diverge when susceptibles are rapidly depleted. Fig. 5 . The graph plots estimates of IDEA d parameter against time during the recent West African Ebola outbreak. Approximate date of the last generation incorporated into estimates is plotted on the X-axis; estimated d is plotted on the Y-axis. d estimates were either derived via IDEA model fitting to "incident" cases (blue diamonds) or cumulative incidence (crosses), or derived by estimating Farr's K and transforming resultant estimates using the relation described by equation (9). When K is estimated using 4-generation series (green diamonds), resultant d estimates are volatile and bear little resemblance to d estimates derived through fitting IDEA. However, estimates of K derived as geometric means of all available K values (red squares) provide a more reasonable approximation of d. Fig. 5 , d estimates were either derived via IDEA model fitting to "incident" cases (blue diamonds) or cumulative incidence (crosses), or derived by estimating Farr's K and transforming resultant estimates. As in Fig. 5 , volatile estimates of K were derived using 4generation series (green diamonds), but estimates of K derived as geometric means of all available K values (red squares) provided a reasonable approximation of d. Fig. 7 . A possible application for raw estimates of Farr's K emerged in analysis of data from the 2014e2015 Western Hemisphere Chikungunya outbreak; here it appears that a multi-wave epidemic is signaled by a sudden surge in K to a value > 1 (red line), indicating that there is renewed exponential growth in cases (blue bars), rather than exponential decline. X-axis, date of most recent generation; left Y-axis, Farr's K; right Y-axis, estimated per-generation Chikungunya case count and transforming resultant estimates. As in Fig. 1 , volatile estimates of K were derived using 4-generation series (green diamonds), but estimates of K derived as geometric means of all available K values (red squares) provided a reasonable approximation of d.