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 this section, we aim to test numerically the validity of approximations (11) and (12). In particular, given a damped SIR model, we explore the parameter space R 0;SIR and r for which these approximations hold. Note that the link between IDEA and Farr's law given by equation (9) is not an approximation, but a genuine equality (subject to the time step used in Farr's law being equivalent to one generation), so there is no need to test it numerically. To measure the performance of the approximation, we consider the distance between the simulated incidence time series. Let N be the number of generations simulated and I SIR ðkÞ (resp. I IDEA ðkÞ) the incidence from the SIR (resp IDEA) model at the k th generation, we define their distance by Fig. 3 shows the values of the distance d for different values of R 0;SIR and r. We see that for a combination of R 0;SIR and r such that the depletion of susceptible is not too large, the approximation is very good. But when the values of R 0;SIR and r generate a depletion of susceptible individuals that is no longer negligible (white area in Fig. 3) , then the incidence curve from the IDEA model diverges from that generated by the damped SIR model. This is demonstrated in Fig. 4 .