SIR model with general distribution function in the infectious period

The Susceptible-Infected-Removed or SIR model, as it was formulated by Kermack and McKendrick, is the key model for epidemic dynamics. Most applications of such a basic scheme use a constant rate for the removal term. However, that assumption corresponds to the rather unrealistic exponential distribution of infectious times. On the other hand, recent approaches, like numerical simulations, frequently assume a fixed and uniform duration for the infectious state—which is unrealistic too. The extreme assumptions in those different schemes are a hurdle that can frustrate any intention of drawing comparison between results from them. In the present contribution we study the delay equations for the SIR model, comparing the solutions for many typical cases with the simulation counterpart and with the standard SIR model. Using delay equations, where each infected individual is removed at a specific time after being infected, the dynamics for the infected and susceptible agree almost exactly with the numerical implementation. Even in the general case of distributed infective periods, the agreement is excellent.