The Kermack–McKendrick model is a set of three differential equations:

,

,

,

which together preserve the total population count,

. The third equation is not necessary, and the first two are simplified by dividing out the constant

, that is, transforming

for all

. Furthermore, the first equation entails a logarithmic derivative, so

, which simplifies the second equation to read

.

This form has only one dependent and one independent variable, but it is not an ordinary differential equation. A more simple alternative is the logistic equation

.

Population count

is a sigmoid similar to

, so

compares naturally to

. A discrete approximation to

can be found by Euler's method,

,

.

According to the overshoot error of Euler's method,

turns out to have asymmetry comparable to

when, say,

. Over this range, a cubic function,

,

does a satisfactory job of converting

, with

. Using this function (and adjusting scale parameters as necessary) a 99.9% accurate, discrete solution of the SIR equations is achieved (see the source code for more details).

This range of solutions gives one answer to the question: how do the SIR equations

** **relate to the logistic equation? For the range of valid parameters, the identity between logistic and SIR solutions allows an SIR-compliant fit function to be computed via the simpler logistic iteration. There is an argument that discrete fit functions are less useful than their continuous counterparts. We have recently shown in another Demonstration [4] that this argument is not too serious. When data has significant variance, it often needs to be binned. Fitting by discrete or continuous functions then produces relatively similar results. The discrete method may even be preferable because it is computationally fast.