Approximate relationships between SIR and logistic models

Infectious epidemics are often described using a three-compartment Susceptible-Infectious-Removed (SIR) model, whose solution can be shown to involve generalizations of the logistic distribution. Using mathematical relationships relating these generalized logistic distributions, the population proportion remaining Susceptible can be approximated using the inverse of a standard cumulative logistic distribution, while the population proportion actively Infectious can be approximated using the density of a logistic or log-logistic distribution. Conversely, the parameters of an underlying SIR model can be approximately inferred from population-based data that have been estimated using logistic and/or log-logistic models.