Generalised logistic function

The generalized logistic function or curve, also known as Richards' curve, originally developed for growth modelling, is an extension of the logistic or sigmoid functions, allowing for more flexible S-shaped curves:
where formula_2 = weight, height, size etc., and formula_3 = time.
It has five parameters:
The equation can also be written:
where formula_17 can be thought of as a starting time, formula_18 
Including both formula_12 and formula_17 can be convenient:
this representation simplifies the setting of both a starting time and the value of Y at that time.
The general model is sometimes named a "Richards' curve" after F. J. Richards, who proposed the general form for the family of models in 1959.
The logistic, with maximum growth rate at time formula_17, is the case where formula_24.
Generalised logistic differential equation.
A particular case of the generalised logistic function is:
which is the solution of the Richards' differential equation (RDE):
with initial condition
where
provided that ν > 0 and α > 0.
The classical logistic differential equation is a particular case of the above equation, with ν =1, whereas the Gompertz curve can be recovered in the limit formula_29 provided that:
In fact, for small ν it is
The RDE models many growth phenomena, arising in fields such as oncology and epidemiology.
Gradient of generalized logistic function.
When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point formula_3 (see ). For the case where formula_33, 
Modeling COVID-19 infection trajectory.
A generalized logistic function, also called the Richards growth curve, is widely used in modelling COVID-19 infection trajectories. Infection trajectory is a daily time series data for the cumulative number of infected cases for a subject such as country, city, state, etc. There are variant re-parameterizations in the literature: one of frequently used forms is
where formula_36 are real numbers, and formula_37 is a positive real number. The flexibility of the curve formula_38 is due to the parameter formula_37: (i) if formula_40 then the curve reduces to the logistic function, and (ii) if formula_37 converges to zero, then the curve converges to the Gompertz function. In epidemiological modeling, formula_42, formula_43, and formula_44 represent the final epidemic size, infection rate, and lag phase, respectively. See the right panel for an examplary infection trajectory when formula_45 are designated by formula_46.
One of the benefits of using growth function such as generalized logistic function in epidemiological modeling is its relatively easy expansion to the multilevel model framework by using the growth function to describe infection trajectories from multiple subjects (countries, cities, states, etc). See the above figure. Such a modeling framework can be also widely called the nonlinear mixed-effects model or hierarchical nonlinear model.
Special cases.
The following functions are specific cases of Richards' curves: