Generalised logistic function

A=0, K=1, B=3, Q=ν=0.5, M=0, C=1

Effect of varying parameter A. All other parameters are 1.

Effect of varying parameter B. A = 0, all other parameters are 1.

Effect of varying parameter C. A = 0, all other parameters are 1.

Effect of varying parameter K. A = 0, all other parameters are 1.

Effect of varying parameter Q. A = 0, all other parameters are 1.

Effect of varying parameter $\nu$. A = 0, all other parameters are 1.

The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.

Definition

Richards's curve has the following form:

$Y(t)=A+{K-A \over (C+Qe^{-Bt})^{1/\nu }}$

where $Y$ = weight, height, size etc., and $t$ = time. It has five parameters:

• $A$: the lower (left) asymptote;
• $K$: the upper (right) asymptote when $C=1$. If $A=0$ and $C=1$ then $K$ is called the carrying capacity;
• $B$: the growth rate;
• $\nu >0$ : affects near which asymptote maximum growth occurs.
• $Q$: is related to the value $Y(0)$
• $C$: typically takes a value of 1. Otherwise, the upper asymptote is $A+{K-A \over C^{\,1/\nu }}$

The equation can also be written:

$Y(t)=A+{K-A \over (C+e^{-B(t-M)})^{1/\nu }}$

where $M$ can be thought of as a starting time, at which $Y(M)=A+{K-A \over (C+1)^{1/\nu }}$. Including both $Q$ and $M$ can be convenient:

$Y(t)=A+{K-A \over (C+Qe^{-B(t-M)})^{1/\nu }}$

this representation simplifies the setting of both a starting time and the value of $Y$ at that time.

The logistic function, with maximum growth rate at time $M$, is the case where $Q=\nu =1$.

Generalised logistic differential equation

A particular case of the generalised logistic function is:

$Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}$

which is the solution of the Richards's differential equation (RDE):

$Y^{\prime }(t)=\alpha \left(1-\left({\frac {Y}{K}}\right)^{\nu }\right)Y$

with initial condition

$Y(t_{0})=Y_{0}$

where

$Q=-1+\left({\frac {K}{Y_{0}}}\right)^{\nu }$

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 $\nu \rightarrow 0^{+}$ provided that:

$\alpha =O\left({\frac {1}{\nu }}\right)$

In fact, for small ν it is

$Y^{\prime }(t)=Yr{\frac {1-\exp \left(\nu \ln \left({\frac {Y}{K}}\right)\right)}{\nu }}\approx rY\ln \left({\frac {Y}{K}}\right)$

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 $t$ (see). For the case where $C=1$,

${\begin{aligned}\\{\frac {\partial Y}{\partial A}}&=1-(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial K}}&=(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial B}}&={\frac {(K-A)(t-M)Qe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial \nu }}&={\frac {(K-A)\ln(1+Qe^{-B(t-M)})}{\nu ^{2}(1+Qe^{-B(t-M)})^{\frac {1}{\nu }}}}\\\\{\frac {\partial Y}{\partial Q}}&=-{\frac {(K-A)e^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial M}}&=-{\frac {(K-A)QBe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\end{aligned}}$

Modeling COVID-19 infection trajectory

Richards curve in epidemiological modeling

Richards's curves were widely used in modeling COVID-19 infection trajectories, daily time series data for the cumulative number of infected cases in a certain geographical area (country, city, state, ...). There are various re-parameterizations in the literature: one of the frequently used forms is

$f(t;\theta _{1},\theta _{2},\theta _{3},\xi )={\frac {\theta _{1}}{[1+\xi \exp(-\theta _{2}\cdot (t-\theta _{3}))]^{1/\xi }}}$

where $\theta _{1},\theta _{2},\theta _{3}$ are real numbers, and $\xi$ is a positive real number. The flexibility of the curve $f$ is due to the parameter $\xi$:

• if $\xi =1$ then the curve reduces to the logistic function, and
• if $\xi$ converges to zero, then the curve converges to the Gompertz function.

In epidemiological modeling, $\theta _{1}$, $\theta _{2}$, and $\theta _{3}$ represent the final epidemic size, infection rate, and lag phase, respectively. The figure on the right shows an example infection trajectory when $(\theta _{1},\theta _{2},\theta _{3})$ are designated by $(10,000,0.2,40)$.

Extrapolated infection trajectories of 40 countries severely affected by COVID-19 and grand (population) average through May 14th

One of the benefits of using Richards's curve as a growth function in epidemiological modeling is its relatively easy expansion to the multilevel model framework if it is used to model growth at multiple levels (city-level, state-level, national level, global level ...), as in the above-mentioned figure. Such a modeling framework is also called the nonlinear mixed-effects model or hierarchical nonlinear model.

Special cases

The following functions are specific cases of Richards's curves:

• Logistic function
• Gompertz curve
• Von Bertalanffy function
• Monomolecular curve

References

• Richards, F. J. (1959). "A Flexible Growth Function for Empirical Use". Journal of Experimental Botany. 10 (2): 290–300. doi:10.1093/jxb/10.2.290.
• Pella, J. S.; Tomlinson, P. K. (1969). "A Generalised Stock-Production Model". Bull. Inter-Am. Trop. Tuna Comm. 13: 421–496.
• Lei, Y. C.; Zhang, S. Y. (2004). "Features and Partial Derivatives of Bertalanffy–Richards Growth Model in Forestry". Nonlinear Analysis: Modelling and Control. 9 (1): 65–73. doi:10.15388/NA.2004.9.1.15171.