The Trans-Gompertz Function: An Alternative to the Logistic Growth Function with Faster Growth
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The Trans-Gompertz Function: An Alternative to the Logistic Growth Function with Faster Growth F. Kozusko1 • M. Bourdeau2
Received: 1 January 2015 / Accepted: 27 June 2015 / Published online: 4 July 2015 Springer Science+Business Media Dordrecht 2015
Abstract The growth characteristics of the recently derived Trans-Gompertz function are compared to those of the Generalized Logistic function. Both functions are defined by one shaping parameter and one rate parameter. The functions are matched at a specified point on the growth curve by equating both the first and second derivatives. Analysis shows that the matched Trans-Gompertz function will have grown at a faster rate with a larger inflection point ratio. Keywords Gompertz Logistic Trans-Gompertz Sigmoid Population Growth Tumor Growth
1 Introduction The S-shaped or sigmoid curves Generalized Logistic and Gompertz are used in numerous scientific applications such as modeling tumor, animal and bacterial growth (Tsoularis 2001). These functions are defined by a single shaping parameter: h, which also specifies the location (inflection point ratio) of the maximum growth rate on the growth curve. In typical variable notation, ðt; NðtÞÞ, where t is the growth age and NðtÞ is the population count, growth is modeled from ð0; N0 Þ to ðt ! 1; N ! N1 Þ where N? is the asymptotic limit of N as t ? ?, called the carrying capacity. Additionally there is a rate parameter, RL , which specifies the growth rate. RL has no effect on the inflection point ratio but merely acts as a time scale factor. In fitting data to a particular growth curve, the values of N0 and N? may be known or postulated. The outcome of the fit is the determination of RL and h. Past and future & F. Kozusko [email protected] 1
Department of Mathematics, Hampton University, Hampton, VA, USA
2
TIAA-CREF, New York, NY, USA
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growth may then be modeled using the sigmoid equation. (Other sigmoids have been created by using more shaping parameters or more involved shaping parameter dependence, see Marusic (1996) for example.) In Kozusko and Bourdeau (2011), we derived a new family of sigmoid growth functions based on modeling the rate change in the subpopulations of N. We designated this family as Trans-Logistics, of which Generalized Logistic and Gompertz are a subfamily. The function we discuss in this paper we called TransGompertz. It has the characteristic of a sigmoid with one shaping parameter and one rate parameter. Trans-Gompertz presents an overall different growth profile when matched with the Logistic curve at any point in the growth. In this paper, we will compare Trans-Gompertz and Logistic growth at different points along a growth curve and show their differences in growth periods and inflection point ratios.
2 The Generalized Logistic Equation The S-shaped solutions to the ODE: " h # N N_ RL 1 ¼ ; N1 h N
Nð0Þ ¼ N0
ð1Þ
sometimes called the Theta Logistic equation, are classified according to the values of h: •
• •
(h [ 0Þ Generalized Logisti
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