On a Versatile Stochastic Growth Model

Growth phenomena are ubiquitous and pervasive not only in biology and the medical sciences, but also in economics, marketing and the computer and social sciences. While on the surface deceivingly simple, the complex interactions that govern growth render the task of producing dependable mathematical models for such phenomena very challenging, indeed. Not surprisingly, the research community has devoted a fair amount of well-deserved attention to producing mathematical models that can predict and explain the extent of growth that we witness in various areas of human endeavor. The plethora mathematical growth models proposed in the literature falls into two broad categories: deterministic growth models and stochastic growth models. The deterministic models, including the well-known exponential, logistic, Gompertz and Bass models are well understood and in widespread use, chiefly because they are mathematically tractable and often reasonably accurate. On the other hand, the stochastic growth models, while often more realistic, typically lead to complicated or even intractable mathematics. This may well be one of the main reasons that stochastic growth models have been less widely used, especially in the biological and biomedical communities. The main contribution of this work is draw attention to a simple stochastic growth model that was somehow overlooked by the research community. Specifically, we introduce a three-parameter version of the classic pure-birth process and to show that, suitably instantiated, this stochastic model can be used to model growth phenomena in many, seemingly unrelated, application domains ranging from yeast production to the spread of Tasmanian sheep population, to the adoption of durable consumer goods and to software reliability. We point out that the three-parameter pure-birth model is computationally attractive since it admits of conceptually simple, closed form solutions for the time-dependent probabilities of the system being in a given state. In summary, we argue that the pure-birth model is a worthwhile addition to the panoply of tools that researchers have at their disposal and, consequently, deserves a place in the growth model toolbox.