On a Versatile Stochastic Growth Model Samiur Arif Old Dominion University

Old Dominion University ODU Digital Commons Computer Science Faculty Publications Computer Science 2012 On a Versatile Stochastic Growth Model Samiur Arif Old Dominion University Ismail Khalil Stephan Olariu Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/computerscience_fac_pubs Part of the Artificial Intelligence and Robotics Commons Repository Citation Arif, Samiur; Khalil, Ismail; and Olariu, Stephan, "On a Versatile Stochastic Growth Model" (2012). Computer Science Faculty Publications. 94. https://digitalcommons.odu.edu/computerscience_fac_pubs/94 Original Publication Citation Arif, S., Khalil, I., & Olariu, S. (2012). On a versatile stochastic growth model. International Journal of Computational Intelligence Systems, 5(3), 472-482. doi:10.1080/18756891.2012.696911 This Article is brought to you for free and open access by the Computer Science at ODU Digital Commons. It has been accepted for inclusion in Computer Science Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. International Journal of Computational Intelligence Systems, Vol. 5, No. 3 (June, 2012), 472-482 On a Versatile Stochastic Growth Model Samiur Arif 1 Ismail Khalil 2 Stephan Olariu1 1 Computer Science, Old Dominion University, 4700 Elkhorn Avenue, Norfolk, VA 23508, USA E-mail: fsarif,[email protected] 2 Institute of Telecooperation, Johannes Kepler Linz University, Altenberger Strasse 69, Linz, Austria E-mail: [email protected] Abstract 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. We introduce a three-parameter version of the classic pure-birth process growth model when suitably instantiated, can be used to model growth phenomena in many seemingly unrelated application domains. We point out that the model is compu- tationally attractive since it admits of conceptually simple, closed form solutions for the time-dependent probabilities. Keywords: Stochastic Growth Models, Pure Birth Process, Time-Dependent Probabilities, Continuous Markov Chain 1. Introduction the first systematic attempt at understanding growth and extinction phenomena noticed in the social sciences that, nonetheless, could not be convincingly Over the centuries, the task of understanding the dy- explained [16,41]. Their work was among the earli- namics of various growth phenomena noticed in na- est systematic attempts at enlisting the help of prob- ture and society had captured the interest of schol- ability theory in modeling, and thus, understanding, ars and philosophers. However, with the exception the dynamics of population growth. While their pi- of D. Bernoulli’s attempt at modeling the outbreak oneering work had focused on a rather narrow prob- of a smallpox epidemic [7], T. R. Malthus’ popu- lem, namely that of accounting for the wholesale lation growth model [28] and P. Verhulst formula- extinction of family names in Great Britain, their tion of the logistic growth model [39], most of the mathematical methods (and, in particular, the by- early efforts, including Gompertz’s human mortality now classic Galton-Watson process) turned out to be model [17], were empirical, the data available was surprisingly powerful and general [18]. highly unreliable and the general sense of evidence was lacking by today’s standards. All this were to Unfortunately, Galton and Watson’s work and change in 1874 when Watson and Galton undertook their mathematical model was neglected for many Published by Atlantis Press Copyright: the authors 472 SA IK SO years, more precisely until 1924, when Yule [46] ap- maximum harvest for agriculture, to understand the plied similar probabilistic machinery to the study of dynamics of biological invasions, and have numer- the dynamics of the proliferation of new species and ous environmental conservation implications. Popu- genera. Yule’s contribution, a linear pure-birth pro- lation models are also used to understand the spread cess, was rediscovered, a few years later by W. H. of parasites, viruses, and disease. The realization Furry [15] in the context of electron physics and by of our dependence on environmental health has cre- Feller and Lotka in population biology [14, 24]. ated a need to understand the dynamic interactions The next major landmark event in modeling of the earths flora and fauna. Methods in population growth model was provided by Kolmogorov and modeling have greatly improved our understanding Dimitriev’s seminal work [27] where they general- of ecology and the natural world [22, 26, 33, 36]. ized the Galton-Watson model, proposing branching processes as a powerful modeling tool for a large 1.1. Our contributions class of growth phenomena that, as it turned out, Among the numerous growth models proposed over sub-summed much of the previous work cited above the decades, two main general-purpose models and set the stage for a systematic look at growth stand out: the deterministic growth models and the models. stochastic growth models [1, 23, 29, 31, 33, 37]. The Not surprisingly, in the following decades a deterministic models, including the well-known ex- plethora of mathematical growth models intended to ponential, logistic, Gompertz and Bass models are capture the essence of natural and social phenomena well understood and in widespread use, chiefly be- ranging from the social sciences to genetics, to bi- cause they are mathematically tractable and, most ology, to epidemiology, to physics, to astronomy, to of the time, fairly accurate. On the other hand, the computer science and macro-economics have been stochastic growth models, while often more realis- proposed in the literature [1,3,8,9,11,18,20,26,31]. tic, typically lead to complicated or even intractable Some of these models are intended to capture the mathematics. We note that for particular instances essence of high-powered, unhindered growth as wit- of the stochastic model closed forms for the vari- nessed, for example, in particle physics and astron- ous transition probabilities can be obtained by us- omy (e.g. the Big Bang theory). By contrast, most ing bivariate probability generating functions and of the growth phenomena that we encounter in bi- by solving a Fokker-Planck partial differential equa- ology and medicine, economics and the social sci- tion. However, it is common knowledge that, in gen- ences involve a close interaction between the phe- eral, the resulting partial differential equation does nomenon under study and its surrounding environ- not admit of a closed form solution and the only re- ment. For example in economics, the merger of course is to employ various approximation schemes companies is subject to internal stimuli and to ex- [3, 13]. This is, no doubt, one of the main reasons ternal pressure (inhibition) coming from the market- that stochastic growth models have been less widely place and competition [34]. In the biological sci- used, especially in the biological and biomedical ences, when resources are plentiful and environmen- communities [29, 30]. tal conditions appropriate, bacteria population can With this in mind, the main contribution of this increase rapidly. However, in most instances re- work is draw attention to a simple stochastic growth sources are not unlimited and environmental condi- model, namely, the pure birth process, that was tions are far from optimal. Climate, food, habitat, somehow overlooked by the research community. water availability, and other similar factors conspire We point out that the pure-birth model admits of to keep population growth in check. Indeed, the en- conceptually simple, closed form solutions for the vironment can only support a limited number of in- time-dependent probabilities of the system being in dividuals in a population before some resource runs a given state. Importantly, these closed-form solu- out and endangers the very survival of those indi- tions can be obtained through a straightforward it- viduals. Population models are used to determine erative approach that lends itself to efficient com- Published by Atlantis Press Copyright: the authors 473 On a Versatile Stochastic Growth Model puter implementation. Our extensive simulations namics must include reproduction. In Subsections have shown that, suitably instantiated, the pure-birth 2.1 and 2.2 we will discuss two important mod- process can be used to model growth phenomena els of population growth based on reproduction of in many, seemingly unrelated, application domains organisms: the exponential and the logistic mod- ranging from yeast production to the spread of Tas- els. In Subsection 2.3 we look briefly at the Gom- manian sheep population, to the adoption of durable pertz model used in a variety of situations ranging consumer goods and to software reliability. from reliability to cancer research [29, 31]. Further In summary, we do not to advocate replacing in Subsection 2.4 we discuss Bass’s growth model, the aforementioned, well-established, growth mod- the immensely popular model used in marketing re- els. Our goal is to argue that being simple and ver- search to predict, among other things, the time of satile, the pure-birth model is a worthwhile addition adoption of durable consumer goods. Finally, to set to the panoply of tools that researchers have at their the stage for our further discussion, in Subsection disposal and, consequently, deserves a place in the 2.5 we look at a simple stochastic

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