HISTORY OF MATHEMATICAL MODELS

For long, scholars from all fields have looked for ways of representing social and scientific phenomena in the form of abstract laws, principles and mathematical forms. Population studies is no exception to this. One of the earliest and most forceful arguments in favour of modelling was, however, given by Keyfitz. In the first volume of Population and Development Review Keyfitz (1975) published, in an article entitled “How do we know the facts of demography?”, he argued that the empirical relationships based on regression analysis can be misleading as they depend heavily on the cases for which data are available. Also, without modelling questions regarding cause and effect, multiple causation, and nature of relationships cannot be answered. In this article, he showed that the empirical relationship between percent aged 65 and over and growth rate of population was dependent on: (a) number of countries for which data are available; and (b) homogeneity among the countries. In such situations analytical and mathematical models can provide a better understanding of the relationship.

The most important of all the demographic models is the stable population model (Smith and Keyfitz, 1977; Keyfitz, 2005). It says that if individuals are born at a constant rate of 1 person per unit of time, and the survival probability of a person aged x is p(x), then at any time the expected size of the population is given by

where E[X] refers to the expected size of population at age x, p(t) refers to chance of survival from birth to age t, and a refers to the upper limit of the age distribution.