On the Generalized "Birth-and-Death" Process

Abstract

The importance of stochastic processes in relation to problems of population growth was pointed out by W. Feller [1] in 1939. He considered among other examples the "birth-and-death" process in which the expected birth and death rates (per head of population per unit of time) were constants, $\lambda_o$ and $\mu_o$, say. In this paper, I shall give the complete solution of the equations governing the generalised birth-and-death process in which the birth and death rates $\lambda(t)$ and $\mu(t)$ may be any specified functions of the time. The mathematical method employed starts from M. S. Bartlett's idea of replacing the differential-difference equations for the distribution of the population size by a partial differential equation for its generating function. For an account of this technique,$^1$ reference may be made to Bartlett's North Carolina lectures [2]. The formulae obtained lead to an expression for the probability of the ultimate extinction of the population, and to the necessary and sufficient condition for a birth-and-death process to be of "transient" type. For transient processes the distribution of the cumulative population is also considered, but here in general it is not found possible to do more than evaluate its mean and variance as functions of $t$, although a complete solution (including the determination of the asymptotic form of the distribution as $t$ tends to infinity) is obtained for the simple process in which the birth and death rates are independent of the time. It is shown that a birth-and-death process can be constructed to give an expected population size $\bar n_t$ which is any desired function of the time $t$, and among the many possible solutions the unique one is determined which makes the fluctuation, Var$(n_t)$, a minimum for all. The general theory is illustrated with reference of two examples. The first of these is the $(\lambda_0, \mu_1t)$ process introduced by N. Arley [3] in his study of the cascade showers associated with cosmic radiation; here the birth rate is constant and the death rate is a constant multiple of the "age, $t$, of the process. The $\bar n_t$-curve is then Gaussian in form, and the process is always of transient type. The second example is provided by the family of "periodic" processes, in which the birth and death rates are periodic functions of the time $t$. These appear well adapted to describe the response of population growth (or epidemic spread) to the influence of the seasons.