This paper first considers a simple deterministic epidemic in a growing population. It then examines a simple deterministic epidemic in several related growing communities, where infection is transmitted between individuals both within and across the communities. An explicit parametric solution is derived for the number of susceptibles in these communities as a function of time.

This paper is concerned with a pure death process, starting with N individuals, with death rates μ n, n = 1, 2, …, N. It is shown that the fates of distinct individuals are positively correlated if μ n/n decreases with n, and negatively correlated if μ n/n increases with n. The application of this result to the problem of variability in compartmental models is elaborated and in particular a conjecture of Faddy (1985) is settled. Further applications to well-known death processes are also briefly described.

Saddlepoint approximations to the probabilities for a general time-homogeneous birth process are derived from the known Laplace transform with respect to time. Numerical computations for the simple (logistic) epidemic show the approximation to have a remarkably low relative error over the useful range of times for populations as small as 5. The merit or otherwise of renormalising the approximate probabilities is discussed. To study the accuracy for moderately large population sizes, a special form of birth rate is used, for which the exact probabilities can be computed directly.

A simple interaction model is proposed for describing variable meeting-rates in a population exposed to a contagious disease. In comparison with the usual assumption of a uniform meeting-rate it predicts, on average, a more rapid spread of infection initially and a slower spread finally; it also predicts that, for a given removal rate, an epidemic could develop among a smaller number of susceptibles.

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.