A drug addiction process in which individuals in a closed population can become addicts or pushers is modelled. Expressions for the state probabilities and factorial moments are obtained. A generalized model is also developed. This generalized drug addiction process can be viewed as a particular case of what may be called a competing death process.

Two-dimensional bounded birth and death processes are considered. State probabilities are given for the generalized case in which the transition rates are arbitrary time-independent functions of the population sizes. Some applications in epidemic theory and species interaction processes are briefly described.

Ever since Lotka (1925) and Volterra (1926), (1931) first considered mathematical formulations for prey-predator processes, the resultant equations have resisted attempts to solve them. However, over the intervening 50 years, standard techniques have allowed a few isolated results to be obtained for some simplified versions of the original process, but the classical equations for the stochastic model have remained unsolved. We give here solutions to the classical process for the case in which interactions occur over a sufficiently short period of time that no births occur. The technique used is one recently developed by Severo (1969a), (1969b), (1971). The approach can be easily generalised to allow solution for the case in which births do occur, as well as for the simplified versions of the original process.

Epidemic models have generally assumed that the population mixes homogeneously. More realistic models can be obtained by dividing the population into sub-populations within which mixing is homogeneous but between which mixing is heterogeneous. We consider such a model for the general epidemic process and present a formulation which permits inroads to be made towards obtaining explicit solutions to the state probabilities.

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.