Distributional properties of a model for the spread or drug abuse

We examine the properties of the distribution of the number of drug abusers in a previously free community assuming that initiators enter the community during a ‘latent’ period in which they randomly infect other members of the community, and that during the subsequent ‘control’ period the spread of abuse follows a linear birth and death process. The form of distribution is shown to be unaltered by a series of steady changes in the birth and death parameters. The distribution can be regarded as the convolution of three distributions:pseudo-binomial or binomial, negative binomial, and Polya-Aeppli. Special cases include the Laguerre series distribution.