Stochastic models for the spread of HIV among intravenous drug users

In this paper we examine the spread of HIV when this disease is transmitted through the random sharing of contaminated drug injection equipment. We first model the spread of disease using a standard set of behavioral assumptions discussed by Kaplan [1]. We demonstrate that deterministic and stochastic models based on these assumptions behave very similarly and use a branching process approximation to show that if the basic reproductive number, R0, is less than or equal to unity then the disease will always become extinct. If R0>1 then, although the disease might take off, it is still possible for it to die out, and we calculate the probability of extinction. This is not of the simple form R0-a, where a is the initial number of infectious addicts, which might have been expected from Whittle's stochastic threshold theorem [2]. We next discuss an extended model that incorporates a three-stage AIDS incubation period and again examine a branching process approximation. We finally explore the extent to which control strategies such as needle exchange and improved needle cleaning can reduce the risk of a HIV epidemic before concluding with a brief discussion.