A steady-state model for the spread of HIV among drug users

This paper proposes a new approach to model the spread of HIV/AIDS among intravenous drug users (IVDUs). The focus is on a group of n IVDUs within which infective contacts occur, and which evolves in discrete time, subject to group splitting, immigration, and emigration. We are interested in finding the probability distribution of the ultimate number Yn of HIV infectives produced by the group as time tends to infinity, and obtain a stochastic recursive equation for it. Although, on the surface, the process resembles a branching process, our results cannot be obtained using techniques from the theory of branching processes. We use the probability metrics approach to obtain limit theorems for the normalized sequence Ln = (Yn-EYn)n−12. Finally, we consider the behavior of Ln under different sets of regularity conditions, when for example Ln = (Yn-EYn)n−1α tends to an α-stable distribution.