Digital Object Identifier (DOI): 10.1007/s002850200140 J. Math. Biol. 44, 561–598 (2002) Mathematical Biology David Greenhalgh · Fraser Lewis The general mixing of addicts and needles in a variable-infectivity needle-sharing environment Received: 20 September 2001 / Revised version: 21 December 2001 / Published online: 17 May 2002 – c  Springer-Verlag 2002 Abstract. In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. The model we discuss focuses on the transmission of HIV through the sharing of contaminated drug injection equipment and in particular we examine the mixing of addicts and needles when the AIDS incubation period is divided into three distinct infectious stages. The impact of this assumption is to greatly increase the com- plexity of the HIV transmission mechanism. We begin the paper with a brief literature review followed by the derivation of a model which incorporates three classes of infectious addicts and three classes of infectious needles and where a general probability structure is used to represent the interaction of addicts and needles of varying levels of infectivity. We find that if the basic reproductive number is less than or equal to unity then there exists a globally stable disease free equilibrium. The model possesses an endemic equilibrium solution if the basic reproductive number exceeds unity. We then conduct a brief simulation study of our model. We find that the spread of disease is heavily influenced by the way addicts and needles of different levels of infectivity interact. 1. Introduction and literature review In existing models of the spread of HIV through the sharing of contaminated injec- tion equipment it is assumed that addicts and needles interact in very specific ways. In one of the first papers to examine the transmission of HIV via the sharing of infec- tious needles, (Kaplan, 1989), it was assumed that infectious addicts always leave previously uninfectious needles infectious after use and that uninfectious addicts can render infectious needles uninfectious. Greenhalgh and Hay (1997) extended this model so that a previously uninfectious needle is not necessarily left in an infectious state after use by an infectious addict. In both these papers assumptions concerning the interaction between addicts and needles play an important role in the evolution of an HIV epidemic. The model by Greenhalgh and Hay contains an adequate description of how addicts and needles interact when it is assumed that the infectiousness of an indi- vidual during the long AIDS incubation period is approximately constant. However a number of authors (Seitz and M¨ uller, 1994, Anderson and May, 1991, Peterson D. Greenhalgh, F. Lewis: Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK. e-mail: david@stams.strath.ac.uk Key words or phrases: HIV – AIDS – Injecting drug users – Three stage infectivity – Needle exchange – Basic reproduction number – Equilibrium and stability analysis – Simulation