CONTROL OF HIV AMONGST INJECTING DRUG USERS David Greenhalgh (1) and Fraser Lewis (2) (1) Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK, (2) Institute of Evolutionary Biology, Ashworth Laboratories, School of Biological Sciences, University of Edinburgh, West Mains Road, Edinburgh EH9 3JT, UK. Keywords: Human Immunodeficieny Virus (HIV), Acquired Immune Deficiency Syndrome (AIDS), Injecting Drug Users, Control, Basic Reproductive Ratio, Deterministic Model, Stochastic Model. 1. INTRODUCTION This talk is concerned with mathematical mod- els for the control of HIV amongst injecting in- travenous drug-users (IDUs), particularly nee- dle exchange programs, HIV testing of IDUs and improved cleaning practices. Our model is based on the work of Kaplan and O’Keefe (1993) who developed basic mathematical models for the spread of HIV amongst IDUs using data from Connecticut, New Haven, USA. 2. HOMOGENEOUS MODEL We start off by outlining a basic existing homoge- neous differential equation model for the spread of HIV amongst IDUs incorporating these con- trol strategies. We next describe an improvement of the way in which HIV testing can be treated in this model. Our model assumes that the total number of drug addicts and the total number of needles re- main constant. If π 1 ,π 2 and β denote the preva- lence of HIV amongst untested addicts, tested ad- dicts and needles respectively at time t then the differential equations which describe the spread of the disease are: dπ 1 dt = (1 - π 1 - π 2 )λ 1 βα(1 - φ) -(μ + δ t + δ)π 1 ; dπ 2 dt = δ t π 1 - (μ + δ)π 2 ; and dβ dt = (1 - β)γ (π 1 λ 1 + π 2 λ 2 ) -βλγ (1 - π 1 - π 2 )(θ + φ - θφ) -βτ. Here λ 1 is the sharing injection rate of addicts who do not know that they are infected; λ 2 is the decreased sharing rate of addicts who know that they are infected; α is the transmission probabil- ity that when a single uninfected addict makes a single injection with an infected needle the addict is infected and φ is the probability that an addict successfully cleans a needle before use. μ is the per capita rate at which addicts leave the sharing, injecting population for reasons other than devel- oping AIDS; δ t is the per capita HIV testing rate, δ is the per capita rate at which HIV-infected ad- dicts progress to AIDS; γ is the ratio of addicts to needles and θ is the flushing probability that when a single uninfected addict uses an infected needle that needle is left uninfected after use. For this improved model a key parameter is the basic reproductive ratio. This is the expected number of secondary cases that a single newly infected addict will cause on entering an entirely susceptible addict population. We find that R 0 = λ 1 α(1 - φ) (μ + δ t + δ)(ˆ τ 1 + λ 1 ˆ θ)  λ 1 + λ 2 δ t (μ + δ)  where ˆ τ 1 = τ/γ and ˆ θ = θ + φ - θφ. We show that the disease will always die out if R 0 ≤ 1, whilst if R 0 > 1, as well as the disease- free equilibrium (which is always possible) there is a unique endemic equilibrium which is locally asymptotically stable. We verify these analytical results by using deterministic computer simula- tion using the computer package SOLVER. Our simulations suggest that if R 0 > 1 then provided only that disease is initially present in either ad-