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Mathematical Medicine and Biology 1997 14(1):11-38; doi:10.1093/imammb/14.1.11
© 1997 by Institute of Mathematics and its Applications
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Mathematical modelling of the spread of HIV/AIDS amongst injecting drug users

DAVID GREENHALGH{dagger}, and GORDON HAY

Department of Statistics and Modelling Science, University of Strathclyde Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK
Centre for Drug Misuse Research, University of Glasgow Lilybank House, Bute Gardens, Glasgow G12 8RT, UK

{dagger} Author to whom all correspondence should be addressed

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. Our work is based on a model originally due to Kaplan (1989, Rev. Inf. Diseases 11, 289–98). We start off with a brief literature survey and review; this is followed up by a detailed description of Kaplan's model. We then outline a more realistic extension of Kaplan's model. Then we perform an equilibrium and stability analysis on this model. We find that there is a critical threshold parameter R0 which determines the behaviour of the model. If R0 ≤ 1 there is a unique disease-free equilibrium, and if R0 < 1 the disease dies out. If R0 > 1 this disease-free equilibrium is unstable, and in addition there is a unique endemic equilibrium which is locally stable. If a certain condition is satisfied (and for Kaplan's model this condition is always satisfied), additional complete global-stability results are shown. These results are confirmed and explored further by simulation.

Keywords: hiv; aids; injecting drug use; shooting gallery; threshold value; equilibrium and stability analysis


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