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Mathematical Medicine and Biology 2001 18(3):225-262; doi:10.1093/imammb/18.3.225
© 2001 by Institute of Mathematics and its Applications
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A mathematical treatment of AIDS and condom use

David Greenhalgh1, Murray Doyle1 and Fraser Lewis1

1 Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK

In this paper we examine the impact of condom use on the sexual transmission of human immunodeficiency virus (HIV) and acquired immune deficiency syndrome (AIDS) amongst a homogeneously mixing male homosexual population. We first derive a multigroup SIR-type model of HIV/AIDS transmission where the homosexual population is split into subgroups according to frequency of condom use. Both susceptible and infected individuals can transfer between the different groups. We then discuss in detail an important special case of this model which includes two risk groups and perform an equilibrium and stability analysis for this special case. Our analysis shows that this model can exhibit unusual behaviour. As normal, if the basic reproduction number, R0, is greater than unity then there is a unique disease-free equilibrium which is locally unstable and a unique endemic equilibrium. However, when R0 is less than unity two endemic equilibrium solutions can also co-exist simultaneously with the disease-free solution which is locally stable. Numerical simulations using realistic parameter values confirm this and we find that in certain circumstances the disease-free solution and one of the endemic solutions are both locally asymptotically stable, while the other endemic solution is unstable. This unusual behaviour has important implications for control of the disease as reducing R0 to less than unity no longer guarantees eradication of the disease. For a restricted special case of this two-group model we show that there is only the disease-free equilibrium for R0 ≤ 1 which is globally stable. For R0 > 1 the disease-free equilibrium is unstable and there is a unique endemic equilibrium which is locally stable. We then attempt to fit the model to HIV and AIDS incidence data from San Francisco, USA. The paper concludes with a brief discussion.

Keywords: HIV; AIDS; condom use; mathematical model; equilibrium and stability analysis; backwards bifurcation

Received 14 November 2000. Revised 18 August 2001.


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