Three stage AIDS incubation period: a worst case scenario using addict-needle interaction assumptions.
In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. We start off with a brief literature survey and review; this is followed by the derivation of a model which allows addicts to progress through three distinct stages of variable infectivity prior to the onset of full blown AIDS and where the class of infectious needles is split into three according to the different levels of infectivity in addicts. Given the structure of this model we are required to make assumptions regarding the interaction of addicts and needles of different infectivity levels. We deliberately choose these assumptions so that our model serves as an upper bound for the prevalence of HIV under the assumption of a three stage AIDS incubation period. We then perform an equilibrium and stability analysis on this model. We find that there is a critical threshold parameter R(0) which determines the behaviour of the model. If R(0)< or =1, then irrespective of the initial conditions of the system HIV will die out in all addicts and all needles. If R(0)>1, then there is a unique endemic equilibrium which is locally stable if, as is realistic, the time scale on which addicts inject is much shorter than that of the other epidemiological and demographic processes. Simulations indicate that if R(0)>1, then provided that disease is initially present in at least one addict or needle it will tend to the endemic equilibrium. In addition we derive conditions which guarantee this. We also find that under calibration the long term prevalence of disease in our variable infectivity model is always greater than in an equivalent constant infectivity model. These results are confirmed and explored further by simulation. We conclude with a short discussion.