Three-stage AIDS incubation period: a best case scenario using addict-needle interaction assumptions

In this paper we extend the 'needles that kill' model discussed in Kaplan & O'Keefe (1993) to allow addicts to progress through three-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 a lower bound for the prevalence of HIV under the assumption of a three-stage AIDS incubation period. We find that there is a critical threshold parameter R0 which determines the behaviour of the model. If R0 > 1 then there is a unique endemic equilibrium which is locally stable if, as is realistic, the timescale on which addicts inject is much shorter than that of the other epidemiological and demographic processes. Simulations indicate that if R0 > 1, then provided that disease is initially present in at least one addict or needle then 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 the 'needles that kill' model is the same as in our three-stage model.