Back-calculation methods have been widely used to reconstruct the past history of the HIV epidemic and to provide short-term predictions of AIDS incidence, on the basis of reported AIDS cases, knowledge of the incubation period distribution and assumptions on the shape of the HIV infection curve.

Within the back-calculation framework, a great variety of different model assumptions and modelling approaches have been employed at each stage of the process (infection, incubation and reporting). Considerable uncertainty exists about the appropriate form for each stage. For example only information on first half of the incubation distribution is available, and knowledge of the effect and extent of AIDS prophylaxis and treatment is still limited. Furthermore, the history of HIV incidence can only be inferred indirectly.

The complexity of the total model has prevented a formal treatment of uncertainty in model formulation and parameter estimation. For example, parameters of the incubation distribution are usually fixed. Further complexities are added through use of other sources of data, such as seroprevalence estimates (and their inherent imprecision). Informal sensitivity analyses and bootstrapping have provided partial answers to the effects of uncertainty on AIDS projections, but a formal treatment of uncertainty demands a new approach to estimation. In particular, a Bayesian framework is indicated, since informative prior distributions on some parameters would allow useful compromises between assuming complete ignorance about their values, and fixing them absolutely.

We describe the basics of AIDS back calculation, reviewing model assumptions and generalisations. We motivate our approach to model building, and propose estimation through Markov chain Monte Carlo (MCMC). We show some results for the epidemic in England and Wales.