There was fascinating dichotomy presented this morning by the papers of Nowak and Taylor. This dichotomy has been given several names during this meeting, and my favourite is the distinction between thought experiments to understand the processes that generate observed patterns, and the analysis of real experimental data. These two papers are essentially addressing the same subject: the pattern of CD4 counts over time, and it appears to me that both approaches would benefit from consideration of the other. On one hand, Taylor explains much of the variability in the observed counts as being derived from an underlying stochastic process, whereas it may well be due to a highly non-linear process changing on a time-scale faster than the sampling interval. On the other hand, Nowak does not use his model to produce predictions of CD4 numbers which may actually be testable by comparison with such data.

There is general problem here with the use of deterministic models, i.e. those that produce a single value or set of single value results for each time point without any measure of variability. Differential equations are an invaluable tool for mathematical descriptions of disease processes, but suffer from the fact that data-derived estimates are required for the processes embedded in the equation system, for example density dependent transmission. There are methods available for fitting equations directly to observations of the system over time, but these tend to regard the variability in data as some form of random error, and the fitting involves simple reduction of the average difference between observation and model.

The incubation period of AIDS is a key characteristic in understanding the HIV and AIDS epidemic, both clinically and epidemiologically. The incubation period distribution (IPD) provides information about the probability of progression to AIDS as it changes with time since infection with HIV. The IPD also provides the link between the HIV infection rate and the occurrence of AIDS cases over time, and is an essential feature in back calculation procedures (Brookmeyer and Gail 1988). Knowledge of the IPD creates the opportunity to make more reliable projections (Hendriks et al. 1992), which are necessary for health-care planning.

Cohort study data relating to development of AIDS are inevitably incomplete in dates of seroconversion (infection) or development of AIDS, or both. This incompleteness has inspired a variety of approaches. We used a multiple imputation procedure, with four related models, each covering different assumptions, to investigate the sensitivity of the estimated IPD regarding the imputation method. The imputation procedure was used to provide the unobserved interval between seroconversion and enrolment for those individuals who were already HIV infected at enrolment. We can exclude observations relating to individuals who received antiviral and or prophylactic treatment designed to delay the onset of AIDS. The results obtained from data such as these will be valuable in future to aid the understanding of the effects of new therapies on the evolution of the AIDS epidemic.

The IPD was estimated using data available at February 1990 from all homosexual and bisexual men with HIV seropositive blood samples (n = 348; aged 25–45 years), who were part of a larger cohort study in Amsterdam.

The epidemiology of infectious human diseases

Understanding and controlling the spread of infections is of vital importance to society, and in the past century the epidemiology of human disease has become a subject in its own right. Theory and applicable techniques have been developed to study both the evolution of disease within individual people and the transmission of infections through populations. Mathematics has an important role to play in these studies, which raise challenging problems ranging from broad theoretical issues to specific practical ones, and in recent years there have been significant advances in developing and analysing mathematical models of disease progression. For example, in human diseases in particular, the problems of modelling population heterogeneity are especially important.

Over the last decade there has been a great deal of work concerned with HIV and AIDS. This has been concentrated mainly in two areas: the statistical estimation of various parameters associated with HIV infection (for example, the probability of vertical transmission; the description of the incubation period from infection to clinical disease; the estimation from reported AIDS cases of the number of people infected), and the description of transmission of HIV within and between populations (for example, the characterisation of networks of risk behaviour; the impact of different control strategies). To an extent, the growth of studies in this area has become divorced from the study of other infections, and therefore one of the primary purposes of this volume is to bring together work on modelling a wide range of human diseases so as to encourage cross-fertilisation between AIDS related research and research of the epidemiology of other infections.