## Abstract

Back-calculation analysis of the variant Creutzfeldt-Jakob disease epidemic in the United Kingdom is used to estimate the number of infected individuals and future disease incidence. The model assumes a hazard of infection proportional to the incidence of bovine spongiform encephalopathy in the United Kingdom and accounts for precautionary control measures and very wide ranges of incubation periods. The model indicates that current case data are compatible with numbers of infections ranging from a few hundred to several millions. In the latter case, the model suggests that the mean incubation period must be well beyond the human life-span, resulting in disease epidemics of at most several thousand cases.

Variant Creutzfeldt-Jakob disease (vCJD) is caused by an agent that is currently indistinguishable from that responsible for bovine spongiform encephalopathy (BSE) in cattle. However, 5 years after the identification of vCJD, great uncertainty remains over how many individuals have been infected with the agent and how many of these individuals will go on to develop clinical disease (1–5).

In the absence of a test for infection, one approach to estimating the number of infected individuals is provided by back-calculation, a statistical technique developed in the context of the HIV/AIDS (human immunodeficiency virus/acquired immunodeficiency syndrome) epidemic (6, 7). This approach makes use of the fact that the number and timing of cases of disease that occur depend on three factors: (i) how many people were infected, (ii) when they were infected, and (iii) how long it takes from infection for disease to become apparent—the incubation period. To use this approach to estimate the number of individuals infected with the vCJD agent, it is necessary to make assumptions about when people were exposed to infection and how long it takes them to develop disease. Previous work has shown that the estimated number of infections/cases produced by this approach is very sensitive to the assumptions made about the incubation period distribution (2, 8,9).

We have developed a family of back-calculation models (10) to explore the prevalence of infection with the vCJD agent and the incidence of clinical vCJD in the UK. The main features of these models are as follows: (i) The hazard of infection was assumed to have been proportional to the incidence of BSE (11). We did not consider onward, human-to-human transmission of the infectious agent. (ii) The incubation period of the disease was assumed to follow an offset generalized *F* distribution, which has five parameters. We also investigated the lognormal, Weibull, and gamma distributions as special cases of this distribution (12). (iii) We assumed that the incubation period was independent of age at infection (13). (iv) The model was restricted to the 40% (approximately) of the UK population assumed to be methionine homozygous at codon 129 of the prion protein (PrP) gene. (All cases of vCJD identified to date have been of this genotype.) (v) To minimize the impact of reporting delays, we fitted the models to the data on the 82 cases with onsets before 2000 that had been identified by 31 December 2000.

The back-calculation model had seven parameters in total [five for the incubation period distribution, one for the hazard of infection, and one for the effect of the specified bovine offals (SBO) ban in 1989]. The model was fitted by the maximum likelihood method, assuming a Poisson likelihood. Because of a very severe parameter identifiability problem, we estimated the incubation period distribution parameters for fixed values of the hazard of infection (corresponding to total numbers of infections ranging from 100 to 12 million) and for fixed effects of the SBO ban ranging from 0 to 90%. Allowing a very flexible incubation period distribution (offset generalized *F*), we found that the cases observed to date were almost equally compatible with any number of infections up to several millions. However, when a very large number of infections was considered, the model indicated that the average incubation period was likely to be extremely long and, in most instances, well beyond the normal human life-span. As a result, the corresponding epidemic sizes (clinical cases) lay within a much narrower range, from a few hundred to a few thousand cases (Table 1).

When making stronger assumptions about the form of the incubation period distribution (i.e., reducing the number of parameters from five), the range for the number of infections with which the observed cases are compatible becomes narrower. For example, using a simple generalized *F* distribution led to a point estimate of the total number of infections of a few hundred, with an upper 95% confidence limit of about 1000 (and upper 99% confidence limit of 7000). Using an offset lognormal distribution again led to a point estimate of the total number of infections of a few hundred. The predicted course of the vCJD epidemic was calculated under different assumptions about the incubation period distribution (14). No matter which incubation period distribution is used, the point estimates obtained from the model suggest that the epidemic of cases of vCJD is very close to its peak. However, the expected numbers of cases corresponding to the upper limits of infection (14) indicate that the data are also compatible with an epidemic whose peak, many years hence, is determined by mortality among infected individuals from competing causes. Table 1also presents approximate prediction intervals (15) for annual numbers of cases at different times in the future. These indicate that the annual incidence of vCJD is unlikely ever to be much more than 100 cases (14).

None of our models suggest that the number of primary cases of vCJD in methionine homozygotes is likely to be more than a few thousand, even though the number of primary infections could be anything from a few hundred to many thousands or even millions. In interpreting these results, and extrapolating them to other codon 129 genotypes, we must bear in mind our model assumptions. Our key finding that, regardless of the number of infections that have occurred, the number of clinical cases is unlikely to exceed a few thousand (in any one genotype) is sensitive to a number of assumptions.

First, we have assumed that in codon 129 methionine homozygotes, the incubation period for vCJD has a unimodal distribution. This is a key assumption that is open to question (16). In mice, there are genetic factors lying outside the coding region of the PrP gene that have an important influence on the incubation period of transmissible spongiform encephalopathies (17–20). It is possible, therefore, that among human codon 129 methionine homozygotes there are other, presently unknown genetic factors that influence the vCJD incubation period. We have used the generalized *F* distribution, which can take a wide range of unimodal forms. If, across the methionine-homozygous population, the mixture of other genetic factors affecting incubation period results in an overall incubation period distribution that is close to unimodal, we would be confident that, broadly, our findings with respect to the numbers of clinical cases hold. If, however, the overall incubation period distribution is strongly multimodal, there might be many more clinical cases of vCJD than our models predict. If the latter is the case, then the development of reliable back-calculation models will be possible only when the relevant genetic factors have been identified and measured in the population. Strong multimodality is most likely to apply if only a small number of other genetic factors are involved and there was little variation between infected individuals in the infecting dose to which they were exposed.

Second, we have assumed that the incubation period distribution does not vary with age at infection. Experimental evidence in mice indicates that, for a fixed dose, incubation period does vary with age at inoculation (21). However, this variation is small, with young mice having incubation periods 7 days longer than older mice, compared with mean incubation periods of several hundred days. If vCJD infections occurred through diet, as we have implicitly assumed, infected individuals may have been exposed to a very wide range of infectious doses whose impact on incubation period is likely to dwarf any small age effects.

Third, to extrapolate from codon 129 methionine homozygotes to other genotypes, we need to assume that across codon 129 genotypes the relation between the mean and the variance of the incubation period distribution does not vary greatly. If other genotypes have longer mean incubation periods but with lower variance, then we might observe larger numbers of cases in these genotypes. It is, however, unusual for the variance of a distribution to decrease as the mean increases. If this is not the case, then to extend our results to include all genotypes one could, as a worst-case scenario, multiply our predictions by about 2.5 to obtain a figure for the whole population.

A further assumption of the model is that infection was essentially through diet and that the amount of infectivity consumed in food during any given period was proportional to the number of BSE cases occurring up until 1996. In the absence of ongoing human-to-human transmission of the vCJD agent, our findings are likely to be much less sensitive to this assumption than they are to the assumptions about incubation period.

The upper limits of our estimates differ from those presented by Ghani*et al*. (1). These authors used simulation to identify a range of scenarios compatible with the actual observed incidence. One advantage of this approach is that it allows the incorporation into the model of several parameters that could not be estimated. However, it does not enable any probability statement to be made about the coverage of the range of epidemics that it produces, and running more simulations can only increase the range of scenarios that are plausible. We believe the most likely explanation for the different ranges of cases coming out of our work and that of Ghani *et al*. is that the coverage probabilities of those intervals are different.

Our models suggest that the number of primary cases of vCJD in methionine homozygotes is unlikely to exceed a few thousand, but that considerably greater uncertainty surrounds the number of primary vCJD infections that have occurred. Whether a few hundred or many more people have been infected has important consequences for public health and, in particular, for the risk of secondary transmission (22). If secondary transmission does occur, the mean incubation period in secondary cases may be much shorter than in primary cases (23). In the absence of a reliable test for asymptomatic infection, considerable uncertainty about the number of infected individuals may remain for a number of years.

↵* To whom correspondence should be addressed. E-mail: jerome.huillard{at}lshtm.ac.uk