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Estimating the scope of the nvCJD epidemic

A furor has erupted over whether the scope of the nvCJD epidemic in humans can be reliably estimated at this point in its course. A Lancet editorial concluded this week that it could not.

By estimating the scope of the epidemic is meant a function v(t) that gives the number of victims as a function of time. The integral of v(t) gives the total number of victims over the whole course of the epidemic. Broadly speaking, we might expect v(t) to increase monotonically, reach a peak, then taper off, again monotonically. From the mean value theorem, v'(t) has a zero (the epidemic's peak), v''(t) two zeroes (so a quartic polynomial will do), and v(t)at v'(t)=0 x (epidemic duration) provides an upper bound for the integral.

In other words, without explicitly knowing v(t), we could make some more or less model-independent estimate if we had an expansion of v(t) in its derivatives (ie its MacLaurin series), relying on the polynomials being dense in the function space of v.

Now, the data comes in as a growing n-tuple of quarterly reports from the CJD Surveillance Unit. Variants of this n-tuple correspond to using suspected, confirmed, age of onset, age of death, and so on. 'Suspected' is the best leading indicator but least reliable. We saw an (accelerating) number of 60 come in from John Lanchester for October 1996. The method of Newtonian differences is a simple approach to differential equations that anyone can use to estimate the necessary derivatives and work out the next yet-to-be reported value for the coming quarter. Let me illustrate this with an example. Suppose the data for the first four quarters comes in as: (0, 15, 35, 60). Subtracting adjacent data points, one obtains (15, 20, 25). Iterating, (5,5). Having reached all constants, we expand and reverse the ladder: (5,5,5, 5), 15, 20, 25, 30, 35), and finally (0, 15, 35, 60, 90, 125) to estimate 90 cases for the fifth quarter and 135, less usefully, for the sixth. Fitting the quartic polynomial mentioned above requires a 5-tuple, though a good 3-tuple would suffice here from symmetry considerations.

And one could go on and on, least squaring, fitting splines, weighting points by reliability, figuring confidence limits and so forth, ending perhaps with some sense of whether the epidemic is of Old Testament proportions (my view, given the molecular biology and the extent of BSE) or merely equivalent to a few years of British road rage.

What's wrong with this picture (and how soon will it get fixed)? Nothing. But quite simply, we are too early on to know what these first numbers mean. We have only the roughest estimates to the incubation time distribution, the infectious titre of the food over time, and the genetic susceptibility range of the population. These early numbers could been from atypical individuals with atypical exposures and have little to do with parameters for the "main" epidemic. (Thnk of the whole epidemic as a linear superpositioning of mini-epidemics.) It might be minimally 2-3 years before we have a clearer picture, as the Lancet suggests, but it could also be 20-30 years. The nvCJD epidemic might take 50 years to play itself out.

One thing is clear: the British have taken on a horrible risk by under-cutting and under-funding research for the last decade. Money was spent, but on industry bail-out rather than research, in a ratio of 1000 to 1 (billions to millions). Even now, policy focus is on rehabilitating beef sales, and facing up to the epidemic if and when it gets totally out of hand. There is some serious ignorance at work here on the time scales needed to make meaningful progress in research and therapies.

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