Estimating the date of infection from individual response times

Abstract

It is characteristic of both naturally occurring and experimental infections that the affected individuals do not fall ill or die at the same time. If one defines the 'response time' of an individual as the interval between the earliest date on which he could have been exposed to infection (as by eating contaminated food) and the date on which he fell ill, then the distribution of individual response times is always skewed with a long tail to the right. The true distribution has often been taken as log-normal, since probit proportion of responses plotted against logarithm of time since exposure approximates to a straight line (Sartwell, 1950(Sartwell, , 1952Meynell & Meynell, 1958;Meynell, 1963). Sartwell (1966) pointed out that, if the true distribution is indeed log-normal, an unknown date of exposure, aL, can be estimated from the dates of the individual responses by the method of quantiles (Aitchison & Brown, 1963, §6.24). This is so but, owing to the actual distributions observed in practice, the earliest date of exposure can be equally well estimated by another method, and it is shown here that the two estimates necessarily disagree.Assuming first that the true distribution is log-normal, then it is clear from Fig. 1 that, because the plot of the distribution gives a straight line, the log individual response times corresponding to, say, 10 % and 90 % responses are symmetrically placed about the median response time for 50 % responses. However, this will only be so when the earliest date of exposure is correctly chosen for, if this is taken as either before or after the real date, the corresponding plots in Fig. 1 are convex or concave upwards (see Aitchison & Brown, 1963, Fig. 6.3). In general, therefore, if b2 is the calendar date of the median response time and bl, b3 are the dates corresponding to per cent responses, q and 100-q, then the earliest date of exposure, aL, is the solution of the equation log (b27aL)-log (bl-aL) = log (b3-aL)-log (b2--aL), which is readily seen to be aL-blb3-b 2 b, +b3-2b2-The method can be illustrated by experiment 30 of Martin (1946) in which mice 9Hyg. 65, 2

Evidence for a two-stage model of microbial infection

“…Because of this, the corresponding normal distributions of t, log t and l/t will be extremely hard to distinguish in practice because their differences are most marked at the tails of the distributions which can rarely be determined precisely because of the large number of hosts required. Thus, normal distributions of log t with A = 1 1 and 1.5 give virtually linear curves when l/t is plotted (Meynell & Williams, 1967) and, as Fig. 3 shows, analogous normal distributions oft appear almost linear when plotted on a logarithmic time scale.…”

“…lb, the scatter is constant once the initial period is past. In each case, it is assumed that log n follows a parabola; that an individual falls ill or dies if n reaches the morbidity or mortality threshold respectively (Williams & Meynell, 1967); and that the mean rate of increase in log n at a given time is the same for all doses (Williams & Meynell, 1967;Mackaness, 1962, fig. 2).…”

“…Values of log n rather than n were used because plots of the individual counts show that log n is clearly more symmetrically distributed than n itself (e.g. Hobson, 1957;Meynell & Meynell, 1958;Williams & Meynell, 1967). The values Of 0°logn were then plotted against time since inoculation ( Fig.…”

“…if the inoculated organisms were randomly distributed amongst susceptible and resistant sites in the host: Meynell & Stocker, 1957;Meynell, 1957), or it could increase progressively as time passed (e.g. if infection followed a birth-death process: Williams, 1965 a, b; Armitage, Meynell & Williams, 1965;Williams & Meynell, 1967;Shortley & Wilkins, 1965). The first possibility was suggested by viable counts on mice infected with Salmonella typhimurium which showed that the counts made only 1-5 hr.…”