Detecting bioterror attack.

To the Editor: In a recent article (), Kaplan et al. addressed the problems in detecting a bioterror attack from blood-donor screening. The main point of this comment is the "early approximation" used by Kaplan et al. to derive the probability of detecting an attack. The simplification used by Kaplan et al. leads to a probability that does not account for the size of the exposed population and can lead to incorrect results and misinterpretations.

Consider a single bioterror attack that infects a proportion p of an exposed population of size N at time τ = 0, such that the initial number of infected is I= Np. The quantity of interest is the probability D(τ) of finding at least one positive blood donation and detecting the attack within time τ. For attacks conducted with contagious agents that could lead to an epidemic, Kaplan et al. used the early approximation solution of the classic epidemic models () to describe the progression of the number of infected persons. Consequently, the resulting probability of attack detection [noted Des(τ)] is dependent only upon the initial size of the release I0 , the basic reproductive number R0 (the mean number of secondary cases per initial index case), and other variables (the blood screening window ω, the mean number k of blood donations per person and per unit of time, and the mean duration of infectiousness 1/r) (Appendix). Early approximation can lead to unreliable results because it is valid only at earlier stages of the epidemics and in the limit where the proportion p of initially infected is much smaller than the intrinsic steady proportion (R0 – 1) / R0 of the epidemics (Appendix). Relaxing this approximation and using the full solution for the progression of the number of infected persons leads to the probability D(τ)that takes into account the size of the exposed population (Appendix). The latter is important because, in contrast to Des(τ)that leads to the same conclusion, D(τ) indicates that the probabilities of detecting an attack within two exposed populations of different sizes, but with the same numbers of initially infected, are not identical. As illustrated in the Figure, when the other variables are fixed, D(τ)decreases as the proportion p of initially infected increases because the epidemic size decreases as p approaches the threshold (R0 – 1) / R0 . These subtleties of a simple epidemic model are even less reliable when using the blood screening to detect a bioterror attack with agents that cause diseases of very short incubation period.

Figure

Probability of attack detection delay for a contagious agent. Dashed line represents the early approximation Des(τ), solid lines the full solution (where the numbers represent the fraction p of the population initially infected), and the symbol "nc" stands for noncontagious agent (R0=0). The parameters are as follows: blood donation rate k = 0.05 per person per year, screening mean window period ω = 3 days, mean duration of infectiousness 1/r = 14 days, basic reproductive number R0=5, and the initial attack size Np = 500. Note that the exposed populations are therefore 5,000 and 625 for p = 0.1 and p= (R-1) / R = 0.8 , respectively.

Nonetheless, detecting a bioterror attack is very similar to detecting the response of pathogen-specific immunoglobulin M antibodies (as an indicator of recent contact of hosts with pathogens) within a population of hosts by using serologic surveys. Therefore, the reasoning developed for a bioterror attack can be extended and applied to detect and time the invasion or early circulation of certain pathogens within a population. In that perspective, it might be useful to develop an analysis that includes more details of the epidemic progression within this framework.

Appendix

Following Kaplan et al. (), the probability D(τ) of finding at least one positive blood donation and detecting the attack within time τ, after a single bioterror attack initially infecting a proportion P of an exposed population of size N, is given by

When R0 > 1, this logistic function increases, remains constant or decreases from the initial value I( towards the steady state I(t →∞) = I∞ = (R0 – 1) N / R0 for

I, respectively. In particular, in the limit of << (R – 1) / R0 , this expression reduces to the early approximation solution, I exp {-(R0 – 1)rt }, and the resulting probability Des(τ)of attack detection is instead,

Des(τ) = 1 – exp{-1I0[aƒ(τ /∞) − bƒ[(R0 – 1) rτ ]]}

[2]

where the function, ƒ(x) = x – 1 + exp (-x), and the constants are, a = k∞ [1 - r ∞(2R0 – 1)] / [1-r∞(R0-1)] and b= kR {r (2R){1-r∞(R1)]}. This Des(τ) increases when any of the parameters increase.

In Reply: As stated and argued throughout our article (), we conducted a best-case analysis under assumptions that favored blood-donor screening to detect bioterror attacks; if such an analysis fails to justify donor screening, no analysis will. Bicout () is concerned about our assumption of exponential infection growth after attack; however, this assumption was one of several we made deliberately as part of our best-case scenario ().

Bicout's calculations actually reinforce rather than refute our analysis. By relaxing our assumption of exponential infection growth and using the well-known logistic solution to the basic epidemic model (equation 1 in Bicout's letter), Bicout shows that more time is required to detect a bioterror attack than when exponential infection growth is assumed (Figure accompanying Bicout's letter). The number of persons infected over time under the logistic model will be fewer than the number of persons infected if exponential growth is assumed; therefore, screening blood donors to detect a bioterror attack is even less attractive than using our best-case assumptions. The take-home message from our article was and is: It makes little sense to screen blood donors to detect a bioterror attack.