Time and size of possible bioterror event estimated in real time.

In the event of a bioterror attack, rapidly estimating the size and time of attack enables short-run forecasts of the number of persons who will be symptomatic and require medical care. We present a Bayesian approach to this problem for use in real time and illustrate it with data from a simulated anthrax attack. The method is simple enough to be implemented in a spreadsheet.

In the event of a bioterror attack, once the biologic agent has been determined, rapidly estimating the size and time of attack enables a forecast of the number of persons who will be symptomatic and will require medical attention over the days (and perhaps weeks) after the attack. Such a forecast could play a key role in determining the response effort required, e.g., surge capacity planning at hospitals, distributing vaccines or antimicrobial agents to the population, as appropriate (

We present a Bayesian approach to the real-time estimation of the size and time of a bioterror attack, from case report data, that is simple enough to implement in a spreadsheet. The model assumes a single-source outbreak caused by a bioterror attack at a particular point in time. Although the model assumes that the infectious agent is not contagious, the analysis still holds for contagious agents until secondary infections have progressed to symptomatic cases. Thus, our model should prove valuable within the first incubation period after an attack has been detected for a contagious agent and for longer time periods in the event of a noncontagious agent. However, in the event of multiple attacks at different points in time or an attack with a rapidly progressing contagious agent, the problem becomes more difficult and similar to the use of back-calculation to recover the incidence of infection over time from symptomatic case reporting (

The key assumptions in our analysis are that the biologic agent used has been identified and that the probability distribution of the incubation time from infection through symptoms is known. The incubation time distribution for anthrax has been estimated by Brookmeyer and colleagues on the basis of the Swerdlovsk outbreak (

Likely ranges for the incubation times of other plausible bioterror agents are available at the Centers for Disease Control and Prevention’s bioterror Web site (

We assume that the attack is detected through the appearance of infected persons with symptoms, and that as cases are identified, patient interview yields the approximate time at which symptoms appeared, a process which avoids the need for explicit estimates of reporting delay. Corrected as such, case reports provide two types of information. The number of cases observed provides a lower bound on the size of the attack. The specific timing of case reports also conveys information that can be better understood when filtered through the agent-specific incubation time distribution.

The mathematical details of our approach are described in the Appendix. We define the time origin as the instant when the first case (and hence the attack) is detected (though the time origin can be reassigned if case investigation indicates that a subsequently reported case had earlier symptoms). At the moment the attack is detected, consistent with Bayesian principles (

As an example, we simulated an anthrax attack that infects 100 persons using the incubation time distribution for anthrax estimated from the Swerdlovsk outbreak (

Absent intervention, the 100 victims in this simulated attack would appear as case-patients in accord with

Simulated actual (open dots) and forecasted (solid curve) cumulative cases in an anthrax bioterror attack that infects 100 persons 1.8 days before the first symptomatic case is observed. The cases were simulated from a lognormal distribution with median 11 days and dispersion 2.04 days, which corresponds to the incubation time estimated for anthrax based on the Swerdlovsk outbreak (

Days past case no. 1 | Total cases | Estimated attack size | Estimated day of attack (before case no. 1) |
---|---|---|---|

1 | 5 | 850 | 1.1 |

2 | 7 | 120 | 1.9 |

3 | 15 | 160 | 1.4 |

4 | 18 | 100 | 1.8 |

5 | 23 | 90 | 1.8 |

^{a}See Appendix for details.

Posterior probability density of the attack size based on the data in

Posterior probability density of the time of attack based on the data in

Given estimates of the initial size and time of attack, one can forecast the occurrence of future cases over time, as shown in

The key assumptions in our model are that the probability distribution of the incubation time from infection through development of symptoms is known and that attack victims can report the times of symptom onset (so we have not explicitly accounted for reporting delay). In an actual bioterror attack, determining the incubation time distribution itself might be necessary. For example, as shown recently by Brookmeyer et al. (

We seek to estimate the initial attack size and the time of the attack from observed cases of infection in real time. The case report data are the (reporting-delay corrected) times at which cases have been reported. We intend this model to be applied once an attack has been discovered and assume that the agent is noncontagious (or in the case of a contagious agent, that no secondary transmission has occurred) and that any interventions mounted (such as vaccination or the administration of antimicrobial agents) have not yet had any effect on the early case reporting data. We define _{j}_{1} = 0). The unknown time from the attack until the first case is observed is denoted by

We treat _{1} = 0. At the time the attack is detected, we quantify our beliefs regarding the size of the attack by the prior probability distribution _{}. Let _{} If _{} thus

_{}

Consequently, the probability that _{}from which the conditional probability density function of

_{}

Equation 2 implies that the joint prior distribution for the size and time of attack when the first case is observed is equal to

_{}

Now, suppose that by time _{} Conditional upon an attack of size

_{}

where _{2}, _{3}, _{4},…, _{k}

_{}

and application of Bayes rule yields the joint posterior distribution of the size and time of attack as

_{}

The posterior distributions of

To obtain a short-run forecast of future cases, note that conditional upon an attack of size _{}. Unconditioning over equation no. 6 yields a simple short-run forecast of the number of future cases expected given all of the data observed to date. An even simpler approximation is obtained by substituting the posterior expected values of

In our examples, we assume that, a priori, the logarithm of the attack size

Our examples also assume that the incubation time from infection through onset of symptoms is distributed in accord with a lognormal distribution with a median of 11 days and a dispersion of 2.04 days. This is the distribution fit to the data from the anthrax outbreak in Swerdlovsk (

The joint prior distribution of

E.H.K. was supported in part by Yale University’s Center for Interdisciplinary Research on AIDS through Grant MH/DA568286 from the U.S. National Institutes of Mental Health and Drug Abuse.

The joint prior distribution of the attack size and time of attack, based on the assumption that the logarithm of the attack size is uniformly distributed from 1 to 10,000.

The joint posterior distribution of the attack size and time of attack. Based on a total of 23 cases at the end of 5 days since the initial case was observed. The marginal distributions of

Dr. Walden has a docentship in applied mathematics from Uppsala University and is currently affiliated with the International Center for Finance at the Yale School of Management. His research focuses on mathematical finance, but he is also interested in the mathematical modeling of social systems, including statistical methods for modeling public health policy issues.

Dr. Kaplan is the William N. and Marie A. Beach Professor of Management Sciences at the Yale School of Management and professor of public health at the Yale School of Medicine, where he directs the Methodology and Biostatistics Core of Yale’s Center for Interdisciplinary Research on AIDS. His interests include the application of operations research, statistics, and mathematical modeling to public health policy problems such as HIV prevention, and more recently, bioterror preparedness and response.