## Two Stage Inverted Models

Hogan et al. (2004a) describe a two-stage inverted model called the Bayesian Aerosol Release Detector (BARD). This model takes as input weather data and counts of visits to the emergency department for respiratory complaints. It outputs

figure 19.5 The inversion process required to construct outbreak-detection algorithms with atmospheric dispersion models. In two-stage inversion, the inverted aerosol effects model computes downwind concentrations for input into the inverted dispersion model. The inverted dispersion model then computes release parameters from downwind concentrations. In one-stage inversion, the inversion of the combined model computes release parameters directly from biosurveillance data-the intervening step of computing downwind concentrations from biosurveillance data is not required.

figure 19.5 The inversion process required to construct outbreak-detection algorithms with atmospheric dispersion models. In two-stage inversion, the inverted aerosol effects model computes downwind concentrations for input into the inverted dispersion model. The inverted dispersion model then computes release parameters from downwind concentrations. In one-stage inversion, the inversion of the combined model computes release parameters directly from biosurveillance data-the intervening step of computing downwind concentrations from biosurveillance data is not required.

the likelihood ratio P(data\H1)/P(dataIH0), where data are counts of visits to emergency departments by zip code (i.e., the biosurveillance data), H1 is the hypothesis that both an anthrax release (of a particular amount at a particular location and time) and usual respiratory disease occurred, and H0 is the hypothesis that usual respiratory disease alon occurred. If the likelihood ratio is greater than a threshold value, BARD also outputs the release location, timing, and quantity.

The inverted dispersion model is an inverted Gaussian plume model (IGPM) that they created. The IGPM uses search to solve the Gaussian plume model equation with the coordinates of, and concentration at, four downwind locations called sensor locations (Hogan et al., 2004b). It does not use probability as a measurement. The measurement it does use is too complicated to describe in detail; essentially, it is the difference between predicted and actual sensor locations. When given exact downwind concentrations produced by the Gaussian plume model, IGPM was able to find the release location to within 1 meter and the release quantity to within 0.18 kilograms. With increasing levels of noise introduced into the downwind concentrations however, the accuracy of the procedure degraded rapidly, off by as much as 7 km with an amount of noise equal to approximately 20% of the actual concentrations.

The second stage (the inverted model of aerosol effects) is a Bayesian model of inhalational anthrax and usual respiratory disease (Hogan et al., 2004a). It models the probability of presenting to the ED for a respiratory complaint due to inhalational anthrax as a function of the number of spores inhaled and time since release. BARD uses Bayesian inversion (i.e., it uses probability as a measurement) of the anthrax model to compute an expectation over the number of spores inhaled and time since release. We discussed Bayesian inversion in Chapter 13 and Bayesian methods in greater detail in Chapter 18. BARD inputs the four zip codes with the highest expected number of inhaled spores and the expected time of release into IGPM (the time since release determines the weather data that the IGPM uses for analysis). BARD then uses the release location and quantity that IGPM estimates to compute the likelihood ratio.

BARD may also support more diagnostic precision at the time of outbreak detection. Current outbreak-detection algorithms cannot distinguish between a windborne outbreak and one resulting from a different mode of transmission. They detect an anomalous level of ED visits for "respiratory'' complaints, for example, and include a line listing of the verbatim chief complaints. However, the differential diagnosis of a putative "respiratory'' outbreak is quite large. Knowing that the mode of transmission is windborne reduces the differential diagnosis of a respiratory outbreak to fewer than 11 agents (Table 3.2 in Chapter 3).

Another benefit of BARD is that it estimates the likely location and date and time of the release. This information could help focus an environmental and criminal investigation, as well as the response effort.

BARD can also simulate windborne outbreaks of inhalational anthrax. We discuss the importance of outbreak simulations in biosurveillance below. Note that the ability to operate a model in either a forward direction for simulation or inverse direction for detection is general to Bayesian models and not unique to BARD.

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