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Appendix C First-Order Model In this appendix, the committee develops a first-order model that esti- mates health outcomes (measured as fraction of exposed individuals who survive) for any prepositioning strategy. For convenience of presentation, the term survival is used instead of saved to refer to those exposed indi- viduals who have been protected from becoming symptomatic by timely prophylaxis with effective medical countermeasures (MCM). The model development is as follows. 1. δ = time between release (and, by assumption, exposure) and decision to dispense (DTD). For any community, estimates of δ should ideally be informed by existing submodels that incorporate the capabilities of cur- rently used (or planned) monitoring and surveillance systems, as well as data from past BioWatch Actionable Results, accidental releases, and drills, to estimate the various times contributing to the value of δ. These include the time: • needed to deliver MCM from the nearest Strategic National Stock- pile (SNS) location, • required to determine clinically that at least one individual has been infected, and • between positive diagnosis and the decision by the responsible public health authority to issue an order to dispense. Note that in what follows, the assumption is made that all individuals will be exposed at the moment of release. This is an “optimistic” assumption, 237

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238 PREPOSITIONING ANTIBIOTICS FOR ANTHRAX in the sense that individuals exposed later will not require prophylaxis as soon as those exposed immediately and thus will fare better as a result of any dispensing campaign. 2. X = for any particular individual, the time from DTD to that person’s prophylaxis. The value of X is not just clearly different for each individual, but is an uncertain quantity for any individual. In other words, for any individual, X is a random variable. 3. The probability distribution Φ(x) for X can be interpreted to be either: Φ(x) = probability that a randomly selected individual will experi- • ence a time X less than or equal to x, or, equivalently, Φ(x) = the fraction of randomly selected individuals who will ex- • perience a time X less than or equal to x. 4. g = goal for the points of dispensing (PODs) for the time from start of dispensing MCM to completion. Using the simplifying assumptions that the size of the dispensing staff is constant, that staff are never idle, and that the service time is constant at the PODs, it can readily be shown that, given the goal g for the time from starting to completing dispensing, the distribution function for X is uniform: x Φ ( x) = , 0≤x≤ g g with associated density function: 1 φ ( x) = , 0≤x≤ g g 5. T = time from exposure to prophylaxis (TTP) = δ + X. It follows from the definition of X that T is a random variable with probability density function p(t), where 1 p (t ) = , δ ≤t ≤δ+g (1) g 6. The survival function f(t) represents, for any particular release scenario, one of the various incubation period curves or values discussed at length in Chapter 2, where t is the time since exposure. As pointed out in Chapter 2,

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239 APPENDIX C data with which to compute this survival function are either uncertain or limited, and the function will depend on many unknown scenario variables. Nevertheless, to obtain insight into the potential health advantages of prepositioning, the committee has taken the liberty of fitting f(t) to the sur- vival data (based on the Sverdlosk release) presented by Wilkening (2006, 2008) and Brookmeyer et al. (2001, 2005). Using these data, f(t) can be well fit, for values of t up to about 200 hours, by 2 f (t ) = e−(.004t ) This function can in turn be approximated (for t up to around 150 hours) by 2 f (t ) = 1 − (.004t ) (2) 7. S = the expected fraction of the population that will survive a release, de- scribed by a particular scenario, using a particular prepositioning strategy. From the definition of f(t) and p(t), S can be computed from S = ∫ f (t ) p (t ) dt In particular, using the uniform distribution for p(t) given by equation (1) yields: δ+ g 1 ∫ f (t ) dt S= (3) g δ Using the approximation for f(t) given by equation (2), S can then be ob- tained analytically (for d + g <150 hours) from equation (3), yielding: (.004) ((δ + g ) − δ 3 ) 2 3 δ+ g 1 ∫ (1− (.004t ) )dt = 1− 2 S= (4) 3g g δ This equation is valid for g >0; since the practical realities of even the most ideal strategy for predispensing to individuals will involve some finite delay, for all practical purposes, g will never be exactly equal to 0.

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240 PREPOSITIONING ANTIBIOTICS FOR ANTHRAX REFERENCES Brookmeyer, R., N. Blades, M. Hugh-Jones, and D. A. Henderson. 2001. The statistical analysis of truncated data: Application to the Sverdlovsk anthrax outbreak. Biostatistics 2(2):233-247. Brookmeyer, R., E. Johnson, and S. Barry. 2005. Statistics in Medicine 24(4):531-542. Wilkening, D. A. 2006. Sverdlovsk revisited: Modeling human inhalation anthrax. Proceedings of the National Academy of Sciences 103(20):7589-7594. Wilkening, D. A. 2008. Modeling the incubation period of inhalational anthrax. Medical Decision Making 28(4):593-605.