In this appendix, the committee develops a first-order model that estimates 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 individuals 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 currently 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 Stockpile (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, 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.
. The value of*X = for any particular individual*, the time from DTD to that person's prophylaxis*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**can be interpreted to be either:*Φ*(*x*) for*X**Φ*(*x*) = probability that a randomly selected individual will experience a time*X*less than or equal to*x*, or, equivalently,*Φ*(*x*) = the fraction of randomly selected individuals who will experience a time*X*less than or equal to*x*.

- 4.
. 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*= goal for the points of dispensing (PODs) for the time from start of dispensing MCM to completion*g*for the time from starting to completing dispensing, the distribution function for*X*is uniform:with associated density function:

- 5.
. It follows from the definition of*T*= time from exposure to prophylaxis (TTP) = δ +*X**X*that*T*is a random variable with probability density function*p*(*t*), where### (1)

- 6.
**The survival function**. As pointed out in Chapter 2, 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*) 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*f*(*t*) to the survival 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, byThis function can in turn be approximated (for

*t*up to around 150 hours) by### (2)

- 7.
. From the definition of*S*= the expected fraction of the population that will survive a release, described by a particular scenario, using a particular prepositioning strategy*f*(*t*) and*p*(*t*),*S*can be computed fromIn particular, using the uniform distribution for

*p*(*t*) given by equation (1) yields:### (3)

Using the approximation for

*f*(*t*) given by equation (2),*S*can then be obtained analytically (for**δ**+*g*<150 hours) from equation (3), yielding:### (4)

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.

## REFERENCES

- Brookmeyer R, Blades N, Hugh-Jones M, Henderson DA. The statistical analysis of truncated data: Application to the Sverdlovsk anthrax outbreak. Biostatistics. 2001;2(2):233–247. [PubMed: 12933552]
- Brookmeyer R, Johnson E, Barry S. Statistics in Medicine. 2005;24(4):531–542. [PubMed: 15678407]
- Wilkening DA. Sverdlovsk revisited: Modeling human inhalation anthrax. Proceedings of the National Academy of Sciences. 2006;103(20):7589–7594. [PMC free article: PMC1564296] [PubMed: 16679412]
- Wilkening DA. Modeling the incubation period of inhalational anthrax. Medical Decision Making. 2008;28(4):593–605. [PubMed: 18556642]

## Publication Details

### Copyright

### Publisher

National Academies Press (US), Washington (DC)

### NLM Citation

Committee on Prepositioned Medical Countermeasures for the Public; Institute of Medicine; Stroud C, Viswanathan K, Powell T, et al., editors. Prepositioning Antibiotics for Anthrax. Washington (DC): National Academies Press (US); 2011 Sep 30. Appendix C, First-Order Model.