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Prepositioning Antibiotics for Anthrax (2012)

Chapter: Appendix C: First-Order Model

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Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
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Appendix C

First-Order Model

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. image = time between release (and, by assumption, exposure) and decision to dispense (DTD). For any community, estimates of image 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 image. 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,

Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×

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 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. 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:

image

with associated density function:

image

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

image

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,

Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×

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 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, by

image

This function can in turn be approximated (for t up to around 150 hours) by

image

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

image

In particular, using the uniform distribution for p(t) given by equation (1) yields:

image

Using the approximation for f(t) given by equation (2), S can then be obtained analytically (for image + g <150 hours) from equation (3), yielding:

image

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.

Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×

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.

Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×
Page 237
Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×
Page 238
Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×
Page 239
Suggested Citation:"Appendix C: First-Order Model." Institute of Medicine. 2012. Prepositioning Antibiotics for Anthrax. Washington, DC: The National Academies Press. doi: 10.17226/13218.
×
Page 240
Next: Appendix D: Commissioned Paper: A Cost and Speed Analysis of Strategies for Prepositioning Antibiotics for Anthrax »
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If terrorists released Bacillus anthracis over a large city, hundreds of thousands of people could be at risk of the deadly disease anthrax-caused by the B. anthracis spores-unless they had rapid access to antibiotic medical countermeasures (MCM). Although plans for rapidly delivering MCM to a large number of people following an anthrax attack have been greatly enhanced during the last decade, many public health authorities and policy experts fear that the nation's current systems and plans are insufficient to respond to the most challenging scenarios, such as a very large-scale anthrax attack. The U.S. Department of Health and Human Services' Office of the Assistant Secretary for Preparedness and Response commissioned the Institute of Medicine to examine the potential uses, benefits, and disadvantages of strategies for repositioning antibiotics. This involves storing antibiotics close to or in the possession of the people who would need rapid access to them should an attack occur.

Prepositioning Antibiotics for Anthrax reviews the scientific evidence on the time window in which antibiotics successfully prevent anthrax and the implications for decision making about prepositioning, describes potential prepositioning strategies, and develops a framework to assist state, local, and tribal public health authorities in determining whether prepositioning strategies would be beneficial for their communities. However, based on an analysis of the likely health benefits, health risks, and relative costs of the different prepositioning strategies, the book also develops findings and recommendations to provide jurisdictions with some practical insights as to the circumstances in which different prepositioning strategies may be beneficial. Finally, the book identifies federal- and national-level actions that would facilitate the evaluation and development of prepositioning strategies.

Recognizing that communities across the nation have differing needs and capabilities, the findings presented in this report are intended to assist public health officials in considering the benefits, costs, and trade-offs involved in developing alternative prepositioning strategies appropriate to their particular communities.

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