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Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
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A

Mathematical Functions

TABLE A-1

Stationary Population Process Model for Population Age j at Year i

Measure

Formulation

1. Population Size (N)

For year i = 1:

Nij = age – specific population size

For year i > 1:

Nij = Ni-1,j-1DWi-1,j-1CPi-1,j-1

2. Target Population (T)

No Vaccine:

Tij = Nij × Proportion Targetij where Proportion Targetij = 0

Vaccine Steady State for year i = 1:

Tij = Nij × Proportion Targetij where Proportion Targetij = 1

Vaccine Introduced:

Tij = Nij × Proportion Targetij with Proportion Targetij = Input (% of N)ij

3. Vaccinated Immune (V)

Vij = Tij × coverage rateij × effectivenessij

4. Vaccinated Susceptible (VS)

VSij = Tij × coverage rateij × (1 – effectivenessij)

5. Not Vaccinated Immune (B)

Bij = (Vij / herd immunityij) – Vij

6. Not Vaccinated Susceptible (BS)

BSij = NijVijVSijBij

7. Total Cases (C)

Cij = (VSij + BSij) × incidence rate

8. Deaths by Disease (D)

Dij = Cij × case fatality rate

Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×

Measure

Formulation

9. Cases: Impairment (CP)

CPij = (CijDij) × proportion cases impaired

10. Cases: Morbidity (CM)

CMij = CijDijCPij

11. Vaccine Complications (A)

Aij = (Vij + VSij) × vaccine complications rate

12. All Cause Deaths (DA) Including Disease

No Vaccine:

DAij = Nij × all cause mortality rate

Vaccine Steady State:

DAij = (Nij × all cause mortality rate) – Deaths averted by vaccine

* Deaths averted by vaccine = Vaccine Steady State DijNo Vaccine Dij

Vaccine Introduced:

DAij = (Nij × all cause mortality rate) – Deaths averted by vaccine

* Deaths averted by vaccine = Vaccine Introduced DijNo Vaccine Dij

13. Cause Deleted Deaths (DE) Excluding Disease

DEij = DAijDij

Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×

TABLE A-2

Health and Economic Values for Population Age j at Year i

1. Premature Deaths Averted per Year

image

2. Incident Cases Prevented per Year

image

3. Quality-Adjusted Life Years (QALYs) Gained

image

4. Disability-Adjusted Life Years (DALYs) Gainedaa

image

5. Net Direct Costs

image

6. Delivery Costs

image

7. Health Care Costs (HC) Averted

image

8. Workforce Productivity (WP) Gained per Year

image

9. One-Time Costs

Cost Research + Cost Licensure + Cost Start Up

aFox-Rushby, J. A., and Hanson, K. 2001. Calculating and presenting disability-adjusted life years (DALYs) in cost-effectiveness analysis. Health Policy and Planning 16(3):326-331.

Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×

TABLE A-3

Detailed Expressions (in Reference to Table A-1 and Table A-2)

Measure

 

Formulation

All Cause Mortality Rate (Derived by Life Table Over Interval)a

  image

QALYsDeath

  image

QALYsImpairment by disease or complication

  image

QALYsMorbidity by disease or complication

  image

Disability-Adjusted Life Years (DALYs) Generalizationb

 

Years of Life Lost (YLL) + Years of Life Lived with Disability (YLD)

YLD or YLL (W = 1)

  image

DALYs Variables

 

K = age weight modulation factor (0 = off, 1 = on)

F = constant (0.1658)

r = discount rate

j = age of death (YLL) or age of onset of disability (YLD)

G = parameter form the age weighting function (0.04)

L = standard expectation of life at age a (YLL) or duration of disability (YLD)

W = disability weight (YLD)

DALYsDeath

  image

DALYsImpairment by disease or complication

  image

DALYsMorbidity by disease or complication

  image

Health Care Costs (HC)Death

  image

HCImpairments by disease or complication

  image

HCMorbidity by disease or complication

  image
Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×

Measure

 

Formulation

Workforce Productivity (WP) Gained DeathDeath

  image

WPImpairment by disease or complication

  image

WPMorbidity by disease or complication

  image

aPreston, S., P. Heuveline, and M. Guillot. 2000. Demography: Measuring and modeling population processes. Chapter 3: The Life Table and Single Decrement Process. P. 46.

bFox-Rushby, J. A., and K. Hanson. 2001. Calculating and presenting disability adjusted life years (DALYs) in cost-effectiveness analysis. Health Policy and Planning 16(3):326–331.

Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×

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Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×
Page 121
Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×
Page 122
Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×
Page 123
Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×
Page 124
Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×
Page 125
Suggested Citation:"Appendix A: Mathematical Functions." Institute of Medicine. 2012. Ranking Vaccines: A Prioritization Framework: Phase I: Demonstration of Concept and a Software Blueprint. Washington, DC: The National Academies Press. doi: 10.17226/13382.
×
Page 126
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As a number of diseases emerge or reemerge thus stimulating new vaccine development opportunities to help prevent those diseases, it can be especially difficult for decision makers to know where to invest their limited resources. Therefore, it is increasingly important for decision makers to have the tools that can assist and inform their vaccine prioritization efforts.

In this first phase report, the IOM offers a framework and proof of concept to account for various factors influencing vaccine prioritization-demographic, economic, health, scientific, business, programmatic, social, policy factors and public concerns. Ranking Vaccines: A Prioritization Framework describes a decision-support model and the blueprint of a software-called Strategic Multi-Attribute Ranking Tool for Vaccines or SMART Vaccines. SMART Vaccines should be of help to decision makers. SMART Vaccines Beta is not available for public use, but SMART Vaccines 1.0 is expected to be released at the end of the second phase of this study, when it will be fully operational and capable of guiding discussions about prioritizing the development and introduction of new vaccines.

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