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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 For year i = 1:
Size (N) Nij = age − specific population size
For year i > 1:
Nij = Ni−1,j−1 − DWi−1,j−1 − CPi−1, j−1
2. Target No Vaccine:
Population (T) 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 Vij = Tij × coverage rateij × effectivenessij
Immune (V)
4. Vaccinated VSij = Tij × coverage rateij × (1 − effectivenessij)
Susceptible
(VS)
5. Not Bij = (Vij / herd immunityij) − Vij
Vaccinated
Immune (B)
6. Not BSij = Nij − Vij − VSij − Bij
Vaccinated
Susceptible
(BS)
7. Total Cases Cij = (VSij + BSij) × incidence rate
(C)
8. Deaths by Dij = Cij × case fatality rate
Disease (D)
121
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122 RANKING VACCINES: A Prioritization Framework
Measure Formulation
9. Cases: CPij = (Cij − Dij) × proportion cases impaired
Impairment
(CP)
10. Cases: CMij = Cij − Dij − CPij
Morbidity
(CM)
11. Vaccine Aij = (Vij + VSij) × vaccine complications rate
Complications
(A)
12. All Cause No Vaccine:
Deaths (DA) DAij = Nij × all cause mortality rate
Including
Vaccine Steady State:
Disease
DAij = (Nij × all cause mortality rate) −
Deaths averted by vaccine
*Deaths averted by vaccine = Vaccine Steady State Dij − No Vaccine Dij
Vaccine Introduced:
DAij = (Nij × all cause mortality rate) −
Deaths averted by vaccine
*Deaths averted by vaccine = Vaccine Introduced Dij − No Vaccine Dij
13. Cause Deleted DEij = DAij − Dij
Deaths (DE)
Excluding
Disease
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123
Appendix A
TABLE A-2
Health and Economic Values for Population Age j at Year i
1. Premature n= 1
∑ (No Vaccine (D ) − Vaccine (D ))
Deaths Averted ij ij
per Year i =1
2. Incident Cases n= 1
∑ (No Vaccine (C ) − Vaccine (C ))
Prevented ij ij
per Year i =1
3. Quality-Adjusted n= 100
∑ (No Vaccine (QALYs
Life Years ij (Death + Impairment + Morbidity − Complications
(QALYs) Gained i =1
(Impairment + Morbidity))
− Vaccine (QALYsij (Death + Impairment +
))
Morbidity − Complications (Impairment + Morbidity))
4. Disability- n= 100
∑ (No Vaccine (DALYs
Adjusted Life ij (Death + Impairment + Morbidity − Complications
Years (DALYs) i =1
(Impairment + Morbidity))
Gaineda − Vaccine (QALYsij (Death + Impairment +
))
Morbidity − Complications (Impairment + Morbidity))
5. Net Direct Costs n= 100
∑ (Delivery Costs − Healthcare Costsij)
ij
i =1
)
6. Delivery Costs
(
Vaccine(Vij + VSij ) − NoVaccine(Vij + VSij )
× doses × (cost per dose + cost to administer )
n= 100
∑ length of immunity
i =1
7. Health Care Costs n= 100
∑ (No Vaccine (HC
(HC) Averted ij (Death + Impairment + Morbidity – Complications
i =1
(Impairment + Morbidity))
– Vaccine (QALYsij (Death + Impairment +
))
Morbidity – Complications (Impairment + Morbidity))
8. Workforce n= 1
∑ (No Vaccine (WP
Productivity ij (Death + Impairment + Morbidity – Complications
(WP) Gained i =1
(Impairment + Morbidity))
per Year – Vaccine (WPij (Death + Impairment + Morbidity
))
– Complications (Impairment + Morbidity))
9. One-Time Costs Cost Research + Cost Licensure + Cost Start Up
Fox–Rushby, J. A., and Hanson, K. 2001. Calculating and presenting disability-adjusted life years (DALYs) in
a
cost-effectiveness analysis. Health Policy and Planning 16(3):326–331.
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124 RANKING VACCINES: A Prioritization Framework
TABLE A-3
Detailed Expressions (in Reference to Table A-1 and Table A-2)
Measure Formulation
All Cause Mortality Rate 1x − 1x + 1
1x ×n
(Derived by Life Table 1 − e
Over Interval)a
QALYsDeath n= 100
∑ (No Vaccine (D ) − Vaccine (Dij)) × Duration
ij ij
i =1
QALYsImpairment by disease or complication n= 100
∑ (No Vaccine (CP ) − Vaccine (CP )) ij ij
× (1 − HUI2Impairment) × Durationij
i =1
QALYsMorbidity by disease or complication n= 100
∑ (No Vaccine (CM ) − Vaccine ij
(CMij)) × Toll × Duration
i =1
Disability-Adjusted Life Years Years of Life Lost (YLL) +
(DALYs) Generalizationb Years of Life Lived with Disability (YLD)
YLD or YLL (W = 1)
KFerj e− ( r +G )( L+ j ) −(r + G)(Lj ) − 1 1 − K
W
)
(
1 − erL
+
[ −(r + G) j − 1] r
2
(r + G)
− ( r +G ) j
−e
K = age weight modulation factor (0 = off, 1 = on)
DALYs Variables
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 n= 100
∑ (No Vaccine (D ) − Vaccine (D )) × YLL
ij ij ij
i =1
DALYsImpairment by disease or complication n= 100
∑ (No Vaccine (CP ) − Vaccine (CP )) × YLDij ij ij
i =1
DALYsMorbidity by disease or complication n= 100
∑ (No Vaccine (CM ) − Vaccine (CM )) × YLD ij ij ij
i =1
Health Care Costs (HC)Death n= 100
∑ (No Vaccine (D ) − Vaccine (D )) × HC
ij ij
Service UnitsDeath × Cost of Services
i =1
HCImpairments by disease or complication n= 100
∑ (No Vaccine (CP ) − Vaccine (CP )) × HC Service
ij ij
UnitsImpairment × Cost of Services × Durationij
i =1
HCMorbidity by disease or complication n= 100
∑ (No Vaccine (CM ) − Vaccine ij
(CMij)) × HC Service UnitsMorbidity ×
i =1
Cost of Services × Duration
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125
Appendix A
Measure Formulation
Workforce Productivity n= 1
∑ (No Vaccine (D ) − Vaccine (D )) × Hourly
(WP) Gained Death ij ij
Wagej × 2000 hours × Duration
i =1
WPImpairment by disease or complication n= 1
∑ (No Vaccine (CP ) − Vaccine (CP )) ×
ij ij
Hourly Wagej × 2000 hours × Duration
i =1
WPMorbidity by disease or complication n= 1
∑ (No Vaccine (CM ) − Vaccine (CM )) ×
ij ij
Hourly Wagej × 2000 hours × Duration
i =1
Preston, S., P. Heuveline, and M. Guillot. 2000. Demography: Measuring and modeling population
a
processes. Chapter 3: The Life Table and Single Decrement Process. P. 46.
Fox-Rushby, J. A., and K. Hanson. 2001. Calculating and presenting disability adjusted life years (DALYs) in
b
cost-effectiveness analysis. Health Policy and Planning 16(3):326–331.
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