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



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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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