**A
Computational Modeling for SMART Vaccines**

*Scott Levin, Ph.D.
Johns Hopkins University*

Estimates for the health and economic measures used to prioritize vaccine candidates in SMART Vaccines 1.0 are produced by a collection of formulas referred to as the computational submodel (see Figure 2-1). The submodel uses as its input data from the following parameters: demographic characteristics, disease burden, and vaccine characteristics (see Table 3-1). This section outlines the formulas and describes the multi-state population process model underlying these estimates. Health measures include premature deaths averted per year, incident cases prevented per year, quality-adjusted life years (QALYs) gained per year, and disability-adjusted life years (DALYs) averted per year. Economic measures include net direct costs (savings) of vaccine use per year, workforce productivity gained per year, one-time costs, and cost-effectiveness in cost per QALY or cost per DALY (see Table S-1).

Central to interpreting each health and economic measure is an understanding of the populations being compared, the time-scale applied, and the estimate formulation. The underlying population process model is described below, followed by a detailed account of the estimation procedures.

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A
Computational Modeling
for SMART Vaccines
Scott Levin, Ph.D.
Johns Hopkins University
Estimates for the health and economic measures used to prioritize vaccine
candidates in SMART Vaccines 1.0 are produced by a collection of formu-
las referred to as the computational submodel (see Figure 2-1). The sub-
model uses as its input data from the following parameters: demographic
characteristics, disease burden, and vaccine characteristics (see Table 3-1).
This section outlines the formulas and describes the multi-state popula-
tion process model underlying these estimates. Health measures include
premature deaths averted per year, incident cases prevented per year,
quality-adjusted life years (QALYs) gained per year, and disability-adjusted
life years (DALYs) averted per year. Economic measures include net direct
costs (savings) of vaccine use per year, workforce productivity gained per
year, one-time costs, and cost-effectiveness in cost per QALY or cost per
DALY (see Table S-1).
Central to interpreting each health and economic measure is an
understanding of the populations being compared, the time-scale applied,
and the estimate formulation. The underlying population process model
is described below, followed by a detailed account of the estimation
procedures.
73

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74 RANKING VACCINES: A Prioritization Software Tool
The Population Process Model
The mathematical framework underlying SMART Vaccines was devel-
oped in Phase I and has been enhanced to result in the present software,
a comprehensive explanation about the model is provided in the second
and third chapters of the Phase I report, Ranking Vaccines: A Prioritiza-
tion Framework (IOM, 2012). The SMART Vaccine 1.0 population process
model uses a cohort component method to project populations forward at
yearly intervals (Preston et al., 2001) The yearly aging process is simulated
for both a baseline population with no vaccine (i.e., the control) and a test
population with either (1) the vaccine in approximated steady state deliv-
ery or (2) the vaccine having been newly introduced, and the two popu-
lations are compared. Table A-1, provides a comprehensive description of
these three population vaccine conditions. It is through comparing a vac-
cinated population to the baseline that the various health and economic
measures are estimated over the appropriate time-scale.
The model assumes a constant number of infants entering the popu-
lation each year, with the number being given by the number of infants (i.e.,
TABLE A-1
Population Comparison in the SMART Vaccines Process Model
Population Description
Baseline The baseline population is the reference for comparison.
Vaccines not yet developed and those used in SMART
Vaccines 1.0 have a baseline population in which
no vaccine has been used. However, in cases where
a vaccine does exist, the baseline population may
reflect the current vaccination state as reference
against which to compare hypothetical newly
developed vaccines for the same disease that have
different (i.e., more desirable) characteristics.
Vaccine in In a population in which the vaccine is being
approximated administered under the steady state approximation
steady state it is assumed that individuals of all ages have had
delivery the opportunity (i.e., accounting for coverage) to
receive the vaccine. For example, in the case of a
vaccine that solely targets infants, individuals of all
ages are assumed to have had the opportunity for
vaccination. Achieving steady state for an infant
vaccination would require many years, unlike the case
with a vaccine designed for delivery to all ages.
Vaccine In the case of a vaccine first being introduced
first being into a population it is assumed that the vaccine
introduced was delivered solely to the target population (i.e.,
accounting for coverage) at model initialization.
No other members of the population will have had
the opportunity to have received the vaccine.

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Appendix A 75
age less than 1) present in the World Health Organization (WHO) popula-
tion life-table input (WHO, 2013). This is operationalized by assuming that
the number of infants (i.e., children under age one) observed at baseline
will remain constant in the future. This assumption eventually produces a
stationary age distribution, but the total population size eventually reached
will be somewhat different from that of a population immediately reach-
ing replacement fertility (Preston et al., 2000). Information on population
below age 1 is taken from WHO population life-table input. Individuals
may exit the population by death caused by disease or by all other causes.
The model does not account for migration of any sort. The committee
chose to use this simplified version of the population process in light of the
constraints present in the early development phase of SMART Vaccines 1.0.
The model minimizes assumptions regarding population dynamics and,
consequently, reduces the burden of data entry, making it easier to use. It is
expected that this simplified version will serve as a foundation upon which
more complex population processes may be constructed as SMART Vac-
cines advances.
The step-by-step computations of the population process model are
as follows. The model is initialized at time zero t0, which corresponds to the
year of the population data that are input into the model (e.g., 2009 for the
vaccine candidates). At initialization, linear interpolation is used to pro-
duce a population age distribution in 1-year increments from the standard
WHO life-table format, which is in 5-year increments. Age-specific prob-
abilities of dying from all causes, nqx, are computed; the subscript x refers
to the age at the beginning of the age interval, and the subscript n refers to
the length of the interval. The probabilities for dying in a given 1-year time
period, 1qx, are derived from the number of individuals alive at each age, lx,
along with the probability of survival and the death rate between ages x and
x+n, written as npx and nMx, respectively. The following equations are used
to estimate yearly nq1 (Preston et al., 2001):
p = lx+n / lx
n x n
Mx = –ln(npx) / n
1
Ma = nMx for x≤a≤x+n-1 q = 1–exp(–1Mx)
1 x
The process model directly computes age-specific population
parameters, including population size N; population size disease-eligible
NE (i.e., that part of the population that has not previously been perma-
nently impaired by the disease); and the subset of the population targeted
to receive vaccination, TI or TS, as seen in Table A-2: Population Param-
eters. At each yearly interval starting with t0 the population is composed
of individuals in mutually exclusive vaccination states. These states are the

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TABLE A-2
76
Population Process Model
Population Parameters Baseline0 With Vaccine1 Impact
Population size (N) n x
N 0 = nNx-10 - nDAx-10 n x
N 1 = nNx-11 - nDAx-11 n
Nx1 - nNx0
Population size disease eligible (NE) n
NEx0 = nNx-10 - nCPx-10 n
NEx1 = nNx-11 – nCPx-11 n
NEx1 - nNEx0
Target population (TI) Vaccine in Steady State n
TSx0 = 0 n
TSx1 = nNEx x 1 n
TSx1
Target population (TS) Vaccine Introduced n
TIx0 = 0 n
TIx1 = nNEx x nTARGx n
TIx1
Vaccination States Baseline0 With Vaccine1 Impact
Vaccinated immune (V) V 0=0
n x n x
V 1 = nTx 1 x nCOVx x nEFFx n
Vx 1
Vaccinated susceptible (VS) n
VSx0 = 0 n
VSx1 = nTx1 x nCOVx x (1 - nEFFx) n
VSx1
Unvaccinated immune (B) n
Bx0 = 0 If ∑ (nCOVx x nNx0) / ∑nNx0 > 80% n
Bx1
n x
B 1 = (nNEx1 -nVx1 -nVSx1) x nEFFx
Else nBx1 = 0
Unvaccinated susceptible (BS) n
BSx0 = nNx0 n
BSx1 = nNEx1 - (nVx1 + nVSx1 + nBx1) n
BSx0 – nBSx1
Health Events Baseline0 With Vaccine1 Impact
0 0 1 1 1
Total cases (C) n
C x
= nNE
x
x nINCx n
C = (nVS + nBS ) x nINCx
x x x n
Cx0 – nCx1
Deaths by disease (D) n
Dx0 = nCx0 x nCFRx n
Dx1 = nCx1 x nCFRx n
Dx0 – nDx1
Cases: Impairment (CP) n
CPx0 = (nCx0 -nDx0) x nIPx n
CPx1 = (nCx1 - nDx1) x nIPx n
CPx0 – nCPx1
Cases: Morbidity (CM) n
CMx0 = nCx0 - nDx0 - nCPx0 n
CMx1 = nCx1 - nDx1 - nCPx1 n
CMx0 – nCMx1
All-cause deaths (including deaths n
DAx0 = nNEx0 x nqx0 n
DAx1 = nNEx1 x nqx0 - (nDx0 - nDx1) n
DAx0 – nDAx1
caused by disease) (DA)
NOTE: CFR = annual case fatality risk; COV = coverage; EFF = effectiveness; INC = annual incidence proportion; IP = annual case impairment risk; TARG = target
proportion.

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Appendix A 77
vaccinated immune, V; the vaccinated susceptible, VS; the unvaccinated
immune, B (i.e., those who may have indirect protection through herd
immunity); and the unvaccinated susceptible, BS (Table A-2: Vaccination
States). No members of the population belong to the vaccinated immune,
vaccinated susceptible, or unvaccinated immune states in a baseline pop-
ulation when a vaccine is not in existence. Various “health events” occur
each year and are computed based on parameters set by user input (e.g.,
disease incidence, vaccination effectiveness, etc.). These events include the
number of disease cases, C; deaths by the disease, D; diseases cases leading
to permanent impairment, CP; disease cases leading to morbidity, CM (i.e.,
complete recovery by year end); and all-cause deaths, DA. The mathemati-
cal formulas for these measures are displayed in Table A-2. The superscript
0 refers to the simulated baseline population, as opposed to the simulated
population with vaccine 1. Vaccinated populations may be either in the
steady state delivery approximation or in the vaccine newly introduced
state.
The model described in Table A-2 has several notable characteris-
tics. First, the calculated value for the population size disease-eligible (NE)
may vary from year to year. The difference from one year to the next is the
number of people in the population who were permanently impaired in
the intervening year. For example, if a member of the population contracts
tuberculosis and, as a result, has permanent lung impairment, that person
is not eligible to contract the same disease in subsequent years (i.e., not
disease-eligible).
Next, the variable T that represents the target population is used
to distinguish populations (see Table A-1). For example, no proportion of
the baseline population receives the vaccine; therefore the corresponding
target multiplier nTARGx equals zero for all ages. This serves as a proxy
for scenarios in which the vaccine does not exist. By contrast, popula-
tions assumed to have reached steady state vaccine delivery have an target
multiplier of one for all ages. This may be interpreted as an initialization
state in which all members of the current population (i.e., all ages) have
had the opportunity to receive the vaccination (i.e., coverage rates apply)
at t0. Alternatively, in populations where the vaccine is newly introduced,
delivery proceeds only for the age-specific target population specified at
input nTARGx. For example, consider the influenza vaccine, for which there
is no difference between steady state delivery and the introduction of a
new vaccine. The two states are equivalent because the vaccine targets the
entire population (i.e., all ages) each year. By contrast, a vaccine candidate
for tuberculosis may be designed to target only infants, which will create a
major difference between initialization states.

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78 RANKING VACCINES: A Prioritization Software Tool
The effect of herd immunity is incorporated into the process model
for contagious diseases with human-to-human transmission. In particular,
the effect is represented by the presence of an unvaccinated immune popu-
lation B that may receive the benefits of indirect protection (see Table A-2).
If the user chooses to apply herd immunity, the overall coverage within
the population is calculated. If this coverage is greater than or equal to 80
percent, the entire unvaccinated susceptible population BS receives indi-
rect protection. Receiving disease immunity through indirect protection is
treated identically to receiving the vaccine and is conferred in accordance
with the vaccine’s effectiveness. If overall coverage is less than 80 percent,
no indirect protection is assumed.
Finally, a connection exists between the baseline and the vaccine
comparison process models. The number of all-cause deaths with disease,
DA, for a vaccinated population is equal to tbe number of all-cause deaths
in the baseline population minus the number of deaths prevented by the
vaccine in that year. The same age-specific all-cause mortality rates are
applied to both populations (i.e., the baseline and the vaccinated), and the
prevented deaths, nDx0 – nDx1, are subtracted out for the case of the vacci-
nated population, as can be seen in Table A-2. Thus, the resultant deaths DA
diverge, and the difference is projected forward each year of the simulation.
Health and Economic Measures
Health and economic estimates are computed using the population process
model comparisons as their basis (Table A-2: Impact). However, under-
standing the populations in comparison and the time scale applied is funda-
mental to interpreting the meaning of each individual measure. Table A-3
displays this context for each of the nine health and economic measures.
Measures that apply to the steady-state delivery approximation are
calculated for the 1-year time period after initialization; each of these mea-
sures is distinguished by a “per year” phrasing (see Table A-3). Cost-effec-
tiveness measures such as cost per QALY gained or cost per DALY averted
are computed over a 100-year time horizon for those populations first being
introduced to the vaccine (see Table A-3). If desired, the user can choose
to apply a lesser time horizon through the SMART Vaccines 1.0 interface.
The one-time costs measure is designed to capture the estimated total costs
for research, development, and licensure of the new vaccine. This is input
directly as more than $1 billion, $500 million to $1 billion, $100 million to
$500 million, or less than $100 million. One-time costs are interpreted as
taking place over the period of time until vaccine adoption, which is pro-
vided by user input.

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Appendix A 79
TABLE A-3
Comparator Populations and Time Scale
Vaccine in Steady State Approximation Vaccine First Introduced
Time Scale: 1-year horizon Time Scale: 100 years, or
a user-defined horizon
less than 100 years
Premature deaths averted per year Cost-effectiveness in
cost per QALY gained
Incident cases prevented per year Cost-effectiveness in
cost per DALY averted
QALYs gained per year
DALYs averted per year
Net direct costs of vaccine use per year
Workforce productivity gained per year
One-time costs
Time scale: Applied over time to adoption (user defined)
NOTE: DALYs = disability-adjusted life years; QALYs = quality-adjusted life years.
The formulas for each health and economic estimate are shown in Table
A-4. The notations correspond to the definitions presented in Table A-2.
Discounting is applied to both the health and the economic measures,
with a default annual rate of 3 percent, and the user can modify the annual
rate or eliminate the discounting altogether by setting the annual rate to
zero. Further aggregate discounting is applied to the cost-effectiveness
measures only in order to account for time to adoption (see Table A-3). The
duration (nDurationx) used to produce the health and economic estimates
(see Table A-4) varies by measure. The durations for QALYs and DALYs
due to morbidity are input by the user and can be no longer than one year.
Individuals with a disease that causes morbidity are assumed to completely
recover by year’s end. The durations used for QALYs due to death or per-
manent impairment are given by the age-specific future life expectancy,
adjusted and discounted by the health utilities index (HUI2). The dura-
tions used for DALYs due to death or permanent impairment similarly use
discounted life expectancy but are based on the standard life expectancy
and remain unadjusted for HUI2. The durations due to morbidity used for
workforce productivity calculations are input by the user and are identi-
cal to the time-periods used for QALYs and DALYs. However, workforce
productivity loss due to death or permanent impairment is assumed to be
six months. This is the average duration over the 1-year projection inter-

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TABLE A-4
80
Health and Economic Formulas
Measure Formula
Premature ∑nDx0 – nDx1
deaths averted
Incident cases ∑ nCx0 – nCx1
averted
Quality-adjusted ∑nQALYx0(Death + Impairment + Morbidity) – ∑nQALYx1(Death + Impairment + Morbidity)
life years
(QALYs) gained
n
QALYx(Death) = (nQALYx0D - nQALYx1D) x nDurationx
n
QALYx(Impairment) = (nQALYx0CP - nQALYx1CP) x nTollx x nDurationx
n
QALYx(Morbidity) = (nQALYx0CM - nQALYx1CM) x nTollx x nDurationx
Disability- ∑nDALYx0(Death + Impairment + Morbidity) – ∑nDALYx1(Death + Impairment + Morbidity)
adjusted life
years (DALYs) DALYs = years of life lost (YLL) + years of life lived with disability (YLD)
averted
(Rushby and KFe rj − r +G L+ j − r +G j 1− K
YLD or YLL (W=1) = W 2
e ( )( ) − ( r + G ) Lj − 1 − e ( ) − ( r + G ) j − 1 +
{ ( ) } 1 − e − rL
(
)
Hanson, 2001) (r + G) r
n
DALYx(Death) = (nDALYx0D – nDALYx1D) x nYLLx
n
DALYx(Impairment) = (nDALYx0CP – nDALYx1CP) x nYLDx
n
DALYx(Morbidity) = (nDALYx0CM – nDALYx1CM) x nYLDx
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)

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Net direct costs ∑nDeliveryCostsx – ∑nHealthcareCostsx
(savings) of
vaccine use
n
DeliveryCostsx= [(nVx1 +nVSx1) x doses x (dose cost x cost to administer)]/length of immunity
n
HealthcareCostsx = nHealthCarex0(Death + Impairment + Morbidity) –
n
HealthCarex1(Death + Impairment + Morbidity)
n
HealthcareCostsx(Death) = (nDx0D – nDx1D) x ∑nCostOfServicesx
n
HealthcareCostsx(Impairment) = (nCPx0D – nCPx1D) x ∑nCostOfServicesx x nDurationx
n
HealthcareCostsx(Morbidity) = (nCMx0D – nCMx1D) x ∑nCostOfServicesx x nDurationx
Workforce ∑nWPx0(Death + Impairment + Morbidity) – nWPx1(Death + Impairment + Morbidity)
productivity
(WP) gained
n
WPx(Death) = (nDx0D – nDx1D) x nHourlyWagex x 2000 hours x nDurationx
n
WPx(Impairment) = (nCPx0D – nCPx1D) x nHourlyWagex x 2000 hours x nDurationx
n
WPx(Morbidity) = (nCMx0D – nCMx1D) x nHourlyWagex x 2000 hours x nDurationx
Cost- QALYs / Net Direct Costs (Savings) or DALYs / Net Direct Costs (Savings)
effectiveness
81

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82 RANKING VACCINES: A Prioritization Software Tool
vals. In short, workforce loss calculations are limited to one year and do not
account for future years lost for those who experience death or permanent
impairment.
Model Limitations and Further Work
The computational model that underpins SMART Vaccines has a few limi-
tations that would benefit from improvements in subsequent versions.
First, closing the population process to migration does not allow the model
to account for population dynamics that may influence health and eco-
nomic measures, especially for cost-effectiveness modeled over a 100-year
time horizon. However, this basic design does reduce user assumptions and
the practical burden of data entry, and it leads to results that can be inter-
preted as vaccine impacts exclusive to the current population, with mini-
mal confounders.
Second, the steady-state approximation does not account for changes
in the population that may occur by the time steady-state vaccine delivery
is reached. As the time to steady state increases (e.g., it is longer for tuber-
culosis than for influenza), so does the potential for inaccurate estimation.
Given this limitation, we chose to model an immediate steady-state proxy
because the purpose of vaccine intervention is to swiftly achieve steady
state and to maximize health benefits. This ultimate purpose was deemed
most significant to the prioritization exercise, so we chose to keep this
exercise free from the complications that may be inevitable during vaccine
ramp-up periods.
Finally, a limitation exists in the level of detail required for the dis-
ease and the vaccine characteristics specified as input. Under some circum-
stances, the input required may appear coarse in order to capture specific
details of a disease. In these cases data input may be altered to produce a
desired average effect, or the use of pre-defined special populations may
be used.
A good understanding of the modeling concepts within SMART
Vaccines 1.0 will allow adept users to treat input data appropriately and
to capture the complexities of different diseases and vaccines that are not
apparent through data input interfaces. Overall, the committee wished to
balance the complexity of the modeling and data requirements against the
model’s capabilities in order to accurately characterize a broad range of
diseases and vaccines and software usability. In short, SMART Vaccines 1.0
is designed to serve as a foundation for further work in this area.

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Appendix A 83
References
IOM (Institute of Medicine). 2012. Ranking Vaccines: A prioritization
framework: Phase I: Demonstration of concept and a software blueprint.
Washington, DC: The National Academies Press.
Preston, H., P. Heuveline, and M. Guillot. 2001. Demography: Measuring
and modeling population processes. Malde, MA: Blackwell.
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.
WHO (World Health Organization). 2013. Life tables. http://tinyurl.com/
cd2lpou (accessed May 11, 2013).

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