Technical Documentation: The Cost-Effectiveness of Routine Childhood
Vaccination for Hepatitis A in the United States

\*David B. Rein, Ph.D., \*Katherine A. Hicks, M.S., \*Kathleen E. Wirth,
B.A., †Kaafee Billah, Ph.D., †Lyn Finelli, Dr.P.H., †Anthony E. Fiore,
M.D., M.P.H., \*Thomas J. Hoerger, Ph.D., †Beth P. Bell, M.D., M.P.H.,
†Gregory L. Armstrong, M.D.

Send all correspondence to:

> David B. Rein, PhD
>
> RTI International
>
> 2951 Flowers Road
>
> Suite 119
>
> Atlanta, GA 30306
>
> Voice: (770) 234-5035
>
> Fax: (770) 234-5030
>
> E-mail: drein@rti.org

^\*^RTI International; ^†^Centers for Disease Control and Prevention,
National Center for Infectious Disease, Division of Viral Hepatitis

This report has been compiled to provide supporting documentation and
information for the manuscript "The Cost-Effectiveness of Routine
Childhood Vaccination for Hepatitis A in the United States." The
technical report consists primarily of reference tables and information,
with text added in some instances to clarify or interpret the
information contained in the tables. This report can also be used to
understand how to modify and apply this model to other contexts. The
report is divided into 13 sections.

I.  **Overall model functionality**

The Hepatitis A (HepA) vaccine cost-effectiveness analysis tool uses a
Markov model to simulate how a birth cohort progresses through states
and transitions between states with respect to HepA. Figure 1.1
describes the basic structure for the model. A cohort progresses between
the states on an annual basis to simulate a 95-year lifetime starting at
birth. Progression is based on probabilities of relevant events, such as
HepA infection or vaccination, and other transitions between states,
such as those caused by passing time or death, encountered by the
portion of the cohort in each state. Each state and event is associated
with costs for medical care, gains for productivity and gains in life
years and quality-adjusted life years (QALYs), as relevant; the model
collects these values over the course of the cohort's lifetime. On each
model run, the cohort's outcomes from both a baseline scenario (i.e. no
vaccine or a more conservative vaccine policy) and then a scenario that
includes the vaccination policy of interest are observed, recorded, and
compared to calculate cost-effectiveness measures.

**States**. The model states are the following:

-   *Susceptible, not immunized* -- never been immunized; eligible for
    infection

-   *Immune: 1, 2,...95 years to immunity loss* -- immunized;
    vaccine-induced immunity not (yet) lost

    -   95 states: 1 for each possible number of remaining years of
        immunity loss

-   *Susceptible, immunized* -- immunized; vaccine-induced immunity
    lost, therefore eligible for infection

-   *Immune (due to disease), immunity unknown* -- immune due to past
    disease exposure; disease was not identified, therefore immunity is
    not known and individual may seek vaccination

-   *Immune (due to disease), also vaccinated* -- immune due to past
    disease exposure; disease was not identified, therefore individual
    sought and received vaccination

-   *Immune (due to disease), immunity known* -- immune due to past
    disease exposure; disease was identified

-   *0, 1,...18+ years since transplant* -- history of liver transplant
    due to HepA; immune due to past disease exposure; disease was
    identified; increased risk of death; reduced quality of life

    -   19 states: 1 for each possible number of years since transplant;
        18 years after and later are grouped

-   *Dead* -- dead due to HepA or non-HepA-related causes; accumulating
    productivity losses

**Transitions between states.** With each annual time step of the model,
the cohort is further divided between the model states based on
transition probabilities. To demonstrate this concept, we can observe
what happens to the portion of the cohort in the *Susceptible, not
immunized* state.

Each year, a certain percentage of that group will be vaccinated and
distributed between the 95 *Immune: 1, 2,...*or *95 years to immunity
loss* states and the *Susceptible, immunized* state.

Of those not vaccinated (in the *Susceptible, not immunized* state), a
certain portion will be infected with HepA. Infected persons will end up
distributed between *Immune (due to disease), immunity unknown*; *Immune
(due to disease), immunity known*; *0 years since transplant*; or *Dead*
based on their particular disease progression paths. Those who develop
fulminant HepA and die either before or after a transplant are moved to
the *Dead* state. Those who develop fulminant HepA then get and survive
the year of a liver transplant end up in *0 years since transplant*. All
others who have icteric HepA that is identified and reported are moved
into *Immune (due to disease), immunity known*. The rest of infected
cases, which all have anicteric or icteric HepA that never goes
reported, end up in *Immune (due to disease), immunity unknown*.

The portion of the cohort that is not vaccinated or infected stays in
*Susceptible, not immunized* for the following year.

Finally, after all of the above transitions have occurred, a percentage
of the cohort in all states transitions to the *Dead* state to account
for non-HepA-related deaths.

This process is repeated in following years with whatever portion of the
cohort is in the *Susceptible, not immunized* state at the time.

**Figure 1.1 Model structure diagram**

In this diagram, rectangular boxes represent "states" between which the
cohort is distributed at the end of every year based on the transitions
and events that occur. Diamonds represent events that affect costs and
the end-of-year state. Disease development stages are represented by
shaded parallelograms. Dotted lines indicate death from HepA-related
causes. Death from non-HepA-related causes occurs at the same rate
between all states (arrows for this transition not drawn).
Hospitalization includes additional outpatient visits.

**Outcomes**. The model calculates cumulative outcomes, as well as
incremental costs and outcomes associated with the vaccination policy of
interest relative to baseline. It then compares these outcomes of the
cohorts under both scenarios to assess the impact of the vaccination
policy as compared to the baseline group using cost-effectiveness,
cost-utility and cost-benefit analysis measures. Costs and QALYs are
discounted using a user-defined. We collect the following outcomes:

-   Cases of each level of HepA

-   HepA-related outpatient visits, hospitalizations, liver transplants

-   HepA-related deaths

-   Childhood and adult immunizations

-   Mild and severe adverse events from HepA immunizations

-   Days of work loss due to:

    -   Care-giving or self-care for HepA-related illness

    -   HepA-related death

    -   Getting HepA vaccinations

    -   Care-giving or self-care for adverse events from HepA
        vaccinations

-   Life years

-   QALYs

-   Costs for medical care of HepA-related illness

-   Childhood and adult HepA immunization costs

-   Productivity losses due to:

    -   Care-giving or self-care for HepA-related illness

    -   HepA-related death

    -   Getting HepA vaccinations

    -   Care-giving or self-care for adverse events from HepA
        vaccinations

-   Public health response costs

**Analyses**. With a few exceptions, values of all parameters can be
varied between model runs, however analyses typically vary by the
following factors:

-   Age of mass childhood vaccination (Age one for the analysis in the
    manuscript, this can be set to any age)

-   Regions considered (in the manuscript these are specified to the
    three regions established by the 1999 ACIP recommendations, but the
    population of these regions can be specified to divide any area into
    at least three groups -- with additional programming, the model
    could be modified to add additional regions)

-   Policy scenario comparison

  ----------------------------- ------------------ ----------------------
  Policy change of interest     Baseline policy    Policy of interest
                                (comparator)       

  Implementing a recommendation No vaccination     Recommend vaccination
  for childhood vaccination                        in high-incidence
  rather than no childhood                         regions only
  vaccination                                      

                                                   Recommend vaccination
                                                   nationwide

  Expanding existing            90% first dose     Recommend vaccination
  recommendation from           coverage in        nationwide
  high-incidence only to also   high-incidence     
  include low-incidence regions regions only       

  Expanding recommendations in  Partial            90% first dose
  a way that increases coverage vaccination        coverage in each
  nationwide.                   coverage based on  region of the country.
                                partial            
                                implementation in  
                                each of the three  
                                regions.           
  ----------------------------- ------------------ ----------------------

II. **Incidence of Infection.**

Incidence in the the Hepatitis A (HepA) vaccine cost-effectiveness
analysis tool is defined as the annual probability of infection with
Hepatitis A virus, given that a patient was not immune from previous
infection. This probability depends on patient age and region of the
country. The actual U.S. Hepatitis A incidence trends are characterized
by approximately 5- to 15-year cycles and by a long-term decline. We
modeled incidence based on the average incidence over the 5-year time
frame of 1990 to 1995. These years were chosen because they immediately
preceeded the introduction of the hepatitis A vaccine in 1996, and the
introduction of the vaccine was thought to alter U.S. patterns of
incidence. Intuitively, we tried to model the level of incidence that
should be expected in each region if a vaccine had never been invented.
In the model, the introduction of vaccine would then protect patients
from this underlying risk of infection.

To do this, we estimated an actual mean reported starting incidence
value based on the average incidence from 1990 to 1995, and then an
average annual rate of decline expected in subsequent years. To do this,
we used a linear regression of the logarithm of reported hepatitis A
rates against year. In this regression the intercept was allowed to vary
across the 3 regions while the rates of decline were assumed to be
equal.^1,2,3^ From the regression we estimated a starting (i.e. 1990)
reported mean incidence of acute hepatitis A of 22.6 cases per 100,000
patients in the region 1, 14.1 cases per 100,000 patients in region 2,
and 6.7 cases per 100,000 patients in region 3. We then estimated
applied an estimated annual rate of decline of 1.4% to each intercept
for each year after 1990 in the analysis. The model assumed that
*without* immunization this rate of decline would have continued from
1990 indefinitely into the future.

Actual incidence varies widely by age from the mean population
incidence. Age-specific incidence was estimated by multiplying the
estimated average incidence for each region by the ratio of age-specific
incidence to overall incidence during 1990 to 1995, the years
immediately preceding the licensing of the HA vaccine.^2^

Reported cases represent only a portion of total incidence. Based on
previous estimations, for each reported case of hepatitis we assumed
there were 3.28 unreported cases.^4^ All reported cases were assumed to
be icteric (symptomatic with jaundice). The number of additional
anicteric cases (asymptomatic or mildly symptomatic without jaundice)
was estimated by applying an age-specific ratio, also determined by
regression.^4^

III. **Loss of vaccine-acquire immunity**

The level of immunity in a vaccinated person is measured by *geometric
mean titre* (GMT) level; higher GMT levels equate to immunity to
contracting HepA. In this model, people lose vaccine-acquired immunity
if their GMT level falls below a certain threshold value; all those with
a GMT above that level have immunity. The model assumes that after
immunization GMT levels reach an initial peak followed by a decline over
time based on an annual percentage loss. Among those within the cohort
that are vaccinated, some are vaccinated with 1 dose and others 2; those
who receive 2 doses generally have higher initial GMT levels. Then,
among people receiving the same number of doses, the initial GMT levels
after immunization vary based on a specified lognormal distribution.
Therefore time to immunity loss also varies. Table 3.1 lists the
parameters and corresponding default values that determine
vaccine-acquired immunity.

When a portion of the cohort is immunized, that portion is assigned
initial GMT levels and then, based on those levels, distributed into one
of 96 states corresponding with 0 to 95 years until immunity loss. Those
with 0 years until immunity loss never had a high enough GMT level to be
immune and go directly to *Susceptible, immunized*; those with more
years until immunity loss progress to one of the 95 *Immune* (after
vaccination) states. Each year individuals who are newly immunized are
introduced to these 96 states.

Those who were immunized in years prior and are still immune progress to
the state corresponding with 1 less year until immunity loss, with the
exception that after *Immune: 1 year to immunity loss* they progress to
*Susceptible, immunized*.

Table 3.1. Parameters related to vaccine-induced immunity

+-------------------------------+-------------------+------------------+
| **Parameter**                 | **Default value** | **Source**       |
+===============================+===================+==================+
| Average GMT level immediately | 1 dose: 141       | Van Herck K, Van |
| after receiving vaccine       |                   | Damme P (2001)   |
|                               | 2 doses: 1120     |                  |
+-------------------------------+-------------------+------------------+
| Lower bound of 95% CI of GMT  | 1 dose: 112       | Van Herck K, Van |
| levels immediately after      |                   | Damme P (2001)   |
| receiving vaccine             | 2 doses: 912      |                  |
+-------------------------------+-------------------+------------------+
| Upper bound of 95% CI of GMT  | 1 dose: 177       | Van Herck K, Van |
| levels immediately after      |                   | Damme P (2001)   |
| receiving vaccine             | 2 doses: 1374     |                  |
+-------------------------------+-------------------+------------------+
| Annual rate of loss of GMT in | 0 yrs since vac:  | Van Damme P      |
| years following vaccination   | 0                 | (personal        |
| (0-1)                         |                   | communication)   |
+-------------------------------+-------------------+------------------+
|                               | 1-4 yrs since     |                  |
|                               | vac: 0.2          |                  |
+-------------------------------+-------------------+------------------+
|                               | 5-95 yrs since    |                  |
|                               | vac: 0.05         |                  |
+-------------------------------+-------------------+------------------+
| GMT level at which immunity   | 20                | Delem A et al    |
| is lost                       |                   | (1993)           |
+-------------------------------+-------------------+------------------+
| CI = confidence interval      |                   |                  |
|                               |                   |                  |
| GMT = geometric mean titre    |                   |                  |
| level                         |                   |                  |
|                               |                   |                  |
| yrs since vac = years since   |                   |                  |
| vaccination                   |                   |                  |
+-------------------------------+-------------------+------------------+

The model conducts this process using the following 3 steps: determine
the number of years of immunity from vaccination associated with the
different initial post-vaccination GMT levels; calculate the shape of
the distribution of GMT levels among newly vaccinated people; and assign
newly vaccinated to states.

**Associated GMT levels with years of immunity.** The model first
determines the GMT levels that are associated with each year of
remaining immunity by using the GMT at which immunity is lost and the
annual rate of loss of GMT after vaccination. Table 3.2 presents
selected GMT levels associated with each year of remaining immunity
given the default values presented above in Table 3.1.

Table 3.2. Relationship between GMT level and years of immunity from
vaccination.

  -------------------------------------- --------------------------------
  **If your GMT level after immunization **Then you have this many
  is at least...\***                     remaining years of immunity...**

  20                                     1

  25                                     2

  31                                     3

  63                                     10

  82                                     15

  105                                    20

  176                                    30

  294                                    40

  820                                    60

  2288                                   80

  4938                                   95
  -------------------------------------- --------------------------------

**Calculate standard deviation of distribution of GMT levels among newly
vaccinated.** The distribution of initial levels is assumed to have a
lognormal distribution; the user specifies the distributional shape by
an average value and 95% confidence interval bounds (first 3 parameters
in Table 3.1 above). The default values are based on findings by the
Belgian researchers Van Herck and Van Damme (2001), who observed GMT
levels after hepatitis A vaccinations.

Decay rates are assumed to vary based on time since immunization;
default values for that parameter are unpublished, but based on data
collected by Van Damme.

The model uses the mean, confidence interval bounds, and sample size of
GMT level data points in Equation 3.1 to calculate the standard
deviation of the initial GMT levels observed. We applied Van Herck and
Van Damme's study sample size of 77 in our calculations.

Standard deviation of GMT levels =
$\frac{(\text{CI}\text{ upper bound } - \text{ CI lower bound})\sqrt{n}}{2 \cdot 1\text{.}\text{96}}$
(3.1)

**Assign newly vaccinated to states.** Once the shape of the lognormal
distribution is established, the model must then allocates the newly
vaccinated portion of the population between the 96 post-vaccination
states. It does this by determining the percentage of the population
that has GMT levels within the range associated with each number of
years of immunity and, in turn, each state. In order to make this
determination, the model makes use of the fact that the natural log of a
lognormally distributed variable is normally distributed and follows the
following process for both 1 and 2 doses:

> 1\. Calculate the mean and standard deviation of the normally
> distributed natural log of GMT levels using Equations 3.2 and 3.3.
>
> Mean of ln(GMT) =
> $\text{ln}\left( \frac{\text{mean GMT}^{2}}{\sqrt{\text{std dev of GMT}^{2} + \text{ mean GMT}^{2}}} \right)$
> (3.2)
>
> Standard deviation of ln(GMT)
> =$\sqrt{\text{ln}\left( \frac{\text{std dev of GMT}}{\text{mean GMT}} \right)^{2} + 1}$
> (3.3)
>
> Based on the default values, the mean of ln(GMT) is 4.33 for one dose
> and 6.71 for two doses; the standard deviation of ln(GMT) is 1.11 for
> one dose and 0.79 for two doses.
>
> 2\. Calculate the natural log of the GMT level defining the upper
> boundary of each post-vaccination state (20, 25, 31, etc.).
>
> 3\. Standardize the natural log of each GMT level by subtracting the
> mean of ln(GMT) and dividing the difference by the standard deviation
> of ln(GMT). Plotting these standardized values forms a standard normal
> distribution.
>
> 4\. The model has a stored table of values specifying the percent of
> the area under the curve within each 0.01-width bin of a standard
> normal distribution. The model uses these percent-under-the-curve
> values to calculate the percent of the newly vaccinated population
> falling within the range of GMT values corresponding to each year of
> immunity remaining (0-95). Table 3.3 reports

Table 3.3. Standardized GMT level and percent of newly vaccinated people
with different years of remaining immunity.

  ------------------------------------------------------------------------------------------
  **Years of     **GMT       **One dose**                    **Two doses**    
  immunity\***   Level**                                                      
  -------------- ----------- ---------------- -------------- ---------------- --------------
                             **Standardized   **% of newly   **Standardized   **% of newly
                             GMT level**      vaccinated**   GMT level**      vaccinated**

  0              0           n/a              11             n/a              0

  1              20          -1.2             5              -4.7             0

  2              25          -1.0             5              -4.4             0

  3-10           63          -0.2             24             -3.3             0

  11-15          82          0.1              9              -2.9             0

  16-20          105         0.3              9              -2.6             0

  21-30          176         0.8              16             -1.9             3

  31-40          294         1.2              11             -1.3             7

  41-50          491         1.7              6              -0.7             18

  51-60          820         2.1              3              -0.4             24

  61-70          1370        2.6              1              0.6              24

  71-80          2288        3.1              0              1.3              15

  81-90          3821        3.5              0              2.0              7

  91-95          4938        3.8              0              2.3              2
  ------------------------------------------------------------------------------------------

\*Only selected years of immunity presented

> 5\. The newly vaccination portion of the population is distributed
> accordingly to the 96 post-vaccination states based on the determined
> years of immunity.

IV. **Vaccine Costs**

Vaccination cost include the costs associated with vaccine acquisition
(including both public and private prices for pediatric and adult doses)
and the costs of vaccine administration (Table 4.1). The vaccine
acquisition costs for pediatric and adult doses were taken from the
Vaccines for Children vaccine price list, which provide both private
sector and CDC-contract public costs per dose.^5^

Based on vaccine purchase data we estimated the proportion of the
pediatric (0-17 years) and adult (18+) population each would receive the
public price for the vaccine. The proportion of children and adults
receiving the public price was determined by net pediatric Hepatitis A
vaccine sales from the Biologics Surveillance data (CDC-contract
purchases during the time period of 1995 to 2003) and proprietary
vaccine sales data, respectively.^6^

Finally, all vaccine prices were adjusted to reflect vaccine wastage of
five percent and do not include federal excise taxes of \$0.75 per dose.

Administration costs per dose were calculated using the outpatient
claims for 8,327 privately insured individuals found in the 2002
MarketScan^®^ Commercial Claims and Encounters Research Database^7^. The
sample was restricted to patients aged 12 years and younger not enrolled
in a Health Maintenance Organization (HMO) with claims for which
Hepatitis A pediatric vaccination was the primary procedure. To
determine the administration cost, we deducted the private sector
vaccine acquisition cost (\$26.66 per dose)^5^ from the mean cost per
claim found in the sample (\$38.65) to produce a per dose cost of \$12.

We assumed the administration cost would be shared equally (\$6) when
the Hepatitis A vaccine was administered concurrently with previously
other existing routine vaccinations at the relevant ages (i.e. Hepatitis
A vaccination during an office visit at age 1 may be shared with Measles
Mumps Rubella (MMR) vaccination)

Table 4.1 summarizes the vaccination acquisition and administration
input costs used in the base case analysis.

  ---------------- ---------------- --------- ----------- ------------ --------------
  Table 4.1.                                                           
  Vaccination,                                                         
  Costs and                                                            
  Probabilities                                                        

                                              Parameter                
                                              Value                    

  Parameter                                   Base Case                Source
                                              Analysis                 

  Proportion                        Ages 0-17             0.59         ^6^
  vaccines                                                             
  purchased at                                                         
  public price                                                         

                                    18-95                 0.1          Assumption

  Acquisition cost Public           Ages 0-17             \$11.15      ^5^

                                    18-95                 \$18.77      ^5^

                   Private          Ages 0-17             \$27.75      ^5^

                                    18-95                 \$59.57      ^5^

  Administration   First dose       Ages 1                \$6          ^7^,
  cost                                                                 Assumption

                                    2-11                  \$12         ^7^

                                    12                    \$6          ^7^,
                                                                       Assumption

                                    13-95                 \$12         ^7^

                   Second dose      Ages 1                \$6          ^7^,
                                                                       Assumption

                                    2-95                  \$12         ^7^

                                                                       
  ---------------- ---------------- --------- ----------- ------------ --------------

V.  **Adult Vaccination**

We assumed a proportion of the non-aged adult population (aged 18 to 64
years) would receive Hepatitis A vaccination in the absence of a
childhood recommendation due to international travel, military personnel
requirements and participation in other high risk activities (i.e.
illegal drug users, men who have sex with men etc) ^8^.

We estimated region-specific (regions determined according to the 1999
ACIP recommendations) probabilities of one dose adult coverage using
proprietary vaccine purchase data as reported in Samandari T et al
(2004) and unpublished dosage information taken from the Defense
Military Surveillance System.^9^ The number of doses purchased privately
and the number of military personnel vaccinated per state per year
(during the time period of 1996 to 2001) was averaged (assuming one-dose
coverage and applying a vaccine wastage rate of five percent to the
vaccine purchase estimates) to create an overall probability of adult
vaccination. We assumed the probability of receiving a second dose given
one-dose coverage to be zero. Table 5.1 summarizes the coverage rates by
region used in the base case analysis.

+----------------+-------------+--------+---+-----------+--------------+
| Table 5.1      |             |        |   |           |              |
| Adult          |             |        |   |           |              |
| Vaccination,   |             |        |   |           |              |
| Probabilities  |             |        |   |           |              |
+----------------+-------------+--------+---+-----------+--------------+
|                |             |        | P |           |              |
|                |             |        | a |           |              |
|                |             |        | r |           |              |
|                |             |        | a |           |              |
|                |             |        | m |           |              |
|                |             |        | e |           |              |
|                |             |        | t |           |              |
|                |             |        | e |           |              |
|                |             |        | r |           |              |
|                |             |        | V |           |              |
|                |             |        | a |           |              |
|                |             |        | l |           |              |
|                |             |        | u |           |              |
|                |             |        | e |           |              |
+----------------+-------------+--------+---+-----------+--------------+
| Parameter      |             |        | B |           | Source       |
|                |             |        | a |           |              |
|                |             |        | s |           |              |
|                |             |        | e |           |              |
|                |             |        | C |           |              |
|                |             |        | a |           |              |
|                |             |        | s |           |              |
|                |             |        | e |           |              |
|                |             |        | A |           |              |
|                |             |        | n |           |              |
|                |             |        | a |           |              |
|                |             |        | l |           |              |
|                |             |        | y |           |              |
|                |             |        | s |           |              |
|                |             |        | i |           |              |
|                |             |        | s |           |              |
+----------------+-------------+--------+---+-----------+--------------+
| Probability    | High region | Ages   |   | 0.0133    | ^9^          |
| adult          |             | 18-64  |   |           |              |
| immunization   |             |        |   |           | ^1^          |
| first dose     |             |        |   |           |              |
+----------------+-------------+--------+---+-----------+--------------+
|                | I           | Ages   |   | 0.0131    | ^9^          |
|                | ntermediate | 18-64  |   |           |              |
|                | region      |        |   |           | ^1^          |
+----------------+-------------+--------+---+-----------+--------------+
|                | Low region  | Ages   |   | 0.0084    | ^9^          |
|                |             | 18-64  |   |           |              |
|                |             |        |   |           | ^1^          |
+----------------+-------------+--------+---+-----------+--------------+
| Probability    | All regions | Ages   |   | 0         | Assumption   |
| adult          |             | 18-64  |   |           |              |
| immunization   |             |        |   |           |              |
| second dose \| |             |        |   |           |              |
| first dose     |             |        |   |           |              |
+----------------+-------------+--------+---+-----------+--------------+
|                |             |        |   |           |              |
+----------------+-------------+--------+---+-----------+--------------+

VI. **Adverse events**

Adverse events resulting from vaccination were classified according to
severity -- mild or severe (Table 6.1). The probability of a mild
adverse event -- defined as fever 38^o^C or higher which interferes with
daily activities -- was calculated by multiplying the proportion of
those vaccinated who reported fever (0.057)^10^ by the proportion of all
vaccine complications determined to interfere with daily activities
(0.089)^11^. We assumed an associated work loss of 0.875 days and no
additional medical care costs.

The probability (0.000001) and associated medical care costs (\$957) of
a severe adverse event -- defined as onset of Guillain--Barre´ syndrome
-- was taken from expert panel estimates reported by Meltzer MI et al
(2001). We assumed an associated work loss of five days.

+------------------------+--------+---+------------+--------------------+
| Table 6.1 Adverse      |        |   |            |                    |
| Events Costs and       |        |   |            |                    |
| Probabilities          |        |   |            |                    |
+------------------------+--------+---+------------+--------------------+
|                        |        | P |            |                    |
|                        |        | a |            |                    |
|                        |        | r |            |                    |
|                        |        | a |            |                    |
|                        |        | m |            |                    |
|                        |        | e |            |                    |
|                        |        | t |            |                    |
|                        |        | e |            |                    |
|                        |        | r |            |                    |
|                        |        | V |            |                    |
|                        |        | a |            |                    |
|                        |        | l |            |                    |
|                        |        | u |            |                    |
|                        |        | e |            |                    |
+------------------------+--------+---+------------+--------------------+
| Parameter              |        | B |            | Source             |
|                        |        | a |            |                    |
|                        |        | s |            |                    |
|                        |        | e |            |                    |
|                        |        | C |            |                    |
|                        |        | a |            |                    |
|                        |        | s |            |                    |
|                        |        | e |            |                    |
|                        |        | A |            |                    |
|                        |        | n |            |                    |
|                        |        | a |            |                    |
|                        |        | l |            |                    |
|                        |        | y |            |                    |
|                        |        | s |            |                    |
|                        |        | i |            |                    |
|                        |        | s |            |                    |
+------------------------+--------+---+------------+--------------------+
| Probability mild       | Ages   |   | 0.0051     | ^10^               |
| adverse event          | 0-95   |   |            |                    |
|                        |        |   |            | ^11^               |
+------------------------+--------+---+------------+--------------------+
| Probability severe     | Ages   |   | 0.000001   | ^12^               |
| adverse event          | 0-95   |   |            |                    |
+------------------------+--------+---+------------+--------------------+
| Severe adverse event   | Ages   |   | \$957      | ^12^               |
| (medical care costs)   | 0-95   |   |            |                    |
+------------------------+--------+---+------------+--------------------+
|                        |        |   |            |                    |
+------------------------+--------+---+------------+--------------------+

VII. **Herd immunity**

Incident infection stems from a reservoir in children. Vaccination of
children has led to disproportionate declines in incidence among
unvaccinated children and adults. Model estimates the herd immunity
benefits that accrue to the unvaccinated members of the child cohort.

Herd immunity is based on the proportion of children 0-18 with
vaccination coverage. For each 1% increase in vaccination in this
cohort, incidence attributed to children declines 3.9% in children (1%
for those vaccinated and an additional 2.9% among the unvaccinated) and
1% among adults.

For children, 90% of incident infections were attributed to other
children. For adults 25% of incident infections were attributed to
children.

To account for herd immunity, the probability of infection applied to
those with no vaccination coverage was estimated as;

I\* Proportion of children 0 to 18 with vaccination coverage\* Herd
Immunity Adjuster\*Proportion of incident infections from children + I
\* Proportion of infections not from children

With a vaccination strategy at age 2, for children aged 0 to 18 this is;

( I\* ((16/18)\*.729 .90\* -.029) + (I \* .10)

For adults 19 to 95 this is;

( I\* ((16/18)\*.729 .25\* -.01) + (I \* .75)

VIII. **Productivity losses**

Productivity losses were calculated by combining the age-specific labor
participation rate^13^ and daily earnings^14^ with the duration of work
loss attributable to Hepatitis A stages of infection, vaccine
administration and adverse events (Equation 1).

> Equation 1.
>
> Productivity Losses (before discounting) =
> (ProbWork)\*(TimeWorkLoss)\*(Wage)
>
> Where ProbWork = Labor participation rate, by age
>
> TimeWorkLoss = Duration of work loss, by infection state
>
> Wage = Daily earnings, by age

For children aged 0 to 15 years, we applied the labor participation rate
and daily earnings of a parent or caregiver (calculated as a weighted
average of 20 to 44-year-olds). In addition, a labor participation rate
of 0 was assumed for individuals aged 65 years and older.

In the case of death attributable to Hepatitis A infection, we assumed
an annual work loss of 250 days (5 days per week for 50 weeks). Table
8.1 presents the parameter values used in the model and their respective
sources for duration of work loss, labor force participation and daily
earnings.

 

+----------+-------------+-----------+-------+---+---------+----------+
| Table    |             |           |       |   |         |          |
| 8.1.     |             |           |       |   |         |          |
| Prod     |             |           |       |   |         |          |
| uctivity |             |           |       |   |         |          |
| Losses   |             |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           |       | P |         |          |
|          |             |           |       | a |         |          |
|          |             |           |       | r |         |          |
|          |             |           |       | a |         |          |
|          |             |           |       | m |         |          |
|          |             |           |       | e |         |          |
|          |             |           |       | t |         |          |
|          |             |           |       | e |         |          |
|          |             |           |       | r |         |          |
|          |             |           |       | V |         |          |
|          |             |           |       | a |         |          |
|          |             |           |       | l |         |          |
|          |             |           |       | u |         |          |
|          |             |           |       | e |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
| P        |             |           |       | B |         | Source   |
| arameter |             |           |       | a |         |          |
|          |             |           |       | s |         |          |
|          |             |           |       | e |         |          |
|          |             |           |       | C |         |          |
|          |             |           |       | a |         |          |
|          |             |           |       | s |         |          |
|          |             |           |       | e |         |          |
|          |             |           |       | A |         |          |
|          |             |           |       | n |         |          |
|          |             |           |       | a |         |          |
|          |             |           |       | l |         |          |
|          |             |           |       | y |         |          |
|          |             |           |       | s |         |          |
|          |             |           |       | i |         |          |
|          |             |           |       | s |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
| Work     | Symptomatic |           |       |   | 3       | As       |
| Days     | anicteric   |           |       |   |         | sumption |
| Lost     | infection   |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Non-h       | Reported  | Ages  |   | 3.7     | ^15^     |
|          | ospitalized |           | 0-12  |   |         |          |
|          | icteric     |           |       |   |         |          |
|          | infection   |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 13-95 |   | 10      | ^16^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             | U         | Ages  |   | 3       | As       |
|          |             | nreported | 0-95  |   |         | sumption |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Hospi       |           | Ages  |   | 33.2    | ^17^     |
|          | talization, |           | 0-95  |   |         |          |
|          | Icteric     |           |       |   |         |          |
|          | Infection   |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Hospi       |           | Ages  |   | 33.2    | ^17^     |
|          | talization, |           | 0-95  |   |         |          |
|          | Fulminant,  |           |       |   |         |          |
|          | no          |           |       |   |         |          |
|          | transplant  |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Liver       |           | Ages  |   | 153.2   |          |
|          | transplant, |           | 0-95  |   |         |          |
|          | Year of     |           |       |   |         |          |
|          | transplant  |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Vaccination | First     | 0-12  |   | 0       |          |
|          |             | dose      |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 13-95 |   | 0.125   |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             | Second    | 0-2   |   | 0       |          |
|          |             | dose      |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 3-95  |   | 0.125   |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Mild        |           | Ages  |   | 0.875   | As       |
|          | adverse     |           | 0-95  |   |         | sumption |
|          | event       |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Severe      |           | Ages  |   | 5       | As       |
|          | adverse     |           | 0-95  |   |         | sumption |
|          | event       |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
| Costs,   | Daily cost  |           | Ages  |   | \$      | ^14^     |
| Parti    | of work     |           | 0-15  |   | 120.2\* |          |
| cipation | loss        |           |       |   |         | As       |
|          |             |           |       |   |         | sumption |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 16-19 |   | \$63.42 | ^14^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 20-24 |   | \$83.4  | ^14^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 25-34 |   | \       | ^14^     |
|          |             |           |       |   | $121.53 |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 35-44 |   | \       | ^14^     |
|          |             |           |       |   | $143.15 |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 45-54 |   | \       | ^14^     |
|          |             |           |       |   | $146.41 |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 55-64 |   | \       | ^14^     |
|          |             |           |       |   | $146.61 |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 65-95 |   | \       | ^14^     |
|          |             |           |       |   | $105.02 |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          | Labor force |           | Ages  |   | 0.815\* | ^13^     |
|          | pa          |           | 0-15  |   |         |          |
|          | rticipation |           |       |   |         |          |
|          | rate        |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 16-19 |   | 0.437   | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 20-24 |   | 0.749   | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 25-34 |   | 0.8263  | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 35-44 |   | 0.836   | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 45-54 |   | 0.8197  | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 55-64 |   | 0.6217  | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
|          |             |           | 65-95 |   | 0       | ^13^     |
+----------+-------------+-----------+-------+---+---------+----------+
| \* Note: |             |           |       |   |         |          |
| Labor    |             |           |       |   |         |          |
| parti    |             |           |       |   |         |          |
| cipation |             |           |       |   |         |          |
| rate,    |             |           |       |   |         |          |
| daily    |             |           |       |   |         |          |
| earnings |             |           |       |   |         |          |
| of       |             |           |       |   |         |          |
| parent   |             |           |       |   |         |          |
| or       |             |           |       |   |         |          |
| c        |             |           |       |   |         |          |
| aregiver |             |           |       |   |         |          |
| (ca      |             |           |       |   |         |          |
| lculated |             |           |       |   |         |          |
| as a     |             |           |       |   |         |          |
| weighted |             |           |       |   |         |          |
| average  |             |           |       |   |         |          |
| of 20 to |             |           |       |   |         |          |
| 44-yea   |             |           |       |   |         |          |
| r-olds). |             |           |       |   |         |          |
+----------+-------------+-----------+-------+---+---------+----------+

IX. **Order in which the Model Considers Simulated Events**

For the model to proceed, the order of simulated events must be
specified. We specified the order of simulated model states as follows:

1.  **Immunity loss.** Those in immune-due-to-vaccination states
    progress to one less year remaining of immunity. Those who had only
    one year of immunity remaining lose immunity and progress to the
    *Susceptible, immunized* state.

2.  **Vaccination.** Vaccination of a portion of cohort in *Susceptible,
    not immunized* and *Immune due to disease, immunity unknown*. Some
    of those vaccinated receive only one dose, others two. Based on
    resulting GMT levels, these newly vaccinated are distributed among
    the states *Immune: 1, 2,...95 years to immunity loss* or
    *Susceptible, immunized*.

3.  **HepA infection.** The model assumes that infections can only occur
    among those in the cohort in the *Susceptible, not immunized* state
    who are not just vaccinated in the current year or anyone in the
    *Susceptible, immunized* state who did not just lose immunity. By
    default, the risk of infection is the same among both groups,
    however the risk in the latter group can be reduced by a
    multiplicative factor ("Reduction in risk of infection \|
    susceptible, immunized") as compared to the risk among the former,
    although this feature was not used in the manuscript. The number of
    infections is therefore calculated using the following formula:

Number of infections = \[Infections from *Susceptible, not
immunized*\] + (9.1)

\[Infections from *Susceptible, immunized*\]

= \[(% of cohort susceptible, not immunized --

\% of cohort just immunized) •

P(infection \| susceptible, not immunized)\] +

\[(% of cohort susceptible, immunized +

\% of cohort that just lost immunity) •

P(infection \| susceptible, immunized)\]

4.  **HepA disease development stages.** The degree of severity of
    hepatitis A varies between infected persons. In Figure 1.1 HepA
    disease development stages are represented by shaded parallelograms.

Table 9.1 Parameters that determine HepA disease development in infected
cases

+-------------------------------------+---------------------+----------+
| **Parameter**                       | **Default value**   | **       |
|                                     |                     | Source** |
+-------------------------------------+---------------------+----------+
| P(acute icteric HepA \| HepA        | Varies from 0 at    | A        |
| infection) (0-1)                    | age 0 to 0.95 at    | rmstrong |
|                                     | age 95              | GL, Bell |
|                                     |                     | BP       |
|                                     |                     | (2002)   |
+-------------------------------------+---------------------+----------+
| P(symptomatic \| anicteric) (0-1)   | 0.5                 | As       |
|                                     |                     | sumption |
|                                     |                     | by GL    |
|                                     |                     | Armstong |
+-------------------------------------+---------------------+----------+
| Total number of icteric cases per   | 4.28                | A        |
| reported case                       |                     | rmstrong |
|                                     |                     | GL, Bell |
|                                     |                     | BP       |
|                                     |                     | (2002)   |
+-------------------------------------+---------------------+----------+
| P(fulminant \| reported acute       | Ages 0-4: 0.0038    | Ca       |
| icteric HepA) (0-1)                 |                     | lculated |
|                                     | Ages 5-14: 0.0005   | from     |
|                                     |                     | NNDSS    |
|                                     | Ages 15-39: 0.0068  | (19      |
|                                     |                     | 90-2004) |
|                                     | Ages 40-59: 0.055   |          |
|                                     |                     |          |
|                                     | Ages 60-95: 0.08    |          |
+-------------------------------------+---------------------+----------+

> *Anicteric HepA (including symptomatic and unsymptomatic)*. All cases
> that are not icteric are classified as *anicteric*. The model assumes
> that the number of symptomatic anicteric cases is equal to a
> percentage, 50% by default, of all anicteric cases.
>
> *Icteric HepA*. The number of icteric cases is calculated as a
> percentage of infected cases. All icteric cases are, by definition,
> symptomatic.
>
> *Fulminant HepA.* The number of fulminant cases is calculated as a
> percentage of reported icteric cases.

5.  **HepA disease-related health outcomes.** The model assumes that
    within each of these stages, patients require different degrees of
    healthcare. Reporting of cases to the public health system also
    varies between stages, as well as within icteric. Costs for health
    care and work loss, either due to illness within the cohort or
    caretakers of those ill within the cohort, may be incurred.

> *Asymptomatic anicteric.* Asymptomatic cases do not require healthcare
> and are all unreported.
>
> *Symptomatic anicteric.* The model assumes that symptomatic cases are
> relatively mild and therefore do not require extensive healthcare.
> Only 50% of symptomatic anicteric HepA cases necessitate outpatient
> visits; of those, only 1 patient visit per person is assumed. All
> symptomatic anicteric cases are unreported.
>
> *Icteric.* Icteric cases are either reported or unreported; the number
> of reported cases is calculated as the number of icteric cases divided
> by the number of icteric cases per reported case (4.28, by default, as
> estimated by Armstrong and Bell, 2003).
>
> The model assumes 95% of all icteric cases require at least outpatient
> visits; the calculation of what percent require only outpatient visits
> (all among unreported cases) is derived based on this assumption and
> the probabilities of needing other healthcare. All reported cases,
> however, are assumed to require healthcare, either outpatient visits
> or hospitalizations. We make this assumption on the basis that, in
> order for a case to be reported, the patient must have sought
> healthcare for disease diagnosis.
>
> In order to capture the phenomena that reported cases are more likely
> to have a higher severity of disease, the model assumes that, among
> those cases that require outpatient visits only, reported cases
> require 3 outpatient visits, while unreported cases require 33% less,
> or 2.
>
> *Fulminant.* All fulminant cases are assumed to require
> hospitalization. Not all fulminant cases receive a liver transplant;
> of those that do not, some die with a specified probability and the
> remaining recover.
>
> Those that receive a transplant encounter a risk of death (i.e. a
> transition probability of death is applied) twice in the year of their
> transplant -- once to represent the risk of death immediately after
> surgery and a second time to represent the remaining months of the
> year following surgery (*P(death \| transplant), yr of trans* in Table
> 9.2). The model then assumes that those patients with a history of a
> transplant are at increased risk of death due to transplant-related
> issues for the rest of their lives. The model calculates the percent
> of the cohort with history of a liver transplant as a sum of those who
> had a history of a liver transplant in the previous year and didn\'t
> die from transplant-related issues and those infections from the
> current year that resulted in a non-fatal transplant.
>
> The number of outpatient visits, therefore, is determined by Equation
> 9.2.

Number of OP visits = \[OP visits among reported icteric\] + (9.2)

\[OP visits among nonreported icteric\] +

\[OP visits among anicteric\]

= \[Reported infections • OP visits per reported icteric\] +

\[Nonreported icteric infections •

P(OP visit \| nonreported icteric) •

OP visits per reported icteric •

(1 - % reduction OP visits unrep'd vs. rep'd icteric)\] + \[Anicteric
infections •

P(symptomatic \| anicteric) •

P(OP visit \| anicteric) •

OP visits per anicteric\]

> The number of hospitalizations is calculated as the product of the
> number of reported infections and the sum of the probabilities of
> hospitalization given a reported infection and fulminant given a
> reported infection.

Table 9.2 Parameters that determine HepA disease-related health outcomes
and costs in infected cases

+----------------------------+------------------------+---------------+
| **Parameter**              | **Default Value**      | **Source**    |
+============================+========================+===============+
| P(outpatient visit & work  | 0.5                    | Assumption    |
| loss \| symptomatic        |                        |               |
| anicteric infection) (0-1) |                        |               |
+----------------------------+------------------------+---------------+
| P(any level of medical     | 0.95                   | Bownds L et   |
| care \| icteric HepA)      |                        | al (2003)     |
| (0-1)                      |                        |               |
+----------------------------+------------------------+---------------+
| P(hospitalization, OP      | Ages 0-4: 0.0465       | NNDSS         |
| visit & work loss from     |                        | (2000-2001)   |
| non-fulminant HepA \|      | Ages 5-14: 0.17        |               |
| reported acute icteric     |                        |               |
| HepA) (0-1)                | Ages 15-39: 0.225      |               |
|                            |                        |               |
|                            | Ages 40-59: 0.185      |               |
|                            |                        |               |
|                            | Ages 60-95: 0.2        |               |
+----------------------------+------------------------+---------------+
| Number of outpatient       | 1                      | Assumption    |
| visits \| anicteric HepA   |                        |               |
| requiring OP visits        |                        |               |
+----------------------------+------------------------+---------------+
| Number of outpatient       | 3                      | Berge JJ et   |
| visits \| reported icteric |                        | al (2000)     |
| HepA requiring OP visits,  |                        |               |
| but not hospitalization    |                        |               |
+----------------------------+------------------------+---------------+
| Percent reduction in       | 0.333                  | Assumption    |
| number of OP visits for    |                        |               |
| unreported vs. reported    |                        |               |
| icteric HepA (0-1)         |                        |               |
+----------------------------+------------------------+---------------+
| P(death w/o transplant \|  | Ages 0-59: 0.145       | Schiodt FV et |
| fulminant HepA) (0-1)      |                        | al (2003)     |
|                            | Ages 60-95: 0.67       |               |
+----------------------------+------------------------+---------------+
| P(transplant \| fulminant  | Ages 0-59: 0.1875      | Schiodt FV et |
| HepA) (0-1)                |                        | al (2003)     |
|                            | Ages 60-95: 0.025      |               |
+----------------------------+------------------------+---------------+
| P(death immediately after  | 0.0648                 | UNOS          |
| transplant) (0-1)          |                        |               |
+----------------------------+------------------------+---------------+
| P(death \| transplant)     | Yr of trans: 0.2419    | UNOS          |
| (0-1)                      |                        |               |
|                            | 1-2 yrs since trans:   |               |
|                            | 0.0299                 |               |
|                            |                        |               |
|                            | 3-4 yrs since trans:   |               |
|                            | 0.0236                 |               |
|                            |                        |               |
|                            | 5-9 yrs since trans:   |               |
|                            | 0.0242                 |               |
+----------------------------+------------------------+---------------+
|                            | 10-15 yrs since trans: | Jain A et al  |
|                            | 0.0262                 | (2000)        |
|                            |                        |               |
|                            | 16+ yrs since trans:   |               |
|                            | 0.0136                 |               |
+----------------------------+------------------------+---------------+
|                            |                        |               |
+----------------------------+------------------------+---------------+

6.  **HepA disease-related healthcare costs.**

> Table 9.3 presents the parameters that determine costs once the model
> determines the cohort's outcomes.

Table 9.3 Parameters that determine HepA disease-related health outcomes
and costs in infected cases

  ------------------------------------------------------------------------
  **Parameter**                 **Default Value**  **Source**
  ----------------------------- ------------------ -----------------------
  Cost for health care \|       Yr of trans:       Hauboldt RA (1999)
  fulminant acute icteric HepA  \$285,900          
  w/ transplant                                    

                                Yrs after trans:   Berge JJ et al (2000)
                                \$25,598           

  Cost for health care \|       \$24,138           Berge JJ et al (2000)
  fulminant acute icteric HepA                     
  w/o transplant                                   

  Cost for health care \|       \$8,111            Dalton CB et al (1996);
  hospitalization and OP visits                    Berge JJ et al (2000)
  for non-fulminant acute                          
  icteric HepA                                     

  Cost for health care \| OP    \$821              Dalton CB et al (1996);
  visits only for non-fulminant                    Berge JJ et al (2000)
  acute icteric HepA                               

  Percent reduction in medical  0.333              Assumption
  costs for unreported vs.                         
  reported icteric HepA (0-1)                      

                                                   
  ------------------------------------------------------------------------

7.  **HepA disease-related productivity losses.** The model assumes that
    within each of the stages, costs for health care and work loss,
    either due to illness within the cohort or caretakers of those ill
    within the cohort, may be incurred.

> The number of days of work loss incurred by each type of patient is
> calculated as follows:
>
> *Anicteric HepA.* The number of anicteric cases for which work loss is
> experienced is equal to the number of symptomatic anicteric cases
> times the probability of employment times the probability of having to
> seek outpatient care or experiencing work loss without other care. The
> total amount of work loss is equal to the number of those cases times
> the number of days of work loss associated with an anicteric case.
>
> *Non-fulminant icteric HepA.* The number of days of work loss among
> non-fulminant icteric cases differs by the severity of the inceric
> case; those unreported incur less work loss than those seeking
> outpatient care only who, in turn, incur less than those who are
> hospitalized. The total amount of work loss from this group is equal
> to the number of each of those types of cases times the probability of
> employment times the number of days of work loss associated with each.
>
> *Fulminant HepA.* The number of days of work loss among fulminant
> cases also differs depending on whether a transplant is performed;
> those that receive transplants require significantly more days. The
> total amount of work loss from this group is equal to the number of
> each of those types of cases times the probability of employment times
> the number of days of work loss associated with each.
>
> Costs for these lost work days depend on the patient's age. If a
> patient is younger than the starting age for work, the productivity
> losses are assumed to be for a caregiver; otherwise they are assumed
> to be incurred by the afflicted patient.

Table 9.3 Parameters determining productivity losses due to sickness
assocated with HepA

+------------------------------------+----------------+---------------+
| **Parameter**                      | **Default      | **Source**    |
|                                    | value**        |               |
+------------------------------------+----------------+---------------+
| P(work loss, but no medical care   | 0.5            | Assumption    |
| \| symptomatic anicteric           |                |               |
| infection) (0-1)                   |                |               |
+------------------------------------+----------------+---------------+
| Number of work days lost \|        | 3              | Assumption    |
| symptomatic anicteric infection    |                |               |
+------------------------------------+----------------+---------------+
| Number of work days lost \|        | 3              | Assumption    |
| unreported icteric HepA            |                |               |
+------------------------------------+----------------+---------------+
| Number of work days lost \| OP     | Ages 0-12: 3.7 | 0-12: Jacobs  |
| visits only for reported           |                | RJ, Meyerhoff |
| non-fulminant HepA                 | Age 13-95: 10  | AS (2001)     |
|                                    |                |               |
|                                    |                | 13-95:        |
|                                    |                | O\'Connor JB  |
|                                    |                | et al (1999)  |
+------------------------------------+----------------+---------------+
| Number of work days lost \|        | 33.2           | Berge JJ et   |
| hospitalization for non-fulminant  |                | al (2000)     |
| HepA                               |                |               |
+------------------------------------+----------------+---------------+
| Number of work days lost \|        | 153.2          | Transplant    |
| fulminant acute icteric HepA w/    |                | Patient       |
| transplant                         |                | Partnering    |
|                                    |                | Pgm           |
+------------------------------------+----------------+---------------+
| Number of work days lost \|        | 33.2           | Berge JJ et   |
| fulminant acute icteric HepA w/o   |                | al (2000)     |
| transplant                         |                |               |
+------------------------------------+----------------+---------------+
| Age at which people start working  | 16             | Assumption    |
| (0-95; age public notifications &  |                |               |
| prod. losses from self begin)      |                |               |
+------------------------------------+----------------+---------------+

X.  **Table of all medical care costs and their sources**

We applied medical care costs taken from multiple studies in the
literature for each of the following Hepatitis A stages of infection:

-   Symptomatic anicteric infection

-   Non-hospitalized icteric infection(outpatient case)

    -   Reported

    -   Unreported

-   Hospitalized icteric infection (inpatient case)

-   Hospitalized fulminant liver disease (without a liver transplant)

-   Hospitalized fulminant liver disease (with a transplant)

For symptomatic anicteric infections, we assumed a healthcare cost
(\$84.16) equivalent to one office visit to the patient's primary care
physician ^18^.

Outpatient and inpatient healthcare costs (and ranges) associated with
icteric infections were calculated using the results reported across
four case studies of Hepatitis A outbreaks in the United States
^17,19-21^. Specifically, we took a weighted average of results reported
by the two studies ^17,19^ which used medical and chart data to
determine the base case estimate of healthcare costs incurred by
outpatient and inpatient cases. The remaining two studies which used
patient self-report for cost and utilization information provided the
upper and lower bounds. Finally, we assumed a 0.333 reduction in
healthcare costs for unreported non-hospitalized (outpatient) icteric
infection.

Table 10.1 presents the observed number of cases and associated medical
care costs reported in each of the four studies considered.

+-------------------+----------+------------+----------+--------------+
| Table 10.1        |          |            |          |              |
| Medical Costs     |          |            |          |              |
| (non-fulminant    |          |            |          |              |
| reported icteric  |          |            |          |              |
| infections)^a^    |          |            |          |              |
+-------------------+----------+------------+----------+--------------+
|                   | Ou       |            | I        |              |
|                   | tpatient |            | npatient |              |
+-------------------+----------+------------+----------+--------------+
| Study             | No.      | Cost       | No.      | Cost         |
|                   | Cases    |            | Cases    |              |
|                   |          | (2004 USD) |          | (2004 USD)   |
+-------------------+----------+------------+----------+--------------+
| Dalton et al      | 41       | \$984      | 2        | \$9,719      |
| (1996)            |          |            |          |              |
+-------------------+----------+------------+----------+--------------+
| Berge JJ et al    | 217      | \$790      | 37       | \$8,024      |
| (2000)            |          |            |          |              |
+-------------------+----------+------------+----------+--------------+
| Bownds et al      | 139      | \$615      | 8        | \$16,593     |
| (2003)            |          |            |          |              |
+-------------------+----------+------------+----------+--------------+
| Sansom et al      | 73       | \$588      | 16       | \$4,620      |
| (2003)            |          |            |          |              |
+-------------------+----------+------------+----------+--------------+
| Average           | \$821    |            | \$8,111  |              |
| (Range)^b^        |          |            |          |              |
|                   | (\$58    |            | (\       |              |
|                   | 8-\$984) |            | $4,620-\ |              |
|                   |          |            | $16,593) |              |
+-------------------+----------+------------+----------+--------------+
| ^a^ All findings  |          |            |          |              |
| converted to 2004 |          |            |          |              |
| United States     |          |            |          |              |
| dollars (USD)     |          |            |          |              |
| using the annual  |          |            |          |              |
| Consumer Price    |          |            |          |              |
| Index for all     |          |            |          |              |
| urban consumers.  |          |            |          |              |
|                   |          |            |          |              |
| ^b^ Base cost     |          |            |          |              |
| determined by a   |          |            |          |              |
| weighted average  |          |            |          |              |
| of estimates      |          |            |          |              |
| reported in       |          |            |          |              |
| Dalton (1996) and |          |            |          |              |
| Berge (2000);     |          |            |          |              |
| Preference for    |          |            |          |              |
| medical and chart |          |            |          |              |
| data over         |          |            |          |              |
| self-report.      |          |            |          |              |
+-------------------+----------+------------+----------+--------------+

We assumed separate inpatient medical care costs -- taken from case
studies in the literature -- for fulminant liver disease both with and
without subsequent liver transplantation surgery^17,22^. For those
patients receiving a liver transplant, we applied a recurring annual
cost in addition to healthcare costs incurred during the transplant year
for up to 18 years post-transplant. Table 10.2 summarizes all medical
care cost estimates and their respective sources used in the base case
analysis.

+---------------+-----------+-------+---+---------+------------------+
| Table 10.2    |           |       |   |         |                  |
| Medical Care  |           |       |   |         |                  |
| Costs         |           |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
|               |           |       | P |         |                  |
|               |           |       | a |         |                  |
|               |           |       | r |         |                  |
|               |           |       | a |         |                  |
|               |           |       | m |         |                  |
|               |           |       | e |         |                  |
|               |           |       | t |         |                  |
|               |           |       | e |         |                  |
|               |           |       | r |         |                  |
|               |           |       | V |         |                  |
|               |           |       | a |         |                  |
|               |           |       | l |         |                  |
|               |           |       | u |         |                  |
|               |           |       | e |         |                  |
+---------------+-----------+-------+---+---------+------------------+
| Parameter     |           |       | B |         | Source           |
|               |           |       | a |         |                  |
|               |           |       | s |         |                  |
|               |           |       | e |         |                  |
|               |           |       | C |         |                  |
|               |           |       | a |         |                  |
|               |           |       | s |         |                  |
|               |           |       | e |         |                  |
|               |           |       | A |         |                  |
|               |           |       | n |         |                  |
|               |           |       | a |         |                  |
|               |           |       | l |         |                  |
|               |           |       | y |         |                  |
|               |           |       | s |         |                  |
|               |           |       | i |         |                  |
|               |           |       | s |         |                  |
+---------------+-----------+-------+---+---------+------------------+
| Percent cost  |           | Ages  |   | 0.333   | Assumption       |
| reduction,    |           | 0-95  |   |         |                  |
| Unreported    |           |       |   |         |                  |
| cases         |           |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
| Symptomatic   |           | Ages  |   | \$84.16 | ^18^             |
| anicteric     |           | 0-95  |   |         |                  |
| infection     |           |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
| Non           |           | Ages  |   | \$821   | ^19^             |
| -hospitalized |           | 0-95  |   |         |                  |
| icteric       |           |       |   |         | ^17^             |
| infection,    |           |       |   |         |                  |
| Reported      |           |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
| Hospitalized  |           | Ages  |   | \$8,111 | ^19^             |
| icteric       |           | 0-95  |   |         |                  |
| infection     |           |       |   |         | ^17^             |
+---------------+-----------+-------+---+---------+------------------+
| Hospitalized  |           | Ages  |   | \       | ^17^             |
| fulminant, no |           | 0-95  |   | $24,138 |                  |
| transplant    |           |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
| Liver         | Year of   | Ages  |   | \$      | ^22^             |
| transplant    | t         | 0-95  |   | 285,900 |                  |
|               | ransplant |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
|               | S         | Ages  |   | \       | ^17^             |
|               | ubsequent | 0-95  |   | $25,598 |                  |
|               | years     |       |   |         |                  |
|               | (0-18     |       |   |         |                  |
|               | years)    |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+
|               |           |       |   |         |                  |
+---------------+-----------+-------+---+---------+------------------+

XI. **Calculation of public health cost per case**

Public health response costs were classified into three major activities
-- Surveillance, Immune globulin (IG) coordination and administration,
and Public Notification based on discussions with CDC public health
experts with state and local health department experience. The costs
associated with each activity per reported case were calculated
separately and then added together using the following formulas
(Equations 2 -- 5) and inputs (Table 11.1).

> Equation 2.
>
> Public health costs =
>
> surveillance per reported case +
>
> Immune globulin (IG) coordination and administration +
>
> public notification
>
> Equation 3.                  
>
> Surveillance per reported case =
>
> (1 / (1 + Discount rate) \^ (t -- Age of cohort at model start - 1))
> \*
>
> (No. cases reported \*
>
> (Prob. follow-up by PHA \*
>
> (Hourly wage PHA \*
>
> (PMDCall \* TimeMDCall + PInfCall \* TimeInfCall + PInfVisit \*
> TimeInfVisit))))
>
> Where PHA = public health advisor
>
> PMDCall = Probability of phone call by PHA to the infected patient's
> physician
>
> TimeMDCall = Length of time (hours), phone call by PHA to the infected
> patient's physician
>
> PInfCall = Probability of phone call by PHA to the infected patient \|
> physician phone call
>
> TimeInfCall = Length of time (hours), phone call by PHA to the
> infected patient
>
> PInfVisit = Probability of visit by PHA to the infected patient \|
> physician and patient phone call
>
> TimeInfVisit = Length of time (hours), visit by PHA to the infected
> patient
>
> Immune globulin (IG) coordination and administration =
>
> (1 / (1 + Discount rate) \^ (t -- Age of cohort at model start - 1))\*
>
> \[(No. cases reported \*
>
> (Prob. follow-up by PHA \*
>
> (Hourly wage PHA \*
>
> (PIGCoord \* TimeIGCoord + TimeIGCall \*
>
> (PFSW \* NumIGCallsFSW + (1 - PFSW) \* NumIGCallsNonFSW))) +
>
> IGCost \* (PFSW \* NumIGFSW + (1 - PFSW) \* NumIGNonFSW))))\]
>
> Where PHA = public health advisor
>
> PIGCoord = Proportion of cases with follow-up for whom IG shots for
> contacts are coordinated by PHA
>
> TimeIGCoord = Length of time (hours), coordination of IG shots for
> contacts
>
> TimeIGCall = Length of time (hours), phone call to contact
>
> PFSW = Proportion of reported cases, Food service workers (FSW)
>
> NumIGCallsFSW = No. phone calls to contacts, FSW cases with follow-up
>
> NumIGCallsNonFSW = No. phone calls to contacts, all other cases with
> follow-up
>
> IGCost = IG cost per dose
>
> NumIGFSW = No. IG shots to contacts, FSW cases with follow-up
>
> NumIGNonFSW = No. IG shots to contacts, all other cases with follow-up
>
>                    
>
> Public Notification =
>
> (1 / (1 + Discount rate) \^ (t -- Age of cohort at model start - 1))\*
>
> No. cases reported \* Hourly wage PHA \*
>
> (PFoodVisit \* TimeFoodVisit + (PPublicNotif \* TimePublicNotif))
>
> Where PHA = public health advisor
>
> PFoodVisit = Proportion of reported cases for whom visit made to
> employer (FSW cases only) by PHA
>
> TimeFoodVisit = Length of time (hours), employer visit (FSW cases
> only)
>
> PPublicNotif = Proportion of reported cases for whom a public
> notification is made
>
> TimePublicNotif = Length of time (hours), public notification

  ---------------- --------------------------------- ----------- ---------- ---------
  Table 11.1                                                                
  Public Health                                                             
  Response                                                                  

                                                     Parameter              
                                                     Value                  

  Parameter                                          Base Case              Source
                                                     Analysis               

  General          Loaded mean hourly wage for                   \$21.3     ^23^
                   licensed practical or licensed                           
                   vocational nurse                                         

                   Proportion of reported cases with             0.55       
                   follow-up by PHA                                         

  Surveillance     Probability of phone call by PHA              1          
                   to the infected patient's                                
                   physician                                                

                   Length of time (hours), phone                 1          
                   call by PHA to the infected                              
                   patient's physician                                      

                   Probability of phone call by PHA              0.8        
                   to the infected patient \|                               
                   Physician phone call                                     

                   Length of time (hours), phone                 1          
                   call by PHA to the infected                              
                   patient                                                  

                   Probability of visit by PHA to                0.25       
                   the infected patient \| Physician                        
                   and patient phone                                        

                   Length of time (hours), visit by              4          
                   PHA to the infected patient                              

  IG coordination  IG cost per dose                              \$23.52    
  & administration                                                          

                   Proportion of cases with                      1          
                   follow-up for whom IG shots for                          
                   contacts are coordinated by PHA                          

                   Proportion of reported cases,                 0.06       
                   Food service workers (FSW)                               

                   No. phone calls to contacts, FSW              73         
                   cases with follow-up                                     

                   No. phone calls to contacts, all              25         
                   other cases with follow-up                               

                   Length (hours) of phone calls to              1          
                   contacts                                                 

                   No. IG shots to contacts, FSW                 29         
                   cases with follow-up                                     

                   No. IG shots to contacts, all                 10         
                   other cases with follow-up                               

                   Length (hours) for coordination               4          
                   of IG shots for contacts                                 

  Public           Proportion of reported cases for              0.02       
  Notification     whom visit made to employer (FSW                         
                   cases only)                                              

                   Length (hours) employer visit                 8          
                   (FSW cases only)                                         

                   Proportion of reported cases for              0.002      
                   whom a public notification is                            
                   made                                                     

                   Length (hours) for PHS to make                160        
                   public notification                                      
  ---------------- --------------------------------- ----------- ---------- ---------

XII. **QALY calculations**

Background health utilities were specified by age such that the number
of QALYs annually accrued for those with no Hepatitis A infection were
0.94 for those 0-4; 0.93 for those 5-17; 0.915 for those 18-34; 0.895
for those 35-44; 0.805 for those 45-54; 0.805 for those 55-64; 0.770 for
those 65-74; and 0.695 for those 75 and older^24^. Relative utility
values for infection states were set to 0.83 for symptomatic anicteric
infection, 0.41 for non-fulminant icteric, 0.42 for fulminant
pre-transplant, and 0.79 for fulminant post-transplant^25-27^.

To calculate an infected individual's health utility, the relative value
of their state was multiplied by their background utility. The QALY
input values are then converted to annualized values based on the
duration of sickness experienced in a given year (Equation 6).

> Equation 6.
>
> Annual loss (before discounting) = (HealthyU -
> (HealthyU\*SickU))\*(TimeSick/365.25)
>
> Where HealthyU = Utility value without Hepatitis A infection, by age
>
> SickU = Utility value for infection states
>
> TimeSick = Duration of sickness (days)

Table 12.1 summarizes the health utility values and their respective
sources used in the model.

+-------------+--------------------+-------+---+--------+--------------+
| Table 12.1  |                    |       |   |        |              |
| Quality of  |                    |       |   |        |              |
| Life        |                    |       |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    |       | P |        |              |
|             |                    |       | a |        |              |
|             |                    |       | r |        |              |
|             |                    |       | a |        |              |
|             |                    |       | m |        |              |
|             |                    |       | e |        |              |
|             |                    |       | t |        |              |
|             |                    |       | e |        |              |
|             |                    |       | r |        |              |
|             |                    |       | V |        |              |
|             |                    |       | a |        |              |
|             |                    |       | l |        |              |
|             |                    |       | u |        |              |
|             |                    |       | e |        |              |
+-------------+--------------------+-------+---+--------+--------------+
| Parameter   |                    |       | B |        | Source       |
|             |                    |       | a |        |              |
|             |                    |       | s |        |              |
|             |                    |       | e |        |              |
|             |                    |       | C |        |              |
|             |                    |       | a |        |              |
|             |                    |       | s |        |              |
|             |                    |       | e |        |              |
|             |                    |       |   |        |              |
|             |                    |       | A |        |              |
|             |                    |       | n |        |              |
|             |                    |       | a |        |              |
|             |                    |       | l |        |              |
|             |                    |       | y |        |              |
|             |                    |       | s |        |              |
|             |                    |       | i |        |              |
|             |                    |       | s |        |              |
+-------------+--------------------+-------+---+--------+--------------+
| Duration of | Symptomatic        | Ages  |   | 3      | Assumption   |
| sickness    | anicteric          | 0-95  |   |        |              |
|             | infection          |       |   |        |              |
| (days)      |                    |       |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             | Non-hospitalized   | Ages  |   | 34.4   | ^17^         |
|             | icteric infection  | 0-95  |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             | Hospitalization    | Ages  |   | 67.8   | ^17^         |
|             |                    | 0-95  |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
| QALY value  | Healthy            | Ages  |   | 0.94   | ^24^         |
|             |                    | 0-4   |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 5-17  |   | 0.93   | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 18-34 |   | 0.915  | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 35-44 |   | 0.895  | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 45-54 |   | 0.865  | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 55-64 |   | 0.805  | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 65-74 |   | 0.77   | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    | 75-95 |   | 0.695  | ^24^         |
+-------------+--------------------+-------+---+--------+--------------+
|             | Symptomatic        | Ages  |   | 0.83   | ^18^         |
|             | anicteric          | 0-95  |   |        |              |
|             | infection          |       |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             | Non-fulminant      | Ages  |   | 0.41   | ^17^         |
|             | icteric infection  | 0-95  |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             | Fulminant          | Ages  |   | 0.42   | ^27^         |
|             | pre-transplant     | 0-95  |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             | Fulminant          | Ages  |   | 0.79   | ^27^         |
|             | post-transplant    | 0-95  |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+
|             |                    |       |   |        |              |
+-------------+--------------------+-------+---+--------+--------------+

XIII. **Probabilistic Sensitivity Analysis**

The probabilistic sensitivity analysis (PSA) is an analysis in which the
model is run repeatedly (each run is called an "iteration"), each time
with different parameter values. Each parameter has an associated
statistical distribution with an expected value equal to the base value
and specified measures of variance. The model uses Excel's random number
generator to conduct random sampling from those distributions; those
sample values are used as parameter values for each iteration. Section
14a describes the statistical distribution shapes that are available in
the PSA. Section 14b describes the protocols that the model uses to
generate random values from each type of distribution.

In order to simplify the PSA specifications, the model has PSA
categories for specifying distributions for groups of parameters rather
than specifying those for each parameter individually. Each parameter is
assigned to a category; each parameter's associated distribution is
determined by its base value and its category's specifications. Those
categories are described in Section 14c.

**13a. Statistical distribution shapes.** The following distribution
shapes can be applied to vary parameter values in the PSA: logistic
normal, normal, triangular, or uniform. Table 14.1 lists the required
and other parameters (referred to as "distribution parameters" hereafter
to distinguish from the parameters that determine model progression)
that must or can, respectively, be specified for each model parameter.

Table 13.1 Statistical distributions and associated parameters

+----------+-------------------------------+---------------------------+
| Dist     | Required distribution         | Other distribution        |
| ribution | parameters                    | parameters                |
+----------+-------------------------------+---------------------------+
| Logistic | -   Mean (automatically set   | -   None                  |
| normal   |     to base value)            |                           |
|          |                               |                           |
|          | -   Implicit lower or upper   |                           |
|          |     bound (% less or % more   |                           |
|          |     than base, respectively)  |                           |
|          |                               |                           |
|          | -   Minimum sample value      |                           |
|          |     limit (% less than base)  |                           |
|          |                               |                           |
|          | -   Maximum sample value      |                           |
|          |     limit (% more than base   |                           |
|          |     or absolute value)        |                           |
+----------+-------------------------------+---------------------------+
| Normal   | -   Mean (automatically set   | -   Minimum sample value  |
|          |     to base value)            |     limit (% less than    |
|          |                               |     base)                 |
|          | -   95% confidence interval   |                           |
|          |     lower bound (% less than  | -   Maximum sample value  |
|          |     base)                     |     limit (% less than    |
|          |                               |     base or absolute)     |
|          | -   95% confidence interval   |                           |
|          |     upper bound (% more than  |                           |
|          |     base)                     |                           |
+----------+-------------------------------+---------------------------+
| Tr       | -   Mode (automatically       | -   Minimum sample value  |
| iangular |     calculated so that the    |     limit (% less than    |
|          |     mean equals base value)   |     base)                 |
|          |                               |                           |
|          | -   Minimum (% less than      | -   Maximum sample value  |
|          |     base)                     |     limit (% more than    |
|          |                               |     base or absolute)     |
|          | -   Maximum (% more than      |                           |
|          |     base)                     |                           |
+----------+-------------------------------+---------------------------+
| Uniform  | -   Mean (automatically set   | -   Minimum sample value  |
|          |     to base value)            |     limit (% less than    |
|          |                               |     base)                 |
|          | -   Minimum (% less than      |                           |
|          |     base)                     | -   Maximum sample value  |
|          |                               |     limit (% more than    |
|          | -   Maximum (% more than      |     base or absolute)     |
|          |     base)                     |                           |
+----------+-------------------------------+---------------------------+

The distribution parameters listed in Table 13.1 include the following:

Mean -- expected value of the distribution; not set by user, but
automatically set to the base value used in standard model runs

Implicit lower or upper bound (logistic normal only) -- the bound of the
95% confidence interval around the mean of a value with a logistic
normal distribution; based either on subjective estimates or objective
data; specified as a percentage difference from the base value

Minimum or maximum sample value limit -- lower or upper bound of the
range of possible values that this model parameter can take; values
derived from random sampling that fall outside this range are discarded
and distribution is resampled; specified as a percentage difference from
the base value for minimum or maximum or as an absolute maximum (e.g.,
probabilities can range from 0 to 1, which would be set as minimum =
100% less than the base value and absolute maximum = 1)

Minimum or maximum (triangular or uniform) -- lower / upper bound of
distribution; specified as a percentage difference from the base value

Mode (triangular only) -- value of the variable at the peak of the
distribution; not set by user, but automatically generated so that the
mean of the distribution is equal to the base value used in standard
model runs; calculated using the equation *Mode = 3 \* Mean --
(Minimum + Maximum)*

**13b. Sampling from distributions**

The methods for sampling from a distribution vary by the distribution
shape. Each method applied in the model are described below.

**13b.1 Normal**

Values are randomly selected from a normal distribution defined by a
mean and variance using the Polar Method, as outlined by Rich Timpone,
adapted from Sheldon Ross\' and Donald Knuth\'s books
([http://psweb.sbs.ohio-state.edu/faculty/rtimpone/computer_resources/
polar_vb5.htm](http://psweb.sbs.ohio-state.edu/faculty/rtimpone/computer_resources/ polar_vb5.htm)).^29^
Minimum and maximum sample value limits may also be applied.

1\. If variance is equal to zero, randomly generated value (*RandNorm*)
= *mean*.

Otherwise...

2\. Generate 2 random numbers, *Rand1* and *Rand2*, between 0 and 1.

3\. Calculate variables *V1*, *V2* and *S* using the following formulas:

*V1* = 2 \* *Rand1* - 1

*V2* = 2 \* *Rand2* - 1

*S* = *V1*^2^ + *V2*^2^

4\. If *S* ≥ 1 return to step 2 to get new values for *Rand1* and
*Rand2*.

Otherwise...

5\. If *S* \< 1, calculate randomly generated value using the following
formulas:

*Z* = randomly generated value from standard normal distribution

= (((-2 \* Log(*S*)) / *S*)^(1\ /\ 2)^) \* *V1*

*Standard deviation* = *variance*^(1\ /\ 2)^

*RandNorm* = m*ean* + *Z* \* *Standard deviation*

6\. If *RandNorm* is below a specified minimum or above a specified
maximum sample value limit, then return to to step 2 to get new values
for *Rand1* and *Rand2*.

**13b.2 Logistic normal**

Values are randomly selected from a logistic normal distribution defined
by a mean (*m*), lower or upper confidence interval bound, and minimum
and maximum sample value limits (*min* and *max*, respectively) using
the following method, as outlined by Doubilet and coauthors (1985).^28^
Minimum and maximum sample value limits may also be applied.

1\. If upper and lower bounds given, assign lower as bound (*b*)

2\. Check if *m* = *b* or *b* = *min* or *b* = *max* or *m* = 0. If any
of those conditions are true, then parameter will not be varied in
sensitivity analysis (user will be notified of error).

Otherwise...

3\. Calculate mean of associated normal distribution (*μ*)

> 3a. Calculate logit transforms of the mean (*LogitM*) and bound
> (*LogitB*)
>
> *LogitM* = Log((*m* -- *min*) / (*max* - *m*))
>
> *LogitB* = Log((*b* -- *min*) / (*max* - *b*))
>
> 3b. Assign value of *μ* if certain special conditions hold:
>
> If (*m* / *max*) = 0.5 Then
>
> *μ* = 0
>
> Otherwise if (*m* / *max*) = 0.025 Or *m* / *max* = 0.975 Then
>
> *μ* = (*LogitM*^2^ + *LogitB*^2^) / (2 \* *LogitB*)
>
> Otherwise...
>
> 3c. Calculate *μ* using the following formulas:
>
> *E* = 1.96 / Φ^-1^(*m* / max) (Φ^-1^ represents the inverse of the
> normal distribution function)
>
> *μ* = (*LogitB* - *E* \* Sqr(*LogitB*^2^ - *LogitM*^2^ + *LogitM*^2^
> \* *E*^2^)) / (1 - *E*^2^)

4\. Calculate standard deviation of associated normal distribution (*s*)
using the following formula:

> *s* = \| *μ* -- *LogitB*\| / 1.96

6\. Generate a random variable (*RandNorm*) from the normal distribution
defined by *μ* and *s* using the **Normal** protocol outlined above.

7\. Calculate randomly generated value from logistic-normal distribution
(*RandLgcnorm*) using the following formula:

*RandLgcnorm* = (*min* + *max* \* Exp(*RandNorm*)) / (1 +
Exp(*RandNorm*))

8\. If *RandLgcnorm* is below a specified minimum or above a specified
maximum sample value limit, then return to to step 6 to get a new value
for *RandNorm*.

**13b.3 Triangular**

Values are randomly selected from a triangular distribution defined by a
mean, minimum (*min*) and a maximum (*max*) using an inverse cumulate
distribution function, as described at
<http://www.brighton-webs.co.uk/distributions/triangular.asp>.^30^
Minimum and maximum sample value limits may also be applied.

1\. Calculate the mode of the distribution using the following formula:

*mode* = 3 \* *mean* - (*min* + *max*)

2\. Generate a random number (*Rand*) between 0 and 1.

3\. If *Rand* = (*mode* - *min*) / (*max* - *mode*) then assign the
value from triangular distribution (*RandTriang*) to *mode*.

Otherwise...

4\. If *Rand* \< (*mode* - *min*) / (*max* - *mode*) then calculate
*RandTriang* using the following formula:

> *RandTriang* = *min* + (*Rand* \* (*max* - *min*) \* (*mode* -
> *min*))^(1/2)^

Otherwise...

5\. Calculate *RandTriang* using the following formula:

> *RandTriang* = *max* - ((1 - *Rand*) \* (*max* - *min*) \* (*max* -
> *mode*))^(1/2)^

6\. If *RandTriang* is below a specified minimum or above a specified
maximum sample value limit, then return to to step 2 to get a new value
for *Rand*.

**13b.4 Uniform**

Values are randomly selected from a uniform distribution defined by a
minimum (min) and maximum (max) using the following protocol. Minimum
and maximum sample value limits may also be applied.

1\. Generate a random number (Rand) between 0 and 1.

2\. Calculate a value from the uniform distribution (RandUnif) using the
following formula:

> RandUnif = min + Rand \* (max - min)

3\. If RandUnif is below a specified minimum or above a specified
maximum sample value limit, then return to to step 1 to get a new value
for Rand.

**13c. PSA categories.** In order to simplify the PSA specifications,
the model has PSA categories for specifying relative and absolute
characteristics of distributions for groups of parameters rather than
specifying those for each parameter individually. Each parameter is
assigned to a category; therefore each parameter's associated
distribution is determined by its base value and category's
specification. Table 14.2 lists the categories applied in the analysis.
Users can add categories and re-assign categories as desired.

Table 13.2. PSA parameter categories

  --------------------------------------------------------------------------------------------------
  **Category**   **Distribution   **CI lower    **CI upper    **Minimum   **Maximum   
                 shape**          bound         bound         limit on    limit on    
                                  (Logistic     (Logistic     sampled     sampled     
                                  normal or     normal or     value: %    value:**    
                                  normal) or    normal) or    less than               
                                  Minimum       Maximum       base**                  
                                  (Triangular   (Triangular                           
                                  or Uniform):  or Uniform):                          
                                  % less than   % more than                           
                                  base value**  base value**                          
  -------------- ---------------- ------------- ------------- ----------- ----------- --------------
                                                                          **% more    **absolute**
                                                                          than base** 

  Discounts      Triangular       33.3%         33%           100%        \--         1

  Do not vary    Do not vary      \--           \--           \--         \--         \--

  Known          Triangular       10%           10%           100%        \--         \--
  confidence                                                                          
  interval (CI)                                                                       

                                  (Applied only                                       
                                  if confidence                                       
                                  interval                                            
                                  bounds not                                          
                                  actually                                            
                                  specified)                                          

  Large          Logistic normal  50%           \--           100%        200%        \--
  healthcare                                                                          
  cost                                                                                

  Other          Logistic normal  10%           \--           100%        100%        \--

  Probability    Logistic normal  10%           \--           100%        \--         1

  Quality of     Uniform          10%           10%           100%        \--         1
  life                                                                                

  Small          Logistic normal  50%           \--           100%        100%        \--
  healthcare                                                                          
  cost                                                                                

  Wages          Logistic normal  20%           20%           100%        100%        \--
  --------------------------------------------------------------------------------------------------

Each category is defined by up to 6 specifications, as shown in Table
14.2. Those specifications are defined as follows:

Distribution shape -- Defines the shape of the distribution. The
following distribution shapes can be applied: logistic normal, normal,
triangular, or uniform (as described in Section 14b.1). Categories can
also be specified here to not be varied (by using "Do not vary").

CI lower bound (Logistic normal or normal) or Minimum (Triangular or
Uniform) -- States a defining lower bound, either of a confidence
interval (in the case of logistic normal or normal) or the distribution
itself (triangular or uniform), relative to the base value. For example,
we estimate that the lower bounds of the 95% confidence intervals around
small healthcare costs are about 50% less than the small healthcare
costs used in the model. If a cost was \$50, then the lower bound of the
confidence interval around that cost would be \$25.

CI upper bound (Logistic normal or normal) or Maximum (Triangular or
Uniform) -- Similarly to the CI lower bound, the CI upper bound states a
defining upper bound, either of a confidence interval (in the case of
logistic normal or normal) or the distribution itself (triangular or
uniform).

Minimum limit on sampled value -- Defines the lower limit of the range
of values that a parameter can reasonably be assigned, relative to the
base value. If the random selection process chooses a value less than
this minimum limit, then the random selection process is repeated until
a value within the reasonable range is selected. This limit prevents the
model from using negative or other nonsensical values from being applied
to costs and other parameters.

Maximum limit on sampled value -- Defines the upper limit of the range
of values that a parameter can reasonably be assigned, either on an
absolute scale or relative to the base value. If the random selection
process chooses a value more than this maximum limit, then the random
selection process is repeated until a value within the reasonable range
is selected. This limit prevents the model from using nonsensical values
from being applied to costs and other parameters, such as
quality-of-life and probability values greater than 1.

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