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Did
the
Transmission Potential
Did Modeling
Overestimate
the Transmission
Potential
Modeling Overestimate
of
(H1N1-2009)?
for
of Pandemic
Pandemic
Size Estimation
Estimation
for
(H1N1-2009)? Sample
Sample Size
Post-Epidemic
Studies
Post-Epidemic Seroepidemiological
Seroepidemiological Studies
1,2,3
4,5
4,6
Hiroshi
Nishiura
*, Gerardo
Chowell
Castillo-Chavez
Hiroshi
Gerardo
Carlos
Castillo-Chavez4'6
Nishiura1'2'3*,
ChoweII4'5,, Carlos
11 PRESTO,
Japan, 2
Theoretical Epidemiology,
The Netherlands,
of Public
Science and
and Technology
2TheoreticaI
of Utrecht,
3 School
School of
Public Health,
PRESTO, Japan
Saitama, Japan,
Utrecht, Utrecht,
Utrecht, The
Netherlands, 3
Health,
Japan Science
Technology Agency,
Agency, Saitama,
Epidemiology, University
University of
The
Mathematical and
Center, School
School of
Evolution
and Social
Social Change,
Change, Arizona
Arizona
The University
of Hong
4Mathematical
and Computational
Sciences
of Human
Human
Evolution
and
China, 4
Center,
University of
Hong Kong,
Kong, Hong
Hong Kong,
Kong, China,
Computational Modeling
Modeling Sciences
State University,
United States
States of
of America,
5 Fogarty
International
National
Institutes
of
United States
State
Center,
Institutes
of Health,
States of
of America,
America,
Arizona, United
America, 5
Center, National
Health, Bethesda,
Bethesda, Maryland,
University, Tempe,
Tempe, Arizona,
Fogarty International
Maryland, United
6
Santa Fe
6Santa
Fe Institute,
Santa Fe,
New Mexico,
United States
States of
of America
America
Institute, Santa
Fe, New
Mexico, United
Abstract
Abstract
studies
before and
and after
after the
of H1N1-2009
þÿH IN I 2009 are
useful for
Background:
before
the epidemic
wave of
are useful
for estimating
eprdemrc wave
Background Seroepidemiological
Seroepidemiological studies
estimating
attack rates
rates
with a
a potential
validate
estimates
of the
number
R
rn modeling
population
with
to validate
early
of
the reproduction
R, in
studies.
population attack
potential to
reproduction number,
early estimates
modeling studies
Methodology/Principal
final epidemic
the proportion
of
in
a population
who become
Since the
the final
size
the
of individuals
rndrvrduals
rn a
become
eprdemrc size,
proportion
population who
Methodology/Prlnclpal Findings:
Flndlngs Since
infected during
an epidemic,
rs not
not the
the result
result of
of a
a binomial
brnomral sampling
because infection
rnfectron events
events
are
not independent
infected
are
not
eprdemrc is
process because
independent
during an
sampling process
of each
each other,
other we
we
the use
use
of an
an asymptotic
drstrrbutron
of the
size
to
95% confidence
of
propose
the
of
of
the final
final size
to compute
confidence
propose
compute approximate
approximate 95%
asymptotic distribution
intervals of
of the
the observed
observed
final
size
This
allows the
of the
observed
final
sizes
based on
intervals
final size.
This allows
the comparison
of
the observed
final sizes
against predictions
on the
the
comparison
predictions based
against
þÿþÿ
I I5 1.40
þÿ40
I
and 1.90),
þÿ90)
I
which
also yields
modeling
and
which also
formulae for
for determining
sample sizes
sizes for
for future
future
(R = 1.15,
simple formulae
modeling study
study (R
yields simple
determining sample
studies. We
We examine
examine
of eleven
eleven published
of H1N1-2009
þÿHN1-2009
I
that
seroepidemiological
aa total
total of
studies of
that took
took
published seroepidemiological
seroepidemiological studies.
seroepidemiological studies
after observing
the peak
incidence
rn a
a number
number
of countries.
countries
Observed
rn srx
place
in
of
Observed
seropositive
six studies
studies appear
appear
place after
peak incidence
seropositive proportions
proportions in
observing the
to be
be smaller
smaller than
than that
þÿ I40 four
of the
srx
studies sampled
serum
less than
month
after
to
that predicted
from R = 1.40;
four of
the six
studies
sampled serum
less
than one
one month
after the
the
predicted from
incidence
The comparison
of the
observed
final
sizes
þÿþÿ
I I5and
and 1.90
þÿ90
I reveals
reveals that
all eleven
eleven studies
reported
The
of
the observed
final sizes
against
that all
studies
reported peak
peak incidence.
comparison
against R = 1.15
not
to be
be significantly
the prediction
þÿ I I5 but
but final
sizes
rn
nine
studies
rndrcate
appear
not
to
from the
with R = 1.15,
final sizes
in
nine
studies indicate
appear
predrctron with
significantly deviating
devratrng from
overestrmatron
the value
value R = 1.90
þÿ90
I is
rs used.
used
overestimation
ifrf the
Conclusions
sizes
of published
studies
were
of model
model
Conclusions:
Sample
of
were too
too small
small toto assess
assess the
the valrdrty
validity of
Sample sizes
published seroepidemiological
seroeprdemrologrcal studies
þÿ90
I was
used We
We recommend
recommend
the
use
of the
rn determining
predictions
when R = 1.90
was used.
the use
of
the proposed
approach in
the sample
sample
predictions except
except when
proposed approach
determining the
size
of post-epidemic
studies
95% confidence
confidence
interval
of
size
of
calculating
the 95%
interval
of observed
observed final
final srze
size, and
and
post eprdemrc seroepidemiological
seroeprdemrologrcal studies,
calculating the
relevant
instead of
of the
use
of methods
methods
that
on
a
brnomral
conducting
hypothesis
the use
of
that rely
a binomial
proportion.
proportion
conducting relevant
hypothesis testing
testing instead
rely on
Cltatlon
Nrshrura H,
H Chowell
Chowell
G Castillo-Chavez
Castrllo Chavez C (2011)
Drd Modeling
Overestrmate
the Transmrssron
Potentral
of Pandemic
Pandemrc
Citation:
Nishiura
G,
the
Transmission Potential
of
(H1N1-2009)?
Sample Srze
Size
(2011) Did
(H1N1 2009)7 Sample
Modelrng Overestimate
Estrmatron
for Post-Epidemic
Post Eprdemrc Seroepidemiological
Studres
PLoS ONE
ONE 6(3):
e17908
dor 10 1371/journal pone 0017908
Estimation
for
PLoS
doi:10.1371/journal.pone.0017908
6(3) e17908.
Seroeprdemrologrcal Studies.
Editor
Alessandro
Indrana University
at Bloomington,
Unrted States
States of
of America
Amerrca
Editor:
Alessandro
Vespignani,
Vesprgnanr Indiana
Un|vers|ty at
Bloomrngton United
Received
December
14 2010;
2010 Accepted
15 2011;
2011 Published
Published
March 24,
24 2011
2011
Received
December
14,
March
Accepted February
February 15,
© 2011
2011 Nishiura
Nrshrura et
et al.
al This
Thrs is
rs an
an
artrcle distributed
drstrrbuted
under the
the terms
terms
of the
the Creatrve
Commons
Attrrbutron
Lrcense
whrch
Copyright:
ß
open-access
article
under
of
Creative Commons
Attribution License,
which permits
Copyright
open access
permrts
unrestrrcted
use
drstrrbutron
and reproduction
rn any
medrum
the original
author and
and source
source
are
credrted
unrestricted
use,
distribution,
and
provided
are
credited.
reproductron in
any medium,
provrded the
orrgrnal author
HN was
was
the Japan
Science and
and Technology
PRESTO program.
GC received
received financial
Liberal Arts
Funding:
supported
Japan Science
Technology Agency
GC
financial support
support from
from the
the College of
of the
the Liberal
Arts
Funding: HN
supported by the
Agency PRESTO
program.
and Sciences
Sciences of
of Arizona
Arizona
State University.
National Science
Science Foundation
Foundation
Grant DMS
DMS - 0502349),
U.S. Department
of Defense
Defense (NSA - Grant
and
State
(NSF - Grant
Grant H98230-06-1-0097),
0502349), U.S.
H98230-06-1-0097),
University. National
Department of
the Alfred
Alfred T. Sloan
Sloan Foundation
Foundation
and the
the Office
Office of
of the
the Provost
Provost
of Arizona
Arizona
State University
CCC's research.
research. The
had no
no
role in
in study
the
and
of
State
The funders
funders had
role
study design, data
data
University support
support CCC’s
collection
and analysis,
decision
to publish,
or preparation
of the
the manuscript.
collection
and
to
analysis, decision
publish, or
preparation of
manuscript.
-
-
-
Interests:
The authors
authors have
have declared
declared
that no
no
interests
exist.
Competing
The
that
competing
exist.
Competing Interests:
competing interests
E-mail: [email protected]
[email protected]
* E-mail:
*
the population attack
attack
rate
estimating
rate (i.e.
(i.e. infected
infected fraction
fraction of
of aa
estimating the
here also
also referred
as
the
referred toto as
the final
final size
size oror the
the
population) [4],
[4], here
of infected
infected individuals
in a
a population at
proportion of
individuals in
the end
end of
of anan
proportion
population at the
epidemic.
In addition,
population-wide seroepidemiological
seroepidemiological sursuraddition, population-Wide
epidemic. In
are
useful
for
in
realveys
are
useful
for
monitoring
epidemiological
dynamics
in
realveys
monitoring epidemiological dynamics
effectiveness
of
time,
of certain
certain interventions
interventions [5],
and
time, assessing
assessingeffectiveness
[5], and
of vaccination
determining prioritization
prioritization strategies
strategies of
vaccination during the
the
course
of an
course
of
an epidemic
epidemic (e.g.
(e.g. identifying
identifying subpopulations
subpopulations that
that should
should
be
at particular times
be vaccinated
vaccinated at
times during anan ongoing epidemic)
epidemic)
[6,7].
[6,7].
Both
and epidemiological
Both
serological
studies have
have
serological and
epidemiological modeling studies
increased
our
of the
the transmission
increased
our
understanding
transmission dynamics
of
understanding of
dynamics of
from
the
of the
the
H1N1-2009
H1N1-2009
from the
beginning of
pandemic [4,8].
In
[4,8]. In
the reproduction number,
the average
R, defined
defined asas the
average
number, R,
particular, the
number
of secondary
cases
number
of
generated by
by aa single
primary case
case
secondary cases
generated
single primary
Introduction
Introduction
Influenza
caused the
the first
first influenza
influenza
of
Influenza
A
(H1N1-2009) caused
pandemic of
A(H1N1-2009)
pandemic
the twenty-first century
A substantial
substantial
fraction
of the
the World
the
fraction
of
world
century [1]. A
has probably
been infected
infected already
with this
this virus,
population has
virus, but
but
probably been
already with
direct estimation
estimation
of the
the infected
infected fraction
of the
the population is not
not
aa direct
of
fraction of
feasible by
on
available
þÿ case data
data
feasible
available
epidemiological
by relying only
only on
epidemiological ‘case’
surveillance
data consisting
of confirmed
confirmed
cases
or
influenza(e.g.
data
cases
or
influenza(e.g. surveillance
consisting of
like illness
illness cases).
In particular, influenza
influenza
known
to
involve
like
is known
to
involve
cases). In
infections
and disease
disease severity
tends to
to be
selfasymptomatic
[2],
be selfasymptomatic infections
[2], and
severity tends
individuals
who
often do
do not
not
limiting
healthy individuals
who
often
require
limiting among
among
medical
attention.
due to
to the
the non-specific
nature
of
medical
attention.
Moreover,
of
Moreover, due
non-specific nature
influenza-like
illness
insufficient
to
confirm
or
symptoms,
influenza-like
illness
is insufficient
to
confirm
or
symptoms,
exclude
the diagnosis
of influenza
influenza
exclude
the
[3]. Therefore,
Therefore, seroepidemiodiagnosis of
seroepidemiostudies before
before and
and after
after an
an
Wave
are
crucial for
logical studies
epidemic
are
crucial
for
epidemic wave
PLoS ONE
ONE || www.plosone.org
PLoS
www.plosone.org
1
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
throughout
of
[9],
was estimated
using
its entire
entire course
course
of infection
infection
estimated
throughout its
[9], was
using
epidemiological
data
during
the
early
stages
of
the
pandemic.
data
the
of
the
One
epidemiological
during
early stages
pandemic. One
of
R is its
potential to
provide early
of the
the important
features of
of R
its potential
to provide
and
important features
early and
crude
predictions
crude
of the
the expected
final epidemic
size [10].
For
predictions of
expected final
epidemic size
[10]. For
instance,
estimate
for
is
the frequently
cited initial
initial
estimate
for H1N1-2009
H1N1-2009
instance, the
frequently cited
R
= 1.40 [8],
R=1.40
and the
the final
final size
size equation
of any
[8], and
equation of
any homogeneously
homogeneously
model
an
mixing
(with
initially
fully susceptible
population)
mixing model
(with an
initially fully
susceptible population)
that 51.1%
51.1%
of the
the population
would
infection
predicts
of
experience
predicts that
population would
experience infection
the end
end of
of the
the epidemic
next
several
by
section).
Nevertheless, several
by the
epidemic (see
(seenext
section).Nevertheless,
studies
have
that
the
infected
seroepidemiological
studies
have
suggested
that
the
infected
seroepidemiological
suggested
fraction
was
to be
be smaller
smaller
than 51.1%
51.1%
result that
that
fraction
was
likely
than
[11],
likely to
[11], aa result
has led
led researchers
researchers
to speculate
on
additional
has
to
additional
(often
speculate on
(often unforeseen)
unforeseen)
mechanisms
or
factors
the transmission
transmission
mechanisms
or
factors
influencing
dynamics.
influencing the
dynamics.
Hence,
play aa key
key role
validating
studies play
role in
in validating
Hence, seroepidemiological
seroepidemiological studies
crude
predictions
R.
the
crude
based on
on
R. Further,
whenever
the observed
observed
Further, whenever
predictions based
(sample)
than
based on
R, the
of
final size
size is smaller
smaller
than that
that based
on
the use
use
of
R,
(sample) final
seroepidemiological
provide indirect
evidence
of
studies may
indirect
evidence
of the
the
seroepidemiologicalstudies
may provide
positive
particular public
effect of
of particular
health interventions.
interventions.
positive effect
public health
A glance
at the
the literature
literature
shows that
that various
A
shows
various seroepidemiological
glance at
seroepidemiological
studies published
so far
far have
have adopted
binomial
studies
sampling
process
published so
adopted aa binomial
sampling process
to
quantify
uncertainty
‘proportion’
to
the
of the
the
of infected
infected
quantify the
uncertainty of
þÿ proportion of
individuals
the confidence
confidence
intervals
of
individuals
(e.g.
intervals
of
(e.g. [12,13]).
[12,13]). Accordingly,
Accordingly, the
the proportion
have also
also been
been derived
derived from
binomial
distribution
the
from aa binomial
distribution
proportion have
exact
or
one
of the
the
using
or
approximate
methods [6,14,15].
of
using exact
approximate methods
[6,14,15]. Perhaps
Perhaps one
main reasons
reasons
for widespread
use
of the
the binomial
in this
this
main
for
of
binomial proportion
proportion in
Widespreaduse
context
can
be attributed
attributed
to a
a well-known
well-known
and simple
context
can
be
to
and
formula for
for
simple formula
the sample
size determination
determination
of the
the binomial
binomial
the
of
proportion
sample size
proportion [16].
[16].
it should
should
be noted
noted that
that H1N1-2009
transmitted
Nevertheless,
be
is transmitted
H1N1-2009
Nevertheless, it
from human
human
to human,
and the
the risk
risk of
of infection
infection
in one
one
individual
from
to
in
individual
human, and
on
other individuals
individuals
in the
the same
same
unit. This
This
depends
other
in
population unit.
depends on
population
highlights
need to
account
for the
so-called ‘‘dependent
the need
to
account
for
the so-called
highlights the
þÿ dependent
happening’’
[17,18].
Moreover,
an
observed
final size
size represents
represents aa
an
observed
final
Moreover,
þÿhappening[17,18].
single
among all
all possible
possible sample
paths of
of the
the
stochastic realization
realization
single stochastic
among
sample paths
epidemic,
to explicitly
account for
for demographic
need to
epidemic, indicating
indicating aa need
explicitly account
demographic
stochasticity.
issues
for aa formal
formal framework
framework for
for
These
issues call
call
for
stochasticity. These
determining
post-epidemic seroepidemiological
seroepidemiological
the sample
size of
of post-epidemic
determining the
sample size
studies.
studies.
The
purpose of
present study
is toto introduce
an approximate
The purpose
of the
the present
introduce
an
study is
approximate
method
the computation
of the
method for
for the
computation of
the uncertainty
bound of
of the
the final
final
uncertainty bound
epidemic
which also
permits us
to
simple methods
methods for
for
also permits
us
to discuss
discuss simple
size, which
epidemic size,
sample
We reanalyze
reanalyze published
published datasets
datasets of
of postpostsize calculations.
calculations.
We
sample size
peak seroepidemiological
studies of
of H1N1-2009
H1N1-2009 and
explicitly test
test
and explicitly
peak
seroepidemiologicalstudies
if
R for
for H1N1-2009
indicated aa biased
biased estimate
estimate
if early
estimates of
of R
indicated
H1N1-2009
early estimates
of
final epidemic
of the
the final
size.
epidemic size.
Materials
and
Materials
and Methods
Methods
Seroepidemiological
data
Seroepidemiological data
As
to motivate
our
As aa way
way to
motivate our
study, we
we start
start by
by presenting
presenting summary
summary
study,
results of
of the
studies of
Table
results
the seroepidemiological
of H1N1-2009.
Table 11
H1N1-2009.
seroepidemiologicalstudies
summarizes
a total
total of
seroepidemiological studies
studies that
that were
were
summarizes
a
of eleven
eleven seroepidemiological
conducted
after
conducted
after
observing peak
peak incidence
incidence of
of H1N1-2009
H1N1-2009 in
in
observing
various
If the
the epidemic
various populations
populations [6,7,11–15,19–22].
curve
[6,7,11-15,19-22] If
epidemic curve
revealed
revealed
aa multimodal
multimodal distribution
distribution with
with clearly
clearly distinct
distinct peaks,
peaks, the
the
datasets can
can
either
post-peak datasets
either be
be after
after the
the first
first wave
wave (e.g.
(e.g. England
England
post-peak
our
interest to
to London
London
and
[14],
but we
we restrict
restrict our
interest
and the
the West
West Midland,
Midland,
[14] but
because
other areas
areas
were
or
after the
second wave
because other
were far
far less
less affected)
affected) or
after
the second
wave
The majority
of studies
studies sampled
(e.g.
USA [13]).
majority of
sampled serum
serum from
from
(e.g. USA
[13]). The
at clinics
clinics or
hospital
laboratory, registered
patients at
or blood
blood donors,
donors,
hospital laboratory,
registeredpatients
defined
cohort
except
for aa defined
cohort
population in
in Singapore
and aa
except for
population
Singapore [22]
[22] and
,
Table
1. Post-peak
studies of
of pandemic
influenza
Table
1.
(H1N1-2009)
(H1N1-2009) among
Post-peak seroepidemiological
pandemic influenza
population.
seroepidemiological studies
among aa general
general population.
location
Survey
Survey location
Subjects{
Subjects]
Sample
Sample size{
size]
before
Prop
Prop before
(%){
(%)i
Prop
after
Prop after
(%){
(%)i
Sampling
Sampling
period{
period]
After
After
Country
Country
peak1
peak§
Vac¥
Vac¥
Lab
Lab
method"
method1]
Australia
Australia
[19]
[19]
New South
South Wales
Wales
New
Clinical chemistry
Clinical
chemistry
laboratories
laboratories
1247
1247
12.8
12.8
28.6
28.6
09
Aug–Sep 09
Aug-Sep
Yes
Yes
No
No
H|240
HI$40
Canada [15]
Canada
[15]
British Columbia
Columbia
British
Patient
sen/ice
Patient
service
center
center
1127
1127
*7.5
*7.5
46.0
46.0
10
May
May 10
Yes
Yes
Yes
Yes
H|240
&
HI$40
&
MN232
MN$32
China (1) [11]
China
[11]
Beijing
Beijing
Blood donors
donors and
and
Blood
Patients
Patients
710
710
*7.5
*7.5
13.8
13.8
Nov-Dec
09
Nov–Dec
09
No
No
Yes
Yes
H|240
HI$40
China (2) [6]
China
[6]
Hong
Hong Kong
Kong
Blood donors,
Blood
donors,
cohort
pediatric
pediatric cohort
2913
2913
3.3
3.3
14.0
14.0
Nov-Dec
09
Nov–Dec
09
Yes
Yes
No
No
MN240
MN$40
Germany
[7]
Germany [7]
Frankfurt
Frankfurt
adults
Hospitalized
Hospitalized adults
225
225
*7.5
*7.5
12.0
12.0
Nov 09
Nov
09
No
No
No
No
H|240
HI$40
India [20]
India
[20]
Pune
Pune
School children
children
&
School
&
general
general population
population
5047
5047
0.9
0.9
15.5
15.5
09
Sep–Oct
Sep-Oct 09
No
No
No
No
H|240
HI$40
Japan
[21]
Japan [21]
entire Japan
entire
Japan
individuals
Healthy
Healthy individuals
6035
6035
7.6
7.6
40.3
40.3
10
Jul–Sep 10
Jul-Sep
Yes
Yes
Yes
Yes
H|240
HI$40
New Zealand
Zealand [12]
New
[12]
Auckland
Auckland
region
region
Registered
Registered patients
patients
1147
1147
11.9
11.9
30.3
30.3
Nov 09-Mar
Nov
09–Mar 10
Yes
Yes
Yes
Yes
H|240
HI$40
Singapore
[22]
Singapore [22]
Singapore
Singapore
Adult cohort
cohort
Adult
727
727
2.6
2.6
13.5
13.5
Oct 09
09
Oct
Yes
Yes
No
No
HI ($4
HI
(24
fold
fold rise)
UK [14]
UK
[14]
England
Patients
Patients
accessing
accessing
health care
care
health
275
275
14.5
14.5
22.5
22.5
09
Sep
Sep 09
No
No
No
No
H|232
HI$32
USA [13]
USA
[13]
Pittsburgh
Pittsburgh
Clinical laboratories
laboratories
Clinical
846
846
6.0
6.0
21.5
21.5
Nov 09
Nov
09
No
No
Yes
Yes
H|240
HI$40
{
size and
and sampling period refer
refer to
to those
those after
after observing
the peak incidence
incidence
of H1N1-2009.
H1N1-2009.
For several
several studies
Subjects, sample size
of
For
studies examining
the
[Subjects,
observing the
examining pre-existing
pre-existing immunity,
immunity, the
same
or additional
additional
before the
the 2009
2009 pandemic were
were
at different
different
time periods, but
but are
are
not included
included
in this
same
or
samples
investigated
time
not
in
this Table.
Table.
samples before
investigated at
iEstimated
before and
and after
after observing
an epidemic
was
in the
used itit as
Estimated proportions
When age-standardized estimate
estimate was
given
the original
we used
as the
the
proportions seropositive
seropositive before
observing an
epidemic peak. When
given in
original study, we
mean.
population
population mean.
*Three studies
studies did
did not
not estimate
estimate
the proportion
before the
the 2009
2009 pandemic, and
and we
we
assume
that 7.5%
7.5% of
immune
based on
*Three
the
assume
that
of the
the population
was initially
based
on aa
proportion seropositive
seropositive before
population was
initially immune
{
crude average
other studies.
studies.
crude
average among
among other
1
§After
column
if the
the sampling
took place longer than
month
after observing
the highest incidence
incidence
of cases.
cases.
After peak column
represents
than 1 month
after
of
represents if
sampling took
observing the
¥
¥Vaccination
column
if a
a population-wide
vaccination
of H1N1-2009
H1N1-2009
took place prior
Vaccination column
represents
campaign
took
to the
the sampling.
represents if
population-wide vaccination
campaign of
prior to
"
11
methods
to determine
determine
inbibition
and MN,
microneutralization
Laboratory methods
to
seropositivity;
assay
assay.
MN, microneutralization
seropositivity; HI, hemagglutination
hemagglutination inbibition
assay and
assay.
doi:10.1371/journal.pone.0017908.t001
doi:10.1371/journal.pone.0017908.t001
PLoS ONE
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22
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
sample
of
Japanese population
population
of study
volunteers
of the
the general
sample of
study volunteers
general _]apanese
[21].
Only
the
Japanese
study
has
not
been
published
the
has
not
been
in English;
[21]. Only
_]apanesestudy
published in
English;
the
based
National
Epidemiological
of
the data
data are
are
based on
on
National
Surveillance
of
Epidemiological Surveillance
Vaccine-Preventable
Diseases
which are
annually
to
Vaccine-Preventable
Diseases which
are
conducted
to
annually conducted
understand
the
epidemiological
number
of
understand
the
of a
a
number
of
epidemiological dynamics
dynamics of
infectious
diseases,
infectious
at least
least 5,400
diseases,involving
5,400 non-randomly
involving at
non-randomly sampled
sampled
individuals
across
all
in
individuals
across
all age-groups
in each
each year
and covering
24
age-groups
year and
covering 24
prefectures
per prefecture)
prefecture) among
aa total
individuals
total of
of 49
49
prefectures (225
(225 individuals
per
among
prefectures
Japan. Other
published serological
Other published
were
prefecturesacross
across_]apan.
serologicalsurveys
surveys were
not
included
in
because they
were
conducted
before
the
not
included
in Table
Table 1,
were
conducted
before
the
1, because
they
observed
or
because
on
confined
observed
epidemic
because they
focused on
aa confined
epidemic peak
peak or
they focused
healthcare
workers
or
population
workers or
military personnel)
population (e.g.
(e.g. healthcare
military
personnel) [5,23–
[5,23but a
a few
few of
of them
them have
have been
discussed elsewhere
elsewhere
27],
been discussed
[4].
27], but
The sample
size of
of the
the eleven
eleven seroepidemiological
The
which
studies, which
sample size
seroepidemiologicalstudies,
recorded
to
6035
recorded
post-peak
from 225
6035
225 to
post-peak seroprevalence,
seroprevalence, ranged
ranged from
individuals.
studies examined
examined
the first
individuals.
Eight
seroprevalence
before the
first
Eight studies
seroprevalencebefore
the proportion
of the
the population
wave,
estimating
with preprewave,
estimating the
proportion of
population with
VVhere indicated,
the sample
size
existing
indicated, the
existing immunity
immunity (Table
(Table 1).
1). Where
sample size
estimation
of those
those studies
studies relied
relied on
on
estimation
of
aa binomial
binomial proportion
proportion [12–
[1214,19].
The post-peak
varied substantially
with,
14,19]. The
post-peak sampling
sampling period
period varied
substantially with,
for
six
studies
sampling
the
post-peak
serum
more
than
for example,
six
studies
the
serum
more
than 1
1
example,
sampling
post-peak
month
after
Five
month
after the
the peak
incidence.
Five studies
studies clearly
stated that
that a
a
peak incidence.
clearly stated
vaccination
H1N 1-2009 had
had
population-wide
campaign
population-wide vaccination
campaign against
against H1N1-2009
taken place
to sampling.
The laboratory
in
taken
laboratory method
method employed
place prior
prior to
sampling. The
employed in
these studies
studies was
was
based on
on
inhibition
these
based
hemagglutination
assays
hemagglutination inhibition
assays (HI)
or
microneutralization
with eight
studies setting
the
or
microneutralization
assays
assays (MN) with
eight studies
setting the
threshold
level at
at HI$40.
HI240.
It is practically
difficult
seropositive
level
It
practically very
very difficult
seropositivethreshold
to determine
determine
the end
end of
of an
an
and thus,
we
the
to
the
epidemic,
regard
thus, we
epidemic, and
regard the
observed
increase in
in seroprevalence
after
the
observed
increase
(i.e.
seroprevalence
after
the
seroprevalence (i.e. seroprevalence
minus that
that before
before the
the peak)
as an
an
estimate of
of the
the fraction
fraction
of
peak
estimate
of
peak minus
peak) as
infected
individuals
the epidemic.
We
used the
the ageinfected
individuals
during
used
during the
epidemic. We
agestandardized
final size
size estimate
estimate
for an
an
entire
standardized
final
for
entire
population
when
population when
in the
the original
instead of
of using
crude estimates
estimates of
of the
the
given
given in
original study
study instead
using crude
fraction.
The 2009
involved
health
seropositive
The
pandemic involved
public health
2009 pandemic
seropositivefraction.
public
transmission
and spatial
interventions,
(e.g.
interventions, heterogeneous
heterogeneous transmission
(e.g. age
age and
spatial
and seasonality,
as the
the first
to stimulate
stimulate
heterogeneities)
first step
aa
but, as
heterogeneities)and
seasonality,but,
step to
relevant
discussion
on
this subject,
the
a
relevant
discussion
on
this
the
present
study
adopts
subject,
present study adopts a
without
homogeneously
time-dependent
homogeneously mixing
mixing assumption
assumption without
time-dependent dydynamics.
we
focus on
on
the difference
difference
between
the
namics.
Specifically,
focus
the
between the
Specifically, we
observed
final sizes
sizes for
for an
an
entire population
and the
the predictions
of
observed
final
entire
population and
predictions of
final size
size yielded
the modeling
the data
data in
in
final
by the
Thus, the
yielded by
modeling approach.
approach. Thus,
Table 1
1 are
are
here under
under the
the assumption
of a
a well-mixed
Table
analyzed
well-mixed
analyzed here
assumption of
It should
should
be noted
noted that,
in the
the absence
absence of
of any
timepopulation.
be
that, in
population. It
any timethe final
size is
is known
known
to depend
on
the
dependent
final size
to
the
dependent factors, the
depend only
only on
number
under the
the homogeneous
reproduction number
R, under
mixing
assumphomogeneous
assumption [9,10].
tion
[9,10].
the earliest
earliest studies
studies in
in Mexico
the estimation
estimation
of
Following the
Mexico [8,28],
of
[8,28], the
R was
was
conducted
the
data
in
different
R
conducted
using
the
early
epidemic
growth
data
in
different
using
early epidemic growth
locations
across
the world
world
estimates
in 2009
locations
across
the
(yielding
published estimates
in
2009
(yielding published
some
reassessed
The
estimated
in different
different
[29–38],
reassessed
[39]).
estimated
R,
R, in
[29-38], some
[39]). The
and subpopulations,
þÿ less than
than 1’’
þÿ1
epidemic
from ‘‘less
epidemic settings
settingsand
subpopulations, ranged
ranged from
to greater
than 2 [28,29,35].
The definition
definition
of R
also varied
[40]
of
R also
varied
[40] to
greater than
[28,29,35]. The
from study to
to study. One
One study, for
for example, incorporated the
the
from
of seasonal
seasonal variations
variations
in the
the force
of infection
infection
impact of
in
force of
[33].
[33]. Among
the earliest
earliest estimate
estimate of
of R
was
derived from
from the
the early
of
these,
R was
derived
these, the
early phase
phase of
the pandemic
the Spring
in Mexico
Mexico
the
using
various
2009 in
pandemic during the
Spring 2009
using various
methods
the posterior
modeling methods
[8]. Using aa Bayesian
posterior
method, the
Bayesian method,
median of
of R
R (and
the 95%
95% credible
credible intervals)
estimated at
at 1.40
1.40
median
was estimated
(and the
intervals)was
Since
the
median
(1.15,
1.90)
[8].
Since
the
posterior
median
crudely
represents
(1.15, 1.90)
crudely represents
of estimates
estimates in
in other
other published
and because
because the
the
mid-point
studies, and
mid-point of
published studies,
lower and
and upper
bounds
to the
the range
of R
R in
in
lower
roughly correspond to
upper bounds
range of
other studies
studies (with R,2),
we
focus on
on
an
estimate
of R
derived
other
focus
an
estimate
of
R derived
R<2), we
from an
an
of cases
cases
in an
an
outbreak
in La
La Gloria,
from
exponential
in
outbreak
in
Gloria,
exponential growth of
Mexico.
we
not only
assess
the prediction based
on
R
Mexico.
Thus,
not
the
based on
R = 1.40,
Thus, we
1.40,
only assess
but also
but
also
on
the lower
lower and
bounds of
of R.
R. Note
Note that
that the
lower
on
the
and upper
the lower
upper bounds
(1.15)
is
smaller
than
the
posterior
median
of
R
obtained
is
smaller
than
the
median
of
R
obtained
(1.15)
posterior
using
methods in
study
coalescent
other
methods
in the
the same
same
using other
study including
including aa coalescent
population genetic
analysis (R:
(R = 1.22).
estimate of
of RR for
for
Given an
an
estimate
population
genetic analysis
1.22). Given
an
initially
fully susceptible
population, and
and assuming
assuming that
that the
the
an
initially fully
susceptible population,
initial
number
of
is sufficiently
smaller than
than the
the total
total
initial
number
of infectives
infectives
sufficiently smaller
population
size,
the
final
epidemic
size
r
satisfies
the
final
size
satisfies
p
population size,
epidemic
bound
bound
^ r),
1{r~exp({
R
1-p=<=XP(-Rn),
which
referred
which is referred
to
the final
to asas the
final size
size equation
Both sides
sides of
of
equation [10].
[10]. Both
the probability
that an
individual
equation
represent the
probability that
an individual
escapes
equation (1)
(1) represent
escapes
infection
throughout
of anan epidemic.
epidemic. Since
presence
infection
the course
course
of
Since the
the presence
throughout the
of
pre-existing immunity
has yet
yet toto be
be clarified
clarified atat the
beginning of
of
of pre-existing
the beginning
immunity has
the 2009
use
to
calculate
the
the
pandemic, we
we use
equation
(1)
to
calculate
the
predicted
2009 pandemic,
equation (1)
predicted
final
size. Iteratively
1.40
final epidemic
solving (1)
for RR being
being 1.15,
1.15, 1.40
epidemic size.
Iteratively solving
(1) for
and
final size
size r
and 76.7%,
respectively.
and 1.90,
the final
51.1% and
1.90, the
24.9%, 51.1%
76.7%, respectively.
p is 24.9%,
We test
these
forecasts against
against the
observed final
final sizes
sizes given
given in
in
We
test
these forecasts
the observed
Table
1. For
For this
this reason,
itit is
Table
1.
is essential
essential toto compute
uncertainty
reason,
compute uncertainty
bounds
95% confidence
confidence
bounds (e.g.
interval)
of the
the observed
observed final
final sizes
sizes in
in
(e.g. 95%
interval) of
seroepidemiological
studies.
seroepidemiologicalstudies.
bound for
binomial
Uncertainty
for aa binomial
proportion
proportion
Uncertainty bound
As
As
to discussing
aa prelude
prelude to
discussing the
the uncertainty
uncertainty bound
bound of
of final
final size,
size, wewe
a
a binomial
binomial proportion,
proportion,
which
has been
used in
in published
which has
been widely
widely used
published seroepidemiological
seroepidemiological
studies shown
shown in
in Table
Table 1.
1. Let
Let X
studies
X be
be aa binomial
binomial random
random variable
variable for
for
size n,
and let
=X/X/n
n be
sample
let r
be the
the sample
sample proportion
proportion positive.
positive.
n, and
p =
sample size
The most
confidence
interval
of
The
most well-knovvn,
well-known, parsimonious,
parsimonious, confidence
interval
of the
the
binomial
binomial proportion,
proportion, employs
employs aa normal
normal approximation
approximation toto
binomial
also referred
binomial distribution,
which isis also
referred toto asas the
the Wald
Wald
distribution, which
confidence
interval.
The
confidence
interval.
The 100(1-2a)%
confidence interval
interval for
for the
the
100(1-2oc)%confidence
as
sample
proportion r
written as
p is written
sample proportion
first
consider
the confidence
confidence
interval
of
first consider
the
interval of
^+za
r
biz.
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
^ (1{^
r
r)
Q nn ,
ð2Þ
<2>
where
denotes 1-a
1-oc quantile
of the
the standard
standard
normal
distribution
where zzua denotes
normal
distribution
quantile of
=
þÿ rules of
of thumb’’
þÿthumb suggest
for aoc =
0.025).
The ‘‘rules
suggest that
that
(e.g.
-1.96 for
a<21.96
(e.g. zþÿzo,
0.025). The
the normal
normal
as
the
approximation
works well
well as
long asas np>5
nr.5 and
and n(1n(1approximation works
long
the rules
rules of
of thumb
thumb
do
not
r).5,
but the
do not
always work
work out
out well
well [41].
p)>5, but
always
[41].
The
score
interval
The
computation of
of the
the Wilson
Wilson score
interval isis aa better
better
not
difficult
and yields
alternative,
which isis not
computationally difficult
and
yields better
better
alternative, which
computationally
of associated
associated uncertainty
coverage
of
Here, we
we focus
focus onon the
the
coverage
uncertainty [42,43].
[42,43]. Here,
Wald
confidence
interval
in the
the present
Wald confidence
interval
in
present study,
study, because
because we
we extend
extend
its principle
to the
the computation of
95% confidence
its
principle to
of the
the 95%
confidence interval
interval of
of
the final
size.
the
final epidemic
epidemic size.
The
idea behind
the Wald
The
idea
behind the
Wald confidence
confidence interval
interval comes
comes from
from
the Wald
test
for
that the
inverting the
Wald test
for r.
the null
null hypothesis
hypothesis
p. Suppose
Suppose that
is tested
to
detect
relevant
H
tested where
where one
one wishes
wishes to
detect aa relevant
0:r = r
Hozp
po0 is
alternative
where
the
value
of
alternative
H
:r?r
,
where
r
is
the
proposed
value
of the
the
Hlzpaépo,
po
1
0
0
In the
the case
case
of the
the prediction with
proportion. In
of
with R:
R = 1.40, po
r0 might
proportion.
be
set at
at 0.511
0.511
that the
be set
(assuming
the final
final size
size follows
follows aa binomial
binomial
(assuming that
The
Wald
statistic
to
distribution). The
Wald statistic
to be
be compared
normal
compared toto aa normal
distribution
distribution
is given
given by
by
=
^{r0
r
5-90
,
s:e:(^
r)
s.e.(fJ)
þÿ
ð3Þ
(3)
standard
error
of r,
s.e.(^
r) is the
the standard
error of
by the
the square
square
p, approximated by
s.e.(f))
root
term
in
root
term
in (2).
The sample
size estimation
estimation
of
The
of aa binomial
binomial proportion
proportion can
can also
also
sample size
if we
m
denote
of error,
a
employ
In fact, if
we let
let m
denote the
the margin of
a
error,
employ (3). In
where
where
=
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ð1Þ
(1)
3
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
summary
of
that
which
of sampling
error
that quantifies
summary
sampling error
quantifies uncertainty,
uncertainty, which
corresponds
to
half
the
width
of
a
confidence
interval
for
the
to
half
the
width
of
a
confidence
interval
for
the
corresponds
proportion
margin of
of
more
than
then a
a desired
desired margin
of error
error
of no
no
more
than m
m
p, then
proportion r,
means
means
of
in that
population satisfies
satisfies ([10]):
of cases
cases
in
that population
([10]):
r
{
-ln ln 1{
1{q
:
R~
r
R=
ð4Þ
(4)
r)ƒm:
zz,s.e.(fJ)
Sm.
a s:e:(^
^(1{^
1 r)
r
A
A
za @
zi
nn
_
2
ƒm
S m2:
_
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
22
u
u
s
r
2 2
ur33(1(1{r)z
r(1{r)
ln
1{
(1- )2ln2
u
m
1{q
s:e:(r)~u
:
2
S_e_(p)=
u
t
r
N
N rz(1{r)
p+(1-p)ln ln 1{
1{q
ð5Þ
(5)
)+(5)
Solving
for n
n gives
Solving equation
equation (5)
(5) for
gives
z 2
n§
"2
a
^ (1{^
r
r),
(§)2i>(1-/3),
m
ð6Þ
<6>
ð10Þ
(10)
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2
^ (1{^
r
r)zzb (^
rzm)(1{^
r{m)
:
m
2_ ð7Þ
(7)
za/f»<1-f»>+z,fm/<f»+m><1-f»-m>
za
and (7),
it is seen
seen
that the
the sample
size n
n based
on
Comparing
that
based on
Comparing (6)
(6) and
(7), it
sample size
to
the case
case
for a
a
of 50%
50%
in (7)
(6)
the
for
power
of
in
(6) corresponds
corresponds to
power
(7) (i.e.
(i.e.
zZ5b = z50.5
0.5 = 0).
0)=
1-L)
1-EIN
Given
of the
the generation
generation time
time are
are now
now known
known for
for
Given that
that q
and the
the CV
CV of
q and
the Wald
interval
can
H1N1-2009,
Wald confidence
confidence interval
can
employ (10)
(10) for
for
H1N1-2009, the
employ
computing
corresponding
confidence interval,
interval, for
for
the
95%
confidence
computing the
corresponding 95%
and for
the minimum
hypothesis
for estimating
estimating the
minimum sample
sample size
size
hypothesis testing
testing and
One should
required
for post-epidemicseroepidemiological
post-epidemic seroepidemiological studies.
studies. One
should
required for
bear in
mind that
estimate isis nevertheless
nevertheless conservative
conservative (i.e.
(i.e.
bear
in mind
that the
the error
error
estimate
to be
the method
likely to
be underestimated),
underestimated), because
because (i) the
method isis based
based onon
likely
normal
normal
approximation,
we ignore
time-dependent dynamics
dynamics
approximation, (ii)
(ii) we
ignore time-dependent
health interventions,
including
public health
and (iii)
we ignore
ignore heterogeheterogeinterventions, and
including public
(iii) we
neous
transmission
neous
transmission
(see Discussion
Discussion for
for (ii)
(ii) and
and (iii)).
(iii)). .N
N isis the
the
(see
size in
in the
the above
above expressions.
to replace
population size
we wish
wish to
replace .N
N by
by
population
expressions.IfIf we
size n,
the binomial
of nn has
has to
sample
n, the
binomial sampling
error of
to be
be accounted
accounted
sample size
sampling error
in the
the calculation
calculation
of the
the variance.
in
of
variance. In
In the
the case
case of
of simple
simple random
random
the resulting
error
is
the sum
sampling,
resulting standard
standard error
is given
by the
sum of
of the
the
sampling, the
given by
of two
two
respective
variance of
independent processes,
processes, i.e.
i.e.
respectivevariance
independent
well-known
formula
for estimating
the minimum
minimum
size n
n
aa well-known
formula for
sample
estimating the
sample size
for a
a
binomial
Since the
the eventual
eventual
unknown
for
binomial
proportion.
r
p is unknown
proportion. Since
before
the
survey,
one
may
set
r
= 0.511 or
use
aa
before
the actual
actual
one
set
or
use
p=0.5ll
survey,
may
estimate.
It should
should
be noted
noted
that
published
It
be
that
published seroprevalence
seroprevalence estimate.
does not
not
account
for Type
II error
error
equation
explicitly
for
(i.e.
equation (6)
(6) does
explicitly account
Type II
(i.e.
of the
the test)
to
the power
in
power
of
incorporate
power in
Hence, to
power
test) [44].
[44]. Hence,
incorporate the
the sample
one
can
the
calculating
can
alternatively
size, one
calculating the
sample size,
alternatively employ
employ the
formula
following
([45]):
following formula
([45]):
n§
"2
ð9Þ
(9)
The estimated
estimated
R
l.l5
to
l.90 in
The
R (e.g.
(e.g. in
in the
the range
of 1.15
to 1.90
in Mexico)
Mexico) isis not
not
range of
in aa fully
fully susceptible
population,
the
basic reproduction
number R0
R0 in
the basic
reproduction number
susceptiblepopulation,
but satisfies
R0 = R/(1-q) [9]. Using
Using the
the estimator
estimator of
R in
the
but
satisfies R0=R/(1-q)
of R
in (9),
(9), the
standard
error
in
can
be
as
standard
error
in (8)
be rewritten
rewritten as
(8) can
By
error,
both sides
sides and
and using
the approximate
standard
error,
By squaring
squaring both
using the
approximate standard
we
have
we
have
2
1-%
:
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n(N{n)
zs:e:2 (r; n)
N3
+s.e.2(p;n)
bound for
for a
a final
final epidemic
size
Uncertainty
epidemic size
Uncertainty bound
An explicit
derivation
of final
size distribution,
An
of
final size
which employs
distribution, which
explicit derivation
employs aa
recursive
has been
been carried
carried
out
the so-called
so-called
recursive
equation,
out
through
equation, has
through the
Sellke construction
construction
in a
a
series of
of stochastic
stochastic
Sellke
in
series
epidemic
modeling
epidemic
studies [46,47].
In addition,
number
of stochastic
stochastic
studies
of
modeling
addition, aa number
[46,47]. In
modeling
studies in
in the
the context
context
of large
have examined
examined
the
studies
of
the
large populations
populations have
distribution
of the
the final
final epidemic
size via
the central
central
asymptotic
of
via the
asymptotic distribution
epidemic size
limit theorem
theorem
Within
stochastic modeling
it
limit
[48,49].
aa stochastic
modeling framework,
framework, it
[48,49]. Within
known
that an
an
outbreak
declines to
to extinction
extinction
without
is known
that
outbreak
declines
without
causing
causing aa
with a
a probability of
extinction
outbreaks
large
large epidemic
epidemic with
probability of extinction
pp (small
(small outbreaks
are
referred to
to as
as minor
minor epidemic).
occurs
with
are
referred
with
epidemic). AA major epidemic
epidemic occurs
An
standard
error
of
the
final
size
of
probability 1-p.
An
approximate
standard
error
of
the
final
size
of
l-lb.
the major
based on
on
the asymptotic
result of
of
the
the
result
major epidemic
epidemic based
asymptotic convergence
convergence
the final
final size
size distribution
distribution
the
is ([50,51]):
([50,51]):
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
2
u
2
2
ur(1{r)zR2 s0 r(1{r)
pu-p>
u
m
t
,
s:e:(r)~
S-ff-(P):
» N ½1{R(1{r)2
p(1-p)+R2(;)
an
approximate
variance of
of the
the binomial
binomial
n(N-n)/N3 is an
approximate variance
the standard
sampling
error, and
and s.e.(r;n)
is the
standard error
error of
of final
final size
size when
when
sampling error,
s.e.(p;n)is
the sampling
error
linked
the
linked toto nn isis ignored(i.e.
ignored(i.e. what
what we
we replace
replace .N
N by
by nn
sampling error
in equation
The introduction
introduction
of sampling
error
also
in
of
also applies
applies toto
equation (10)).
(l0)). The
sampling error
the standard
standard
error
of the
the
error
of
the binomial
binomial proportion
proportion in
in (2),
(2), but
but this
this term
term is
is
then negligibly
negligibly
usually
for very
very large
large .N
N (because
n(N-n)/N3 is then
usually ignored for
(becausen(.N-n)/_/V3
under an
an
small)
assumption that
that the
the randomly
selected individuals
individuals
small) under
assumption
randomly selected
the entire
entire
sufficiently
population. Thus,
we use
use only
only
Thus, we
sufficiently represent
represent the
in the
the following
involves non-negligible
s.e.(r;n)
following analyses.
fraction
s.e.(p;n)in
analyses.IfIf nn involves
non-negligiblefraction
of .N
one
use
the above
above expression
of
N (e.g.
.5%), one
may use
the
expression (l(11)
introduce
(e.g.>5%),
may
l) oror introduce
the so-called
so-called
finite
correction
factor
the
finite population correction
factor (FPC)
(FPC) for
for the
the
calculation
of the
the error
calculation
of
error [52].
[52].
Given
an
observed
final
% confidence
Given
an
observed
final size
size r,
the 100(1-2a)%
confidence
p, the
100(1-Qoc)
interval
for
calculated
as
interval
for r
as
p is calculated
where
where
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
22
u
u
^
s
r
2 2
3
ui>3(1-i>)+
^ (1{^
^(1{^
r
r) ln 1{
r)z
ur
m
1{q
^+za u
r
:
2
biz.
u.
t
^
r
n
^ z(1{^
n r
r) ln 1{
1{q
ð8Þ
(8)
A
i»<1-f»>21n2(1-§
ð12Þ
(12)
;»+(1-;»)1n(1-&I)l
where r
now
the observed
observed
final size
size and
and possibly
the
where
represents
final
possibly the
p now
represents the
solution to
to (1)
in case
case
of an
an
unique positive
of
initially
positive solution
(l) in
initially fully susceptible
susceptible
R is the
the reproduction number
number
while
and s
U denote
denote
population. R
while m
pl and
the mean
mean
and standard
standard
deviation
of the
the generation
time (and
the
and
deviation
of
generation time
(and thus,
the coefficient
coefficient
of variation
variation
and .N
the population
s/m
of
(CV)),
N is the
U/pl is the
(CV)), and
size. This
This approximation has
has been
been evaluated
evaluated
elsewhere
If
size.
elsewhere
[50,51].
[50,5l]. If
of the
the
the
aa proportion
population is initially
immune, the
proportion qq of
initially immune,
number
R estimated
estimated
from an
an
reproduction number
R
from
exponential
exponential growth
growth
PLoS ONE
ONE || www.plosone.org
PLoS
www.plosone.org
ð11Þ
(11)
that we
an
unbiased
estimate of
and aa known
Suppose
we have
have an
unbiased estimate
of qq and
known CV
CV
Supposethat
of the
the generation
time (e.g.
of
(e.g. from
from separate
To compare
compare
generation time
separate datasets).
datasets).To
the observed
observed
final
size r
the prediction based
the
final size
against the
based onon R:
R = 1.40,
l.40,
p against
0.5l l, with
would be
be 0.511,
with the
the Wald
Wald statistic
statistic compared
compared toto aa normal
normal
r
po0 would
distribution
distribution
given by
by
4
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
Results
Results
^ {r0
r
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
:
2
u
u 3
^
r
s
2
2
ur
^ (1{^
(1{^
r)z
r
r) ln 1{
-f»>+
u^i>3(1
1{q
m
u
2
u
^
r
t
^z(1{^
nn r
r) ln 1{
1{q
_
Confidence
intervals
Confidence
intervals
ð13Þ
(13)
Table
the
of eleven
eleven seroepidemiseroepidemiTable 2
summarizes
the empirical
results of
2 summarizes
empirical results
ological
proportion infected
ological studies of H1N1-2009. The sample
sample proportion
£1
f»<1-i»>21n2(1
ranged from 4.5% to 38.5%. The smallest three final sizes resulted
-
studies
of H1N1-2009.
infected
The
ranged from 4.5% to 38.5%. The smallest three final sizes resulted
from samples
within 11 month
month after
after observing
peak incidence,
incidence, and
and the
the
from
sampleswithin
observingpeak
largest three
involved aa population-wide
population-wide vaccination
vaccination campaignprior
campaign prior
three involved
largest
fl-l-(1-fJ)ln(1-£I>T
pﬃﬃﬃ
Let
n. The
sample
Let n(r)~s:e:(r)
The minimum
minimum
size which
which explicitly
v(p)=S.e.(p)/H.
sample size
explicitly
accounts
for
is calculated
from
accounts
for only
error
calculated
from
only Type
Type II error
z 22
Zua
n(^
r)22 :
n§
"2
m V(/»>
(M)
to
Whereas the
95% confidence
confidence interval
of the
the binomial
binomial
to the
the survey.
the 95%
interval of
survey. VVhereas
narrow
with
the standard
standard
errors
proportion was
was narrow
with the
errors
ranging from
from 0.6%
0.6%
proportion
ranging
to
95% confidence
interval of
final size
was much
much broader
broader
to 1.6%,
the 95%
confidence
interval
of final
size was
1.6%, the
ranging
from
6.6%
to
76.9%,
which
led
to
include
0%
within
the
from
6.6%
to
which
led
to
include
0%
within
the
76.9%,
ranging
confidence
limits
of seropositive
confidence
limits of
seropositive in
in nine
nine studies,
studies, calling
for ad-hoc
ad-hoc
calling for
truncation
truncation
(or calling
for anan alternative
alternative method
method of
of computation
that
(or
calling for
computation that
include
the F
F distribution).
may include
the
The broader
broader uncertainty
uncertainty bound
bound
may
distribution). The
from the
model-based final
final size
than the
binomial proportion
proportion can
can be
be
from
the model-based
size than
the binomial
as
follows.
the smallest
analytically
demonstrated as
follows. First,
First, the
smallest standard
standard
analytically demonstrated
error
in (12)
seen
when
the CV
CV of
the generation
error
in
when the
of the
time is 0,
0, i.e.,
i.e.,
(12) is seen
generation time
ð14Þ
<14 >
A
1
_
If
account
for
we have
If we
we
account
for both
both Type
and II
II errors,
we
have
errors,
Type II and
zzav(/>)+2,ev(/>+m)
r)zzb n(^
rzm) 2
a n(^
:
n§
nz
m
m
A
A
_
ð15Þ
(15)
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
^A33 (1{^
11/ r)
r
^+za u
r
u 2 :
i>iZ<»
t
^
r
nn r
^z(1{^
r) ln 1{
1{q
-
It should
should be
be noted
noted that
that the
the method
used to
to account
account
for
the power
It
method used
for the
power
can
examine the
the range
of r,1-q-m
the
(equation
only
because the
p< 1-q-mbecause
(equation (15))
(15))can
only examine
range of
standard
error
of final
final size
size includes
includes
the logarithmic
approximate
error
of
the
approximate standard
logarithmic
function.
function.
Application
Application
ð16Þ
(16)
;»+(1-p)1n(1-&I))
Because 0#r#1
and 0#q#1,
have
Because
we have
05,051 and
0SqS1, we
and illustration
illustration
and
^
r
§1:
21.
^
r
^
r
z(1{^
r
)
ln
1{
fJ+(1-f>)ln
1{q
To highlight
the importance
of explicitly
for the
the
To
highlight the
importance of
explicitly accounting
accounting for
variance
of the
the final
final size
size distribution,
the following
two
exercises
variance
of
following two
exercises
distribution, the
are
we
examine
are
performed.
examine
post-peak
First, we
performed. First,
post-peak seroepidemiological
seroepidemiological
studies of
of H1N1-2009,
the 95%
95% confidence
confidence
intervals
studies
intervals
H1N1-2009, comparing
comparing the
two
binomial
and asymptotic
generated
methods;
proportion
methods; binomial
generated by
by two
proportion and
asymptotic
final
size distribution.
distribution.
For
this reason,
when
the
final
size
For
this
when calculating
reason,
calculating the
we
the data
data as
as if
if they
were
uncertainty
regard
generated
bounds, we
uncertainty bounds,
regard the
they were
generated
from
binomial
or
the
final
size of
of a
a
from
aa binomial
process
or
the
final
epidemic
process
epidemic size
For simplicity,
assume
that
homogeneously
population. For
we assume
that
homogeneously mixing
mixing population.
simplicity, we
we
have an
an
unbiased estimate
estimate of
of the
the proportion
of population
we
have
unbiased
proportion of
population with
with
based
on
the
observed
pre-existing
on
the
observed
seropositive
pre-existing immunity
immunity based
seropositive
to the
the epidemic
in Table
Table
1. We
consider
proportion
wave in
1.
We consider
proportion prior
prior to
epidemic wave
of the
the observed
observed
final size,
to the
the
uncertainty
final
which corresponds
size, which
uncertainty of
corresponds to
difference
in infected
infected fraction
before and
and after
after observing
the peak
difference
in
fraction before
peak
observing the
incidence.
we
test
the significance
of the
the observed
observed
incidence.
Subsequently,
test
the
Subsequently,we
significance of
final size
size against
model predictions
51.1%
and 76.7%
76.7%
final
and
24.0%, 51.1%
against model
predictions (i.e.
(i.e. 24.0%,
based on
on
1.40
and
The
mean
and
based
R
=
1.15,
1.40
and
1.90,
respectively).
The
mean
and
R=1.15,
1.90, respectively).
standard
deviation
of the
the generation
time are
are
fixed at
at 2.7
and 1.1
1.1
standard
deviation
of
fixed
2.7 and
generation time
the CV
CV is 0.41)
on
contact
days,
based on
contact
tracing
so, the
days, respectively
respectively(and
(and so,
0.41) based
data in
in the
the Netherlands
Netherlands
To address
address
the uncertainty
data
[40].
the
with
[40]. To
uncertainty with
to the
the shape
and scale
scale of
of the
the generation
time distribution,
respect
distribution,
respect to
shape and
generation time
we
also consider
consider hypothesis
of two
two
other scenarios
scenarios in
in which
we
also
testing of
other
which
hypothesistesting
the CV
CV
constant
and
the
is 0 (i.e.
generation
11 (i.e.
(i.e. aa constant
generation time) and
(i.e.
distributed
exponentially
generation
exponentially distributed
generation time).
time).
as
of the
the
selected
Second, as
sensitivity
selected
empirical
sensitivity analysis
analysis of
we
the desired
desired
minimum
size of
of
illustrations,
present
minimum
sample
illustrations, we
present the
sample size
final
size by
the approximate standard
standard
final
epidemic
epidemic size
by employing the
error
of the
the final
final
size. Examining various
various
of error
error
error
of
size.
margins of
from 0%
0% to
to 50%
50% with
with R
1.40 and
and 1.90
1.90 and
and
ranging from
R being
being 1.15,
1.15, 1.40
the CV
CV of
of the
the generation
time ranging from
from 0
0 to
to 1,
the above
above
the
1, the
generation time
mentioned
formulae
and (15)
are
used with
mentioned
formulae
(14)
used
with significance
(14) and
(15) are
significance
level at
at a
oc=0.05
for the
the latter
latter
the power
is set
set at
at
level
= 0.05 and, for
formula, the
is
formula,
power
for
this
the
12b
=
0.80.
Moreover,
for
this
sensitivity
analysis
the
proporMoreover,
1-/3=0.80.
sensitivity analysis
proportion of
of the
the population with
with pre-existing
at
tion
fixed at
pre-existing immunity
immunity qq is fixed
which corresponds
to the
the mean
based on
on
7.5%,
mean based
eight published
7.5%, which
corresponds to
studies
in
Table
1. Subsequently,
we
also
examine
the
studies
in
Table
1.
also
examine
the
Subsequently, we
of the
the minimum
minimum
size required as
as
sensitivity
sample
aa function
function
sensitivity of
sample size
of R
R and
and q.
of
q.
PLoS ONE
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it is
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Therefore,
is proven
proven that
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r
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1-£TI))
The equality
holds when
1.
The
when pr = 1.
equality holds
=
Hypothesis
testing
Hypothesis testing
of the
time at
Assuming CV
CV of
the generation
generation time
at 0.41,
0.41, six
six serological
serological
Assuming
studies appeared
to have
have yielded
studies
yielded significantly
smaller final
final sizes
sizes
appeared to
significantly smaller
than that
that predicted by
1.40 (Table
than
by R:
R = 1.40
(Table 2).
Nevertheless, four
four of
of
2). Nevertheless,
the six
six studies
studies sampled
the
serum within
within 11 month
month after
after observing
observing
sampled serum
and four
of the
studies
with
peak incidence,
four of
the remaining
five studies
with
incidence, and
peak
remaining five
result
serum
than
month
after
the
insignificant
result
sampled
serum
longer
than
1
month
after
the
insignificant
sampled
longer
association
between
peak (no
between the
the significant
significant test
test result
result
peak
(no significant association
and sampling
the peak;
and
within 11 month
month after
after the
peak; p=0.24,
p = 0.24, þÿFisher
Fisher’ss
sampling within
exact
in four
the six
studies with
exact
test).
four of
of the
six studies
with significantly
significantly
test). Populations
Populations in
smaller
final
unvaccinated
smaller
final sizes
sizes were
were unvaccinated
prior toto sampling,
and
prior
sampling, and
three
of the
the
five
studies
with
three
of
five studies
with insignificant results
results involved
involved
vaccination
Taken
vaccination prior
prior toto the
the survey
survey (p
= 0.57, þÿFisher
Fisher’s). s).
Taken
(p=0.57,
of the
six studies
together,
five of
the six
studies with
with significantly smaller
smaller final
final
together, five
sizes sampled serum
serum
within
after peak
sizes
within 11 month
month after
peak incidence
incidence oror
examined
unvaccinated
examined
unvaccinated population, while
while all
all the
the five
five remaining
studies with
test
results conducted
studies
with insignificant test
results
conducted sampling longer
longer
than
than
11 month
month after
after the
the peak
peak oror the
the population
population involved
involved
vaccination
VVhen
vaccination (p =
0.55, þÿFisher
Fisher’s). s).
When comparing observed
observed final
final
=0.55,
sizes against
R:
results
of
all
studies
were
not
found
sizes
R
=
1.15,
results
of
all
studies
were
not
found toto be
be
1.15,
against
different.
studies indicated
indicated
that
the observed
significantly different.
Eight studies
that the
observed
final
sizes were
smaller
than
final sizes
were significantly smaller
than that
that predicted by
by
R:
1.90. Varying
CV of
the generation time
R = 1.90.
Varying the
the CV
of the
time from
from 00 toto 11
with
CV=0= 0 did
with R:
R = 1.40,
the significance
significance levels
levels with
with CV
did not
not vary
vary
1.40, the
from
those of
of CV
the results
from those
= 0.41, but
but the
results with
with CV=1
CV = 1 indicate
indicate
CV=0.41,
5
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
Table
2.
and
the post-peak
seroepidemiological studies
studies of
(H1N1-2009).
Table
2. Uncertainty
bounds
and hypothesis
of the
of influenza
influenza
(H1N1-2009).
post-peak seroepidemiological
Uncertainty bounds
hypothesis testing
testing of
Country
Country
Sample
Sample
size{
sizeT
Prop
Prop
infected
infected
(%){
(%)T
95%
CI
95%
CI of
of
binomial
binomial
prop
(%){
prop
(%)1
95%
CI
95%
CI
of
final
of final
size (%){
size
(%)i
After
After
peak1
peak§
Vac¥
Vac¥
P-values$
P-values$
fR
1.40
1.40
1.15
1.15
1.90
1.90
1.40
1.40
1.40
1.40
0.41
0.41
0.41
0.41
0.41
0.41
0
0
1
1
T
CV
Australia
[19]
Australia
[19]
1247
1247
15.8
15.8
13.8,
17.9
13.8, 17.9
0,
50.2
0, 50.2
Yes
Yes
No
No
*0.02
*0.02
0.30
0.30
*,0.01
**0.010.07
*0.01
*0.01
0.07
0.07
Canada
Canada [15]
[15]
1127
1127
38.5
38.5
35.7,
41.4
35.7, 41.4
16.5,
60.6
16.5, 60.6
Yes
Yes
Yes
Yes
0.13
0.13
0.89
0.89
*,0.01
*0.110.21
0.11
0.11
0.21
0.21
China
China (1)
(1) [11]
[11]
710
710
6.3
6.3
4.5,
8.1
4.5, 8.1
0,
46.8
0, 46.8
No
No
Yes
Yes
*0.02
*0.02
0.18
0.18
*,0.01
**0.010.05
*0.01
*0.01
0.05
0.05
China (2)
China
(2) [6]
[6]
2913
2913
10.7
10.7
11.8
9.6,
9.6, 11.8
67.8
0,
0, 67.8
Yes
Yes
No
No
0.08
0.08
0.31
0.31
*0.01
*0.01
0.07
0.07
0.15
0.15
Germany
[7]
Germany [7]
225
225
4.5
4.5
7.3
1.8,
1.8, 7.3
56.0
0,
0, 56.0
No
No
No
No
*0.04
*0.04
0.22
0.22
**0.030.09
*,0.01
*0.03
*0.03
0.09
0.09
India [20]
India
[20]
5047
5047
14.6
14.6
15.6
13.6,
13.6, 15.6
30.1
0,
0, 30.1
No
No
No
No
*0.10***
*,0.01
0.10
0.10
***
*,0.01
**
*,0.01
**,0.01
Japan
[21]
Japan [21]
6035
6035
32.7
32.7
33.9
31.5,
31.5, 33.9
19.8,
45.6
19.8, 45.6
Yes
Yes
Yes
Yes
*0.88***0.02
*,0.01 0.88
0.88
***0.02
*,0.01
**0.02
*,0.01
*0.02
*0.02
New Zealand
Zealand [12]
New
[12]
1147
1147
18.4
18.4
20.6
16.1,
16.1, 20.6
81.4
0,
0, 81.4
Yes
Yes
Yes
Yes
0.15
0.15
0.42
0.42
*0.03
*0.03
0.13
0.13
0.23
0.23
Singapore
[22]
Singapore [22]
727
727
10.9
10.9
13.1
8.6,
8.6, 13.1
94.4
0,
0, 94.4
Yes
Yes
No
No
0.17
0.17
0.37
0.37
0.06
0.06
0.15
0.15
0.24
0.24
UK [14]
UK
[14]
275
275
8.0
8.0
11.2
4.8,
4.8, 11.2
35.3
0,
0, 35.3
No
No
No
No
*0.11***0.01 0.11
*,0.01
0.11
***0.01
*,0.01
**0.01
*,0.01
*0.01
*0.01
USA [13]
USA
[13]
846
846
15.5
15.5
17.9
13.1,
13.1, 17.9
100.0
0,
0, 100.0
No
No
Yes
Yes
0.32
0.32
0.21
0.21
0.31
0.31
0.37
0.37
0.45
0.45
{
size refers
refers to
to the
the number
number of
of enrolled
enrolled
to measure
measure
the seroprevalence
after observing
an epidemic
infected
is given
Sample size
subjects
the
epidemic peak.
is
the proportion
[Sample
subjects to
seroprevalence after
observing an
peak. Proportion
Proportion infected
given by
by the
proportion
after observing
minus the
the proportion
before the
the peak
in Table
1.
after
Table 1.
observing peak
peak minus
proportion before
peak in
195%
confidence
intervals (CI) show
show lower
lower and
and upper
confidence
intervals of
of the
the proportion.
95% CI
binomial
is derived
derived from
normal approximation
95% confidence
intervals
intervals
The 95%
CI of
of binomial
proportion
from aa normal
approximation
upper confidence
proportion. The
proportion is
{
to binomial
binomial
while the
the 95%
95% CI
CI of
of final
final size
size is similarly
derived from
from the
Wald method
method
result of
to
distribution,
the Wald
employing
asymptotic convergence
result
of final
final size
size distribution.
distribution.
distribution, while
similarly derived
employing asymptotic
convergence
§After
column
if the
the sampling
took place
month
after observing
the highest
incidence
of cases.
cases.
After peak
represents
than 11 month
after
of
peak column
represents if
sampling took
place longer
longer than
observing the
highest incidence
¥Vaccination
column
if a
a population-wide
vaccination
of H1N1-2009
H1N1-2009
took place
Vaccination column
represents
campaign
took
to the
the sampling.
represents if
population-wide vaccination
campaign of
place prior
prior to
sampling.
$
are
based on
on
two-sided
Wald test
test employing
the approximate
standard
error
of final
final epidemic
p-values are
based
two-sided
Wald
error
of
size.
$p-values
employing the
approximate standard
epidemic size.
the
estimated
number
in
Mexico
which
we
would
like
to
test
our
the
coefficient
of variation
of the
R,
variation of
the generation
time. Significant
Significant
R, the estimated reproduction
CV, the coefficient of
reproduction number in Mexico against
against which we would like to test our hypothesis;
hypothesis; CV,
generation time.
difference
indicated
mark followed
followed
difference
is indicated
by
by
by * mark
by p-value.
p-value.
doi:10.1371/journal.pone.0017908.t002
doi:10.1371/journal.pone.0017908.t002
1
¥
*
that only
three
observed
final
sizes were
were
smaller
that
observed
final
sizes
significantly
only three
significantly smaller
than that
that predicted
R:
1.40.
than
by
R
=
1.40.
predicted by
=
with
with pre-existingimmunity
pre-existing immunity qq (with
(with fixed
fixed RR =
1.40).
the
1.40).Interestingly,
Interestingly, the
minimum
size
hits
the
value
around
minimum sample
size
hits
the
largest
value
around
qq=0.20.
= 0.20. For
For
sample
largest
size estimation
estimation
Sample
Sample size
CV = 0.
0. This
example,
yielded the
the largestsample
largest sample size
size with
with CV
This
0.212 yielded
example, qq = 0.212
can
be
and second
can
be inspected
by taking first
first and
second derivatives
derivatives of
of (16)
(16) with
with
inspectedby
to q
to:
respect
the CV=
CV = 0),
0), leading
leading to:
respect to
q (with
(with the
=
shows the
the minimum
minimum
sizes required
Figure
sample
for postpostFigure 11 shows
sample sizes
required for
studies to
to test
test
the final
final size
size against
epidemic
the
epidemic seroepidemiological
seroepidemiologicalstudies
against
R: = 1.15,
1.40 and
and 1.90
1.90 with
with CV
CV being
0.41 and
and 1.
1. Whereas
VVhereas
R
being 0,
1.15, 1.40
0, 0.41
median
lower and
and upper
size of
of empirical
median
(and
(and lower
upper quartiles)
quartiles) sample
sample size
studies in
in Table
Table 1
1 was
was
such sample
sizes can
can
studies
1127
only
1127 (710,
(710, 2913),
2913), such
sample sizes
only
a difference
difference
from the
the prediction
of R:
1.90 at
at a
a
explicitly
from
prediction of
R = 1.90
explicitly prove
prove a
of error
error
5%. To
To argue
the significant
difference
from
margin
5%.
the
from
margin of
argue
significant difference
based on
on
R=1.40
the identical
identical
of error
error
prediction based
R
= 1.40 with
with the
margin of
and with
with varying
CV of
of the
the generation
time 0.41
0.41 (range:
we
and
0, 1),
varying CV
generation time
(range: 0,
1), we
need 8665
8665 (range:
individuals
at the
the power
of
ideally
at
7215, 15947)
ideally need
(range: 7215,
15947) individuals
power of
50% and
and 16121
individuals
at
the
of
80%.
50%
29680)
individuals
at
the
power
of
80%.
16121 (13423,
(13423, 29680)
power
At the
the margin of
of error
error
these numbers
numbers
are
reduced
to 2167
At
10%,
are
reduced
to
2167
10%, these
and 3715
3715 (3093,
As R
R gets
closer
(1804,
(1804, 3987)
3987) and
(3093, 6841),
6841), respectively.
respectively.As
gets closer
to
the lower
lower
and as
as
the variance
of the
the
to
the
uncertainty bound,
bound, and
the
variance of
time
becomes
relative
to
the
the
generation
becomes
larger relative
to
the
mean, the
mean,
generation time
larger
minimum
size required increases.
increases.
minimum
sample
sample size
examines the
the sensitivity
of the
the minimum
size to
to
QA examines
Figure
minimum sample
Figure 2A
sensitivityof
sample size
the reproduction number
number
R.
the
R. Ignoring pre-existingimmunity
pre-existing immunity (q
(q= 0),
R = 2 with
with the
the CV
CV of
ofthe
time 0.41
0.41 (0,
at least
R
the generation
least
generationtime
(0, 1)
1) requires
requiresat
individuals
at power
of 50%
50% and
and 317
317 (281,
201 (177,
201
at
of
(177, 320)
320) individuals
power
(281, 500)
500)
individuals
at power
of 80%.
80%. As
As R
R is reduced
reduced
and approaches
the
individuals
at
and
power of
approaches the
critical level, much
much greater
sizes are
are
For instance,
critical
required. For
greater sample
sample sizes
the minimum
minimum
size for
for R:
than 2-fold
1.2 is more
2-fold higher
the
sample
R = 1.2
more than
sample size
than that
that required for
for R
R: = 1.4.
1.4. Figure
illustrates the
the relationship
than
2B illustrates
Figure 2B
relationship
between minimum
minimum
size and
and the
the proportion of
ofthe
between
sample
the population
sample size
qmax ~1{
^
r
,
^
r
1_
1{exp
^{1
r
f1m.x=1- .
ð19Þ
(19)
@>
_
which
the
which is the
most
difficult
situation
most difficult
situation in
in which
which the
the hypothesistesting
hypothesis testing
the
against
predicted final
final size
size requires
collect anan
against the
predicted
requires usus toto collect
leads the
the
unrealistically
large number
number of
of blood
blood samples.
samples. qqmax
max leads
unrealistically large
denominator
of the
the approximate
standard
error
in
to
be
0.
denominator
of
standard
error
in
(16)
to
be
0.
approximate
(16)
Discussion
Discussion
We
have introduced
introduced
We have
to compute
aa framework
framework to
the uncertainty
uncertainty
compute the
size that
the
Wald
epidemic
that employs
Wald
epidemic size
employs the
an
the absence
approximation, an
approach
motivated by
by the
absence of
of aa
approach motivated
available
to estimate
estimate the
readily available
methodology to
the sample
size of
of postpostmethodology
sample size
Published
epidemic
studies. Published
seroepidemiologepidemic seroepidemiological
seroepidemiologicalstudies.
seroepidemiological studies
studies of
of H1N1-2009
H1N 1-2009 so
so
far
the confidence
ical
far have
have computed
computed the
confidence
interval
of the
the observed
observed
final
size as
interval
of
final size
as ifif itit were
were aa binomial
binomial
the data
proportion. However,
However, the
data generating
generating process
process behind
behind the
the
proportion.
infectious
diseases
involves
dynamics
of infectious
diseases involves
dependence between
between
dynamics of
dependence
infected
individuals
infected
individuals
[17],
which does
does not
not lead
lead toto aa binomial
binomial
[17], which
the observed
size represents
proportion. Moreover,
Moreover, the
observed final
final size
single
proportion.
represents aa single
stochastic
realization
all
stochastic
realization
among
all possible
possible sample
sample paths (i.e.
(i.e. all
all
among
bounds
bounds
=
=
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=
6
of
of
the
the
final
final
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
Size for
for Post-Epidemic
Studies
Sample Size
Post-Epidem ic Serological
Serological Studies
8000
Q
7000
8000
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35
40
45
50
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1. Minimum
Minimum
sizes required
for post-epidemic
studies
of final
size as
Figure
sample
studies
of
final size
as aa function
function of
of the
the margin
Figure 1.
sample sizes
required for
post-epidemic seroepidemiological
seroepidemiological
margin
the reproduction
and the
the coefficient
coefficient
of variation
variation
of the
& B) Sample size
error,
the
number,
and
of
of
the generation
time. (A &
size with
with three
three different
different
error,
number,
reproduction
generation time.
numbers
as a
a function
function
of the
the margin
of error.
error.
an
estimation
formula
based Type
error
alone
reproduction
as
of
(A) employs
estimation
formula based
Type I error
alone (at aoc=0.05),
= 0.05), while
while (B)
reproduction numbers
margin of
employs an
accounts
for both
both Type
and II errors
errors
and 12b
of error
error
random
around
accounts
for
(at aoc=0.05
= 0.05 and
= 0.80). The
The margin
represents
sampling
around which
which the
the reported
error,
represents random
reported
Type I and
1-;'3=0.80).
margin of
sampling error,
would include
include the
the true
true
Since (A) is a
a special case
case
of (B) (with
1.40 in
in (A) is also
also shown
line in
in (B). The
percentage
percentage.
of
= 0.50), RR== 1.40
shown as
as dotted
dotted line
The
(with b
percentage would
percentage. Since
/}=0.50),
coefficient
of variation
variation
of the
the generation
time and
and the
of population
are
fixed
coefficient
of
(CV) of
the proportion
with pre-existing
fixed atat 40.7%
40.7% and
and 7.5%,
7.5%,
proportion of
population with
generation time
pre-existing immunity
immunity are
& D) Sample size
size with
with three
three different
different
coefficients
of variation
as a
a function
of the
of error.
accounts
for
respectively. (C &
coefficients
of
variation as
function of
the margin
error. (C) accounts
for Type
Type I error
error alone
alone
margin of
while (D) accounts
accounts
for both
both Type
and II errors
errors
and 12b
number
and
of population
(a
= 0.05), while
for
(a
= 0.05 and
= 0.80). The
The reproduction
and the
the proportion
with
(oc=0.05),
(ac=0.05
reproduction number
proportion of
population with
Type I and
1-fi=0.80).
are
fixed
at 1.40
1.40 and
and 7.5%,
CV =
=00 corresponds
constant
pre-existing
fixed at
to aa constant
generation
time, whereas
whereas CV=1
CV = 1 represents
an
7.5%, respectively.
corresponds to
represents an
pre-existing immunity
immunity are
respectively. CV
generation time,
distributed
time. In
In (B) and
and (D), several
several lines
lines are
are
due to
to impossibility
account
for
in the
exponentially
generation
truncated, due
to account
for larger margins
of error
error in
the
truncated,
exponentially distributed
generation time.
impossibility to
margins of
estimation
formula.
estimation
formula.
doi:10.1371/journal.pone.0017908.g001
studies published
seroepidemiological
published toto date
date did
did not
not necessarily
necessarily
seroepidemiologicalstudies
an
of prediction based
overestimation
an
overestimation
of
based onon R=1.40,
R = 1.40, and
and
all the
sizes did
did not
moreover, all
the observed
observed fir1al
final sizes
not reveal
reveal significant
significant
moreover,
the lower
l.l5.
Published
deviation from
deviation
from prediction With
with the
lower limit
limit R:
R = 1.15.
Published
studies
that
the
bound
R:
1.90
seroepidemiological
studies
agree
that
the
upper
bound
R = 1.90
seroepidemiological
agree
upper
other published
estimates of
of R>2
(and
published estimates
R.2 [29,30]) was
was likely anan
(and thus, other
still speculate
that R:
1.40 may
overestimation
overestimation
[39]. One
One may
may still
R = 1.40
may Well
well
speculatethat
be
an
all
of
the
final
sizes
overestimation
observed
were
be an
overestimation
(because
all
of
the
observed
final
sizes
were
(because
of the
the epidemic),
us
to
possible
probabilistic trajectories
to
possibleprobabilistic
trajectories of
epidemic), requiring us
consider
stochastic
the data.
data. To
To account
account
for
these
variations
consider
stochastic
variations in
in the
for these
the approximate standard
standard
error
of the
the final
We
issues,
employed
error
of
final
issues, we
employed the
size given
as
result of
of a
a
size
aa convergence
result
homogeneously
given as
convergence
homogeneously mixing
stochastic
calculation
of the
the standard
standard
error
stochastic
epidemic
model. The
The calculation
of
error
epidemic model.
to be
be simple
to compute
are
was
shown to
was
shown
programs are
simple to
compute (spreadsheet
(spreadsheetprograms
the proposed
of final
sufficient).
bound of
final
sufficient). By applying the
proposed uncertainty bound
size to
to influenza
influenza
also shown
that all
all the
the
have also
shown that
size
(H1N1-2009),
we have
(HlNl-2009), We
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indicate
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77
March 2011
6 | Issue
Issue 3
March
2011 | Volume
Volume 6
3 | e17908
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
1000000
1000000
A
100000
Coefficient
B
Coefficient
of
variation
ofthe
generation time
100000
A
_
_
2
_
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g
1.00
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-
§
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.
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10
20
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Reproduction
40
50
0
60
5
10
15
20
25
30
35
40
45
50
with þÿDl' ¬- ¬XlSIlnQ
PFODOFUOIT
lmlT1Uiilty(%)
number
2. Sensitivity
of minimum
minimum
size for
for post-epidemic
studies
to
number
and
Figure
sample
seroepidemiological
studies
to the
the reproduction
number
and the
the
sample size
post-epidemic
reproduction
Figure 2.
Sensitivity of
seroepidemiological
of population
with
minimum
size with
three different
different
coefficients
proportion
of
with
pre-existing
The minimum
sample
with three
coefficients of
of variation
variation (CVs)
as aa
(A). The
(CVs) as
proportion
population
sample size
pre-existing immunity.
immunity. (A).
function
of the
the reproduction
number. (B).
The minimum
minimum
size with
with three
CVs as
as a
a function
of the
of population
function
of
sample
three CVs
function of
the proportion
with pre-existing
(B). The
reproduction number.
sample size
proportion of
population with
pre-existing
ln (A),
the proportion
of population
with pre-existing
is fixed
at 0,
and the
10%
immunity.
fixed at
the estimates
estimates correspond
correspond toto the
the margin
of error
error of
of 10%
(A), the
0, and
proportion of
population with
immunity. In
pre-existing immunity
immunity is
margin of
and Type
and II
ll errors
errors
at a
oc=0.05
and 12b
In (B),
the reproduction
number is fixed
Hxed at
at 1.40,
and the
and
at
= 0.05 and
= 0.50, respectively.
the estimates
estimates correspond
to the
the
(B), the
1.40, and
reproduction number
correspond to
Type I and
1-/}=0.50,
respectively. In
of error
error
of 10%
10% and
and Type
and II
ll errors
errors
at a
oc=0.05
and 12b
margin
of
Type II and
at
= 0.05 and
= 0.50, respectively.
margin of
1-/}=0.50,
respectively.
doi:10.1371/journal.pone.0017908.g002
doi:10.1371/journal.pone.0017908.g002
conservative
conservative
uncertainty
bounds. The
The proposed
proposed method
method has
has aa
uncertainty bounds.
potential for
for explicitly
discussing aa posteriori
posteriori effectiveness
effectiveness of
of
potential
explicitly discussing
interventions
the direct
direct comparison
observed
final
interventions through
through the
of observed
final sizes
sizes
comparison of
in
different
in different
settings. Hence,
Hence, we
we believe
believe that
that the
the proposed
proposed
settings.
calculation
of the
the 95%
calculation
of
95% confidence
confidence interval
interval will
will greatly
help
greatly help
this area
area
of
should also
also be
progressing this
of research.
research. ItIt should
be noted
noted that
that the
the
progressing
use
of the
the proposed
an
use
of
proposed uncertainty
uncertainty bounds
bounds plays
plays an
important role
role
important
especially
for influenza
influenza transmission
transmission with
with R<2
R,2 (Figure
(Figure 2A).
especiallyfor
2A).
Our illustration
of the
the proposed
Our
illustration of
proposed method
method posed
posed four
four technical
technical
the computation
the uncertainty
challenges
for the
of the
bound of
of final
final
challengesfor
computation of
uncertainty bound
the coefficient
coefficient
of variation
size;
of
variation of
of the
the generation
time has
has toto be
be
size; (i) the
generation time
the proportion
known, (ii)
proportion of
of pre-existing
pre-existing immunity
immunity before
before anan
known,
(ii) the
the bounds,
epidemic
critically influences
influences the
bounds, (iii)
(iii) sampling
of several
several
epidemic critically
sampling of
studies took
an
seroepidemiological
took place
place shortly
after an
epidemic
seroepidemiological studies
shortly after
epidemic
and (iv)
peak and
vaccination and
and other
other public
public health
health interventions
interventions
peak
(iv) vaccination
the course
course
of an
the observed
during the
of
an epidemic can
can modify the
observed final
final
size. As
the present
critical
need
size.
As for
for (i),
present study
study demonstrates
demonstrates aa critical
need toto
(i), the
estimate
the variance
of the
the generation
estimate
the
variance of
time in
in addition
addition toto the
the
generation time
mean.
That is, the
distribution
of
mean. That
the distribution
of the
the generation
generation time
time plays
plays aa key
key
role
also in
role not
not only in
in estimating
estimating RR [53,54]
but also
in characterizing
characterizing the
the
[53,54] but
variance
of final
size. With
variance of
final epidemic
With respect
respect toto (ii), although we
we did
did
epidemic size.
not
to
the
not include
include seroepidemiological
studies prior
prior to
the 2009
2009
seroepidemiological studies
have shown
shown that
that such
we have
such aa survey
survey of
of qq isis aa
pandemic [24,25,27],
[24,25,27], we
to
determine
the sample
after the
key to
determine
the
size after
the epidemic
epidemic [55].
In
key
sample size
[55]. In
smaller than
than 51.1%),
the sample
sizes of
of published
smaller
but the
published seroepide51.1%), but
sample sizes
seroepidestudies turned
turned out
out to
to be
be too
too small
small to
to answer
answer
this question.
miological
this
miological studies
question.
formulae
for variance
variance of
of the
the final
size distribution
distribution
Although
for
final size
(i.e.
Although formulae
(i.e.
the square
root
of which
which we
as an
an
standard
the
of
we regarded
approximate
square root
regarded as
approximate standard
has been
been known
known
stochastic
error)
among
stochastic
modeling experts
error) has
among
modeling
experts [50],
[50],
the present
extended
its use
use
to the
the computation
of the
the 95%
95%
the
its
to
present study
study extended
computation of
confidence
interval
of the
the observed
observed
final size
size by
the
confidence
interval
of
final
by replacing
replacing the
number
its estimator.
estimator.
This also
also led
us
to consider
consider
reproduction
by
This
led us
to
reproduction number
by its
Wald test
test
and sample
size estimation.
estimation.
What
the
aa parsimonious
and
What the
parsimonious Wald
sample size
for
present
post-epidemic
present study
study suggests
suggests for
post-epidemic seroepidemiological
seroepidemiological
studies is to
to employ
the proposed
formula
to calculate
calculate
the
studies
(12)
the
employ the
proposed formula
(12) to
95% confidence
confidence
interval
and (14)
or
to help
determine
the
95%
interval
and
(15)
the
(14) or
(15) to
help determine
size for
for seroepidemiological
For the
the latter,
the
sample
For
latter, the
sample size
seroepidemiological surveys.
surveys.
of (14)
useful:
following simplification of
might be
be useful:
(14) might
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
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r
s
2 2
3 (1{^
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r
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ln
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r
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nz
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^
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^
s:e:(^
r
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r
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The standard
standard
error
calculated
the specified
The
error
s.e.(^
r) is calculated
by
s.e.(f))
by using
using the
specified
confidence
interval
twice
the margin of
of error) and
and
the
confidence
interval
(i.e.
the
the
(i.e. twice
confidence
level (i.e.
nominal
For instance, if
if
confidence
level
coverage
probability).
(i.e. nominal
coverage
probability). For
the margin of
of error
error
5% and
and the
the confidence
confidence
level is 95%,
the
the
is 5%
level
95%, the
standard
error
0.05/1.96
the standard
standard
error
0.025. Similarly, the
standard
error
is 0.05/1.96
= 0.025.
error
0.030 and
and 0.020
at the
the confidence
confidence
levels
of 90%
90% and
and 99%,
is 0.030
levels of
0.020 at
99%,
It is worth
worth stressing
that the
the purpose
of post-epidemic
respectively.
of
post-epidemic
respectively.It
stressingthat
purpose
studies is not
not
to test
test
the observed
observed
seroepidemiological
necessarily
the
seroepidemiologicalstudies
necessarilyto
final
size against
but
includes
real-time
final
size
predicted value, but
includes
real-time
against aa predicted
of an
an
and various
various
considerations
of public
monitoring of
epidemic
considerations
of
epidemic and
health
interventions.
As long as
as
there is no
alternative
health
interventions.
As
there
no better
better alternative
method
for computing the
the uncertainty, the
the proposed
method
for
proposed approach
approach
should
also be
be used
used
for
those
other
to
calculate
should
also
for
those
other
purposes to
calculate
purposes
addition
to the
the estimation
estimation
of
should be
addition
to
of qq itself, itit should
be noted
noted that
that our
our
an
that the
the pre-existing
adopted an
assumption
pre-existing immunity
immunity
assumption that
offered
offered
aa complete
protection from
from infection
infection (i.e.
all-or-nothing
complete protection
(i.e. all-or-nothing
If the
the pre-existingimmunity
protection). If
pre-existing immunity isis imperfect and
and described
described
the so-called
by the
so-called leaky
leaky protection
protection (e.g.
(e.g. partial
partial reductions
reductions in
in
by
contact
and
susceptibility
per contact
and in
in infectiousness
infectiousness upon
susceptibility per
upon
infection),
those quantifications
those
will be
be required in
in addition
addition toto the
the estimation
estimation
quantifications will
of the
the proportion
the initially
of
proportion of
of the
initially immune
immune population. Issues
Issues (iii)
and (iv)
technical
and
pose further
further technical
challenges toto precisely
precisely estimate
estimate
(iv) pose
challenges
of seroprevalence
studies. Given
Given
uncertainty bounds
bounds of
in empirical studies.
seroprevalencein
method
method
=
PLoS ONE
ONE || www.plosone.org
PLoS
www.plosone.org
88
March 2011
2011
6 | Issue
Issue 3
3 | e17908
March
| Volume
Volume 6
e17908
Sample
for Post-Epidemic
Studies
Size for
Sample Size
Post-Epidemic Serological
Serological Studies
that
of
is given
that the
the observation
observation
of incidence
incidence
is
in every
discrete time
time
given in
every discrete
unit,
way
forward
may
be
to
employ
a
forward
be
to
a parsimonious
unit, aa possible
possible way
may
employ parsimonious
discrete
time
model (e.g.
or
chain
discrete
time stochastic
stochastic
model
or
chain
(e.g. branching
branching process
process
binomial
model)
to
binomial
which may
well enable
enable us
us
to draw
draw the
the 95%
95%
model) [56],
[56], which
may well
confidence
interval
in
by conditioning
confidence
interval
in a
a given reporting interval
given reporting interval by
conditioning
the
to
intervals. Proposing
the distribution
distribution
to previous
previous reporting
reporting intervals.
Proposing simple
simple
methods
to
part of
future
methods
to address
address these
these issues
issues is part
of our
our
future studies.
studies.
Our
relied
the
Our method
method
relied on
on
the homogeneous
and
homogeneousmixing
mixing assumption
assumptionand
ignored
time dependent
factors that
that include
include seasonality
and public
ignored time
dependent factors
seasonalityand
public
health
interventions.
In this
this sense,
the proposed
health
interventions.
In
the
proposed uncertainty
is
sense,
uncertainty is
regarded
an
underestimate,
the
time-dependent
as
an
because
the
underestimate, because
regarded as
tirne-dependent
variations
in
potential
increase
variance
variations
in the
the transmission
transmission
can
increase the
the variance
potential can
of the
the final
final
size distribution,
and also
also because
because
of
size
heterogeneous
distribution, and
heterogeneous
transmission
(e.g.
also
increase
transmission
can
also
increase
(e.g. age-dependent
age-dependent mixing)
mixing) can
variance
an
could
variance
(e.g.
epidemic
with extremely
(e.g. an
epidemic with
extremely high
high assortativity
assortativity could
multimodal
final size
size distribution
distribution
for
an
entire population
generate
final
for an
entire
population
generate multimodal
If an
an
intervention
focused only
on
of cases
cases
or
if
[57]).
intervention
is focused
aa portion
or
if
[57]). If
only on
portion of
disease-induced
deaths occur
occur
in non-negligible
not
the
disease-induced
deaths
in
only
order, not
non-negligible order,
only the
variance
but
for
final size
relation (our
variance
but also
also the
the formulae
formulae
for the
the final
size relation
(our
have
to
be
reassessed
in
the
equation
(1))
have
to
be
reassessed
[58–60].
Moreover,
in the
Moreover,
equation
[5f%60].
of strong
deterministic
has
presence
of
modeling
presence
strong seasonality,
seasonality,aa deterministic
modeling study
study has
demonstrated
limited
of R
alone in
demonstrated
aa very
predictive
performance of
R alone
in
very limited
predictive performance
the final
final epidemic
size [61,62].
Given that
that seroepianticipating
anticipating the
epidemic size
[6l,62]. Given
seroepistudies tend
tend to
to stratify
demiological
by age-group
(to
demiological studies
stratify population
population by
age-group
(to
the
of
the
risk
of
and
capture
the
age-dependency
of
the
risk
of
infection),
and
capture
age-dependency
infection),
that the
the final
final size
size of
of age-structured
can
be
considering
models can
be
considering that
age-structured models
different
from that
that of
of homogeneous
further
work
different
from
work
homogeneous population
population [63],
[63], further
could at
at least
least incorporate
the
could
by employing
incorporate heterogeneous
heterogeneousmixing
mixing by
employing the
similar
result
of the
the final
final size
size distribution
distribution
existing
convergence
result of
existing similar
convergence
using
multitype epidemic
model (e.
(e.g.
model). An
An
using aa multitype
epidemic model
g. age-structured
age-structured model).
elegant
formula
for
the
asymptotic
final
size
distribution
of
formula
for
the
final
size
distribution
of
elegant
asymptotic
multitype epidemic
models has
has been
been derived
derived by
by Ball
Ball and
and Clancy
Clancy
multitype
epidemic models
[64],
yielding aa variance
variance matrix
matrix (which
similar to
but aa little
little
to but
[64], yielding
(which isis similar
more complicated
than
that
discussed
in
the
present
study).
more
than
that
discussed
in
the
complicated
present study).
Nevertheless, itit should
be noted
noted that
that the
the elements
elements of
of the
the nextnextshould
be
Nevertheless,
the reproduction
generation
matrix (or
(or the
reproduction matrix)
matrix) would
would be
be included
included
generation matrix
as
the
of the
the final
final size
equation for
for multitype
multitype models
models
as
the solution
solution
of
size equation
those cannot
cannot
be
estimator
of
[64,65],
and those
be simply
replaced by
by the
the estimator
of RR
[64,65] and
simply replaced
using
final size
was done
in the
the Ppresent
study
using8
usin 8 final
size (i.e.
i.e. as
as
was
done
in
resent
stud
Y usin
homogeneous
model), and
thus,
the computation
computation of
of 95%
95%
and
thus, the
homogeneous model),
confidence
interval may
may well
well require
require full
full quantification
quantification of
of the
the
confidence
interval
next-generation matrix
matrix (in
to
observation of
of final
final sizes
sizes for
for
addition
to observation
next-generation
(in addition
each type).
each
type).
Each
issues should
be addressed
addressed in
in the
the
Each of
of the
the abovementioned
abovementioned
issues
should be
the context
of empirical
future, ideally
ideally in
in the
context of
applications. Until
Until that
that
future,
empirical applications.
time,
rather than
than relying
relying onon aa binomial
binomial proportion,
proportion, wewe
time, rather
recommend
the use
use
of
the approach
recommend the
of the
introduced in
in this
this study
study ifif
approach introduced
the
to
determine
the
the
goal
is to
determine the
sample
size of
of post-epidemic
post-epidemic
goal is
sample size
calculate
the
seroepidemiological
the 95%
95% confidence
confidence
studies, toto calculate
seroepidemiological studies,
interval
of observed
observed
final
conduct
relevant
interval of
final size,
size, oror toto conduct
relevant hypothesis
hypothesis
testing.
testing.
,
Author
Contributions
Author
Contributions
Conceived
and designed
the experiments:
HN. Performed
Conceived
and
Performed the
the experiments:
experiments:
designedthe
experiments:HN.
HN. Analyzed
the data:
data: HN
GC. Contributed
Contributed
materials/
HN.
Analyzed the
HN GC.
reagents/materials/analysis
reagents/
analysis
tools: HN.
HN. Wrote
the paper:
HN GC
GC C-CC.
C-CC.
tools:
Wrote the
paper: HN
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2011
6 | Issue
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3 | e17908
March
| Volume
Volume 6
e17908
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