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ResearchCite this article: Yaari R, Katriel G, Stone L,

Mendelson E, Mandelboim M, Huppert A. 2016

Model-based reconstruction of an epidemic

using multiple datasets: understanding

influenza A/H1N1 pandemic dynamics in Israel.

J. R. Soc. Interface 13: 20160099.

http://dx.doi.org/10.1098/rsif.2016.0099

Received: 2 February 2016

Accepted: 8 March 2016

Subject Category:Life Sciences – Mathematics interface

Subject Areas:biomathematics

Keywords:disease transmission model, age structure,

serology, school vacations, absolute humidity,

vaccine allocation

Author for correspondence:A. Huppert

e-mail: amith@gertner.health.gov.il

Electronic supplementary material is available

at http://dx.doi.org/10.1098/rsif.2016.0099 or

via http://rsif.royalsocietypublishing.org.

& 2016 The Author(s) Published by the Royal Society. All rights reserved.

Model-based reconstruction of anepidemic using multiple datasets:understanding influenza A/H1N1pandemic dynamics in Israel

R. Yaari1,3, G. Katriel4, L. Stone3,5, E. Mendelson2, M. Mandelboim2

and A. Huppert1,6

1Bio-statistical Unit, The Gertner Institute for Epidemiology and Health Policy Research, and 2Central VirologyLaboratory, Ministry of Health, Chaim Sheba Medical Center, Tel-Hashomer 52621, Israel3Zoology Department, Tel-Aviv University, Ramat Aviv 69778, Israel4Department of Mathematics, ORT Braude College, Karmiel 21610, Israel5School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, Victoria 3001, Australia6Sackler Faculty of Medicine, Tel-Aviv University, Ramat Aviv 69778, Israel

Intensified surveillance during the 2009 A/H1N1 influenza pandemic in Israel

resulted in large virological and serological datasets, presenting a unique

opportunity for investigating the pandemic dynamics. We employ a con-

ditional likelihood approach for fitting a disease transmission model to

virological and serological data, conditional on clinical data. The model is

used to reconstruct the temporal pattern of the pandemic in Israel in five

age-groups and evaluate the factors that shaped it. We estimate the reproduc-

tive number at the beginning of the pandemic to be R ¼ 1.4. We find that the

combined effect of varying absolute humidity conditions and school vacations

(SVs) is responsible for the infection pattern, characterized by three epidemic

waves. Overall attack rate is estimated at 32% (28–35%) with a large variation

among the age-groups: the highest attack rates within school children and the

lowest within the elderly. This pattern of infection is explained by a combi-

nation of the age-group contact structure and increasing immunity with

age. We assess that SVs increased the overall attack rates by prolonging the

pandemic into the winter. Vaccinating school children would have been the

optimal strategy for minimizing infection rates in all age-groups.

1. IntroductionIn recent years, there has been significant progress in employing influenza sur-

veillance data, together with epidemiological modelling, in order to address

some of the important questions regarding the human-to-human transmission

of influenza and the factors that affect it [1]. An especially promising approach,

which also raises methodological challenges, is incorporating different types of

datasets, each with its own strengths and weaknesses, in order to piece together

a picture of the unfolding of an epidemic and the underlying mechanisms [1]. In

this work, we employ several types of data available from the 2009 A/H1N1 pan-

demic in Israel, together with model-fitting techniques, in order to obtain insights

that could not have been obtained from each of these datasets in isolation.

The novel A/H1N1 influenza strain (H1N1pdm) was first identified in Israel at

the end of April 2009 [2,3]. By June, an epidemic was already underway across the

country. Clinical diagnoses of influenza-like illness (ILI) collected during the pan-

demic showed a pattern characterized by three apparent waves of infection

(figure 1a). This pattern presents an opportunity for investigating the changing fac-

tors affecting the transmission rates of influenza within a population. Weather

conditions, in particular temperature and absolute humidity (AH), are the main

factors currently used to explain the seasonal pattern of interpandemic influenza

outbreaks in temperate regions [4–8]. With regards to the 2009 pandemic, several

studies have found that varying AH conditions throughout the pandemic period

0

5

10

15

20

25

30(a)

wee

kly

ILI

inci

denc

e pe

r 10

000

0

5

10

15

20

25

30

daily

ave

rage

AH

(g

m–3

)an

d te

mpe

ratu

re (

°C)

June July Aug Sep Oct Nov Dec Jan Feb Mar

June July Aug Sep Oct Nov Dec Jan Feb Mar

20

40

60

80

100

inte

rest

ove

r tim

e in

‘fl

u’

(b)

B

AC D

Figure 1. (a) Weekly ILI incidence rates per 10 000 (solid line), average daily absolute humidity (dashed line), average daily temperature (dotted line) and vacationperiods (vertical dotted lines) during the pandemic in Israel. (b) Results of the search term ‘flu’ (in Hebrew) using Google Trends between June 2009 and March2010. Alphabet letters mark key events during the pandemic in Israel coinciding with peaks in the public interest in the flu. (A) WHO declares ‘swine flu’ a globalpandemic (12 June 2009), (B) first cases of mortality from ‘swine flu’ in Israel (27 July 2009, 3 August 2009), Israeli Ministry of Health orders the acquisition ofvaccinations for the entire Israeli population (30 July 2009), (C) beginning of national vaccination campaign for individuals belonging to risk groups (2 November2009), (D) opening of vaccination campaign to the entire population (18 December 2009). (Online version in colour.)

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help to explain the spatio-temporal dynamics of the pandemic

[9,10], whereas other studies indicated that regional differences

in AH conditions can help to explain the observed variation in

the pandemic dynamics across different geographical regions

[11,12]. Maximum levels of temperature and AH in Israel

were reached during the month of August (figure 1a), which

could serve to explain the decline in ILI incidence during

August and the subsequent rise at the start of September.

Reduced contact rates during school vacations (SVs) is another

factor thought to have a role in the seasonality of influenza

[13,14]. Evaluating the effect of SVs on the spread of influenza

is difficult, as typically the influenza season does not overlap

with the school summer vacation. The early initiation of the

pandemic has enabled one to examine and confirm the curtail-

ing impact of SVs on influenza transmission [10,12,15–20]. In

Israel, the early initiation resulted in the pandemic overlapping

not only with the summer vacation during July and August, but

also with the Jewish New Year holidays in October (figure 1a).

From a research perspective, another opportunity pre-

sented by the pandemic relates to the unique datasets

collected during this period. In the wake of the pandemic,

the Israeli Ministry of Health enhanced its typical influenza

surveillance programme, by carrying out more virological

tests of suspected influenza cases and by conducting a large

seroepidemiological study [21]. Virological tests provide an

indication of the fraction of actual influenza cases among

patients with influenza-like symptoms. Serological surveys

make it possible to assess the actual proportions of the disease

spread in the population, and in so doing account for the

undiagnosed cases. The availability of these types of data to

complement the clinical diagnosis data is particularly impor-

tant under the circumstances of a worldwide pandemic.

As public awareness and attitude towards H1N1pdm changed

in time over the pandemic, so did the tendency to consult with

physicians upon the appearance of clinical symptoms, which

in turn affected the time series of clinical diagnoses [19]. Exam-

ining the time series obtained from a Google Trends search for

‘flu’ in Israel during the pandemic period (figure 1b) reveals a

different wave pattern from the one observed in the ILI data

(figure 1a), including a disproportionate temporary leap in

the public interest in flu at the beginning of August, which

could serve as an alternative explanation to the observed

peak in ILI incidence during this period. Combining all avail-

able types of epidemiological and social data should help

facilitate a more accurate evaluation of the pandemic’s

dynamics in Israel and the factors that shaped it.

Several recent studies introduced new frameworks for

integrating multiple types of surveillance data, including vir-

ological and/or serological data, with a disease transmission

model in order to examine different aspects of influenza epi-

demics [19,20,22–24]. Here, we present an approach based on

conditional likelihood, for fitting and parameter estimations

of dynamical models using a variety of datasets. Our trans-

mission model incorporates weather data, as well as data

on social contacts among age-groups, to project the number

of infected individuals in different age-groups. Our obser-

vation model uses clinical diagnosis data while attempting

to give a realistic description of the processes by which the

virological and serological datasets were collected. The com-

bination of the two models is used to fit the uniquely large

and detailed virological and serological datasets collected

during the pandemic in Israel. By fitting the virological and

serological datasets, we infer maximum-likelihood estimates

for the transmission model parameters, assessing the

Table 1. List of the data sources used by the model.

data description values source/reference

Tjþ(t), Tj(t) number of positive/total virology tests of eILI

diagnosed patients of age-group j on

week t of the pandemic

see electronic supplementary

material, table S1

Israeli Center for Disease Control (ICDC)

and the central virology laboratory

at Sheba medical center

Yjþ(t), Yj(t) number of positive/total serology tests of

individuals in age-group j on week t of

the pandemic

see electronic supplementary

material, table S2

Central Virology Laboratory at Sheba

Medical Center [18]

eILIj(t) number of ILI, URI, viral infection and fever

diagnoses for Maccabi HMO members in

age-group j on week t of the pandemic

see electronic supplementary

material, table S3

Maccabi HMO and the ICDC

Mcij

(1 � i, j � 5)

contact matrices of 5-year band age-groups

for eight European countries

see source ref. POLYMOD study [22]

school vacations

periods

dates of school vacations see electronic supplementary

material, table S5

Israeli Ministry of Education

Temp(t),

RH(t)

mean temperature and relative humidity on

day t for Israel. Used to calculate the

mean absolute humidity for Israel AH(t)

see electronic supplementary

material, figure S3. Data are

available in source ref.

Israel Meteorological Service [23]

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impact of SV and weather conditions on the reproductive

number as well as the relative susceptibility of individuals

in the different age-groups. Using the model, we reconstruct

the temporal pattern of the pandemic in Israel and the overall

infection rates in different age-groups. We show that the

timing of SVs, in relation to the seasonality induced by

weather, can have dramatic effect on the outcome of an epi-

demic. Finally, we investigate the potential effectiveness of

vaccination in mitigating the pandemic and assess optimal

vaccine allocation among different age-groups.

2. Methods2.1. The dataThe following is a short description of the different datasets

used in the modelling (table 1). Detailed information regard-

ing the data can be found in the electronic supplementary

material, S1A. The virological data used here are the results of

RT-PCR tests conducted on throat and nasal swabs collected

from suspected influenza cases visiting clinics belonging to the

Israeli Ministry of Health’s sentinel network. The serology data

contain the results of haemagglutination inhibition (HI) assays

performed on serums collected from individuals prior to and

during the 2009 pandemic [21]. The clinical surveillance data

consist of diagnoses made by physicians belonging to the

Macabbi health maintenance organization (HMO), which has a

nationwide coverage of approximately 25% of the Israeli popu-

lation. The dataset is comprised of four different diagnoses that

together cover more than 90% of the diagnoses given to patients

sampled in the sentinel clinics for virological testing [25]. We use

here the term extended influenza-like illness (eILI) to refer to

the combined sum of the four diagnoses. The virological, serolo-

gical and clinical data were aggregated into weekly data in

five age-groups: 0–4, 5–19, 20–44, 45–64 and 65þ (electronic

supplementary material, tables S1–S3).

We calculated a 5 � 5 age-group contact frequency matrix

for the population of Israel (electronic supplementary material,

table S4) based on results of a survey of the number of contacts

among age-groups in eight European countries [26]. We found

that after accounting for differences in demographics and scaling,

the variance obtained from the European matrices is verysmall (elec-

tronic supplementary material, figure S2). By taking the mean of

these matrices and fixing it according the demographics of the Israeli

population, we obtain, up to a scaling factor, an age-group contact

matrix for Israel. The SV dates used in the model are based on the

official vacations schedule for the 2009/2010 school year in the

Jewish sector (electronic supplementary material, table S5). Daily

mean temperature and AH data for Israel used in the model were

obtained by averaging measurements made by meteorological

stations of the Israeli Meteorological Society [27] in four Israeli

cities representing four geographical regions (electronic supplemen-

tary material, figure S3). We used Google Trends data in some model

variants for modelling trends in the tendency to visit a physician

over time upon experiencing influenza-like symptoms. These

data (available at [28]) were extracted by searching Google Trends

for the term ‘flu’ (in Hebrew) over the pandemic period (June

2009–March 2010) in Israel (figure 1b).

2.2. Model formulationFigure 2 summarizes the modelling scheme used here, which will

be described in detail below. In general, the scheme involves a

deterministic transmission model that characterizes the trans-

mission of influenza in the Israeli population as a whole. The

transmission model estimates the number of new infectives each

day over the course of the pandemic. This information is fed into

a stochastic observation model that makes predictions about the

results of serological and virological tests of the population. The

likelihood function is defined based on the probability of obtaining

the laboratory results (the serological and virological data at our

disposal) given the clinical surveillance data and the transmission

model output.

2.2.1. Transmission modelFor the description of the infection process, we use the determinis-

tic version of a discrete-time, age-of-infection model framework

[29]. The model is extended to incorporate multiple age-groups.

The model follows the general SIR framework that assumes each

transmissionmodel

demographics(Nj)

age-group contacts(M)

school vacationperiods

temperature,humidity

observationmodel

weekly eILIincidence(eILIj(t))

monitoredpopulation

(qj)

declared physicanvisiting rates

(xj)

estimated weeklyinfluenza incidence

(Zj(t))

estimated weeklypositive eILI rates (pj(t))

estimated weeklyseropositive rates (pj(t))

weekly positivevirological test rates

(Tj+(t)/Tj(t))

weekly positiveserological test rates

(Yj+(t)/Yj(t))

pre-pandemicseropositive rates

(cj)

susceptibility factors(lj)

generation timedistribution (Pt)

effect of schoolvacations (k)

effect of weather(d )

initial number ofinfected (I0)

Figure 2. A diagram describing the components of the modelling scheme. Ovals represent model components. Rectangles represent data used by the model.Trapezoids represent model parameters that are estimated by the model fitting. Rounded rectangles represent model outputs. The transmission model outputfeeds into the observation model whose two types of output are compared with the available virological and serological data. (Online version in colour.)

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individual at any time is in one of three classes: susceptible,

infected or recovered. The model equations are

ijðtÞ ¼ lj � wðtÞ � Sjðt� 1Þ

�Xm

k¼1

M jk � s jkðtÞNk

�Xd

t¼1

Pt � ikðt� tÞ !

þ i 0j ð2:1aÞ

SjðtÞ ¼ Sjðt� 1Þ � ijðtÞ: ð2:1bÞ

Here, ij(t) is the number of newly infected from age-group j (1 �j � m) and Sj(t) is the number of susceptible individuals from

age-group j on day t. A detailed description of the model par-

ameters and the derivation of the model equations can be

found in the electronic supplementary material (§B). Briefly,

the model parameters are (see also tables 2 and 3):

— Nj, the population size of group j.— Pt(1 � t � d ), the infectivity profile or generation time

distribution.

— Mjk, the contact frequency matrix (§2.1).

— sjk(t), the effect of SVs on the contact structure. For the most part,

we assume SVs affect contacts among school children only, with

the parameter k measuring the percentage change in the

number of contacts among school children during SV periods

(see equation S2 in electronic supplementary material, B).

We also explored other options for modelling the effect of SVs

(see Discussion and electronic supplementary material, G).

— w(t), the effect of weather conditions on the probability of

infection. Two alternative forms are considered using either

AH or temperature. In both cases, the parameter d measures

the strength of the seasonal forcing related to AH or temperature

(see electronic supplementary material, B equation S3).

— lj, the susceptibility of individuals from age-group j which

is proportional to the probability of a susceptible indivi-

dual from age-group j being infected upon contact with

an infected person. The value of each of these terms by

itself is not very informative. However, by comparing these

values, the relative susceptibility of the different age-groups

can be assessed. We define blj ¼ lj=maxðljÞ as the relative

susceptibility of the age-groups.

— i0j , the daily number of imported influenza cases of age-group j.

Given the model formulation, one can obtain the formula for the

reproductive number on day t – R(t), measuring the mean

number of infections caused by one infected individual through-

out the individual’s period of infectivity, given the conditions on

day t (electronic supplementary material, C). In our case, there

are two external factors that cause the value of R to vary in

time: weather conditions and the effect of SVs. In addition, R is

reduced in time owing to the depletion of susceptible individuals

in the population.

2.2.2. Observation modelThe observation model links the incidence of influenza given by

the transmission model to the observed virological and serologi-

cal data. We start by formulating the relationship between

influenza incidence and the virological data. We set Zj(t) to be

the weekly incidence of influenza by summing up the daily inci-

dence of newly infected from age-group j—ij(t)—given by the

transmission model. Only a proportion of the individuals

infected in the full population are reported and assigned with

one of the clinical diagnoses in our dataset. This proportion

depends on both the fraction of monitored population in each

Table 2. List of the fixed model parameters.

parameter description value source

Pt (1 � t � d ) generation time distribution gamma distribution with d ¼ 7

days, mean of 2.7 days and

variance of 1.8 days

based on data collected during the

initial stages of the pandemic [2,3]

Mij (1 � i, j � 5) contact frequency matrix for the five

modelled age-groups in Israel

see electronic supplementary

material, table S4

generated using the eight POLYMOD

matrices (see electronic

supplementary material, §A.4)

Nj population size of age-group j at the end

of 2009

see electronic supplementary

material, table S6

Israel Central Bureau of Statistics [30]

uj fraction of monitored population (Maccabi

HMO members) in age-group j at the

end of 2009

see electronic supplementary

material, table S6

social security report on HMO

membership for 2009 [31]

jj fraction of individuals of age-group j that

declare to would consult their

physician when experiencing influenza-

like symptoms

see electronic supplementary

material, table S6

results of a survey conducted in Israel in

2011 [28]

i0j average daily number of imported cases

in age group J detected in Israel

during June 2009

see electronic supplementary

material, table S6

data collected during the pandemic’s

initial stages [2,3]

hj distribution of confirmed cases among

age-groups detected in Israel up to

beginning of June 2009

see electronic supplementary

material, table S6

data collected during the pandemic’s

initial stages [2,3]

xj baseline pre-pandemic fraction of

seropositive individuals in age-group j

see electronic supplementary

material, tables S2 and S6

results of serology tests taken prior to

the pandemic [18]

S0j initial fraction of susceptible individuals

in age-group j

set to either 12xj or 1 for all

age-groups

according to assumption regarding

protection of pre-pandemic

seropositives (§§2.2.3, 2.2.4)

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age-group—uj ; Nmonitored j=Nj—which is known (electronic sup-

plementary information, table S6), and a reporting rate—rj(t)—which is the fraction of infected individuals of age-group j who

consult their physician on week t and are diagnosed with eILI.

The number of infected in age-group j who were monitored and

diagnosed with eILI on week t is thus uj � rjðtÞ � ZjðtÞ. For the

main part of this paper, we assume constant reporting rates in

time rj, setting rj ¼ jj, where jj are the fraction of individuals from

age-group j that declare they would consult their physician when

experiencing influenza-like symptoms, according to the results of

a survey conducted in Israel in 2011 [32]. We performed a sensitivity

analysis to assess the effect of uncertainty regarding the used

reporting rates on the model outcome (electronic supplementary

material, F). We also examined several model variants to assess

the effect of varying reporting rates in time as well as varying frac-

tion of symptomatic cases in the different age-groups, on the model

outcome (electronic supplementary material, G,H).

The virological dataset consists of Tj(t)—the number of tests

performed on week t on individuals of age-group j; and

Tþj ðtÞ—the number of positive tests for the H1N1pdm influenza

strain. These tests were conducted almost exclusively on individ-

uals diagnosed with one of the clinical diagnoses included in our

eILI dataset, which is why we need to use this extended dataset

and not just the ILI dataset. The probability that an individual of

age-group j who is diagnosed with eILI on week t is infected is

given by pjðtÞ ¼ uj � rjðtÞ � ZjðtÞ=eILIjðtÞ. Therefore, the prob-

ability of obtaining Tþj ðtÞ positive cases out of the Tj(t) eILI

cases which are virologically tested is given by the binomial

distribution

Tþj ðtÞ � binomial TjðtÞ, pjðtÞ ¼uj � rjðtÞ � ZjðtÞ

eILIjðtÞ

� �: ð2:2Þ

Next, we formulate the relation between influenza incidence and

the serological data. These data consist of Yj(t)—the number of

serological tests performed on week t on individuals of age-

group j; and Yþj ðtÞ—the number of seropositive tests for

H1N1pdm. The state of seropositives in the population on

week t is assumed to be represented by the results of the serolo-

gical tests on week t þ 2, as there is a gap of two weeks on

average between infection and the rise of antibodies in the

serum to a detectable level known as the seroconversion period

[33]. The number of seropositives in the population on each

week is determined by the cumulative number of infected indi-

viduals. In addition, prior to the pandemic, some individuals

were already seropositive to the pandemic strain owing to

cross reactivity with previously circulating influenza strains.

From the serological tests of serums taken between April 2008

and April 2009, we have Y0j and Y0þj , the number of pre-

pandemic total/positive tests for each age-group (electronic

supplementary material, table S2). The ratio xj ¼ Y0þj =Y0j is

used as an estimate of the fraction of pre-pandemic seropositives

within the population.

Table 3. Maximum-likelihood estimates (MLE) and bootstrapped 95% CI for the parameters together with results obtained from fitting the basic model, undertwo alternative scenarios: assuming pre-pandemic seropositive individuals were protected completely or not at all from the pandemic strain (see §§2.2.2 and2.2.3 for details). We present here the values of the estimated relative susceptibility of the age-groups lj ; lj=max (lj ). The values of lj themselves aregiven in electronic supplementary material, table S8.

parameter/result description

MLE [95% CI] assumingpre-pandemic seropositiveswere fully protected

MLE [95% CI] assumingpre-pandemic seropositiveswere not protected

lj relative susceptibility of individuals of age-group j

compared to individuals of the most susceptible

age-group

l1 ¼ 1.0 [1.0 – 1.0]

l2 ¼ 0.81 [0.71 – 0.93]

l3 ¼ 0.69 [0.58 – 0.84]

l4 ¼ 0.40 [0.31 – 0.49]

l5 ¼ 0.21 [0.11 – 0.34]

l1 ¼ 1.0 [1.0 – 1.0]

l2 ¼ 0.68 [0.58 – 0.78]

l3 ¼ 0.63 [0.52 – 0.76]

l4 ¼ 0.38 [0.30 – 0.45]

l5 ¼ 0.17 [0.09 – 0.25]

d strength of seasonal forcing in the transmission rates

related to absolute humidity conditions

0.12 [0.09 – 0.15] 0.10 [0.08 – 0.13]

k percentage change in number of contacts among

school children during school vacations

20.18 [(20.28) – (20.08)] 20.20 [(20.33) – (20.10)]

I0 total number of infected individuals on 1 June 2009 3035 [2223 – 3785] 3789 [2932 – 4412]

AR overall attack rates AR1 ¼ 0.41 [0.35 – 0.47]

AR2 ¼ 0.48 [0.45 – 0.50]

AR3 ¼ 0.30 [0.26 – 0.33]

AR4 ¼ 0.13 [0.10 – 0.15]

AR5 ¼ 0.03 [0.02 – 0.05]

AR1 ¼ 0.42 [0.35 – 0.48]

AR2 ¼ 0.54 [0.49 – 0.56]

AR3 ¼ 0.31 [0.28 – 0.34]

AR4 ¼ 0.13 [0.11 – 0.15]

AR5 ¼ 0.03 [0.02 – 0.04]

R reproductive number at the initiation of the pandemic 1.43 [1.38 – 1.47] 1.40 [1.35 – 1.44]

negative log-likelihood 731 691

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Because it is difficult to assess to what extent being seroposi-

tive prior to the pandemic is a correlate of immunity to the

pandemic strain [34], two extreme alternatives were examined:

(i) there is complete correlation, so that a pre-pandemic seroposi-

tive individual could not be infected by the pandemic strain, and

(ii) there is no correlation, so that a pre-pandemic seropositive

has the same probability of being infected as a pre-pandemic

seronegative individual. In the first case, the probability of

obtaining Yþj seropositive tests out of a total of Yj tests on

week t þ 2 of the epidemic may be modelled as

Yþj ðtþ2Þ� binomial Yjðtþ2Þ,pjðtÞ¼xjþPt

t¼1 ZjðtÞNj

!: ð2:3aÞ

Here, xj is the probability an individual of age-group j was sero-

positive prior to the pandemic, and the attack rate Stt¼1ZjðtÞ=Nj is

the probability an individual of age-group j was infected by

week t of the pandemic, and so became seropositive by week

t þ 2. Because, in this case, we assume the two are mutually

exclusive their sum pj(t) is the probability an individual is sero-

positive by week t þ 2. In the second case, we model the same

probability as follows:

Yþj ðtþ2Þ� binomial Yjðtþ2Þ,pjðtÞ¼ xjþð1�xjÞ �Pt

t¼1 ZjðtÞNj

!:

ð2:3bÞ

Here, pj(t) (the probability an individual is seropositive by week

t þ 2) is the sum of xj and the attack rate Stt¼1ZjðtÞ=Nj multiplied

by the factor (1 2 xj) to exclude the proportion of infected indi-

viduals that were already seropositive before the pandemic

started. In addition, the two models differ in their initial

conditions regarding the proportion of susceptible individuals

at the start of the pandemic (see below).

2.2.3. Initial conditionsWe estimate I0—the initial total number of infected at the point of

time in which the fit to the data begins (1 June). Dividing the I0

infected individuals among the different age-groups was done

according to hj—the distribution of confirmed H1N1pdm cases

among the five age-groups up to that point in time [2,3]. How-

ever, we also investigated other possibilities to divide I0

(e.g. according to the size of each age-group) and found out

that generally the exact distribution of the I0 infected among

the age-groups has no major effect on the outcome. When

using equation (2.3a) in the observation model, the initial fraction

of susceptible individuals in each age-group (S0j) is set to 1 2 xj,

as it is assumed in this case that the pre-pandemic seropositives

are immune to the pandemic strain. When using equation (2.3b)

in the observation model, we set S0j ¼ 1, so that all individuals

have the potential of being infected by the pandemic strain. We

also investigated a model variant in which we estimated S0j,

while assuming the same probability of infection upon contact

for each age-group (see electronic supplementary material, G).

2.2.4. Model variantsWe have examined fitting several model variants, distinguished

by the incorporation of different assumptions affecting the

number of model parameters being fitted (see electronic sup-

plementary material, G), as well as the type of weather data

employed to modulate the transmission rates (AH or

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temperature). The results of the model variants were compared

using Akaike information criterion (AIC) [35].

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2.3. InferenceWe use a maximum-likelihood approach for estimating model

parameters. Our likelihood function is a conditional likelihood:

the probability of obtaining the observed results of the virologi-

cal and serological tests from the transmission and observation

models, given the clinical surveillance data. Conditional likeli-

hoods are used when part of the relevant data is not modelled

(see [36]). In our case, the eILI surveillance data, which depend

also on the circulation of other viruses leading to ILI symptoms,

are not modelled, so that the relevant likelihood is that of the vir-

ological and serological data conditional on the eILI data, which

are used to compute the probability pj(t) of a virological test

being positive (see equation (2.2)). Thus, for a set of parameter

values c ¼ fd, k, l1::5, I0g the likelihood function is given by

the product of the two binomial probabilities described in

equations (2.2) and (2.3a or 2.3b):

LðcÞ ¼Ymj¼1

YTt¼1

PðTþj ðtÞjTjðtÞ, pjðtÞÞ � PðYþj ðtþ 2ÞjYjðtþ 2Þ, pjðtÞÞ

¼Ymj¼1

YTt¼1

TjðtÞTþj ðtÞ

!pjðtÞT

þj ðtÞð1� pjðtÞÞTjðtÞ�Tþj ðtÞ

�Yjðtþ 2ÞYþj ðtþ 2Þ

!pjðtÞY

þj ðtþ2Þð1� pjðtÞÞYjðtþ2Þ�Yþj ðtþ2Þ

:

ð2:4Þ

Our conditional likelihood approach differs from previous works

integrating virological, serological and clinical data [19,22,24],

which incorporated the probability of observing the clinical

data into their likelihood function. The difficulty in incorporating

the ILI surveillance data into the likelihood stems from the fact

that, because the ILI data include a large proportion of non-

influenza cases, the number of ILI cases at each point in time

depends also on the current incidence of other viruses, which

are likely to follow their own dynamics. This introduces the

need to include many extra parameters in the likelihood function,

one for each time point: the number of non-influenza ILI cases,

or, equivalently, the fraction of ILI cases which are true influenza

cases. In [19], the availability of clinical and virological data from

other regions of England was used to provide an estimate of the

fraction of non-influenza-related cases within the clinical cases in

London throughout the pandemic. Such external data are not

available to us. In [22,24], the estimation of the many extra

parameters was avoided by assuming a prior distribution for

these parameters (in the prior distribution, the parameters are

independently distributed according to a gamma distribution),

and using this prior distribution to integrate out the extra par-

ameters, so that only the two parameters of the gamma prior

need to be estimated. In our approach, we opted to avoid the

problem of the unknowns introduced by the ILI data by not fit-

ting the clinical data altogether and therefore use the conditional

likelihood approach as described above.

In a previous work, we have focused on modelling the sto-

chasticity of the infection process described by the transmission

model and the stochasticity in the reporting of clinical diagnoses

[8]. Here, for the sake of simplicity, the stochasticities involved in

these processes are ignored, and the likelihood function is based

on the stochasticity of the sampling processes involved in the lab-

oratory testing, as it is assumed that the latter are dominant,

owing to the small populations involved in these data. This

assumption was verified by testing the fitting procedure on

simulated data incorporating process and observation noise

(see electronic supplementary material, D).

An optimization program written in Matlab was used to find

the maximum-likelihood parameter estimates. The program used

a combination of a nonlinear optimization algorithm (fminunc

function) and a simulated annealing algorithm (simulannealbnd

function). To ensure that the global maximum is found, the pro-

gram was run numerous times, each time using a random set of

initial parameter values. The program was tested on simulated

data to verify the identifiability of the model parameters, and

was found to work well with these data (electronic supplemen-

tary material, D). 95% CI for the parameter estimates were

calculated using parametric bootstrapping (electronic sup-

plementary material, D) and likelihood profiles (electronic

supplementary material, E). We also conducted sensitivity ana-

lyses to assess the effect of uncertainty related to our fixed

parameter values on the outcome of the model fit (electronic

supplementary material, F).

2.4. Testing the effect of vaccinationThe transmission model used in the data fitting (equation (2.1))

does not include vaccination as the vaccination stocks for the

pandemic strain reached Israel late in the epidemic, and the vac-

cination coverage attained was too small to have any tangible

impact on the epidemic (see electronic supplementary material,

figure S12). Nevertheless, after fitting the model to the data, we

wish to test the hypothetical effect of vaccination using the

model. For this purpose, equation (2.1b) of the transmission

model was extended to incorporate vaccination in the following

manner:

SjðtÞ ¼ Sjðt� 1Þ � ijðtÞ � vjðtÞ � z: ð2:5Þ

Here, vj(t) is the number of vaccinations given to individuals

of age-group j on day t and z is the efficacy of the vaccine

which was set to 70% [37]. In testing the effect of vaccination,

the transmission model parameters were set to the maximum-like-

lihood estimates as obtained by the basic model fit (table 3).

Assuming vaccine coverage C in the total population, we searched

for the optimized allocation of vaccinations within the different

age-groups—cj ¼ SjvjðtÞ � ðSjcj ¼ C � SjNjÞ—that minimizes the

total attack rate, and evaluated the effect of the vaccinations on

the pandemic dynamics. We either assumed that the vaccinations

were given immediately at the beginning of the pandemic

vjðtÞ ¼cj t ¼ 10 t . 1

�� �, or examined other scenarios in which vac-

cination is conducted later in the pandemic and the allocation of

the vaccines is spread evenly over a period of time (e.g. we set

vjðtÞ ¼cj=30 92 � t � 121

0 otherwise

�to model allocation of vaccines

during September 2009). When examining the effect of vaccination

during the pandemic we need to revise equation (2.5) further in

order to take into consideration ‘wasting’ of vaccines on individ-

uals who are already protected (after being infected) at the time

of vaccination (see electronic supplementary material, J).

3. Results3.1. Model fitIn the following, we present the results of fitting our basic

model variant, described in the Methods section, to the viro-

logical and serological datasets, while using AH as the

weather factor modulating the transmission rates. Table 3

gives the maximum-likelihood estimates and 95% confidence

intervals (CIs) obtained while assuming pre-pandemic sero-

positives were fully protected and while assuming they

were not protected any more than seronegative individuals

(§§2.2.2 and 2.2.3). The results obtained in the two scenarios

0

0.5

1.0(a)

0

0.5

1.0(b)

0

0.5

1.0(c)

wee

kly

posi

tive

eIL

I ra

tes

0

0.5

1.0(d)

June July Aug Sep Oct Nov Dec Jan Feb Mar0

0.5

1.0(e)

0

0.5

1.0( f )

0

0.5

1.0(g)

0

0.5

1.0(h)

wee

kly

sero

posi

tive

rate

s

0

0.5

1.0(i)

June July Aug Sep Oct Nov Dec Jan Feb Mar

June July Aug Sep Oct Nov Dec Jan Feb Mar June July Aug Sep Oct Nov Dec Jan Feb Mar

June July Aug Sep Oct Nov Dec Jan Feb Mar June July Aug Sep Oct Nov Dec Jan Feb Mar

June July Aug Sep Oct Nov Dec Jan Feb Mar June July Aug Sep Oct Nov Dec Jan Feb Mar

June July Aug Sep Oct Nov Dec Jan Feb Mar June July Aug Sep Oct Nov Dec Jan Feb Mar

0

0.5

1.0( j)

Figure 3. Observed (dots and bars) and estimated (curved lines) weekly rates of influenza-positive eILI cases (a – e) and weekly seroprevalence rates ( f – j) in thefive age-groups: 0 – 4 (a,f ), 5 – 19 (b,g), 20 – 44 (c,h), 45 – 64 (d,i) and 65þ (e,j). The observed data are the results of the virological and serological tests (given inelectronic supplementary material, tables S1 and S2). Binomial 95% CI for the observed data were calculated using the Clopper – Pearson method. The estimatedcurves were obtained from the model fit using equations (2.2) and (2.3b). (Online version in colour.)

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are very similar, the most notable difference being the attack

rate in school children estimated as a little lower in the first

scenario. As the differences between the two alternatives

are relatively minor, we focus from this point forward on

the results of the second scenario (i.e. assuming a pre-

pandemic seropositive has the same probability of being

infected as a pre-pandemic seronegative individual), which

obtained the higher likelihood among the two scenarios.

We found that employing AH as the seasonal driver yields

better results than using temperature, regardless of the

model variant used (see electronic supplementary material,

table S10). Models incorporating more parameters, such as

reporting rates that vary in time, managed in some cases to

obtain a better fit to the data in terms of their AIC score,

while not having any major impact on the parameter

estimates and the estimated pandemic dynamics obtained

from this simpler model (electronic supplementary material,

table S10). In addition, sensitivity analyses showed no

major impact on the outcome of the model fit resulting

from uncertainty in the fixed model parameters (electronic

supplementary material, table S9).

Figure 3a–e shows the fit between the fraction of weekly

eILI incidence that is influenza related obtained from

the model fit ( pj(t) in equation (2.2)) and the proportion of

influenza-positive eILI cases in the virological dataset (elec-

tronic supplementary material, table S1), including 95%

binomial CIs calculated using the Clopper–Pearson method.

Figure 3f–j shows the fit between the estimated seroprevalence

rates given by the model fit (pj(t) in equation (2.3b)) and the

weekly fraction of seropositives in the population according

to the serological tests (electronic supplementary material,

table S2), including 95% binomial CI. As the figures show,

for the most part, the estimates for all age-groups are found

within the 95% CI of the observed data. Figure 4a shows the

estimated influenza incidence rate given by the transmission

model in the five age-groups. The total estimated attack rate

is 32% (bootstrapped 95% CI 28–35%), considerably larger

than the attack rate of a seasonal influenza epidemic, which

is estimated at 5–20% [38]. However, we find large variation

in the attack rates between the different age-groups, with the

highest attack rates among school children and the lowest

among the elderly (figure 4b). We also find that susceptibility

to the pandemic strain decreased considerably with age

(figure 4c).

3.2. Effect of absolute humidity and school vacationsOur basic model employs the effect of both AH and SVs.

Results of fitting model variants that employ only AH or

only SV reveal that each of these factors by itself is able to

produce similar dynamics to the one obtained by the basic

model fit (figure 4a), with small but visible differences in

June July Aug Sep Oct Nov Dec Jan Feb Mar0

100

200

300

400

500

wee

kly

infl

uenz

a in

cide

nce

rate

per

10

000(a)

0−4 5−19 20−44 45−64 65+0

0.1

0.2

0.3

0.4

0.5

0.60.6

atta

ck r

ate

(b)

0−4 5−19 20−44 45−64 65+0

0.2

0.4

0.6

0.8

1.0

rela

tive

susc

eptib

ility

(c)

0−45−1920−4445−6465+

Figure 4. Results of the model fit (using equations (2.2) and (2.3b), see also table 3). (a) Weekly influenza incidence rate in the five age-groups. (b) Attack rate inthe five age groups with bootstrapped 95% CI. (c) Relative susceptibility of individuals in the five age groups with bootstrapped 95% CI. (Online version in colour.)

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the age-group incidence between the three model variants

(electronic supplementary material, figure S8). While the

two factors present a competing explanation for the obtained

pandemic dynamics, the comparison of the three model var-

iants using AIC (electronic supplementary material, table S10)

as well as the likelihood surface with respect to the two factors

(electronic supplementary material, figure S5) clearly indicates

that employing both factors is better in terms of fitting the

combination of virological and serological data. Furthermore,

tests conducted on simulated data confirmed that the two par-

ameters measuring the effect of AH and SV (d and k) are

simultaneously identifiable when fitted to the type and

amount of data available to us (electronic supplementary

material, D).

Given the maximum-likelihood estimates of the basic

model employing both factors, we analyse the impact of

each of the factors, AH and SV, on the pandemic dynamics.

Figure 5a shows the estimated influenza incidence rate for

the whole population according to this model with and with-

out the effect of AH and/or SV (i.e. setting the parameter d

and/or k values to zero). According to this exercise, without

the effect of AH or SV, the epidemic would have had a single

peak during August. Only the combination of AH and SV has

managed to delay the epidemic enough, so that the bulk

of the epidemic occurred during November–December.

Figure 5b shows the estimated value of the reproductive

number R as it changed over the duration of the pandemic

(electronic supplementary material, equations S4 and S5),

for the four configurations used in figure 5a. At the beginning

of the pandemic, R is estimated to be 1.4 (1.35–1.44). Varying

AH conditions were responsible for a reduction of up to 20%

during summer and an increase of up to 25% during winter

in the value of R (dashed line compared with solid line in

figure 5b), whereas SVs reduced R by 12% (dotted line com-

pared to solid line in figure 5b). The latter is attributed to an

estimated 20% (8–31%) reduction in contacts among school-

aged children during SVs. Whenever R declines below the

threshold R ¼ 1, the incidence of influenza cases starts to

decrease. The declines in the epidemic during the summer

and the beginning of October were the result of the combined

effect of SV and unfavourable AH conditions, whereas the

decline in the third wave during November is attributed by

the model to the depletion of the susceptible population.

In terms of total attack rates, SV alone (without the effect

of AH), would have reduced the attack rates from 34% to

27%. However, with the effect of AH, SV increased the overall

attack rates from 24% (had there been no vacations) to 32%.

That is, by delaying the epidemic and extending it into

winter, the overall epidemic size with SVs turned out to be

larger than it would have been otherwise. Similar results

can be seen in figure 6a which shows the outcome of the pan-

demic according to the model if the same SV periods were

shifted in time, so that the summer vacation starts either on

the first of June, July (the actual dates), August, September

or October. When vacations start at June or July, the bulk

of the epidemic is postponed to wintertime, and when

vacations start at September or later the big epidemic wave

occurs in summer. When summer vacations start in August,

there are two big waves, in summer and in late winter.

Once more, postponing the epidemic to winter increases the

overall epidemic size.

3.3. Effect of optimal vaccine allocationUsing the model with the estimated parameters given by the

maximum-likelihood estimates (table 3), we tested what

would have been the effect of different vaccination campaigns

on the pandemic. Figure 6b shows the results of a hypothetical

vaccination campaign during September 2009, after the first

wave of the pandemic. With up to 20% coverage in the

June July Aug Sep Oct Nov Dec Jan Feb Mar0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

R

(b)

June July Aug Sep Oct Nov Dec Jan Feb Mar0

100

200

300

400

500

600

700

800

wee

kly

infl

uenz

a in

cide

nce

rate

per

10

000

(a)

none AH SV AH + SV

0.20

0.25

0.30

0.35

atta

ck r

ate

none SV only AH only AH + SV

Figure 5. (a) The combined and separate effect of absolute humidity (AH) and school vacations (SV) on influenza incidence in the population as a whole. Using theparameter estimates obtained by the model fit incorporating both factors (equations (2.2) and (2.3b)), we project the dynamics of the pandemic without the effectof one or both factors by running the model while setting the relevant parameter(s) (k and/or d) to zero. The inset shows the overall attack rates in each of thescenarios. (b) The combined and separate effect of AH and SV on the reproductive number R in the four scenarios described in (a). (Online version in colour.)

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population as a whole, the optimal allocation strategy, which

reduces the total attack rate to a minimum, is to vaccinate

almost only school children (5–19). Vaccinating just 15% of the

population using the optimal allocation strategy (vaccinating

58% of all school children) would have completely mitigated

the winter wave of the outbreak.

To illustrate the advantage of vaccinating children over

adults, we have compared the outcome of vaccinating 20% of

the population, where all the vaccinations are either given to

children 0–19, young adults 20–44 or to adults 45þ years

old. In each case, we have calculated the overall attack rate

and its distribution among all five age-groups (electronic sup-

plementary material, figure S15). Vaccinating the children

outperforms vaccinating the adults by a huge margin. An

important finding is that vaccinating the children is advan-

tageous not only from an overall perspective, but also from

the adults’ perspective, as the attack rates in the adult age-

groups are smaller when the children are vaccinated compared

with when the adults themselves are vaccinated. This finding

was obtained while assuming the efficacy of the vaccine is the

same for all age-groups, whereas typically the efficacy is smaller

in the elderly [23,39], which would only strengthen this result.

4. DiscussionIn this study, we reconstructed the dynamics of the 2009 pan-

demic in Israel in five age-groups by fitting a dynamic model

to unique virological and serological datasets. Our model has

managed to fit the results of the virological and serological

tests in five age-groups well (figure 3), while estimating a

relatively small number of parameters. Although the model

was not fitted to the ILI data, the reconstructed dynamics fol-

lows a similar trend suggested by the ILI data (figure 1a),

with two small waves during summer and autumn followed

by a large wave during winter (electronic supplementary

material, figure S10). Analysis of the model shows that

these waves were triggered by the combination of varying

weather conditions and SVs. The estimated attack rate for

the pandemic in the whole population of Israel is 32%, with

a large variation in the attack rates among different age-

groups (figure 4b). Our estimation of the attack rate in the

different age-groups resembles results of previous studies

[19,20,22,33,40]. The distribution of infectives among the

age-groups is explained in part by the age-group contact

patterns, with school children having the highest number

of contacts and the elderly the least. However, the model

indicates increasing immunity with age to be another

significant factor that is essential in reproducing the observed

pattern of attack rates in the different age-groups (figure 4c).

This result reaffirms similar conclusions made by previous

serological studies [40–42] as well as modelling studies of

the pandemic in other countries [20,43–45]. Nonetheless, it

should be noted that the differences in susceptibility between

individuals in the 0–4 age-group and individuals in the 5–19

and 20–44 age-groups, as estimated by our model, might

June July Aug Sep Oct Nov Dec Jan Feb Mar Apr May0

50

100

150

200

250

300

wee

kly

infl

uenz

a in

cide

nce

rate

per

10

000

(b)

June July Aug Sep Oct Nov Dec Jan Feb Mar Apr May0

50

100

150

200

250

300

350

400

450

500

wee

kly

infl

uenz

a in

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rate

per

10

000

(a)

June July Aug Sep Oct0.1

0.2

0.3

atta

ck r

ate

summer vacation start month

JuneJulyAugSepOct

0 5 10 15 200

0.10.20.30.4

vaccine coverage (%)

atta

ck r

ate0%

5%10%15%20%

Figure 6. (a) Effect of timing of school vacations on the pandemic dynamics. Using the parameter estimates obtained by the model fit (equations (2.2) and (2.3b)),we project the dynamics of the pandemic if the timing of school vacations were shifted, so that summer vacation would start on 1 June, 1 July (actual), 1 Augustand so forth. The inset shows the overall attack rates in each of the scenarios. According to the model, postponing the epidemic to winter increased the overallepidemic size. (b) Effect of optimal vaccine allocation on the pandemic dynamics with hypothetical vaccination campaigns during September 2009, after the firstwave of the pandemic in Israel. Using the parameter estimates obtained by the model fit (equations (2.2) and (2.3b)), we project the dynamics of the pandemicassuming overall vaccine coverage of 0 – 20% of the population. The inset shows the overall attack rates in each of the scenarios. According to the model, with acoverage of 15% using optimal allocation consisting of vaccinating almost only school children, the outbreak would have been mitigated (see electronicsupplementary material, figure S14 for more details). (Online version in colour.)

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also be related to behavioural differences that render young

children more likely to be infected when contacting an

infectious individual.

We obtained an estimate of R ¼ 1.4 at the beginning of the

pandemic, which is in the range of other estimates from differ-

ent parts of the world (1.2 � R � 2.2) [10,12,16,19,20,22,24,44].

In the beginning of July, owing to the combined effect of SVs

and increasing AH levels, our estimate for R is just above 1

(figure 5b), which is in line with previous estimates made

based on incidence of confirmed cases at the initial phase of

the pandemic in Israel [3]. Our formulation of R takes into

account the existing prior immunity at the beginning of the

pandemic (as given by the estimated parameters lj, which

encompass the probability of being infected upon contact).

We can calculate the value of the basic reproductive number

R0 if we assume young children (0–4) were completely sus-

ceptible to the pandemic strain, by plugging l1 in place of

each lj in electronic supplementary material, equation (S1) at

time t ¼ 0, leading to an estimate of R0 ¼ 2.2. Notably, this

estimate of R0 for the pandemic is considerably lower than

an estimate (R0 ¼ 3.25) obtained from fitting 11 years of

seasonal influenza epidemics in Israel [8].

Our estimate of the impact of AH on the transmission of

influenza is similar to a previous estimate based on modelling

11 years of seasonal influenza epidemics in Israel [8]. As in the

seasonal influenza study, we have found that AH better

explains the seasonality of influenza in Israel than tempera-

ture. The estimated 12% reduction in R during SVs is around

the low range of the 14–50% reduction estimated by modelling

studies in other countries [10,12,13,15–20]. Obviously, the

effect of SVs is a social-related phenomenon that may vary

from population to population. In the basic model formulation

presented here, SVs can affect contacts among school children

only. In the electronic supplementary material, we present a

model variant in which contacts of school children with

other age-groups could also be affected during SVs. Using

this variant, we obtained a larger reduction in the number of

contacts among school children during SVs when compa-

red with the reduction estimated in the basic model variant

(40% compared with 20%), paired with an increase (of

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approx. 60%) in the number of contacts between school chil-

dren and other age-groups during these periods (see

electronic supplementary material, G). This increase may be

attributed to increasing within-families contacts during SVs.

This model variant has managed to improve the overall fit

of the model (as seen in its improved AIC score) while not

having any major effect on the rest of our results.

Our pandemic model suggests that the interplay of season-

ality in the transmission and mitigation efforts can lead to

some complex and unexpected outcomes, which should be of

consequence when considering the use of school closures as a

measure to mitigate an ongoing epidemic. The model demon-

strates that having SVs at the time they occurred actually

increased the magnitude of the pandemic in Israel, by delaying

the epidemic and extending it into winter when better con-

ditions for influenza prevail (figures 5a and 6a). The only

scenario in which closing the schools early in the epidemic

would have been advantageous in terms of reducing the

overall size of the epidemic is if the time gained by delaying

the epidemic were used to vaccinate a larger part of the popu-

lation. This result was not reported in other modelling works

that projected the course of the pandemic without the effect

of SVs [10,20]. These modelling works either did not model

the effect of weather conditions [20] or found the effect of

weather conditions in their settings to be minimal [10]. Without

the effect of weather, SVs would always reduce the magnitude

of the epidemic. Only when combined with seasonality

induced by weather (or some other seasonal forcing) can SVs

cause an increase in the overall attack rate.

The actual vaccination campaign against the pandemic

strain in Israel was ‘too little too late’ owing to the late arrival

of the vaccine to Israel to have any effect on the pandemic

dynamics (see electronic supplementary material, J). Using

the model, we investigated what would have been the effect

of vaccinations had they been delivered early enough to influ-

ence the epidemic. The model demonstrates that vaccinating

just 15% of the population during September 2009, after the

first wave of the outbreak, using an optimal vaccine allocation

which includes vaccination of mostly school children, would

have mitigated the winter wave of the outbreak entirely

(figure 6b). In an unrealistic scenario in which a vaccination

campaign could have taken place prior to the initiation of the

pandemic, a campaign covering 20% of the population with

an optimal allocation consisting of mostly school children

(78% of all school children) would have prevented the outbreak

completely (electronic supplementary material, figure S13). In

optimizing the allocation of vaccines, we considered only the

effect of vaccination on the total attack rate. However, we

have found that vaccinating children is advantageous over

vaccinating adults even from the perspective of reducing the

infection rates among adults (electronic supplementary

material, figure S15). This result is in line with previous model-

ling studies [23,46–49] and a randomized controlled trial study

[50] that have shown that vaccinating children can be an effec-

tive method to protect the elderly from influenza. In our case,

the combination of higher contact rates in children and lower

probability of infection in the elderly means that concentrating

on vaccinating the children would have been the optimal strat-

egy in reducing both morbidity and mortality. Because it

appears that increased protection for older individuals com-

pared with younger individuals is a hallmark of influenza

pandemics [51], vaccinating children to protect the whole

population should probably be the preferred course of action

in the case of a future pandemic.

Authors’ contributions. R.Y., G.K., L.S. and A.H. conceived and designed thestudy. R.Y. implemented the study and wrote the first draft. G.K., L.S.and A.H. revised and provided critical comments on the draft. E.M.and M.M. collected and analysed the virological and serological data.

Competing interests. We declare we have no competing interests.

Funding. R.Y. and A.H. were supported by the Israel National Institutefor Health Policy Research.

Acknowledgements. The authors acknowledge three anonymous refereeswhose helpful review helped to improve this paper. The authors alsoacknowledge Prof. David Earn, Prof. Pejman Rohani and Prof. JacobOleson for their helpful comments and suggestions on a preliminaryversion of the paper appearing in Dr Yaari’s PhD thesis.

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