Empirical datasets
In order to quantify our statistical model, we used data on imported laboratory-confirmed cases of MERS as of 26 June 2015 (the latest date on which our data analysis was conducted), especially focusing on the date on which the first diagnosed case arrived in each importing country. The date of entry of imported MERS cases is hereafter referred to as the arrival time and the corresponding information was retrieved from secondary data sources including the European Centre for Disease Prevention and Control (ECDC) [
19] and the World Health Organization [
20]. Since it was sometimes difficult to determine if a case was the result of an importation event instead of local transmission (e.g., spillover event or exposure to an undiagnosed case), original case reports were also tracked, especially among diagnosed cases in Middle East countries [
21‐
23]. Qatar and the Kingdom of Saudi Arabia were excluded from countries at risk of importation, because indigenous cases with the history of exposure to dromedary camels have been recurrently reported after the identification of the first case in 2012.
In addition to the arrival time of MERS case importations; three pieces of further information were retrieved. First, weekly incident counts of MERS in Saudi Arabia [
24] were used to mirror the force of infection among travelers. Second, the number of flight routes between pairs of countries was obtained from the airline transportation network data. The total number of flight routes between each pair of countries has an approximate dimension of 3 times 4,600 (or with 230 nodes and 4,600 edges) and was obtained from the Global Flights Network [
25] derived from the OpenFlights database as on 10 November 2014 [
26]. Third, dichotomous data to identify the major religion of each country which is in common with Saudi Arabia and Qatar was obtained from the literature [
27]: a country in which more than 30 % of the population is Muslim was defined as a Muslim majority country.
Building risk models
Here we describe the proposed model aimed to predict the risk of importation in each country. Let
F
j,t
be the cumulative distribution function representing the probability that MERS has already been imported to country
j by discrete day
t. The day
t = 0 corresponds to the date of the illness onset of first identified MERS case, and throughout the present study, we set 3 September 2015 as day zero (i.e. the date on which the initially identified case experiences symptoms) [
28]. Although a few earlier cases were confirmed as MERS by inspecting laboratory data a number of days after their dates of death, dates of illness onset among deceased cases were unavailable and had to be discarded when calculating the arrival time. Using
F
j,t
, the probability that a country
j has not yet imported MERS by day
t is
$$ {S}_{j,t}=1-{F}_{j,t}. $$
(1)
The daily risk of importing MERS in country
j on day
t is defined as
$$ {\lambda}_{j,t}=\frac{F_{j,t+1}-{F}_{j,t}}{S_{j,t}}. $$
(2)
Thus, we have
$$ {S}_{j,t}={S}_{j,0}{\displaystyle {\prod}_{k=0}^{k=t-1}\left(1-{\lambda}_{j,k}\right)}. $$
(3)
We parameterized the daily risk
λ by examining the statistical performance of different types of model parameterizations. In all models that we examined, we use the so-called “effective distance”, initially proposed by Brockmann and Helbing [
29]. The metric is derived from the airline transportation network, originally based on itinerary data, by using the transition matrix and length of paths between countries. The effective length of a path {
n
1,
n
1, ⋯,
n
L
} is given by
$$ L- \log {\displaystyle {\prod}_{k=1}^{L-1}{P_{n_{k+1}}}_{n_k}}, $$
(4)
where
P
ji
denotes the conditional probability that an individual that left
i moves to
j. (Note that ∑
j
P
ji
= 1). Assuming that the number of passengers is identical among all international flights, the transition matrix is calculated as
\( {P}_{ji}=\frac{m_{ji}}{{\displaystyle {\sum}_k}{m}_{ki}} \), where
m
ki
is the number of direct flights from
i to
k per unit time derived from open source data [
25]. Finally, the effective distance
m
j
of a country
j from Saudi Arabia is calculated as the minimum of the effective lengths of all paths that go from Saudi Arabia to the country
j. The effective distance, as calculated from the abovementioned process, has been known to exhibit strong linear correlation with the arrival time of SARS and H1N1-2009 across the world [
29].
Assuming that the effective distance is a critical indicator of the risk of disease spread, the simplest model 1 that we examined was parameterized as
$$ {\lambda}_{j,t}=\frac{k}{m_j}, $$
(5)
where
k is a constant. Namely, the hazard is an inverse function of the effective distance. As an alternative model, the information of Muslim majority countries is added, labeling corresponding countries at greater risk as compared with other countries, because countries sharing the religion with Saudi Arabia may be at greater risk of exposure to cases (e.g. through Hajj). As a consequence, a linear weight
α is given on Muslim majority countries, i.e.,
$$ {\lambda}_{j,t}=\left\{\begin{array}{c}\hfill \frac{\upalpha k}{m_j}\kern1.5em \mathrm{if}\ j\ \mathrm{is}\ \mathrm{Muslim}\ \mathrm{country},\hfill \\ {}\hfill \frac{k}{m_j}\kern1.5em \mathrm{otherwise}.\kern5.25em \hfill \end{array}\right. $$
(6)
In models 3 and 4, we additionally use the incidence data of MERS in Saudi Arabia over time. It is natural to assume that the daily risk of importation is proportional to the force of infection in Saudi Arabia, and thus, the incidence of MERS, i.e.,
$$ {\lambda}_{j,t}=\frac{k}{m_j}{I}_t, $$
(7)
as the model 3, where
I
t
is the incidence of MERS in Saudi Arabia on day
t. Since the original incidence data were recorded weekly (while our model is written on the daily basis), the weekly incidence was transformed to the daily data assuming uniform distribution of incidences within each week. Model 4 incorporates both of abovementioned factors into the model, i.e.,
$$ {\lambda}_{j,t}=\left\{\begin{array}{c}\hfill \frac{\upalpha k{I}_t}{m_j}\kern1.5em \mathrm{if}\ j\ \mathrm{is}\ \mathrm{Muslim}\ \mathrm{country},\hfill \\ {}\hfill \frac{k{I}_t}{m_j}\kern1.5em \mathrm{otherwise}.\kern5.25em \hfill \end{array}\right. $$
(8)
Statistical estimation and assessment
To estimate model parameters, a maximum likelihood method was employed. For the countries which have already imported MERS by 26 June 2015, we used the arrival time
t
j to fit the probability mass function of time at which the first importation event occurs, given by the product of
\( {\lambda}_{j,{t}_j} \) and
\( {S}_{j,{t}_j} \). Countries that have not imported MERS cases were dealt with as the censored observation. The total likelihood was
$$ L\left(\boldsymbol{\uptheta}; {\mathbf{t}}_a\right)={\displaystyle {\prod}_{j\in I}{\lambda}_{j,{t}_j}{S}_{j,{t}_j}{\displaystyle {\prod}_{j\in U}{S}_{j,{t}_m},}} $$
where
I is the set of index of countries which imported MERS at arrival time
t
j and
U is the set of index of countries which are MERS-free by the date of analysis of 26 June 2015 (
t
m = 1039 days after MERS onset). Assuming that the dichotomous information of Muslim majority country was always available, the penalized likelihood is comparable between models 1 and 2 and also between models 3 and 4. We calculate the Akaike Information Criterion (AIC) for these comparisons [
30].
Once the risk model was quantified, we assessed the diagnostic performance of our model in predicting the risk of importation by employing the receiver-operating characteristic (ROC) curve and measuring the Area under the curve (AUC) [
31]. For each model, the optimal cut-off value of estimated risk was calculated in predicting the importation as on
t
m =1039 days using the Youden index, and sensitivity and specificity were estimated. In addition, a prediction of the risk of importation across countries for 3 years since the emergence (
t = 1095) was computed for illustration.