Dataset and global airline network

The dates of first onset of mpox case for each location (including country, city and/or name of hospital) were extracted from open-access database developed in response to the multi-country outbreak [Reference Kraemer32, 33] at 5 September 2022. The data included 175 countries or territories. Two authors, RK and DY, independently checked the validity of data against official announcements from WHO and each governmental report. The extracted case information was then matched with the airport information based on the nearest neighbourhood approach. To construct the airline transportation network, we used the ADS-B exchange data [34]. All flights included in the ADS-B exchange data within a single day of 1 December 2019 (before the COVID-19 pandemic) was extracted. The dataset provides a graph of global travel information consisting of 1724 nodes (corresponds to each airport) and 21 704 edges with edge-weights (corresponds to direct flights between two airports with its PV. The PV on a certain flight was estimated as the reported (maximum) number of seats of the airplane. Then, the PV was multiplied by 0.93 for domestic travel and 0.69 for international travel [18].

Effective distance

To model the impact of airline network on mpox transmission, an idea of effective distance, which was introduced by Brockmann and Helbing [Reference Dirk and Dirk35] and frequently used in previous studies for forecasting the global spread of emerging infectious disease such as SARS, influenza H1N1-2009, MERS and COVID-19 [Reference Otsuki and Nishiura22, Reference Shi29], was estimated from the airline network. The effective distance is defined from the minimum distance on the adjacency matrix of the network, incorporating the PV-weighted path length and the degree of each node. In other words, the effective distance is a metric quantifying the network distance between each nodes with the weighted edge being proportional to the PV, irrespective of the physical distance between locations.

The effective distance, d _ij, between London Heathrow airport (ICAO code: EGLL) and the ith airport in the jth country is defined as the minimum length among all possible effective paths. The effective paths from Heathrow airport to the ith airport with a sequence of l transit airports {a _Heathrow, a ₁, …, a _l−1, a _i} is given by

(1)$$m_l^{( {i, j} ) } = 1-\log \left({\mathop \prod \limits_{k = 1}^{l-1} P_{k + 1, \;k}} \right), \;$$

where P _l,m denotes the transition probability matrix from the lth to the mth airport. Each element in P _l,m is estimated by $P_{l, m} = \left({w_{lm}/\left({\mathop \sum \nolimits_n w_{ln}} \right)} \right), \;$ where w _lm is the PV that moved from the lth to the mth airport. Lastly, d _ij is defined as the minimum effective paths, which is given by

(2)$$d_{ij} = \mathop {\min }\limits_l m_l^{( {i, j} ) }.$$

As we will discuss in the next section, since the network structure and its associated effective distance changed after the travel restrictions due to the change in the PV, the effective distance d _ij is dynamically changed in the assumed three scenarios described below.

Modelling with effective distance: hazard-based approach

We modelled the risk of importing mpox by estimating the survival probability in each country. Let T be a random variable indicating the survival time in each airport from the first case in the UK (6 May 2022) to importation at the airport. Also define the survival probability at time of t as F(t) = P(T > t) with the probability density function (pdf) of f(t) = −dF(t)/dt. t indicates day, and t = 0 is 6 May 2022. The associated hazard function at time of t for importation of mpox from the UK for the ith airport in the jth country is modelled in the form of Weibull regression model, which is given by

(3)$$\eqalign{\lambda _{ij}( t )& = \mathop {\lim }\limits_{{\rm \Delta }t\to + 0} \displaystyle{{P( {t \le T < t + {\rm \Delta }t{\rm \vert }T \ge t} ) } \over {{\rm \Delta }t}} = \displaystyle{{\,f_{ij}( t ) } \over {F_{ij}( t ) }} \cr & = \alpha p( {\alpha t} ) ^{\,p-1}\exp \left({\,f\left({\displaystyle{\beta \over {d_{ij}\;}}} \right) + {\boldsymbol X}_{\boldsymbol j}{\boldsymbol \gamma }} \right), \;}$$

where p > 0, α > 0, γ is a vector of regression parameters, β is also a regression parameter of interest to measure the impact of the effective distance, f() is a penalised smoothing spline function with the degree of freedom of 4, and X_j is a covariate vector. By using the parameter, p, in Equation (3), the Weibull distribution can flexibly model the survival probability even when it is non-convex shape. In addition, other distributions including log-normal and log-logistic distributions were compared and we confirmed that the Weibull distribution can provide the best performance in terms of AIC. This formulation makes the hazard function and the estimated median time of importation be proportional to the effective distance d _ij, which is consistent with Shi et al. (2021) and Otsuki and Nishiura (2016) [Reference Otsuki and Nishiura22, Reference Shi29]. The covariates X_j include: income per capita at 2022 [36], the proportion of working age at 2020 [37], the proportion of sexual minorities concealing their sexual orientation [Reference Pachankis and Bränström38], the proportion of MSM population [39, 40], Socio Demographic Index at 2019 [36] and WHO region (i.e. African Region (AFR), Eastern Mediterranean Region (EMR), European Region (EUR), Region of the Americas (AMR), South-East Asian Region (SEAR), Western Pacific Region (WPR)). Complete case analysis was considered, and thus airports with incomplete covariate information were excluded from the estimation.

The parameters were estimated by the maximum likelihood approach. Then, the future hazard function at 31 December 2022, was predicted by extrapolating the time variable t. In order to capture the importation risk from the UK, the following countries currently designated as endemic of mpox is not included in the parameter estimation: Benin, Cameroon, the Central African Republic, the Democratic Republic of the Congo, Gabon, Ghana, Ivory Coast, Liberia, Nigeria, the Republic of the Congo, Sierra Leone and South Sudan [2]. Further, to check the goodness-of-fitting in the model, we calculated the concordance index, which measures the agreement between an observed response and the predictor.

Lastly, we conducted the sensitivity analysis by weakening the assumption in the effective distance: i.e. since the effective distance strongly depends on the assumption that mpox cases has been spreading from the UK, we checked how our result would be changed when weakening this assumption (i.e. not specifying the starting point of the virus spread). More precisely, we used a ‘closeness centrality index’ on the airline network, which measures how attractive a certain airport is in the PV sense, instead of the effective distance in the model. The closeness centrality for the ith airport is defined as

(4)$$C_i = \displaystyle{1 \over {\mathop \sum \nolimits_{\,j\in S_i, \;\;j\ne i} w_{\,ji}}}, $$

where w _ji is the PV that moved from the jth to the ith airport and S _i is the set of airports that are connected to the ith airport. Then, instead of d _ij in Equation (3), C _i is used. C _i does not assume the starting point of the virus spread, and in that sense we do not make the assumption that the infection is spreading from the UK in this sensitivity model.

Hypothetical scenarios to estimate relative risk reduction due to travel restriction

To estimate the impact of travel restrictions on the importation risk, we calculated the (observed) cumulative risk at time of t, defined by

(5)$$R_{ij}^{{\rm obs}} ( t ) = 1-\exp \left({-\mathop \smallint \limits_0^t \lambda_{ij}( u ) du} \right).$$

Then we compared $R_{ij}^{{\rm obs}} ( t )$ with the cumulative risk of following hypothetical scenarios: (H1) Reduce the current PV from/to infected countries by 50%, (H2) Increase the current PV to ‘pre-COVID-19 level in 2019’ (cf. the current PV was assumed 93% for domestic and 69% for international travel, respectively, compared with the pre-COVID-19 level) [18], and further reduce the PV from/to infected countries by 90%, and (H3) Reduce the current PV to ‘the level at the most severe travel restrictions in 2021’ (i.e. the PV is assumed 61% for domestic and 27% for international travel, respectively, compared with compared with the pre-COVID-19 level), and further reduce the PV from/to infected countries by 50% [18]. Once the regression parameters in Equation (3) was estimated, the cumulative risks based on these scenarios can be calculated by plugging the scenario-specific d _ij into Equation (3): i.e. in a similar way with the calculation of $R_{ij}^O$, the cumulative risks in (H1)–(H3) are given by

(6)$$R_{ij}^h = 1-\exp \left({-\mathop \smallint \limits_0^t \lambda_{ij}( {u, \;\;d_{ij}^h } ) du} \right), \;$$

where h = 1, 2, 3 indicates scenarios (H1)–(H3), respectively, and $d_{ij}^h$ is the effective distance based on each scenario assumption. Lastly, we estimated the relative risk change as follow:

(7)$$\eqalign{&{\rm Relative \,risk \,change \,risk \,under \,the \,assumption \,of \,the \, {\it h}th \,scenario}\cr & \quad = R_{ij}^h /R_{ij}^O{\rm .}}$$

To measures the proportion of expected risk reduction between observed and the hypothetical scenarios the relative risk change was calculated at t = 239 (31 December 2022).