Impact of international travel and border control measures on the global spread of the novel 2019 coronavirus outbreak

Significance To contain the global spread of the 2019 novel coronavirus epidemic (COVID-19), border control measures, such as airport screening and travel restrictions, have been implemented in several countries. Our results show that these measures likely slowed the rate of exportation from mainland China to other countries, but are insufficient to contain the global spread of COVID-19. With most cases arriving during the asymptomatic incubation period, our results suggest that rapid contact tracing is essential both within the epicenter and at importation sites to limit human-to-human transmission outside of mainland China.

where n i is the number of airports within the country i that have a direct flight connection to/from China. After the enforcement of travel lockdown on January 23, we removed the flights from Wuhan and recalculated this weight. Thus, we utilize all airline data in the weights for arrivals before January 23, 2020, and the adjusted weight for arrivals on January 23,2020 and after. We do not consider the number of airline routes in our baseline analysis, as these likely have changed since the last update of the dataset in 2014. However, the number of airports within the country that have a direct flight connection to/from mainland China should be relatively comparable.
Estimation of the probability of travel.
In our model fitting, we used daily incidence of COVID-19 from December 8, 2019 to February 15, 2020 (3)(4)(5)(6)(7)(8)(9) . These cases were disaggregated into cases reported in Wuhan, Hubei, the rest of China, and internationally outside mainland China. For each infected case, we sampled an incubation period and the time from symptom onset to hospital admission from the maximum likelihood distributions. We first calculated the probability that infected individual i from the epicenter travels by plane over the course of the incubation period where t S is the minimum of the time of symptom onset and travel restriction (for region appropriate case), is the weight for all flights out of mainland China, and t I is the time of (t) ω infection (i.e. exposure). We do not account for the daily travel of an individual in their incubation period, as the data used for fitting is based on symptomatic individuals. We then calculated the daily probability of travel between symptom onset and first medical visit as: where t m is the minimum of the time of first medical visit and travel ban. This equation expresses the probability that a symptomatic individual i travelled on day t and not before. Thus, we calculated the expected number of symptomatic cases for a given day as: where T i,t = 1 if the date of symptom onset for individual i is t , otherwise T i,t = 0. The number of cases travelling prior to their incubation period and then exhibiting symptoms after arrival is: We repeated this process 1,000 times and calculated the average trend.
We assumed that all reported infected cases acquired infection within mainland China, thus neglecting any reported cases of human-to-human transmission outside of China in the fitting. We estimated the probability of travel from the time of infection to the time of first medical visit, accounting for the travel lockdowns in Wuhan (January 23) and Hubei (January 25) and allowing for exportation from the rest of mainland China. To estimate the probability of travel per day during the course of the outbreak, we fitted the expected number of symptomatic cases outside of China (Eq S5) to the reported daily incidence outside of China and the number cases exhibiting symptoms after arrival (Eq S6) to incidence outside of China based on the time of symptom onset.
For a probability of travel per day in the calibrated range 0.003 to 0.03 (increasing at increments of 10 -4 ), we computed the log-likelihoods for the daily incidence outside of China using a Poisson distribution and the average trend in exported daily incidence. Based on arrival dates and dates of symptom onset of 30 cases, 20 exhibited symptoms after arrival (Table S5). Thus, we weighted the log-likelihood for the number cases exhibiting symptoms after arrival (Eq S6) by ~67% and the log-likelihood for the expected number of symptomatic cases outside of China (Eq S5) by ~33%. We determined the maximum likelihood estimate of the probability of travel per day and constructed a confidence interval using likelihood weighted sampling. Using this approach, we estimated that the probability of travel per day is 0.0068 (95% CI: 0.0059 -0.0079). Assuming that individuals traveled up to the point of hospitalization, our estimate is 0.0060 (95% CI: 0.0052 -0.0070).

Exportation probability and impact of border control measures
We assumed that no infected individuals travelled from Wuhan after the travel lockdown enforced on January 23, 2020 (1,(10)(11)(12)(13)(14) . Additional lockdowns in other cities in Hubei followed the one in Wuhan, with some reports indicating that public transportation was halted by 2:00 pm on January 24, 2020 (1, 10-14) . As people potentially travelled out from these cities before 2:00pm on January 24, we restricted the travel for cases in Hubei, outside of Wuhan starting from January 25, 2020.
To estimate the exportation probability from mainland China, we used daily incidence of COVID-19 from December 8, 2019 to February 15, 2020, disaggregated into cases reported in Wuhan, Hubei, the rest of China, and internationally outside mainland China (3)(4)(5)(6)(7)(8)(9) . We sampled a time from symptom onset to first medical visit (after January 1) to generate a time of symptom onset for each SARS-CoV-2 case whose date of symptom onset was not specified.
For our baseline analysis, we sampled the incubation period and duration from symptom onset to first medical visit for each infected individual from the baseline distribution and fixed the probability of travel (see Estimation of the probability of travel section for further details). We calculated the daily probability that an infected case would be exported from mainland China. We make the simplifying assumption that the individual either travels or not ( e.g. they cannot fly within China the day before and then fly internationally). If the person does travel, we consider only the flights outside of China. For an infected case i , the probability of travel on day t was calculated as: where t I is the time the individual i was infected, t S is the time of symptom onset, t M is the time in which the individual seeks medical attention, and p ( t ) is the probability of traveling that day.
The expected number of cases exported outside of China for day t was calculated as: and the cumulative number of cases exported from China was calculated as: To estimate the number of individuals traveling during the incubation period at time t ( G t ), we evaluated D i,t only for the time between exposure and symptom onset. Thus, the number of symptomatic people that traveled outside China was calculated as F t -G t , which provides us with an upper bound for the effectiveness of screening for symptoms at the airport.
We calculated the daily probability that at least one of these infected cases is exported as: and the probability that at least one case has been exported since the start of the outbreak was calculated as: Using the daily probability that at least one of these cases is exported, we estimated the expected time of the first exportation event as: where We repeated this process 1,000 times and calculated the mean exportation probability under the baseline assumptions. We then repeated the entire process 1,000, bootstrapping the average duration of the incubation period, as well as the probability of travel per day, to calculate the 2.5 and 97.5 percentiles for our credible intervals.

Quarantine at the epicenter to curb exportation
We calculated the expected probability of exportation when individuals in the incubation period are quarantined after contact tracing. We estimate the expected probability that an infected case is exported as: where c is the time from infection to quarantine, is assumed to be the constant weight for flights ω out of mainland China, and is assumed to be a constant probability of travel per day (see p Estimation of the probability of travel section for further details)

Identification of individuals travelling in their incubation period through health questionnaires .
To identify individuals travelling in their incubation period through health questionnaires , we examined the time since their last exposure, which we assumed to be the time of infection. For a given duration inquiry q , the probability of identifying an individual travelling in their incubation period is: where i is the duration of the incubation period , is assumed to be the constant weight for flights ω out of mainland China, and is assumed to be a constant probability of travel per day (see p Estimation of the probability of travel section for further details).

Time to first infection after arrival in country
With no information of the generation time of the COVID-19, we used the serial interval (time from symptom onset in index patient to time in symptom onset in secondary case) as a proxy to estimate the time to the first infection event. For the distribution of the serial interval, we used a negative binomial (a discrete analogues of the gamma distribution estimated in published work on COVID-19). For a given incubation duration m in the secondary case, we can evaluate the average time to the transmission event from the index case to the secondary case where b (.) is the probability distribution function and B (.) is the cumulative probability distribution function for the serial interval. We then used these expected times and the distribution of the incubation period to provide an estimate for the average time from arrival to the first infection event.
Similarly, we calculated the expected time from arrival to symptom onset. For a given arrival time k , we calculated the average time from arrival to symptom onset by where i is the duration of the incubation period , is assumed to be the constant weight for flights ω out of mainland China, and is assumed to be a constant probability of travel per day (see p Estimation of the probability of travel section for further details) .
To evaluate the uncertainty in the time from arrival to the first transmission event, we bootstrapped from the distributions of the time from arrival to symptom onset (Eq S17) and the time from symptom onset in the index case to the first transmission event (Eq S16).

Validation
We validated our estimates of the probability of travel further by examining this value based on the amount of data included in the fitting process ( Figure S3). We obtained a comparable value for the probability of travel based on data of time of symptom onset after January 30, 2020, which is approximately 78% of the full dataset up to February 15, 2020.
We also examined the robustness of our estimates by evaluating the time of the first importation events using weights specified by the number of routes rather than the number of airports. We found that our likelihood estimate of the arrival time is robust to the change in the airline weight ( Figure S4, Table S4).