Conceived and designed the experiments: NB PC. Performed the experiments: PC DP. Analyzed the data: NB PC DP. Contributed reagents/materials/analysis tools: PC DP. Wrote the paper: NB PC.
The authors have declared that no competing interests exist.
The time delay between the start of an influenza pandemic and its subsequent initiation in other countries is highly relevant to preparedness planning. We quantify the distribution of this random time in terms of the separate components of this delay, and assess how the delay may be extended by nonpharmaceutical interventions.
The model constructed for this time delay accounts for: (i) epidemic growth in the source region, (ii) the delay until an infected individual from the source region seeks to travel to an atrisk country, (iii) the chance that infected travelers are detected by screening at exit and entry borders, (iv) the possibility of inflight transmission, (v) the chance that an infected arrival might not initiate an epidemic, and (vi) the delay until infection in the atrisk country gathers momentum. Efforts that reduce the disease reproduction number in the source region below two and severe travel restrictions are most effective for delaying a local epidemic, and under favourable circumstances, could add several months to the delay. On the other hand, the model predicts that border screening for symptomatic infection, wearing a protective mask during travel, promoting early presentation of cases arising among arriving passengers and moderate reduction in travel volumes increase the delay only by a matter of days or weeks. Elevated inflight transmission reduces the delay only minimally.
The delay until an epidemic of pandemic strain influenza is imported into an atrisk country is largely determined by the course of the epidemic in the source region and the number of travelers attempting to enter the atrisk country, and is little affected by nonpharmaceutical interventions targeting these travelers. Short of preventing international travel altogether, eradicating a nascent pandemic in the source region appears to be the only reliable method of preventing countrytocountry spread of a pandemic strain of influenza.
The emergence of a pandemic strain of influenza is considered inevitable
In this paper we demonstrate how the delay to importation of an epidemic of pandemic strain influenza may be quantified in terms of the growing infection incidence in the source region, traveler volumes, border screening measures, travel duration, inflight transmission and the delay until an infected arrival initiates a chain of transmission that gathers momentum. We also investigate how the delay is affected by the reproduction number of the emerged strain, early presentation of cases among arriving passengers, and reducing traveler numbers. As noted in previous simulation modeling
Some issues of the delay distribution, such as the natural delay arising in the absence of intervention and the effect that reducing traveler numbers has on this delay has been studied previously
This paper adds to previous work
Consider a region in which a new pandemic strain of influenza has emerged, and a region currently free from the infection. We refer to these as the
For an epidemic to take off in an atrisk country, a series of events need to occur. First, the epidemic needs to get underway in the source region. Second, an intending traveler needs to be infected shortly before departure. Third, the infected traveler must actually travel and successfully disembark in the atrisk country. Fourth, the infected traveler, or fellow travelers infected during the flight, must initiate an epidemic in the atrisk country with the infectiousness that remains upon arrival. Finally, the epidemic needs to reach a sufficient number of cases to begin predictable exponential growth.
International spread of the emerged pandemic strain of influenza may occur when a recently infected person travels. By ‘recently infected’ we mean that their travel is scheduled to occur within ten days of being infected. We assume that the number of individuals traveling from the source region to the atrisk country each day is known. The probability that a randomly selected traveler is a recentlyinfected person is taken to be equal to the prevalence of recentlyinfected people in the source region on that day. The incidence of infection in the source region is assumed to grow exponentially initially, with the rate of exponential growth determined by the disease reproduction number (the mean number of cases a single infective generates by direct contact) and the serial interval (the average interval from infection of one individual to when their contacts are infected) (
The process through which a pandemic is imported. (A) The prevalence in the source region, which determines the probability that a randomly selected traveler is infected at scheduled departure. (B)–(D) Density functions of the time since infection during the early stages of the epidemic in the source region for infected travelers (B) before and (C) after departure screening, and (D) after arrival screening for clinical symptoms. In (B), the step illustrates the probabilistic removal of travelers who have completed their incubation period. In (D), the distribution of time since infection in (C) will have shifted to the right by an amount equal to the flight duration, and cases incubated inflight may be detected by symptomatic screening, as will those symptomatic cases that were not detected previously. Screening sensitivity for this illustration is 60% on both departure and arrival. (E) Upon entering the community undetected, an infected traveler may initiate a minor (inconsequential) or major epidemic, depending on the characteristics of the disease and public health policy.
The time since infection of a recentlyinfected traveler is a key component of the calculations, because it affects the chance of positive border screening, the chance of inflight transmission and the infectivity remaining upon arrival in the atrisk country. The time since infection at the time of scheduled departure is random and the dependence of its probability distribution on the exponential growth rate of infection is illustrated by
It is assumed that individuals detected by departure screening are prevented from traveling. To be detected by screening an infected traveler must be symptomatic and positively screened. An individual is assumed to become symptomatic 48 hours after being infected (cf.
The instantaneous rate at which susceptible contacts are infected depends on the time since infection, and is described by an infectiousness function (
Travelers infected during flights of less than 12 hours duration are asymptomatic at arrival and will not be detected by screening. The probability that an arriving traveler who was infected in the source region is detected on arrival is computed from the distribution of the time since infection on arrival. This distribution is obtained from the curve in
Authorities are assumed to implement one of two control options when detecting an infected traveler by arrival screening. Under option one (individualbased removal), all passengers who test negative are released immediately and only passengers who test positive are isolated. Under the second option (flightbased quarantining), authorities prevent all passengers from dispersing into the community until the last person has been screened from that flight. Should any one passenger be detected as infected then all passengers will be quarantined, as previously recommended
Transmission chains can be initiated in the atrisk country by infected travelers who mix within the community upon arrival. Suppose now that a flight arrives with one, or more, infected passengers who mix within the community. We classify these infected arrivals into those who are ‘presymptomatic’ and those who are ‘symptomatic’ at entry. It is assumed that the ‘symptomatic’ infected arrivals do not recognize their symptoms as pandemic influenza and will not present to medical authorities. In other words, they spend the remainder of their infectious period mixing in the community. On the other hand, the ‘presymptomatic’ infected arrivals, including all individuals infected during flight, are assumed to mix freely in the community only from entry until they present to medical authorities after some delay following the onset of symptoms.
Not all infected travelers entering the community initiate a ‘major’ epidemic, even when the reproduction number (
The probability that a typical infective generates a local epidemic is computed by using a branching process approximation
We calculate the probability distribution of
Once successfully initiated, an epidemic may initially hover around a handful of cases before reaching a sufficient number of cases for its growth to become essentially predictable. As mentioned, 20 concurrent cases is our criterion for an epidemic to have gathered momentum. We determine the distribution of
For the illustrative purposes, we chose values of 1.5, 2.5 and 3.5 for
The probability that a recently infected traveler evades screening is substantial even if screening reliably detects symptomatic travelers (
Effects of border screening and early presentation. (A) The effects of screening sensitivity andon the probability of escaping detection on both departure and arrival during a 12 hour transit. (B) The effects of screening sensitivity and travel duration on the probability than an infected traveler escapes detection during transit and initiates an epidemic after arrival (assuming no other symptomatic individuals on the same flight are identified).
As the duration of travel approaches the disease incubation period, effective symptomatic screening substantially reduces the likelihood that a traveler evades screening and initiates an epidemic (
The delay contains a fairly substantial
Components of delay until initiation and effects of border screening. (A) The number of infected people successfully arriving and entering the community of an atrisk country (
Components of the delay in atrisk country following initiation. (A) Results of 10,000 simulations (bars) and fitted shiftedGamma distribution of delay time (
Without screening, the daily probability that an epidemic is initiated (
Although flightbased quarantining is effective in preventing the entry of infected travelers during the height of the epidemic, a substantial cumulative risk of initiation has already occurred before this from the handful of infectives that have slipped through undetected (
The natural component of the delay is highly sensitive to the disease reproduction number (
Effects of interventions on the total delay

Intending travelers (per day)  5^{th} Percentile  Median  95^{th} percentile 
1.5  10  63  83  103 
100  47  66  85  
400  38  57  76  
2.5  10  26  34  41 
100  20  27  35  
400  17  24  31  
3.5  10  19  24  28 
100  15  20  24  
400  12  17  22 
It is assumed that the pandemic is identified and declared when there are 10 concurrent cases in the source region attributed to humantohuman transmission, and that screening is applied at both departure and arrival. The time between screening events is assumed to be 12 hours and infected travelers are not isolated following the onset of symptoms.
The delay is quite insensitive to the rate of transmission inflight. For example, with
In general, the additional delay achieved by introducing nonpharmaceutical border control measures is generally small in comparison with the natural delay (
We have formulated a model of the importation of an infectious disease from a source region to an atrisk country that permits a comprehensive analysis of the effect of border control measures. Our results are most relevant to the early stage of a pandemic when most cases are contained within a single source region. Once the pandemic has spread to several countries, models with greater complexity and ability to more realistically model global mixing patterns
The nature of the next pandemic influenza virus, and particularly its reproduction number, is uncertain. If its reproduction number is low (
The additional delay from isolating individuals detected by border screening is merely a few days under most plausible scenarios, even if both departure and arrival screening is introduced and screening detects every symptomatic traveler. While the extra delay is more than quadrupled if flights with a detected case(s) are quarantined, the effect remains modest (weeks at most) and it is questionable whether the extra delay achieved warrants the disruption created by such a large number of quarantined passengers.
Inflight transmission is a commonly raised concern in discussions about the importation of an infection, so inclusion of inflight transmission is an attractive feature of our model. Events of substantial inflight transmission of influenza have been documented
Early presentation by infected arrivals not detected at the borders was found to add only a few days to the delay. To some extent this arises due to our assumption that presymptomatic transmission can occur, for which there is some evidence. In contrast, Ferguson
Of the border control measures available, reducing traveler numbers has the biggest effect on the delay and even then it is necessary to get the number of travelers down to a very low number. An equivalent control measure is to quarantine all arriving passengers with near perfect compliance.
Our results indicate that short of virtually eliminating international travel, border control measures add little to avoiding, or delaying, a local epidemic if an influenza pandemic takes off in a source region. All forms of border control are eventually overwhelmed by the cumulative number of infected travelers that attempt to enter the country. The only way to prevent a local epidemic is to rapidly implement local control measures that bring the effective reproduction number in the local area down below 1, or to achieve rapid elimination in the source region, in agreement with other recent studies
Estimating the daily probability of epidemic initiation
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We thank James Wood, Katie Glass and Belinda Barnes and an anonymous reviewer for helpful comments.