### Hypothetical setting

To clarify the optimal length of quarantine, we first consider a hypothetical setting where infected travellers are flying from a nation with an epidemic (somewhere in Asia, given the data on the origin of seasonal influenza [13]) to a disease-free small island nation (e.g., New Zealand or smaller South Pacific and Caribbean islands). Specifically, we consider a situation when the disease-free country is fortunate enough to be informed about the possible emergence of the influenza pandemic at the source, sufficiently in advance of its arrival to implement border control measures. Given that the possible emergence is still uncertain and very recent news, we assume that the disease-free island nation is not ready or willing to completely shut down all its airports, but that quarantine is immediately instituted at the border. Before closing all the airports we assume that the island nation still permits the arrival of 20 aircraft with a total of 8000 incoming individuals (i.e., each with 400 individuals including airline staff on board) who were potentially exposed to influenza at the source country or on the aircraft. For this population of travellers we explore the question – how long should we place them in quarantine?

We assume that all incoming individuals are placed into routine quarantine on arrival in the island nation and are monitored for onset of symptoms during the quarantine period. We also assume that all infected individuals who develop influenza symptoms are successfully detected (e.g., through self-report questionnaires, reporting by ground staff, specific interview assessment by trained health personnel and/or thermal scanning). The impact of imperfect detection on the effectiveness of quarantine is examined in the Appendix. Optimistically, symptomatic cases are assumed to be immediately isolated in a designated facility at symptom onset, and assumed not to result in any secondary transmissions [14]. Similarly, those who developed symptoms en-route are also assumed to be successfully isolated upon arrival (and we ignore these individuals in the following analyses as the detection is owing to the entry screening). We assume that quarantine security would be fully effective and that no secondary transmission would occur in the quarantine facility. Successful detection during quarantine relies largely on onset of influenza-like symptoms, but, as a possible option, we also consider adding rapid diagnostic testing to improve the sensitivity of case detection.

### Epidemiologic characteristics of influenza

To theoretically and quantitatively examine the effectiveness of quarantine, we use several parameters describing the epidemiologic characteristics of seasonal influenza – which we then use for considering pandemic influenza. The most important of these characteristics is the cumulative distribution of the incubation period (i.e., the time from infection to onset) of length *t*, *F*(*t*). The incubation period has been very useful in suggesting the optimal length of quarantine for many diseases [15], because arbitrarily taking the 95th or 99th percentile point as the quarantine period could ensure the absence of symptomatic infection with probability of 95% or 99% [12, 16–21]. However, it is difficult to directly apply this concept to influenza [12], because the conditional probability, *α*, of developing symptomatic disease (given infection) has been suggested to be 66.7% [22, 23], and detection through quarantine is not relevant for asymptomatic infected individuals who account for the remaining 33.3%. Thus, we consider the effectiveness of quarantine as the reduction of the risk of introducing "infectious" individuals into the community and, thus, additionally use the cumulative distribution of the generation time (i.e., the time from infection of a primary case to infection of a secondary case by the primary case) of length *t*, *G*(*t*). Further, to simulate the key ripple benefit of quarantine (the predicted number of secondary transmissions caused by released infectious individuals), we assume that the reproduction numbers of symptomatic (*R*
_{s}) and asymptomatic cases (*R*
_{a}), i.e., the average numbers of secondary transmissions caused by a single symptomatic case and an asymptomatic case are 2.0 and 1.0, respectively. The basic reproduction number, *R*
_{0}, is therefore *αR*
_{s}+(1-*α*)*R*
_{a} = 1.67 which corresponds to an estimate in a previous study [24]. Moreover, the estimate is also within the estimated range of community transmission in another study which explored various historical data [25].

Distribution of the incubation period, which was assumed to follow a gamma distribution, was extracted from a published dataset [26]. Since the original data showed daily frequency of onset only, we fitted the cumulative distribution of the incubation period to the observed data, minimising the sum of squared errors. We did not identify more detailed data and note that the obtained frequency did not deviate much from outbreak data on an aircraft [27, 28], a historical study of Spanish influenza [15, 29], and from data in a published meta-analysis [22]. Similarly, the generation time was retrieved from a previous study of volunteers infected with influenza [22], which assumed that infectiousness is proportional to viral shedding, and we obtained the parameter estimates by minimising the sum of squared errors. A lognormal distribution was employed to model the generation time. Strictly speaking, the viral shedding curve alone does not inform the generation time, but our outcome measure (i.e., the probability of releasing infectious individuals) is reasonably analysed using virological data (as we are dealing with infectiousness), assuming that the frequency of contact is independent of time since infection. Furthermore, we favoured the use of this dataset as it would give a more conservative result since the right-tail is fatter than those assumed previously [30, 31].

### Effectiveness of quarantine

Although secondary transmission on aircraft is probably relatively rare due to the functioning of ventilation systems [32, 33], a previous transmission event has been reported in this setting [27]. Therefore, we use arrival time as the latest time of possible infection (i.e., *t* = 0). In other words, we conservatively argue the quarantine period as if all infected incoming individuals experienced this infection upon arrival. In reality, earlier acquisition of infection would increase the probability of non-infection after quarantine and therefore increase the effectiveness of quarantine. Although our worst case scenario potentially overestimates the optimal length of quarantine, a more realistic scenario requires the exact time of infection for all incoming infected individuals, which is in principle impractical (see Appendix for more detailed insights into this issue).

We considered the effectiveness of quarantine, *ε*(*t*), as a relative reduction of the risk of introducing infected individuals into the community as a function of time since infection *t*, i.e.,

\epsilon (t)=1-\frac{{r}_{1}(t)}{{r}_{0}(t)}

(1)

where *r*
_{1}(*t*) and *r*
_{0}(*t*) are the risks of releasing infected individuals into a new community in the presence and absence of the quarantine measure, respectively. Since all infected individuals enter the community without quarantine, we assume *r*
_{0}(*t*) = 1 for any *t*. If the risk in the presence of quarantine, *r*
_{1}(*t*), is regarded as the risk of releasing "symptomatic infected" individuals (regardless of infectiousness) after quarantine of length *t*, *r*
_{1}(*t*) is given by 1-*F*(*t*). Therefore, only the incubation period determines the effectiveness, i.e., *ε*(*t*) = *F*(*t*), which has been the fundamental concept in previous studies [12, 15–21]. However, we further consider the infectiousness for influenza, emphasising the importance of asymptomatic infection, because the proportion 100×(1-*α*) is as large as 33.3%. We thus regard the risk *r*
_{1}(*t*) as the probability of releasing "infectious" individuals into the community after quarantine of *t* days.

To comprehensively discuss this issue we decompose *r*
_{1}(*t*) into the sum of symptomatic and asymptomatic individuals (denoted by *r*
_{1s}(*t*) and *r*
_{1a}(*t*), respectively). For those who will eventually develop symptoms, the probability of release, *r*
_{1s}(t), is

*r*
_{1s
}(*t*) = *α*(1 - *F*(*t*))(1 - *G*(*t*))

where *F*(*t*) and *G*(*t*) are, respectively, the cumulative distributions of the incubation period and generation time. Because of the absence of adequate data, we assume independence between the incubation period and generation time, which most likely yields conservative estimates of the effectiveness (compared to that explicitly addressing dependence between these two distributions). For those who remained asymptomatic throughout the entire course of infection, the probability *r*
_{1a}(t) is

*r*
_{1a
}(*t*) = (1-*α*)(1-*G*(*t*))

because the incubation period is not relevant to the detection of asymptomatic infected individuals. Due to the absence of data, it should be noted that we assume that the length of generation time among asymptomatic individuals is identical to that among symptomatic cases, an assumption that has been used by others [24, 25]. As the assumption adds an uncertainty to the model prediction, we examine the potential impact of differing generation times between symptomatic and asymptomatic infected individuals (see Appendix). Consequently, the effectiveness of quarantine, *ε*(*t*), is given by subtracting *r*
_{1s}(*t*) and *r*
_{1a}(*t*) from 1: i.e.,

*ε*(*t*) = 1 - [*α*(1 - *F*(*t*))(1 - *G*(*t*)) + (1 - *α*)(1 - *G*(*t*))]

We further investigate the additional benefit of testing for the pandemic influenza virus using rapid diagnostic testing during quarantine. A key assumption made is that the currently available diagnostic tests would perform as well with the new pandemic strain of virus (and be supplied to the islands in time). We assume that the sensitivity (S_{e}) and specificity (S_{p}) of the rapid diagnostic test are 69.0% and 99.0%, respectively [34]. Since our effectiveness measure is conditioned on infected individuals, the risk of releasing infectious individuals in the presence of quarantine with use of rapid diagnostic testing is obtained by multiplying a factor (1-S_{e}) to *r*
_{1}(t) which represents a proportion of cases that are missed even following rapid diagnostic testing. Thus, we get the effectiveness *ε*
_{d}(*t*) as

*ε*
_{
d
}(*t*) = 1 - (1 - S_{e}) [*α*(1 - *F*(*t*)) (1 - *G*(*t*)) + (1 - *α*) (1 - *G*(*t*))]

Due to the absence of more detailed data, we assume that both the sensitivity and specificity of the rapid diagnostic test are independent of time since infection. Considering that the sensitivity may well decline in later stages of illness (by implicitly assuming that the diagnostic test is correlated with viral load), it should be noted that the results associated with equation (5) are probably most valid only for those in the early stage of illness (which is consistent with our particular interest in quarantine period). We stress that the estimated effectiveness *ε*
_{d}(*t*) for a long quarantine period (e.g., longer than 8 days) should be treated cautiously. Since the sensitivity S_{e} of asymptomatic infected individuals may be smaller than that among symptomatic cases (due to lower virus shedding titres among asymptomatic individuals), we examine the effectiveness of quarantine with differing S_{e} between symptomatic and asymptomatic infected individuals (see Appendix).

### Sensitivity analysis and preventive performance

We also examined the sensitivity of our effectiveness measures (4) and (5) to different lengths of quarantine and prevalence levels at the source by means of simulations. First, the sensitivity was assessed using the number of released infectious individuals after quarantine of length *t*. We examined plausible prevalence levels of 1%, 5% and 10% at the source, which respectively indicate that there were 80, 400 and 800 infected individuals among a total of 8000 incoming individuals. The highest prevalence, 10%, may represent transmission events within an airport of the country of origin or on an aircraft. The analysis was made by randomly simulating the incubation period (*F*), the generation time (*G*), the presence of any symptoms (*α*) and the sensitivity of the rapid diagnostic test (S_{e}) where *F* and *G* randomly follow the assumed gamma and lognormal distributions, respectively. The two dichotomous variables (i.e., the presence of symptoms and sensitivity of the rapid diagnostic test) were randomly simulated with uniform distributions (i.e., drawing random real numbers from 0 to 1) and using cut-off points at *α* = 0.667 and S_{e} = 0.690. The random sampling was performed for the number of infected individuals (80, 400 and 800 times) in each simulation, and the simulation was run 100 times for each length of quarantine and prevalence level. To show the ripple benefit, we also investigated the number of secondary transmissions caused by released infectious individuals. This estimate was achieved by further randomly simulating the numbers of symptomatic and asymptomatic secondary transmissions. Both numbers were assumed to follow Poisson distributions with mean *R*
_{s}(1-*G*(*t*
_{d}))(1-*F*(*t*
_{d})) and *R*
_{a}(1-*G*(*t*
_{d})), respectively, for each of the released symptomatic and asymptomatic infectious individuals after the quarantine of length *t*
_{d} days.

Finally, we examined the preventive performance of quarantine combined with rapid diagnostic testing. When the combination scheme is employed, those testing negative to the rapid diagnostic test following quarantine of length *t* would be the population of interest, as they are then released into the community. Let *p* be the prevalence level at the source (0 ≤ *p* ≤ 1). Among infected individuals (who account for 100*p*% of the travellers), the fraction of those who are detected or lose infectiousness following quarantine of length *t* (i.e., true positives) is (1-*r*
_{1}(t)). Of the remaining infected individuals *r*
_{1}(t), the fraction of those testing positive, S_{e}
*r*
_{1}(t), to the rapid diagnostic test are placed into isolation and, thus, are added to the true positives. Consequently, the remaining fraction (1-S_{e})*r*
_{1}(t) are false negatives and are released into the community (Figure 1). Among uninfected individuals (i.e., 100(1-*p*)% of the travellers), the length of quarantine does not influence the preventive performance (because they are not infected and their quarantine is irrelevant to the loss of infectiousness). Thus, among the total number of incoming travellers, the fractions (1-*p*)(1-S_{p}) and (1-*p*)S_{p} will be testing positive (false positives) and negative (true negatives), respectively, to the rapid diagnostic test. Consequently, positive predictive value (PPV) of quarantine combined with rapid diagnostic testing is

\text{PPV}=\frac{p[1-(1-{\text{S}}_{\text{e}}){r}_{1}(t)]}{p[1-(1-{\text{S}}_{\text{e}}){r}_{1}(t)]+(1-p)(1-{\text{S}}_{\text{p}})}

(6)

whereas negative predictive value (NPV) is

\text{NPV}=\frac{(1-p){\text{S}}_{\text{p}}}{p(1-{\text{S}}_{\text{e}}){r}_{1}(t)+(1-p){\text{S}}_{\text{p}}}

(7)

PPV measures the preventive performance of quarantine policy to correctly place infected individuals in quarantine (or isolation) during their infectious period (i.e., how efficiently are we placing infectious individuals in the quarantine facility, among a total of those who are diagnosed as positive either by quarantine of length *t* or rapid diagnostic testing). NPV measures the preventive performance of the release policy (i.e., how large is the fraction of true negatives among a total of those who are diagnosed as negative after the quarantine of length *t* and rapid diagnostic testing). We numerically computed both PPV and NPV for different prevalence levels (from 0–15%) and different lengths of quarantine (from 0 to 10 days). All analyses were made using the statistical software JMP ver. 7.0 (SAS Institute Inc., Cary, NC).