Estimating ascertainment of COVID-19 cases
We use the monthly number of observed COVID-19 infections amongst travellers arriving into Australia from a given country to estimate the country’s true incidence rate. We account for Australia’s ascertainment in a similar way by estimating the true incidence rate with the observed incidence in Australian travellers arriving into New Zealand.
Let
γ be the true incidence rate of COVID-19 in a given country. Then, in a sufficiently large sample of the population, we expect to find
nγ infected individuals, where
n is the size of the sample, i.e. the arrivals from the given source country. Assuming that the number of infected individuals follows a Binomial distribution with unknown parameter
γ, the maximum-likelihood estimate of
γ is given by
\(\hat {\gamma }=x/n\), where
x is the number of infected individuals in the sample population. We construct the 95% Agresti-Coull interval [
16] to ensure that the interval falls within the parameter space. The estimated incidence rates and their 95% confidence intervals are shown and compared to those reported by the ECDC in Additional file
2.
In addition to adjusting the ECDC data to account for under-reporting, we estimate the disease onset date of all reported cases. From the data published in a recent study [
17], we find that the delay between a case showing symptoms and being reported follows a Gamma distribution. To adjust the data we subtract
X days from the date of report for each recorded COVID-19 case, where
X∼ Gamma(1.76, 4.47).
The importation model
The importation model requires as input the date of arrival into the country under investigation (in our case Australia), the country of stay prior to arrival, the duration of the overseas stay, daily incidence rates of COVID-19 in the country of origin and the lengths of the latent and infectious periods.
Upon arrival into Australia each individual is required to fill in an arriving passenger card, recording the country of stay prior to arrival into Australia, the length of stay and the date of arrival amongst other information. The length of stay is recorded in the following ranges: less than a month, one to three months, three to six months, six to twelve months and more than twelve months. Since our model requires the exact length of stay, we draw uniformly at random from the range recorded for each individual.
Country-level daily incidence rates of COVID-19 are reported by the ECDC and adjusted to account for under-ascertainment and disease onset dates as described in the previous sub-section.
If a traveller is not infected with COVID-19 upon return, the traveller either never contracted the disease or contracted the disease and recovered. The probability of not contracting the disease is given by
$$ q_{c} = \prod_{d=d_{a}}^{d_{r}}(1-\beta_{d}) $$
(1)
where \(\phantom {\dot {i}\!}\beta _{d}=1-e^{-\gamma _{d}}\), γd is the incidence rate of COVID-19 in the country of origin on date d, da is the date the traveller arrives in the source country and dr is the date of return to, in our case, Australia.
The probability of recovering before the arrival date is given by
$$ \begin{aligned} q_{r} = \left\{ \begin{array}{ll} 1 - \prod_{d=d_{a}}^{d_{c}} (1-\beta_{d})& \text{if}\quad d_{a}< d_{c}\\ 0&\text{otherwise}, \end{array}\right. \end{aligned} $$
(2)
where dc denotes the date n−1 days prior to the arrival date and n is the sum of the latent and the infectious period.
The probability of being infected upon arrival is then given by
$$ p = 1 - (q_{c} + q_{r}). $$
(3)
Our model also considers in-flight transmission of the disease. To transmit the disease to others, the infected individual must be within the infectious period. If an individual is not infectious while travelling, the individual either never contracted the disease, contracted the disease and recovered or contracted the disease and is within the latent period. The probability of being within the latent period is given by
$$ q_{l} = \left\{ \begin{array}{ll} \left[\prod_{d=d_{a}}^{d_{l}}(1-\beta_{d})\right]\!\left[1\,-\,\prod_{d=d_{l}}^{d_{r}}(1-\beta_{d})\right]& \text{if}\quad d_{a}< d_{l}\\ 0&\text{otherwise}, \end{array}\right. $$
(4)
where dl denotes the date l days prior to the arrival date and l is the length of the latent period.
The probability of being infectious while travelling is then given by
$$ r = 1 - (q_{c}+q_{r}+q_{l}). $$
(5)
Following recent studies of the infectiousness profile of COVID-19, we set the infectious period to eleven days, beginning three days prior to the onset of symptoms [
18,
19]. The incubation period describes the time between contracting a disease and showing symptoms. There is general agreement that the incubation period of COVID-19 is between five and six days [
20‐
22] and is approximately three days longer than the latent period [
18]. We draw the length of the incubation period from a log-normal distribution with mean equal to 1.621 and standard deviation equal to 0.418 [
21]. To find the latent period we subtract three from the incubation period.
The number of individuals that an infectious traveller infects while on a plane is drawn from a Poisson distribution with mean
λ=
tR0/
s [
23], where
R0 denotes the basic reproduction rate,
s is the length of the infectious period and
t is the duration of the flight. We set
R0=14.8, following the results presented in a study that estimates the basic reproduction rate of COVID-19 on a cruise ship [
24]. We assume
R0 on a ship to be similar on a plane where the population is almost fully mixed.
The expected number of importations within a given time period is then given by
$$ I = \sum_{i} p_{i} + X\sum_{i} r_{i}, $$
(6)
where X∼ Poisson (λ), pi is the probability that individual i is infected and ri is the probability that individual i is infectious during the flight. The sums run over all individuals who arrive during the period of interest.