It is important to anticipate the importation of the infection into Australia. Suppose Australia has no border control in place when WHO declares that a new influenza strain capable of human-to-human transmission has emerged. A decision on when Australia should respond has a large political component, but it is useful to have a way of assessing the risk of importation objectively. Here is one suggestion of how one might be guided in practice.
Consider a response as soon as overseas disease incidence and travel to Australia are such that
Prob(a recently-infected traveler arrives from the source region within the next 14 days) > 0.05.
(The duration 14 days and probability 0.05 are chosen here for illustration. Alternative values may be preferred, after careful consideration.)
A way to proceed from the daily disease incidence data in the source region to Inequality 1 is as follows: Observe that the number of recently-infected travelers into Australia on day t is random. Its probability distribution may be taken as Poisson with mean kt πt , where πt is the prevalence of recently-infected individuals in the source region and kt is the number of travelers from that region into Australia on day t.
Data on the travel volumes kt are available for the immediate future. The prevalence πt can be projected into the immediate future by fitting a model that specifies s exp(ρt) for the disease prevalence in the source region on day t. Estimates of α and ρ can be updated daily, as new data become available, and Australia would initiate response on the first day when s and ρ are such that Inequality (1) is satisfied.
The set of s and ρ values that trigger this response are shown in Figure 3.2, on the assumption that 1 per 12,500 of the individuals from the source region travels to Australia per day (e.g. the source region has 5 mil. inhabitants and 400 travel to Australia every day). In other words, Australia’s response is triggered on the first day that the point defined by the estimated µ and ρ values falls above the solid curve of Figure 3.2, if a critical probability value of 0.05 is used in Inequality (1). The contour corresponding to a critical probability value of 0.01 is also shown in Figure 3.2 (dashed line).
Figure 3.2 The probability that a recently-infected traveler arrives from the source region within the next 14 days exceeds 0.05 if the point (α,ρ) lies above the solid curve [e.g. the point marked (B) triggers a response], where α exp(ρt) describes the growth of the epidemic in the source region. The points (A) and (B) correspond to data in Figure 3.3(a) and Figure 3.3(b), respectively.
Figure 3.3 illustrates how α and ρ could be estimated in the event of an outbreak. The parameter ρ is obtained by fitting an exponential curve to the incidence data in the source country. From this incidence data, the prevalence is estimated by adding the incidences over the most recent five days, and dividing by the population size. The parameter α is the mean number of infected people traveling, so α = prevalence x number of travelers. For the purpose of converting the incidence data in Figure 3.3 into values of α and ρ that can be used in Figure 3.2, the number of travelers was assumed to be 400/day, and the total size of the population was 5 million.
The number of cases that have occurred overseas at the time when Australia responds is of interest because it indicates how much is known about the pathogen and its disease characteristics at that time. This number depends on the strategy used to trigger response and the number of travelers from the source region to Australia per day. In the above illustration, assuming that day 40 is the day on which response is triggered, there were 110 cases in the source region by the time response is triggered in Australia. A comprehensive study to determine the likely number infected by the time Australia responds would be useful once a criterion for responding, such as Inequality (1), has been chosen.
Figure 3.3 Assuming data on incidence is available from the source region for (a) the 30 days preceding the decision point, and (b) the preceding 40 days. Also shown is a curve of the form α exp(ρt ) fitted to each data set. Prevalence at any time is based on the incidences of new cases over the preceding five days. The estimated parameters for the above two illustrative data sets are
(a) α =0.00136, ρ =0.0459, and (b) α =0.0032, ρ =0.0623.
The corresponding two points (α,ρ) = (0.00136, 0.0459) and (α,ρ) = (0.0032, 0.0623) are marked (A) and (B), respectively, in Figure 3.2.
We now consider the effect of various border control measures on the delay until a local outbreak gathers momentum in Australia.