## Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease

The delay from the time when WHO declares that a new influenza strain capable of human-to-human transmission has emerged, until an epidemic is established in Australia depends on three components. They are:

- the delay until a recently-infected person travels to Australia;
- the potential delay arising from the fact that not every infected traveler initiates a transmission chain that takes off;
- the time it takes for an Australian transmission chain to gather momentum.

We now illustrate the sort of delay one expects when there are no interventions in place, taking each of the above factors into account. For these calculations we used “20 infectious cases on a single day” to indicate that transmission has gathered momentum in Australia. The choice of “20 cases” is made because both the theory of branching processes (Athreya and Ney, 1972) and our simulations indicate that when an outbreak reaches that number of infections on a single day then the growth of the epidemic has almost certainly has reached its exponential growth phase and proceeds deterministically, i.e. without a substantial chance component.

We assume that there are 10 concurrent infected cases in the source region when WHO declares that a new pandemic strain has emerged, and that the epidemic subsequently grows exponentially in the source region. The number of people within the infected source region is assumed reasonably small (5 million) and the number of travelers per day attempting to travel to Australia from the source region is set to 10, 100 and 400 per day unless stated otherwise. It will be seen that the inter-country transmission of infection occurs during the early exponential phase of the epidemic in the source country, hence the results are reasonably insensitive to the number of people in the source region. We assume a duration of travel between attempted departure and possible arrival of 12 hours, which approximates the travel time from south-east Asia where the next pandemic is considered most likely to be initiated. It is assumed that the aircrafts ventilation and filtration systems are functioning correctly, so although we allow infected travelers to transmit infection during the flight, it is assumed to occur at the same rate as at other times.

## Effects of *R*_{0} and traveler numbers</h2>]

In Figure 3.1 we show the probability distribution for the delay assuming an SEIR epidemic with *R*_{0 }=1.5 and 3.5 in both the source region and Australia. The graphs are shown for three volumes of travel, namely 400, 100 and 10 travelers from the source region into Australia per day. A comparison of the graphs for *R*_{0 }=1.5 and 3.5 illustrates that the delay decreases as *R*_{0 }increases, and that the magnitude of this change is noteworthy.

Comparing the distributions for the three travel volumes indicates how the delay depends on the amount of travel between the source region and Australia and, in particular, indicates the effect that limiting travel has on this delay.

A moderate reduction in the daily number of intending travelers departing the source country has a small though noticeable effect on the delay distribution. For example if we assume that *R*_{0}=1.5, then decreasing the numbers of intending travelers originating from the source country from 400 to 100 per day increases the median delay until 20 concurrent cases in Australia from 56 to 65 days. Reducing the number of intending travelers further to 10 per day extends this delay to 81 days (Figure 3.1). For *R*_{0}=3.5 the median delay is 17, 19 and 23 days for 400, 100 and 10 travelers per day originating from infected areas, respectively (Figure 3.1).

**Figure 3.1** The effect of *R*_{0} (1.5, 3.5) and daily traveler numbers (10, 100, 400 per day) on the distribution of the time delay until an epidemic reaches 20 concurrent cases in Australia following identification in the source country (assumed to occur when there are 10 concurrent cases in infected areas). Calculations assume the peaked infectiousness function and no border screening or early presentation.

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