Table 4.

A list of the assumptions and characteristics of our model that give rise to the linear relationship between the importation rate and the mean outbreak size. The linear relationship is that I_tot = λ_vI₁, where I_tot is the mean total number of cases, λ_v is the importation rate, and I₁ is the mean number of cases that arise from one importation.

model assumption	example where the model assumption is violated	effect of violating the assumption on outbreak size
Mixing between individuals in the population is homogeneous.	A group of travellers, all of whom are infected, fail to self-isolate, but also travel everywhere together and contact all of the same people.	No matter what the size of the group, the resulting outbreak will be of similar size since the contacts of group members are redundant. Here, the mean outbreak size is not linearly related to the importation rate because a larger group would correspond to a larger number of importations, yet the resulting outbreak would not be much larger.
Homogeneous mixing means than an infected person is equally likely to contact every susceptible person in the population.	Mixing is non-homogeneous because group members are constrained to have contacts only among the same individuals as the other group members, and not all individuals in the population.
The number of susceptible people is relatively unchanged during the timeframe of interest.	The susceptible population is small, or infection control measures are few.	Infected individuals that arrive later will generate smaller infection chains due to fewer susceptible people to infect. Therefore, the total outbreak size cannot be calculated by summing the size of the outbreaks per importation, since the timing of the importation affects the outbreak size due to that importation.
The number of people an infected person contacts is unchanged during the timeframe of interest.	Waning compliance with public health measures; school re-openings.	As above, outbreak sizes per importation cannot be added to determine the total outbreak size because the timing of the importations affects the value of the outbreak size per importation.
Infectivity does not change over time.	Seasonality	See above.
model characteristic	a different characteristic	effect of considering the different characteristic
Few ‘prior’ cases: cases that are not attributable to importations (see Methods—Output variables).	High infection prevalence in the absence of importations.	The relationship between travel-related cases and the importation rate will be linear, but total infections are the sum of prior cases and travel-related cases, such that the linear relationship will not hold.
The quantity of interest is the mean outbreak size.	The quantity of interest is the median or a different quantile.	The linear relationship with the importation rate applies only to the mean outbreak size. As can be observed in table 2, the linear relationship does not apply to the median or 95% prediction intervals.