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

The reproduction number (*R*) is often used to reflect how infectious a disease is. We will, in part, use this quantity to assess alternative interventions to control an outbreak, because *R* is changed by control measures. The *basic* reproduction number (*R*_{0}) is the reproduction number when there is no immunity from past exposures or vaccination, nor any deliberate intervention in disease transmission. We refer to *R* as an effective reproduction number when there is some immunity or some intervention measures are in place.

It is useful to recall some of the characteristic of a reproduction number, because its interpretation is not always straightforward.

When individuals are homogeneous and mix uniformly, *R* is defined as the mean number of infections generated during the infectious period of a single infective. Individuals may differ in the number they infect, due to chance, but the mean number infected is *R.* Epidemics of an SEIR infection can not occur when *R* is less than 1 at the start of the outbreak and established outbreaks will fade out if either interventions maintain *R* below 1 or the susceptible part of the population has been depleted sufficiently to maintain *R* below 1.

The basic reproduction number may vary across locations because contact rates among people may differ due to differences in population density and cultural differences. The effective reproduction number may vary, as well, because the communities in different locations may differ in their level of immunity.

What *basic* reproduction number should we use in our assessment of proposed interventions against pandemic influenza? It may be that recent exposures to currently circulating strains of influenza, or vaccinations to protect against them, provide some immunity against a newly emerged strain; see Jordan *et al.* (1958), Spicer and Lawrence (1984), Mills *et al.* (2004). Then the appropriate reproduction number, with reference to which we should judge any proposed intervention in our community, is the effective reproduction number corresponding to the population as it is initially, complete with its initial level of immunity arising from exposure to influenza strains that have circulated previously but in the absence of any deliberate interventions. For convenience we shall refer to this baseline reproduction number as the basic reproduction number and denote it by *R*_{0} in this document.

There are two aspects of infectious disease transmission that *R* does not capture well. One is the rate of transmission in calendar time. To illustrate this, consider two SIR infections with the **same** *R*_{0}. Suppose that in one of these infections individuals are highly infectious over a short infectious period. For the other infection individuals are less infectious, but over a longer infectious period. Both will result in the same eventual attack rate, but the former epidemic will take off more quickly, will have a higher incidence at the peak of the epidemic and will be much shorter; see Section 2.4.

The other aspect that *R* does not capture well in general is *q,* the probability that an imported outbreak gathers momentum and becomes large, as distinct from fading out after relatively few people are infected. This probability is fundamental to the issues of containment and delaying a local epidemic, so we need to be mindful of this fact when assessing interventions on the basis of *R.*

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