Sub- or supercritical transmissibilities in a finite disease outbreak: Symmetry in outbreak properties of a disease conditioned on extinction

This work is concerned with the transmissibility of a disease, on observation of an outbreak of limited size. When such an outbreak occurs, an accurate estimate of the transmissibility of the responsible pathogen is essential for an appropriate response to future outbreaks. Transmissibility is usually characterised in terms of the reproduction number, R, which is the mean number of new cases of infection produced by a single infectious individual. A subcritical reproduction number (R < 1) guarantees that an outbreak will eventually die out of its own accord. By contrast, a supercritical reproduction number (R > 1) does not guarantee spread of the disease, since even with appreciable transmissibility, an outbreak may become extinct due to stochastic effects associated with a small number of infected individuals. Once the number of infectious individuals is conditioned on extinction, we show that an exact symmetry of the underlying theory ensures two distinct values of R, one larger than unity, the other smaller than unity, for which all outbreak properties are identical, with no signature of difference. Therefore a disease with a subcritical R, or its supercritical counterpart, when conditioned on extinction, have, for a given outbreak, identical individual likelihoods. In the full likelihood, this symmetry is lost, since the individual likelihood for a subcritical R is weighted by an extinction probability of unity, but the individual likelihood of a supercritical R is weighted by a sub-unity extinction probability. However, the inference can still benefit from the underlying symmetry, since it yields a mapping of all supercritical reproduction numbers onto the subcritical domain (R < 1), thereby speeding up evaluation of the likelihood profile. The symmetry holds in the standard situation, where the distribution of secondary cases is Poisson, as well as where this distribution has a negative binomial form and super-spreading can occur.