The impact of dispersion in the number of secondary infections on the probability of an epidemic

Heterogeneity in communities is a key factor to consider when modelling the transmission of an infection or implementing strategies to control its transmission. Heterogeneity in the level of infectiousness of individuals can arise in a number of ways. For example, it can arise through the structure of a community with, say, households, through the nature of a disease that may include superspreaders and others who are hardly infectious (like SARS), or through mixing and behaviour patterns that can be altered by interventions. Lloyd-Smith et al. (Lloyd-Smith, Schreiber, Kopp and Getz (2005)) observed that, under certain specific assumptions, greater heterogeneity, leads to a greater probability of disease elimination. In this paper we explore the impact of heterogeneity on the probability of disease elimination more generally. We show that, for many commonly arising distributions in ecology and epidemiology, an increase in heterogeneity, when the mean is fixed, leads to a reduction in the probability of a local outbreak. This result has important consequences for health care strategies, such as choosing strategies that increase heterogeneity for the same mean level of infectivity thereby delaying, or possibly preventing, an outbreak. However, while broadly true, including for most offspring distributions common to epidemic and ecological situations, the result is not in general true as we show by counter–examples for each of the types of heterogeneity considered. We conjecture a general principle determining when the result holds, but it remains an open question precisely when greater heterogeneity leads to an increase in the probability of extinction.