  One clear aspect of behaviour in the COVID-19 pandemic has been people's
focus on, and response to, reported or observed infection numbers in their
community. We describe a simple model of infectious disease spread in a
pandemic situation where people's behaviour is influenced by the current risk
of infection and where this behavioural response acts homeostatically to return
infection risk to a certain preferred level. This model predicts that the
reproduction rate $R$ will be centered around a median value of 1, and that a
related measure of relative change in the number of new infections will follow
the standard Cauchy distribution. Analysis of worldwide COVID-19 data shows
that the estimated reproduction rate has a median of 1, and that this measure
of relative change calculated from reported numbers of new infections closely
follows the standard Cauchy distribution at both an overall and an individual
country level.
