In the middle of 2020, with its borders tightly closed to the rest of the world, Australia almost achieved the local elimination of COVID-19 and subsequently maintained 'COVID-zero' in most parts of the country for the following year. Australia has since faced the relatively unique challenge of deliberately 'undoing' these achievements by progressively easing restrictions and reopening. Exploring the role of mathematical modelling in navigating a course through the pandemic through qualitative interviews with modellers and others working closely with modelling, we argue that each of these two significant phases of Australia's COVID-19 experience can be understood as distinct forms of 'model society'. This refers at once to the society enacted through the governance of risk, and to the visions of societal outcomes - whether to be sought or to be avoided - that are offered up by models. Each of the two model societies came about through a reflexive engagement with risk facilitated by models, and the iterative relationship between the representations of society enacted within models and the possibilities that these representations generate in the material world beyond them.
Keywords: COVID-19; COVID-zero; Evidence; elimination; evidence-based policy; mathematical models; risk.