Since the outbreak of the Covid-19 pandemic economists have turned to the SIR model and its subsequent variants for the study of the pandemic's economic impact. But the SIR model is lacking the optimising behaviour of economic models, in which agents can inuence future transitions with their present actions. We borrow ideas and modelling techniques from the Mortensen-Pissarides (1994) search and matching model and show that there is a well-defined solution in line with the original claims of Kermack and McKendrick (1927) but in which incentives play a role in determining the transitions. There are also externalities that justify government intervention in the form of imposing more restrictions on actions outside the home than a decentralised equilibrium would yield.