  Faced with a dangerous epidemic humans will spontaneously social distance to
reduce their risk of infection at a socio-economic cost. Compartmentalised
epidemic models have been extended to include this endogenous decision making:
Individuals choose their behaviour to optimise a utility function,
self-consistently giving rise to population behaviour. Here we study the
properties of the resulting Nash equilibria, in which no member of the
population can gain an advantage by unilaterally adopting different behaviour.
We leverage a new analytic solution to obtain, (1) a simple relationship
between rational social distancing behaviour and the current number of
infections; (2) new scaling results for how the infection peak and number of
total cases depend on the cost of contracting the disease; (3) characteristic
infection costs that divide regimes of strong and weak behavioural response and
depend only on the basic reproduction number of the disease; (4) a closed form
expression for the value of the utility. We discuss how these analytic results
provide a deep and intuitive understanding into the disease dynamics, useful
for both individuals and policymakers. In particular the relationship between
social distancing and infections represents a heuristic that could be
communicated to the population to encourage, or "bootstrap", rational
behaviour.
