  We analyze equilibrium behavior and optimal policy within a
Susceptible-Infected-Recovered epidemic model augmented with potentially
undiagnosed agents who infer their health status and a social planner with
imperfect enforcement of social distancing. We define and prove the existence
of a perfect Bayesian Markov competitive equilibrium and contrast it with the
efficient allocation subject to the same informational constraints. We identify
two externalities, static (individual actions affect current risk of infection)
and dynamic (individual actions affect future disease prevalence), and study
how they are affected by limitations on testing and enforcement. We prove that
a planner with imperfect enforcement will always wish to curtail activity, but
that its incentives to do so vanish as testing becomes perfect. When a vaccine
arrives far into the future, the planner with perfect enforcement may encourage
activity before herd immunity. We find that lockdown policies have modest
welfare gains, whereas quarantine policies are effective even with imperfect
testing.
