  We study the problem of a policymaker who aims at taming the spread of an
epidemic while minimizing its associated social costs. The main feature of our
model lies in the fact that the disease's transmission rate is a diffusive
stochastic process whose trend can be adjusted via costly confinement policies.
We provide a complete theoretical analysis, as well as numerical experiments
illustrating the structure of the optimal lockdown policy. In all our
experiments the latter is characterized by three distinct periods: the epidemic
is first let freely evolve, then vigorously tamed, and finally a less stringent
containment should be adopted. Moreover, the optimal containment policy is such
that the product "reproduction number x percentage of susceptible" is kept
after a certain date strictly below the critical level of one, although the
reproduction number is let oscillate above one in the last more relaxed phase
of lockdown. Finally, an increase in the fluctuations of the transmission rate
is shown to give rise to an earlier beginning of the optimal lockdown policy,
which is also diluted over a longer period of time.
