  When effective medical treatment and vaccination are not available,
non-pharmaceutical interventions such as social distancing, home quarantine and
far-reaching shutdown of public life are the only available strategies to
prevent the spread of epidemics. Based on an extended SEIR
(susceptible-exposed-infectious-recovered) model and continuous-time optimal
control theory, we compute the optimal non-pharmaceutical intervention strategy
for the case that a vaccine is never found and complete containment
(eradication of the epidemic) is impossible. In this case, the optimal control
must meet competing requirements: First, the minimization of disease-related
deaths, and, second, the establishment of a sufficient degree of natural
immunity at the end of the measures, in order to exclude a second wave.
Moreover, the socio-economic costs of the intervention shall be kept at a
minimum. The numerically computed optimal control strategy is a
single-intervention scenario that goes beyond heuristically motivated
interventions and simple "flattening of the curve." Careful analysis of the
computed control strategy reveals, however, that the obtained solution is in
fact a tightrope walk close to the stability boundary of the system, where
socio-economic costs and the risk of a new outbreak must be constantly balanced
against one another. The model system is calibrated to reproduce the initial
exponential growth phase of the COVID-19 pandemic in Germany.
