  In this paper, we provide insights on how much testing and social distancing
is required to control COVID-19. To this end, we develop a compartmental model
that accounts for key aspects of the disease: 1) incubation time, 2)
age-dependent symptom severity, and 3) testing and hospitalization delays; the
model's parameters are chosen based on medical evidence, and, for concreteness,
adapted to the German situation. Then, optimal mass-testing and age-dependent
social-distancing policies are determined by solving optimal control problems
both in open loop and within a model predictive control framework. We aim to
minimize testing and/or social distancing until herd immunity sets in under a
constraint on the number of available intensive care units. We find that an
early and short lockdown is inevitable but can be slowly relaxed over the
following months.
