Are the upper bounds for new SARS-CoV-2 infections in Germany useful?

At the end of 2019, an outbreak of a new coronavirus, called SARS–CoV–2, was reported in China and later in other parts of the world. First infections were reported in Germany by the end of January and on March 16th the federal government announced a partial lockdown in order to mitigate the spread. Since the dynamics of new infections started to slow down, German states started to relax the confinement measures as to the beginning of May. As a fall back option, a limit of 50 new infections per 100,000 inhabitants within seven days was introduced for each city or district in Germany. If a district exceeds this limit, measures to control the spread of the virus should be taken. Based on a multi– patch SEAIRD–type model, we will simulate the effect of choosing a specific upper limit for new infections. We investigate, whether the politically motivated bound is low enough to detect new outbreaks at an early stage. Subsequently, we introduce an optimal control problem to tackle the multi–criteria problem of finding a bound for new infections that is low enough to avoid new outbreaks, which might lead to an overload of the health care system, but is large enough to curb the expected economic losses.