WIAS Preprint No. 2748, (2020)

Beyond just ``flattening the curve'': Optimal control of epidemics with purely non-pharmaceutical interventions


  • Kantner, Markus
    ORCID: 0000-0003-4576-3135
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412

2010 Mathematics Subject Classification

  • 92D30 37N25 37N40 93C10 49N90 34B15


  • Mathematical epidemiology, optimal control, non-pharmaceutical interventions, effective reproduction number, dynamical systems, COVID-19, SARS-CoV2




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.

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