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 spreading of epidemics. Based on an extended SEIR model and continuous-time optimal control theory, in this paper the optimal non-pharmaceutical intervention strategy is presented for the case that a vaccine is never found  . In this case, the optimal control must meet two competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of herd immunity at the end of the measures, in order to exclude a second wave. The presented control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". The characteristic time course of the computed optimal control is the following: After an intense but short initial lockdown, a dynamical equilibrium is enforced during which the number of actively infected is held on a constant level for a longer period of time. At the same time, the measures can be gradually relaxed with an ever increasing speed. Two simple rules for the optimal control of the effective reproduction number are presented, see Eqs. (13)-(14), which ensure a stable route out of the initial lockdown. The rules are directly applicable in practice, as they only depend on a precise knowledge of the cumulative number of cases and a reliable estimate of the effective reproduction number. The obtained results could serve as a guideline for the implementation and gradual relaxation of non-pharmaceutical intervention measures also in other scenarios, e.g., when the development of a vaccine is expected soon . The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 epidemic in Germany.
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 model and continuous-time optimal control theory, in this paper the optimal non-pharmaceutical intervention strategy is presented 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 resulting evolution of the time-dependent effective reproduction number 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 epidemic in Germany.
R
spreading
spread
A
and complete containment (eradication of the epidemic) is impossible
D
two
R
herd
natural
R
The presented
Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal
R
The characteristic time course of the computed optimal control is the following: After an intense but short initial lockdown, a dynamical equilibrium is enforced during which the number of actively infected is held on a constant level for a longer period of time. At the same time, the measures can be gradually relaxed with an ever increasing speed. Two simple rules for the optimal control of the
Careful analysis of the resulting evolution of the time-dependent
R
are presented, see Eqs. (13)-(14), which ensure a stable route out of the initial lockdown. The rules are directly applicable in practice, as they only depend on a precise knowledge of the cumulative number of cases and a reliable estimate of the effective reproduction number. The obtained results could serve as a guideline for the implementation and gradual relaxation of non-pharmaceutical intervention measures also in other scenarios, e.g., when the development of a vaccine is expected soon
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