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Figure 5 | Journal of Mathematics in Industry

Figure 5

From: Beyond just “flattening the curve”: Optimal control of epidemics with purely non-pharmaceutical interventions

Figure 5

(a) Optimal time evolution of the transmission control function \(u (t )\) for different values of \(C_{0}\). The value of \(C_{0}\) is color-coded. In all scenarios, the interventions start with a strict lockdown, where \(u (t )\) is reduced below \(\mathcal{R}_{0}^{-1}\) for about 10 to 12 days. This initial lockdown is followed by a long “critical period” during which the measures are gradually relaxed. The length of this period is determined by the peak number of simultaneously critically infected \(C_{0}\). Towards the end of the intervention, a moderate tightening of the NPIs is required. (b) Same as (a), but zoomed on the region with \(u (t )<\mathcal{R}_{0}^{-1}\). (c) By optimal transmission control, the number of patients in a critical state C is kept below the limiting value \(C_{0}\) at all times. (d) Characteristic time span \(T_{\text{FWHM}}\) of the critical period during which the peak number of simultaneously infected must be held constant. The dashed line shows the analytical approximation \(T_{\text{crit}}\) given in Eq. (17). (e) Total number of disease-related deaths (solid lines) and total costs of the measures (dashed lines) at the end of the epidemic vs. the control parameter \(\mathcal{P}\) (see Sect. 3). The optimized transmission function minimizes the number of disease-related deaths to a \(C_{0}\)-independent value for \(\mathcal{P}\to \infty \), but to a high cost in the case of low \(C_{0}\). The squares indicate the minimal values of \(\mathcal{P}\) that guarantee \(C(t)< C_{0}\) for all times

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