Figure 4

Optimal transmission control for \(C_{0}=30{,}000\) available ICUs. (a) Temporal evolution of the optimally controlled epidemic. The susceptible population terminates slightly below the critical value \(\mathcal{R}_{0}^{-1}\), which guarantees herd immunity and rules out a second wave of the epidemic. Moreover, the optimal control ensures that the available number of ICUs is not exceeded by the critically ill: \(C (t )< C_{0}\) for all \(t\in [0,\infty )\). A more detailed plot of the ICU load is given in Fig. 5(c). (b) Effective reproduction number (6) corresponding to the optimally steered intervention. The optimal mean contact reduction \(u (t )\) is shown for comparison. (c) Comparison of the trajectories of the uncontrolled (dashed lines) and the optimally controlled epidemic (solid lines) in different projections of the state space. The arrows indicate the direction of time. The grey shaded region highlights the critical period