Optimal single-interval control for SIR-type systems

Although modeling studies are focused on the control of SIR-based systems describing epidemic data sets, few of them present a formal dynamic characterization in terms of the two main indexes: the infected peak prevalence (IPP) and the final epidemic size (EFS). These indices are directly related to equilibrium sets and stability, which are crucial concepts to understand what kind of non-pharmaceutical interventions (social distancing, isolation measures, mask-wearing, etc.) can be implemented to handle an epidemic. The objective of this work is to provide a theoretical single-interval control strategy that simultaneously minimizes the EFS while maintaining the IPP arbitrary low, according to health system capacity limitations. Several simulations illustrate the true role of the herd immunity threshold and provide new insight into the way authorities may act.