Optimal one-round interventions for an SIR epidemic under two distinct time horizons

Non-pharmaceutical interventions play a vital role in mitigating the outbreak of infectious diseases. During the COVID-19 pandemic, numerous countries and regions imposed multiple rounds of intervention measures to curb the spread of the disease. To get insights into optimal implementation of one-round interventions, two optimal control problems were examined under distinct time horizons, based on whether the herd immunity was reached. Here, one-round interventions were defined as interventions that operated under a budget constraint. Both of optimal control policies were proved to be purely bang-bang with a single switch, but their timings of onset were different. These results may provide some theoretical support for developing intervention strategies.