Multiple epidemic waves in a switching system with multi-thresholds triggered alternate control

In this study, we propose a switching model with multi-thresholds, considering alternately trigger on/off enhanced control interventions (ECIs) following the evolution of the epidemic of emerging infectious diseases. Despite the high non-smoothness of the proposed model, we obtain the generalized formulations of the peak sizes of all the epidemic waves generated by the switching control, and also the final epidemic size as infections tend to zero after finite epidemic waves. We therefore provide the critical strength of control interventions to control the peak or final size below a target value. Furthermore, we analyze the impacts of four specific alternate control modes of the proposed model, namely persistent, one-shot, fixed-mode alternate and adaptive alternate control, on the final and maximum peak sizes, and further analyze the optimal thresholds in terms of minimizing four epidemic indexes, i.e. maximum peak size, final size, duration of control, and switching times. Our results reveal that to minimize the maximum peak size (or avoid the runs on medical resources), the key is to design optimal threshold conditions or switching mode to average all the infections into multiple epidemic waves. It seems hard to achieve the goal of simultaneously minimizing the four indexes. Alternatively, we consider to search the optimal thresholds in terms of one index with constrained conditions to other indexes, and conclude that the adaptive alternate control by adjusting the threshold conditions following the evolution of epidemic can be an excellent strategy.