more stringent requirements expressed in MTL specifications, and control in early phases of COVID-19 spread can generally mitigate the effects more efficiently.

Related Work
============

**COVID-19 epidemic modeling and control strategies**: Ever since the outbreak of COVID-19, there has been numerous research focusing on the modeling of COVID-19 epidemic based on data collected from both the epicenters and other places [@Bertozzi_Franco2020]. Among the various models, *compartmental models* such as SEIR and SUQC models have been used frequently for the analysis of COVID-19. There has also been work in analyzing or predicting COVID-19 using artificial intelligence models [@Zheng_AI2020], stochastic intensity models [@Chen2002], etc. The models we use in this paper are based on the SEIR (both the standard and with shield immunity) and SUQC models, but we have replaced some essential parameters (e.g., the shield strength in shield immunity, the quarantine rate in quarantine control) with control inputs which can be synthesized to vary on a daily basis.

**Optimal control of epidemic models**: There is a rich literature in designing vaccination control for the SEIR or SIR models of epidemics. However, such methods have not been applied in the setting of COVID-19 mitigation. Besides, there has been no work in optimal 