the recovery rate of infectious individuals (the reciprocal is the infectious period).

Note that this model is slightly different from the model in [@Alonso2012] as we use $V$ as a separate term instead of $VS$. We found it more convenient for computational purposes in the optimization problems for control synthesis in later sections.

![Block diagram of the COVID-19 SEIR model with vaccination control.[]{data-label="diagram1"}](diagram_covid.pdf)

COVID-19 SEIR Model with Shield Immunity
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*Shield immunity* is a strategy recently proposed in [@Weitz_nature] to limit the transmission of the disease. The basic idea of this strategy is to increase the proportion of interactions with recovered individuals as opposed to the other individuals in the population. The effectiveness of this strategy is based on the assumption that recovered individuals (virus-negative and antibody-positive) can safely interact with both susceptible and infectious individuals without getting infected with the disease. See Figure \[diagram2\] as an illustration.

As the model used in [@Weitz_nature] is modified from a SIR model, we consider a corresponding SEIR model with shield immunity and we use the shield strength as control inputs. Specifically, we consider the following model: $$\begin{aligned}
\begin{cases}
\dot{I} = \epsilon E - (\gamma+\mu+\alpha)I ;\\
\dot{E} = \beta SI/(N+\chi R) - (\mu+\epsilon)E;\\
