  We propose a model predictive control (MPC) approach for minimising the
social distancing and quarantine measures during a pandemic while maintaining a
hard infection cap. To this end, we study the admissible and the maximal robust
positively invariant set (MRPI) of the standard SEIR compartmental model with
control inputs. Exploiting the fact that in the MRPI all restrictions can be
lifted without violating the infection cap, we choose a suitable subset of the
MRPI to define terminal constraints in our MPC routine and show that the number
of infected people decays exponentially within this set. Furthermore, under
mild assumptions we prove existence of a uniform bound on the time required to
reach this terminal region (without violating the infection cap) starting in
the admissible set. The findings are substantiated based on a numerical case
study.
