  Compartmental models have long served as important tools in mathematical
epidemiology, with their usefulness highlighted by the recent COVID-19
pandemic. However, most of the classical models fail to account for certain
features of this disease and others like it, such as the ability of exposed
individuals to recover without becoming infectious, or the possibility that
asymptomatic individuals can indeed transmit the disease but at a lesser rate
than the symptomatic.
  In the first part of this paper we propose two new compartmental
epidemiological models and study their equilibria, obtaining an endemic
threshold theorem for the first model. In the second part of the paper, we
treat the second model as an affine control system with two controls:
vaccination and mitigation. We show that this system is static feedback
linearizable, present some simulations, and investigate of an optimal control
version of the problem. We conclude with some open problems and ideas for
future research.
