  This work introduces a novel epidemiological model that simultaneously
considers multiple viral strains, reinfections due to waning immunity response
over time and an optimal control formulation. This enables us to derive optimal
mitigation strategies over a prescribed time horizon under a more realistic
framework that does not imply perennial immunity and a single strain, although
these can also be derived as particular cases of our formulation. The model
also allows estimation of the number of infections over time in the absence of
mitigation strategies under any number of viral strains. We validate our
approach in the light of the COVID-19 epidemic and present a number of
experiments to shed light on the overall behaviour under one or two strains in
the absence of sufficient mitigation measures. We also derive optimal control
strategies for distinct mitigation costs and evaluate the effect of these costs
on the optimal mitigation measures over a two-year horizon. The results show
that relaxations in the mitigation measures cause a rapid increase in the
number of cases, which then demand more restrictive measures in the future.
