  Since the start of the still ongoing COVID-19 pandemic, there have been many
modeling efforts to assess several issues of importance to public health. In
this work, we review the theory behind some important mathematical models that
have been used to answer questions raised by the development of the pandemic.
We start revisiting the basic properties of simple Kermack-McKendrick type
models. Then, we discuss extensions of such models and important
epidemiological quantities applied to investigate the role of heterogeneity in
disease transmission e.g. mixing functions and superspreading events, the
impact of non-pharmaceutical interventions in the control of the pandemic,
vaccine deployment, herd-immunity, viral evolution and the possibility of
vaccine escape. From the perspective of mathematical epidemiology, we highlight
the important properties, findings, and, of course, deficiencies, that all
these models have.
