  It is widely accepted that the number of reported cases during the first
stages of the COVID-19 pandemic severely underestimates the number of actual
cases. We leverage delay embedding theorems of Whitney and Takens and use
Gaussian Process regression to estimate the number of cases during the first
2020 wave based on the second wave of the epidemic in several European
countries, South Korea, and Brazil. We assume that the second wave was more
accurately monitored and hence that it can be trusted. We then construct a
manifold diffeomorphic to that of the implied original dynamical system, using
fatalities or hospitalizations only. Finally, we restrict the diffeomorphism to
the reported cases coordinate of the dynamical system. Our main finding is that
in the European countries studied, the actual cases are under-reported by as
much as 50\%. On the other hand, in South Korea -- which had an exemplary and
proactive mitigation approach -- a far smaller discrepancy between the actual
and reported cases is predicted, with an approximately 17\% predicted
under-estimation. We believe that our backcasting framework is applicable to
other epidemic outbreaks where (due to limited or poor quality data) there is
uncertainty around the actual cases.
