  COVID-19 has forced quarantine measures in several countries across the
world. These measures have proven to be effective in significantly reducing the
prevalence of the virus. To date, no effective treatment or vaccine is
available. In the effort of preserving both public health as well as the
economical and social textures, France and Italy governments have partially
released lockdown measures. Here we extrapolate the long-term behavior of the
epidemics in both countries using a Susceptible-Exposed-Infected-Recovered
(SEIR) model where parameters are stochastically perturbed to handle the
uncertainty in the estimates of COVID-19 prevalence. Our results suggest that
uncertainties in both parameters and initial conditions rapidly propagate in
the model and can result in different outcomes of the epidemics leading or not
to a second wave of infections. Using actual knowledge, asymptotic estimates of
COVID-19 prevalence can fluctuate of order of ten millions units in both
countries.
