  This work provides an overview on deterministic and stochastic models that
have previously been proposed by us to study the transmission dynamics of the
Coronavirus Disease 2019 (COVID-19) in Europe and USA. Briefly, we describe
realistic deterministic and stochastic models for the evolution of the COVID-19
pandemic, subject to the lockdown and quarantine measures, which take into
account the time-delay for recovery or death processes. Realistic dynamic
equations for the entire process have been derived by adopting the so-called
"kinetic-type reactions approach". The lockdown and the quarantine measures are
modelled by some kind of inhibitor reactions where susceptible and infected
individuals can be "trapped" into inactive states. The dynamics for the
recovered people is obtained by accounting people who are only traced back to
hospitalised infected people. To model the role of the Hospitals we take
inspiration from the Michaelis-Menten's enzyme-substrate reaction model (the
so-called "MM reaction") where the "enzyme" is associated to the "available
hospital beds", the "substrate" to the "infected people", and the "product" to
the "recovered people", respectively. The statistical properties of the models,
in particular the relevant correlation functions and the probability density
functions, have duly been evaluated. We validate our theoretical predictions
with a large series of experimental data for Italy, Germany, France, Belgium
and United States, and we also compare data for Italy and Belgium with the
theoretical predictions of the logistic model. We found that our predictions
are in good agreement with the real world since the onset of COVID 19, contrary
to the the logistics model that only applies in the first days of the pandemic.
In the final part of the work, we can find the (theoretical) relationships that
should be satisfied to obtain the disappearance of the virus.
