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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., Comment: 51 pages, 39 Figures. Review/Research Manuscript on modelling the dynamics of SARS-CoV 2 Infection. arXiv admin note: text overlap with arXiv:2101.05596, arXiv:2012.01869

A new coronavirus disease, called COVID-19, appeared in the Chinese region of Wuhan at the end of last year; since then the virus spread to other countries, including most of Europe. We propose a differential equation governing the evolution of the COVID-19. This dynamic equation also describes the evolution of the number of infected people for 13 common respiratory viruses (including the SARS-CoV-2). We validate our theoretical predictions with experimental data for Italy, Belgium and Luxembourg, and compare them with the predictions of the logistic model. We find that our predictions are in good agreement with the real world since the beginning of the appearance of the COVID-19; this is not the case for the logistic model that only applies to the first days. The second part of the work is devoted to modelling the descending phase, i.e. the decrease of the number of people tested positive for COVID-19. Also in this case, we propose a new set of dynamic differential equations that we solved numerically. We use our differential equations parametrised with experimental data to make several predictions, such as the date when Italy, Belgium, and Luxembourg will reach a peak number of SARS-CoV-2 infected people. The descending curves provide valuable information such as the duration of the COVID-19 epidemic in a given Country and therefore when it will be possible to return to normal life. The study of the the dynamics of COVID-19 when the population have been subject to less restrictive measures is beyond the scope of this work and it will be matter of future works., Comment: 34 pages, 19 figures, 6 tables