Abstract
A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ap- peared in the Chinese region of Wuhan at the end of 2019. Since then, the virus spread to other countries, including most of Europe and USA. This work provides an overview on deterministic and stochastic models that have previously been proposed by us to study the transmission dy- namics 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 quar- antine measures, which take into account the time-delay for recovery or death processes. Realistic dynamic equations for the entire process are derived by adopting the so-called kinetic-type reactions approach. The lockdown and the quarantine measures are modelled by some kind of in- hibitor 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 peo- ple, respectively. In other words, everything happens as if the hospitals beds act as a catalyser in the hospital recovery process. The statistical properties of the models, in particular the relevant correlation functions and the probability density functions, have duly been evaluated. We val- idate 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 have found that our predictions are in good agreement with the real world since the onset of COVID 19, contrary to 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 (corresponding to a value of the effective reproduction number of the infection lower than 1).