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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., 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

Here, we propose a new tool to estimate the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighboring values in a time series. This makes it possible to describe the signal as efficiently as prior proposed parameters such as permutation entropy (PE) or modified permutation entropy (mPE), but (1) reduces the size of the sample that is necessary to estimate the parameter value, and (2) enables the use of the Kullback-Leibler divergence to estimate the distance between the time series data and random signals., 10 pages, 6 figures

In recent years, the use of conformal transformation techniques has become widespread in the literature on gravitational theories alternative to general relativity, on cosmology, and on nonminimally coupled scalar fields. Typically, the transformation to the Einstein frame is generated by a fundamental scalar field already present in the theory. In this context, the problem of which conformal frame is the physical one has to be dealt with and, in the general case, it has been clarified only recently; the formulation of a theory in the ``new'' conformal frame leads to departures from canonical Einstein gravity. In this article, we review the literature on conformal transformations in classical gravitational theories and in cosmology, seen both as purely mathematical tools and as maps with physically relevant aspects. It appears particularly urgent to refer the analysis of experimental tests of Brans-Dicke and scalar-tensor theories of gravity, as well as the predictions of cosmological inflationary scenarios, to the physical conformal frame, in order to have a meaningful comparison with the observations., LaTeX, 54 pages, no figures. To appear in Fundamentals of Cosmic Physics