## 2 The \(\mu\)-SEIR Model  
We will explore the dynamics of a compartmental model with four classes to separate the population, corresponding to the subpopulation susceptible to be infected (S) and the infected, which is composed of the subpopulation of individuals which have been exposed (E) with the disease but have not shown the symptoms yet, the symptomatic subpopulation of infected individuals (I) diagnosed with the disease and/or showing symptoms and the recovered subpopulation (R), which presents an immunity which can or cannot be temporary and where individuals are presumed fully immune. The dynamics of the subpopulations can be visualized in the Figure 1:  
The model dynamics may be rewritten with the following equations:  
\[S_{i+1}=(1-\beta_{I}I_{i}-\beta_{E}E_{i})S_{i} \tag{1}\] \[E_{i+1}=(\beta_{I}I_{i}+\beta_{E}E_{i})S_{i}+(1-\mu)E_{i}\] (2) \[I_{i+1}=\mu E_{i}+(1-\gamma)I_{i}\] (3) \[R_{i+1}=\gamma I_{i}+R_{i} \tag{4}\]  
for any integer \(i\in\mathbb{Z}_{0^{+}}=\mathbb{Z}_{+}\cup\{0\}\) and any given initial non-negative conditions \(S_{0}\geq 0\), \(E_{0}\geq 0\), \(I_{0}\geq 0\) and \(R_{0}\geq 0\). The infectivity rates \(\beta_{I}\) and \(\beta_{E}\) are defined through the force of infection \(\lambda_{i}\), which is the rate at which susceptible individuals contract the infection per capita. The value of new infected individuals would be \(\lambda_{i}S_{i}\), being \(S_{i}\) the susceptible subpopulation at that moment. The number of infectious individuals will directly influence the force, so the infectivity rates \(\beta_{I}\) and \(\beta_{E}\) will be introduced in the definition of the force of infection as \(\lambda_{i}=\beta_{I}I_{i}+\beta_{E}E_{i}\). These rates will depend on multiple factors, such as the average number of contacts any individual of the susceptible subpopulation encounters during certain time with individuals of the exposed and infectious subpopulations, respectively, and the probability of transmission of the disease in a contact between a susceptible individual and an infectious or exposed one ([PERSON] and [PERSON], 2015). The parameter \(\mu\) will represent the average transition rate from the a non-asymptomatic infected subpopulation to the diagnosed/symptomatic one. It is assumed that all exposed individuals recover after experimenting observable symptoms, so the transition to recovered comes uniquely from the infected subpopulation at an average rate \(\gamma\). The model will be discrete as the information regarding the spread of the disease is taken and published periodically. Note that (a) the rationale of the sampling period interpretation is that it is unity, typically one day or one week for a correct practical use of the model. (b) the exposed along the incubation period are asymptomatic, including those who then become symptomatic. (c) both the exposed and the infectious produce contagions on the Susceptible which reflects what, in fact, happens with the COVID-19 disease transmission mechanism. The model parameters should be expressed in values of dimensionality being the inverse of the sampling period units. We have chosen the daily period because the available data in the countries we intend to study is released at that rate. It is argued that this model may be of interest for the evaluation of the COVID-19 disease propagation in its blowing-up phase because of the following reasons:  
Figure 1: Scheme of the SIR Model. * It is of simple structure and of a discrete nature. Furthermore, it does not need the incorporation of a discretization modulation parameter to guarantee the non-negativity of the sequence solution as other epidemic discrete models usually need. See, for instance, References [24; 25] and some references therein. * It has two potentially distinct coefficient transmission rates (\(\beta_{E}\) and \(\beta_{I}\)) which allow to consider infectivity of the susceptible subpopulation from both infective subpopulations: the exposed and infectious subpopulations.