## 4 Data Processing
### Processing the Available Data
#### 4.2.1 Estimation of the Exposed Subpopulation  
As we can see in Equations (46)-(48) and Equations (1)-(4) and (26)-(29) respectively, when calculating the values of \(\beta_{I}\) and \(\beta_{E}\), the value of the exposed subpopulation is as important as the infectious one. However, using only the official repositories, in which the different authorities share their data related to COVID-19, we cannot obtain the number of individuals incubating the virus at any given time, as it is obviously something which is not directly observed and thus not registered. In order to extrapolate such number from the available data for the extended model, we will assume:  
* That each infected individual comes from a exposed individual. * That each exposed individual will eventually be registered as infected. This assumes that they will be tested due to developing symptoms or being in close contact to someone who has. * That each exposed individual will become infected individual in a range of days from \(k_{0}\) to \(k_{1}\), with a probability of transition based on a normal distribution around a central value [55]. * The density function of the probability of transition will be defined from experimental observations [56; 57] by the mean value (4 days) an the variance (3 days), as seen as in [58]. The estimation of the exposed subpopulation will be done for all dates in a single step, using the known values of new cases of all dates. Now, let \(E_{k}\) and \(C_{k}\) be the number of exposed individuals and the new infected cases at the day \(k\), respectively. Then, the new infected individuals at day \(k\) will depend on the probability of transition and the number of exposed individuals on previous days. In this case, as:  
\[C_{k}=g_{k_{0}}E_{k-k_{0}}+\ldots+g_{k_{1}}E_{k-k_{1}}=\sum_{m=k_{0}}^{k_{1}} g_{m}E_{k-m} \tag{49}\]  
with the fraction of individuals that transient to the infected individuals \(g_{i}=\frac{\varphi_{i,\sigma}(i)}{\sum_{j=k_{0}}^{j-k_{1}}\varphi_{i,\sigma }(i)}\), being \(\varphi_{i,\sigma}(x)\) the value of the probability density function of a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) at specific values \(i\in[k_{0},k_{0}+1,\ldots,k_{1}]\). Observe that these values of \(g_{k}\) correspond to a special case of the extended model in which \(\sum\mu_{i}=1\). Then we will build a database with \(n\) data and a \((n-k_{0})\times(n-k_{0})\) matrix \(\mathbf{A}\) with the values \(g_{k_{0}},\ldots,g_{k_{1}}\) that multiplying the vector of historical exposed \(\mathbf{E}=(E_{1},\ldots,E_{n-k_{0}})^{T}\), we would obtain the vector of new cases with a delay of \(k_{0}\) days, \(\mathbf{C}=(C_{k_{0}+1},\ldots,C_{n})^{T}\). As the known vector is \(\mathbf{C}\) we will invert the matrix \(\mathbf{A}\), so that multiplying its inverse to the right to the vector \(\mathbf{C}\), we get the desired vector \(\mathbf{E}\) as described in the following equation  
\[ \begin{pmatrix}C_{k_{0}+1}\\ \vdots\\ C_{n-1}\\ C_{n}\end{pmatrix} =\mathbf{A} \begin{pmatrix}E_{1}\\ E_{2}\\ \vdots\\ E_{n-k_{0}}\end{pmatrix} \Longrightarrow \begin{pmatrix}E_{1}\\ E_{2}\\ \vdots\\ E_{n-k_{0}}\end{pmatrix} =\mathbf{A}^{-1} \begin{pmatrix}C_{k_{0}+1}\\ \vdots\\ C_{n-1}\\ C_{n}\end{pmatrix}  \tag{50}\]  
In this approximation, we will assume that there is no exposed individuals before day \(1\), so the infected individuals for the first days must come from these initial transitions. This assumption is reasonable because the number of cases at these moments is relatively low, so the value of exposed individuals must also be low. Consequently, the transitions of the first rows from \(1\) to \(k_{1}-k_{0}\) will be normalized, so that every row in \(\mathbf{A}\) equals to \(1\). This way, we define the matrix \(\mathbf{A}=<a_{i,j}>\) with:\[a_{i,j}=\left\{