## 2 Materials and Methods
### Estimation of the Parameters
In (a), (b), and (c) we can see the estimation of the \(I\) function, \(R\) function, and the \(D\) function, respectively, where there are some errors plotted with the filled area. Because the residuals needs to be determined numerically, we examined the accuracy of the normality of the estimation of the parameters. In order to carry out this examination, we generated the residuals of the infected, recovered and dead populations for data of Morocco. First, residuals are plotted in Figure 4, where we can see ranges of \(l_{k}-I_{k}^{data}\) in (a) and \(R_{k}-R_{k}^{data}\) in (b) and \(D_{k}-D_{k}^{data}\) in (c). It can be seen from the figure that residuals of the infected population are in the range \([-50,50]\), and residuals of the recovered population are in the range \([-30,50]\), while residuals of the dead population are in the range \([-6,6]\). Second, Figure 5 displays the corresponding histograms of \(l_{k}-I_{k}^{data}\) in (a) and \(R_{k}-R_{k}^{data}\) in (b) and \(D_{k}-D_{k}^{data}\) in (c) for the estimation of the parameters for Morocco with their corresponding normal distributions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\Gamma_{i}\) & \(\delta N_{i}\) \\ \(\beta_{1}\) & 0.892559835359739 \\ \(\beta_{2}\) & 0.0268308310475789 \\ \(\alpha\) & 0.899989478657192 \\ \(\gamma_{1}\) & 4.38815800908638 \(\times\) 10\({}^{-6}\) \\ \(\gamma_{2}\) & 0.028964892404635 \\ \(\gamma_{3}\) & 0.048799035924354 \\ \(\mu_{1}\) & 0.00633092058125248 \\ \(\mu_{2}\) & 0.000866774799367013 \\ \(T\) & 0.0483013932330731 \\ \(u\) & 0.000293106579352138 \\ \(v\) & 0.529909144905706 \\ \(\rho_{1}\) & 4.5368569540136 \(\times\) 10\({}^{-7}\) \\ \(\rho_{2}\) & 0.171682998125492 \\ \(\rho_{3}\) & 0.0465276743678475 \\ \(d\) & 0.0586180610122414 \\ \hline \(S_{0}\) & 34,500,000 \\ \(E_{0}\) & 2000 \\ \(l_{0}\) & 1000 \\ \(R_{0}\) & 81 \\ \(E_{0}^{q}\) & 2000 \\ \(I_{0}^{q}\) & 959 \\ \(D_{0}\) & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimated parameters’ values and the initial conditions for SEIRQD Model. In order to make valid inferences during regression analysis, it is necessary to check the assumptions, in order to be able to trust the results. One of the most useful assumptions for regression analysis is that the residuals are normally distributed. It can be seen from these figures that the residuals follow a normal distribution, furthermore, they remain within small ranges, thus the estimated parameters are reliable and correctly predict the observed data of Morocco, especially the number of infections and deaths. The objective of this part of the article is not to predict what will happen in the coming months, but to validate the proposed model by fitting the model's outputs to real data. Therefore, our next objective is to suggest optimal control strategies to make the national lockdown more effective in reducing new infections and saving more lives. Figure 4: Residuals of the (**a**) active infected cases, (**b**) recovered individuals and (**c**) dead individuals. Figure 3: The data fitting of \(I\) in (**a**), \(R\) in (**b**) and \(D\) in (**c**) to estimate the parameters of the model.