## 3 Results
### Model Assessment
#### 3.2.1 Comparisons  
Figure 5 shows the state trajectories of the SEIR-H model along with reported data points. The trajectories are obtained by solving the system of differential Equations (1)-(4) and (7)-(9), with \(\beta^{\text{opt}}\) and \(\lambda_{s}^{\text{opt}}(t)\) (\(*=I\to H,I\to F,H\to U,H\to F,U\to F\)) parameters obtained from the parameter estimation explained in Section 2.3. The trajectories shown (from the top to bottom) are the daily reported \(dR^{\beta\text{opt},\lambda_{s}^{\text{opt}}}\), normal bed occupation in hospitals \(H^{\beta\text{opt},\lambda_{s}^{\text{opt}}}\), occupation in ICU \(U^{\beta\text{opt},\lambda_{s}^{\text{opt}}}\), and daily death \(dF^{\beta\text{opt},\lambda_{s}^{\text{opt}}}\). As happened with the proportions \(\lambda_{s}\), the simulated trajectories of hospitalised, ICU and daily death are plotted starting on 5 October 2020, which is when the \"new normality\" measures were enforced. The simulation trajectories stop on 1 April 2021 and the posterior distributions estimated for that day are used to advance trajectories for an additional month with the percentile 95 credibility intervals for the trajectories. Figure 4: The proportions for the dynamics in the hospitals: the optimal values with the 95 credibility interval. From Figure 5, we observe that the trajectories from the simulation can follow very well the data. Actually, there is a lot of variability in the data, which is due to the limited testing regime and delays in reporting of deaths. On the weekends, the amount of testing and also the number of reports of the deaths are reduced. Thus, higher variability is observed in daily reported and daily deaths. The relative mean deviation of the data can be defined using a moving standard deviation and moving average of length 7 days. The moving standard deviation of the data \(D\) at time \(t_{j}\) is expressed as:  
\[\text{movstd}_{D}(t_{j})=\sqrt{\frac{1}{6}\sum_{k=-3}^{3}{(D(t_{j+k})-\text{ movave}_{D}(t_{j}))^{2}}}, \tag{24}\]  
where \(\text{movave}_{D}(t_{j})\) is the moving average of the data at time \(t_{j}\), which can be defined as:  
\[\text{movave}_{D}(t_{j})=\frac{1}{7}\sum_{k=-3}^{3}{D(t_{j+k})}. \tag{25}\]  
The relative mean deviation of the data is expressed as:  
\[\text{rel\_dev}_{D}=\frac{1}{j_{\text{end}}-W-j^{*}}\sum_{j=j^{*}}^{j_{\text{ end}}-W}\frac{\text{movstd}_{D}(t_{j})}{\text{movave}_{D}(t_{j})}. \tag{26}\]  
Figure 5: The trajectories from the simulations using \(\beta^{\text{opt}}(t)\) and \(\lambda_{*}^{\text{opt}}(t)\) compared to the data. Using the posterior distribution estimated on 1 April 2021, regions for trajectories are shown for one month ahead with percentile 95 credibility intervals. For the hospitalised and ICU, the relative mean deviation are \(\text{rel\_dev}_{\text{D}^{H}}\approx 0.039\) and \(\text{rel\_dev}_{\text{D}^{H}}\approx 0.036\), respectively, meanwhile, for the daily reported and daily deaths are \(\text{rel\_dev}_{\text{D}^{R}}\approx 0.413\) and \(\text{rel\_dev}_{\text{D}^{F}}\approx 0.204\), respectively. Data from \(j^{*}=1\) to \(t_{j_{\text{end}}}-W\) is used in the computation of \(\text{rel\_dev}_{\text{D}^{R}}\), and from \(j^{*}=j_{h}\) to \(t_{j_{\text{end}}}-W\) for the rest. In order to assess the model, the deviation of the relative error between the data and the simulation results is computed. For this task, the relative error between the data and the simulation results at time \(t_{j}\) is defined as:  
\[\text{rel\_error}_{\text{X}}(t_{j})=\frac{D(t_{j})-X(t_{j})}{X(t_{j})}, \tag{27}\]  
where \(X\) is just a representation of any simulation results, whether it is daily reported, hospitalised, ICU or daily death.