## 3 Results
### Model Assessment
#### 3.2.2 Projection Error under the Assumption of Nearly Constant Parameter  
This section presents a statistical analysis of the behaviour of the error as a function of the projection horizon. An issue criticised to epidemiological models during the pandemic was the error in the projections. These errors can have their origin in different factors, e.g., the difficulty in detecting positive cases due to a limited testing rate, delay and difficulties in confirming deaths related to COVID-19, the assumptions that social behaviour will be homogeneous and stationary, among others. These sources of error are dynamic, therefore they force a continuous update of the model parameters and limit the projection horizon for which the parameters estimated at a given moment will be appropriate. Furthermore, from the perspective of a decision-maker, it is important to be informed about the limitations of a modelling framework to anticipate scenarios. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **rel\_dev** & **dev\_rel\_error** \\ \hline Daily Reported & \(\approx\)0.413 & \(\approx\)0.668 \\ Hospitalised & \(\approx\)0.039 & \(\approx\)0.047 \\ UCI & \(\approx\)0.036 & \(\approx\)0.050 \\ Daily Death & \(\approx\)0.204 & \(\approx\)0.234 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between the simulation trajectories and data. We estimate the error in the projection under the assumption of nearly constant values for the parameters. For the projection from the time \(t_{j+W}\), a Sigmoid function is used, where the range is from the parameter value at \(t_{j}\) to an average in \(2W=28\) days of the parameter, with a variation width of about \(W/2=7\) days starting from \(t_{j+W}\). For the transmissibility parameter, the mentioned Sigmoid function can be expressed as:  
\[\beta_{\text{proj}}^{t_{j}}(t)=\tilde{\beta}+\frac{\beta(t_{j})-\tilde{\beta}}{ 1+\exp\left(\frac{W-2}{W}(t-W)-\frac{W}{4}\right)}\qquad\text{for }t\geq t_{j+W}, \tag{29}\]  
where \(\tilde{\beta}\) is obtained by averaging the values of transmissibility for the last \(2W=28\) days: from \(t_{j-W}\) to \(t_{j+W}-1\). Using the available data until \(t_{j+W}\), \(\beta\) can be determined until \(t_{j}\), i.e., \(\beta(t_{j})\) gives the better fit of data \(D_{j}^{R}\). Thus, the average transmissibility \(\tilde{\beta}\) for the last \(2W\) days is composed of \(W\) days with \(\beta\) varying daily from \(t_{j-W}\) to \(t_{j-1}\), and \(\beta\) is constant with value \(\beta(t_{j})\) from \(t_{j}\). This average can be expressed as:  
\[\tilde{\beta}=\frac{\sum_{l=-W}^{-1}\beta(t_{j+l})+W\beta(t_{j})}{2W}. \tag{30}\]  
The Sigmoid function is chosen because, in one hand, the parameter value \(\beta(t_{j})\) fit better to data for the last \(W\) days, but in the other hand, \(\tilde{\beta}\) gives a representative transmissibility for longer-term if there is no change in the containment policies or behaviour on the population scale. The Sigmoid function is just a simple way to join smoothly these two values of the parameters. Similar functions are used for the parameters of proportions. Thus, the projections are obtained using these parameters, and solving the system of differential equations. The results obtained are named as \(dR^{j}\), \(H^{j}\), \(U^{j}\), and \(dF^{j}\). The relative error (Equation (27)) is iteratively computed using the results obtained from the simulations \(dR^{j}\), \(H^{j}\), \(U^{j}\), and \(dF^{j}\), with \(j\) such that, \(t_{j_{k}}=5\) October 2020 until 1 March 2021. Thus, vectors of relative error are obtained, which are denoted as \(\text{rel\_error}_{X^{j}}(t_{j+k})\) for \(k=1,2,\ldots,W+W_{p}\): composed of \(W=14\) days of training and \(W_{p}=30\) days of test or projection. These vectors are visualised using box plots for each \(k=1,2,\ldots,W+W_{p}\). In the analysis, the days are remapped in order that the projection days could start from 1. Thus, the day \(n\) is related to \(k\) by \(n=k-14\), where \(k=1,2,\ldots,W+W_{p}\).