## 2 Materials and Methods
### Estimation of Model Parameters
#### 2.3.3 Estimation of the Parameters for the Dynamics in the Hospitals  
As mentioned above, the different proportions \(\lambda_{*}\), with \(*\in\{I\to H,I\to F,H\to U,H\to F,U\to F\}\) for the dynamics in the hospitals, are estimated as of \(t_{j_{k}}=5\) October 2020 (Algorithm 1, line 5). In order to estimate the proportions \(\lambda_{*}(t_{j_{k}})\), the initial condition for the SEIR part is obtained using the optimal estimated transmissibility at time \(t_{j_{k}}-1\) (\(\beta_{j_{k}-1}^{\text{opt}}\)), and the initial state of the compartments \(H\), \(U\) and \(F\) are set using the 7-days average (from \(t_{j_{k}}-3\) to \(t_{j_{k}}+3\)) of the data of hospitalisation (normal bed), ICU and daily death, respectively. For the estimation of the proportions \(\lambda_{*}\) at time \(t_{j}=t_{0}+j\) (\(j>j_{h}\)), the state \((S(t_{j}),E(t_{j}),I(t_{j}),R(t_{j}),H(t_{j}),U(t_{j}),F(t_{j}))\), obtained using the optimal values of the estimated parameters \(\beta_{j-1}^{\text{opt}}\) and \(\lambda_{*j-1}^{\text{opt}}\) at day before, is adopted as initial condition. The parameters associated with the average lengths of stay in the hospitals are set to be constant. These times are estimated from individualised case record data provided by the DVGS (see Table 1 for the specific values). The estimation of the proportions \(\lambda_{*}\) is performed using also the Bayesian method, maintaining constant the parameter \(\beta_{j}^{\text{opt}}\) obtained beforehand, and it is expressed as:  
\[P(\lambda_{*j},\sigma_{j}^{H},\sigma_{j}^{U},\sigma_{j}^{F}|D_{j}^{H},D_{j}^ {U},D_{j}^{F})\;\propto\;P(D_{j}^{H},D_{j}^{U},D_{j}^{F}|\lambda_{*j},\sigma_ {j}^{H},\sigma_{j}^{U},\sigma_{j}^{F})\;P(\lambda_{*j},\sigma_{j}^{H},\sigma_ {j}^{U},\sigma_{j}^{F}). \tag{19}\]  
The observed data are: \(D_{j}^{H}=\{D^{H}(t_{j}+k)|k=1,\dots,W\}\), \(D_{j}^{U}=\{D^{U}(t_{j}+k)|k=1,\dots,W\}\), \(D_{j}^{F}=\{D^{F}(t_{j}+k)|k=1,\dots,W\}\), where these data are assumed to be normal-distributed around the simulation results of hospitalisation, UCI and daily death, with standard deviations \(\sigma_{j}^{H}\), \(\sigma_{j}^{U}\) and \(\sigma_{j}^{F}\), which are restricted to the intervals \([0,1000]\), \([0,200]\) and \([0,200]\), respectively. Also, it is assumed that the data \(D_{j}^{H},D_{j}^{U},D_{j}^{F}\) are independent,thus the likelihood in the Equation (19) is a product of three likelihoods: \(P(D_{j}^{H}|\lambda_{*j},\sigma_{j}^{H})\), \(P(D_{j}^{H}|\lambda_{*j},\sigma_{j}^{H})\), and \(P(D_{j}^{F}|\lambda_{*j},\sigma_{j}^{F})\). The likelihood functions can be expressed as:  
\[P(D_{j}^{H}|\lambda_{*j},\sigma_{j}^{H}) =\prod_{k=1}^{W}\frac{1}{\sqrt{2\pi}\sigma_{j}^{H}}\text{exp} \left[\frac{\left(D^{H}(t_{j}+k)-H^{\lambda_{*j}}(t_{k})\right)^{2}}{2(\sigma_ {j}^{H})^{2}}\right], \tag{20}\] \[P(D_{j}^{H}|\lambda_{*j},\sigma_{j}^{H}) =\prod_{k=1}^{W}\frac{1}{\sqrt{2\pi}\sigma_{j}^{H}}\text{exp} \left[\frac{\left(D^{H}(t_{j}+k)-U^{\lambda_{*j}}(t_{k})\right)^{2}}{2(\sigma_ {j}^{H})^{2}}\right],\] (21) \[P(D_{j}^{F}|\lambda_{*j},\sigma_{j}^{F}) =\prod_{k=1}^{W}\frac{1}{\sqrt{2\pi}\sigma_{j}^{F}}\text{exp} \left[\frac{\left(D^{F}(t_{j}+k)-dF^{\lambda_{*j}}(t_{k})\right)^{2}}{2(\sigma _{j}^{F})^{2}}\right], \tag{22}\]  
where, \(H^{\lambda_{*j}}\) and \(U^{\lambda_{*j}}\) are the hospitalised (in normal bed) and in ICU from the simulation using \(\{\beta_{j}^{\text{opt}},\lambda_{*j}\}\). The daily deaths from the simulation \(dF^{\lambda_{*j}}\) are obtained from:  
\[dF^{\lambda_{*j}}(t_{k})=F^{\lambda_{*j}}(t_{k})-F^{\lambda_{*j}}(t_{k}-1), \qquad\text{for }j<k\leq j+W, \tag{23}\]  
where \(F^{\lambda_{*j}}\) is the accumulated count of death, obtained from the simulation using \(\{\beta_{j}^{\text{opt}},\lambda_{*j}\}\).