## 2 Materials and Methods
### SEIR-H Model
#### 2.1.2 Reproduction Number  
The dynamics of disease transmission can be expressed using a single parameter called the reproduction number \(R_{0}\). This number can be defined as the expected number of secondary cases that arise after introducing an infectious individual into a susceptible population. This reproduction number is useful to define a threshold value to be maintained for a certain time, for the epidemic to persist (\(R_{0}>1\)), or be extinguished (\(R_{0}<1\)) [15; 27; 28].  
The reproduction number is affected by various biological, socio-behaviour and environmental factors [27]. The numerical value depends on the estimation method, and the specific value obtained from the mathematical models based on differential equations should be understood as a threshold rather than as the number of secondary cases [28]. The numerical value can be considered a function of three parameters: the duration of the infectious period, the average probability of transmission of the virus in each contact and the average rate of contacts [27].  
\begin{table}
\begin{tabular}{c l c c} \hline \hline
**Parameter** & **Description** & **Type \({}^{*}\)** & **Value \({}^{**}\)** \\ \hline \(\beta\) & Transmissibility. & T-D & Est. \\ \(\alpha\) & Reciprocal of average latent period. & C & \(1/3\)\({}^{(a)}\) \\ \(\gamma\) & Reciprocal of average infectious period. & C & \(1/7\)\({}^{(b)}\) \\ \(\lambda_{I\to H}\) & Proportion of infected that are hospitalised. & T-D & Est. \\ \(\lambda_{I\to F}\) & Proportion of infected that are death. & T-D & Est. \\ \(\lambda_{H\to U}\) & Proportion of hospitalised that are admitted in ICU. & T-D & Est. \\ \(\lambda_{H\to F}\) & Proportion of hospitalised that are death. & T-D & Est. \\ \(\lambda_{U\to F}\) & Proportion of admitted in ICU that are death. & T-D & Est. \\ \(\delta_{H\to U}\) & Reciprocal of average stay in hospitals bed before & C & \(1/7\)\({}^{(c)}\) \\  & addition in ICU. & & \(1/9\)\({}^{(c)}\) \\ \(\delta_{H\to F}\) & Reciprocal of average stay in hospital bed before death. & C & \(1/11\)\({}^{(c)}\) \\ \(\delta_{H\to O}\) & Reciprocal of average recovery period in hospital bed. & C & \(1/11\)\({}^{(c)}\) \\ \(\varphi_{U\to F}\) & Reciprocal of average stay in ICU before death. & C & \(1/11\)\({}^{(c)}\) \\ \(\varphi_{U\to\omega}\) & Reciprocal of average recovery period in ICU. & C & \(1/12\)\({}^{(c)}\) \\ \hline \hline \end{tabular} \({}^{*}\) T-D. Time-Dependent; C: Constant. \({}^{**}\) Est.: indicates that the values will be estimated using the methodology described in Section 2.3; all periods of time in this table are considered in days. \({}^{(a)}\) See [18; 19; 20; 24; 25]. \({}^{(b)}\) See [18; 20; 25; 26]. \({}^{(c)}\) Obtained from the data provided by DGVS.  
\end{table}
Table 1: SEIR-H parameters.  
The reproduction number of the classic SEIR model can be expressed in terms of the parameters \(\beta\) and \(\gamma\), as follows:  
\[R_{0}=\frac{\beta}{\gamma}. \tag{5}\]  
As the spread of disease is evolving, the number of susceptible individuals declines monotonically. Thus, the number in which effectively works as an epidemic threshold is:  
\[R_{e}(t)=\frac{\beta(t)}{\gamma}\frac{S(t)}{N}, \tag{6}\]  
also known as the effective reproduction number. This expression is obtained by adding the right-hand sides of the Equations (2) and (3) and then making it equal to zero.  
From these equations, it can be observed that the value of the reproduction number \(R_{0}\) or \(R_{e}(t)\) can be altered by interventions on parameters \(\beta\) and \(\gamma\). Containment and social distancing measures can affect the parameter \(\beta\), by reducing the average rate of contacts. However, hygiene recommendations and proper use of face masks can also affect \(\beta\) by reducing the average probability of virus transmission in each contact. While the parameter \(\gamma\) depends on the disease itself, it can be altered by effective tracing, testing and isolation of infected individuals.