## 2 Materials and Methods
### SEIR-H Model
#### 2.1.1 Dynamics of The Spread  
The spread dynamics of the disease, highlighted in yellow in Figure 1, is similar to the SEIR model [15], and is given by:  
\[\frac{dS}{dt} =-\beta\frac{S}{N}I, \tag{1}\] \[\frac{dE}{dt} =\beta\frac{S}{N}I-\alpha E,\] (2) \[\frac{dI}{dt} =\alpha E-\gamma I,\] (3) \[\frac{dR}{dt} =\gamma I+R_{im}. \tag{4}\]  
In this model, the Equations (1)-(4), respectively, show the dynamics of the compartments of: the susceptible \(S\), the exposed \(E\), the infectious \(I\) and the reported \(R\). The term \(R_{im}\) represents a rate of reported cases from the travellers, which was relevant at the beginning of the dynamics of COVID-19 in Paraguay. The susceptible group \(S\) represents those who have not yet contracted the disease and could contract it in the future.  
The exposed group \(E\) represents those who were exposed to the infection and have been infected, but are in a latent period not yet capable of infecting others. The latent period refers to the time between exposure to the virus and the moment when the individual becomes contagious and is one of the epidemiological characteristics of COVID-19 [8; 19; 20; 21]. While in the classic SEIR model, compartment \(E\) could include those in the incubation period [22], in this work, however, we only include in \(E\) those in the latent period. It is crucial to establish this distinction between the latent period and the incubation period, since COVID-19 is characterised by a pre-symptomatic infection [9; 10; 23]. Furthermore, the inclusion of the E-compartment can resemble the Erlang distribution in the infectious period [11].  
Figure 1: Schematic diagrams of the SEIR-H model: the region in yellow indicates the transmission dynamics given by a SEIR with external reported cases, and the region in green represents the dynamics related to hospital resources. A more detailed version of the green region is used to show that the \(R\) compartment contains the dynamics related to the health system, represented mainly by \(H\), \(U\) and \(F\).  
The infectious compartment \(I\) represents individuals who become contagious. The key feature which distinguishes compartments \(I\) from \(E\), is the onset of infectivity. Individuals in \(I\) may transmit the disease to susceptibles in \(S\), even without presenting symptoms.  
The compartment \(R\) is composed of those who are reported with the confirmation of COVID-19 positive diagnosis. In the classic SEIR model, the \(R\) compartment is associated with those who are recovered or removed from the dynamics after a certain time. In this work, the \(R\) compartment also has the meaning of removed, given the assumption that those who are reported are also isolated and removed from the transmission dynamics. However, since \(R\) will be fitted to the officially positive confirmations, we rename it as the reported compartment. The population \(N\) is maintained constant throughout the time, equal to the initial population. This condition differs from the actual population that varies due to the reported cases from the travellers \(R_{im}\), and also because of the individuals who are travelling abroad (which are not considered in the proposed model).  
Three parameters appear in this model: (i) \(\beta(t)\) is the rate at which susceptible individuals acquire the disease, also known as transmissibility, which is time-dependent in this work, (ii) \(\alpha\) is the rate at which an exposed individual becomes infectious, also defined as the reciprocal of the average latent period, and (iii) \(\gamma\) is the rate at which an infectious individual is reported, expressed as the reciprocal of the average time an individual remains contagious to other individuals. Table 1 shows the parameters of the SEIR-H model, with its descriptions and values, and whether it is considered to be constant or time-dependent.