## 2 Materials and Methods
### Estimation of Model Parameters  
This section presents the methodology to estimate the parameters of the proposed model. For this purpose, we use data provided by the Ministry of Public Health and Social Welfare, and the Health Monitoring Centre. The available data are daily records of:  
1. diagnosed with positive confirmed cases or daily reported cases \(D^{R}(t_{j})\), excluding the count of confirmed cases from travellers,
2. diagnosed with positive confirmed cases in the isolated groups who were travellers from abroad \(R_{im}(t_{j})\),
3. occupation in general bed in the hospitals by the patients of COVID-19 \(D^{H}(t_{j})\),
4. occupation in ICU by patients of COVID-19 \(D^{Ut}(t_{j})\),
5. confirmed daily deaths by COVID-19 \(D^{F}(t_{j})\). These data in a specific day are assumed to be independent of each other. These data are from \(t_{1}=7\) March 2020 until \(t_{j_{\text{end}}}=22\) May 2021. From the data provided by the Health Monitoring Centre, it was determined: (i) the average time of stay in general beds in the hospitals by the COVID-19 patients, and (ii) the average time of ICU stay by COVID-19 patients. The estimation of the model parameters is mainly performed in two steps: (i) the estimation of the transmissibility \(\beta\) of the disease for the spread dynamics using the Equations (1)-(4), and (ii) the different portions \(\lambda_{*}\), with \(*\in\{I\to H,I\to F,H\to U,H\to F,U\to F\}\), for bed occupations in hospitals using the Equations (7)-(9) in addition with the Equations (1)-(4). The estimation of transmissibility \(\beta\) is performed from the date of the first confirmed positive case in Paraguay, at \(t_{1}\). Due to the initial strong containment, the amount of hospital bed occupations by COVID-19 patients remained very low for several months. Because of this conditions, and due to the high intermittency of the available data, there were difficulties in estimating the portions \(\lambda_{*}\), thus these values are estimated only as of \(t_{\hat{h}}=5\) October 2020, when the \"new normality\" policy measures are applied. Consequently, the states of \(H\), \(U\) and \(F\), which are associated with these proportions are simulated from \(t_{\hat{h}}\). The time-varying parameters are determined using the moving time-window. In this work, we fixed the size of the time-window to \(W=14\) days. In the classic SEIR model, the time-window size would correspond to the entire epidemic cycle, keeping the transmissibility \(\beta\) invariant over the cycle. This hypothesis is a reasonable assumption in seasonal flu epidemics, because most of the population do not change their behaviour during the season. However, this condition is not the case with the COVID-19 pandemic, because behavioural changes of the population are continuous and vary mainly in response to official containment measures. Thus, in order to capture the relevant dynamics of transmission, a mechanism is needed to approach a time-varying transmissibility. The strategy of using a moving time-window in the parameter estimation resembles the moving average for time-series data, i.e., a smaller time-window size will better fit daily noisy data with higher oscillations, and on the other hand, a larger time-window size will offer smoother results and reveal trends. Thus, parameters obtained using time-windows with size \(W=14\) days can be understood as an average during this time span. This size was chosen mainly because of two reasons: (i) according to the officials at the Health Monitoring Centre in Paraguay (DGVS), the delay from the day when a contagion occurs until the detection is normally at least 10 days, and (ii) the strategy of containment measures in Paraguay is usually dictated in a two-week time span. Algorithm 1 shows schematically the procedure for parameter estimation.