## 2 Materials and Methods
### Estimation of Model Parameters
#### 2.3.1 Estimation of Initial Condition  
The population of Paraguay is estimated at \(N=7\),252,671 according to the National Institute of Statistics (INE: Instituto Nacional de Estadistica [[https://www.ine.gov.py/default.php?publicacion=2](https://www.ine.gov.py/default.php?publicacion=2)]([https://www.ine.gov.py/default.php?publicacion=2](https://www.ine.gov.py/default.php?publicacion=2)), accessed on 20 May 2021). The initial time is set on \(t_{0}=6\) March 2020, one day before the day when the first case is reported. On this day, the reported group is set as \(R(t_{0})=0\), and the initial conditions for: the exposed \(e_{0}=E(t_{0})\), the infectious \(i_{0}=I(t_{0})\) and the transmissibility \(\beta(t_{0})\) are estimated using \(D_{0}^{R}=\{D^{R}(t_{0}+k)\mid k=1,\ldots,W\}\), available data of daily reported or confirmed cases in the first \(W=14\) days (Algorithm 1, line 1). The initial susceptible population is \(S(t_{0})=N-E(t_{0})-I(t_{0})-R(t_{0})\). Some parameters that do not depend on the particularities of each country or region but on the disease itself are set according to the values presented in the literature. The average of the latent period is \(1/\alpha=3\) days [18; 19; 20; 21; 24; 25] and the average of the infectious period is \(1/\gamma=7\) days [18; 20; 25; 26] (see Table 1). Although the infectious period can be reduced by early tracing and isolation (quarantine) strategies for infectious persons, after the movement restriction period and throughout much of the pandemic in Paraguay, early tracing and isolation of infectious persons was difficult. The estimation is performed using the Bayesian method, which can be expressed as:  
\[P(e_{0},i_{0},\beta_{0},\sigma_{0}^{R}|D_{0}^{R})\ \propto\ P(D_{0}^{R}|e_{0},i_{0}, \beta_{0},\sigma_{0}^{R})\ P(e_{0},i_{0},\beta_{0},\sigma_{0}^{R}), \tag{10}\]  
where \(P(e_{0},i_{0},\beta_{0},\sigma_{0}^{R}|D_{0}^{R})\) is the posterior probability distribution of the parameters \(\{e_{0}\), \(i_{0}\), \(\beta_{0}\), \(\sigma_{0}^{R}\}\) given the observation of the data \(D_{0}^{R}\), \(P(D_{0}^{R}|e_{0},i_{0},\beta_{0},\sigma_{0}^{R})\) is the probability of observing the data \(D_{0}^{R}\) given specific values of the parameters, which is also known as likelihood function, and \(P(e_{0},i_{0},\beta_{0},\sigma_{0}^{R})\) is the prior probability distribution of the parameters. We assumed that the observed data \(D_{0}^{R}\) are normal-distributed around the daily reported cases \(dR^{e_{0},i_{0},\beta_{0}}\) from the simulations results using the parameter set \(\{e_{0},i_{0},\beta_{0}\}\) with standard deviation \(\sigma_{0}^{R}\). Thus, the likelihood function can be expressed as:  
\[P(D_{0}^{R}|e_{0},i_{0},\beta_{0},\sigma_{0}^{R})=\prod_{k=1}^{W}\frac{1}{ \sqrt{2\pi}\sigma_{0}^{R}}\text{exp}\left[\frac{\left(D^{R}(t_{0}+k)-dR^{e_{0},i_{0},\beta_{0}}(t_{k})\right)^{2}}{2(\sigma_{0}^{R})^{2}}\right], \tag{11}\]  
where given a set of values of the parameters \(\{e_{0},i_{0},\beta_{0}\}\), the simulated daily reported cases \(dR^{e_{0},i_{0},\beta_{0}}(t_{k})=R^{e_{0},i_{0},\beta_{0}}(t_{k})-R^{e_{0},i_ {0},\beta_{0}}(t_{k}-1),\qquad\text{for }0<k\leq W,\) (12)  
where \(R^{e_{0},i_{0},\beta_{0}}\) is the accumulated count of reported cases from the simulation results. This standard deviation \(\sigma_{0}^{R}\) is restricted to the interval [0,100]. The priors are assumed as restricted uniform distributions, with a search range of \([0,100]\) for both initial exposed and infectious populations, \(e_{0}\) and \(i_{0}\), and \([0,0.8]\) for the transmissibility \(\beta\) (see Table 3). The library stan[31] in the platform R[32] is used for estimating the posterior distributions of the parameters. In the estimation, stan uses the Hamiltonian Monte Carlo (HCM) algorithm of Markov chain Monte Carlo method (MCMC) [33].