# Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control
## 2 Compartmental Model of the Spread of the COVID-19 Epidemic in Hungary
### Computing the Reproduction Number  
To give meaningful estimates from an epidemiological perspective, we computed the basic and the time-dependent effective reproduction numbers. The basic reproduction number, namely the average number of new infections generated by a single infected individual in a fully susceptible population, can be given by the following closed-form expression [21]:  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Description** & **Nominal Value** & **Uncertainty** \\ \hline The population of Hungary & \(\mathbf{N}=9.8\times 10^{6}\) & \\ \hline Inverse of... (1/day) & & \\ latent period & \(\alpha=1/2.5\) & \(\pm 20\%\) \\ presymptomatic infectious period & \(\zeta=1/3\) & \(\pm 30\%\) \\ infectious period of symptomatic individuals & \(\rho_{l}=1/4\) & \(\pm 25\%\) \\ infectious period of asymptomatic individuals & \(\rho_{A}=1/4\) & \(\pm 25\%\) \\ average length of hospitalization & \(\lambda=1/10\) & \(\pm 10\%\) \\ \hline Relative infectiousness of asymptomatic & \(\delta=0.75\) & \(\pm 10\%\) \\ Probability of developing symptoms & \(\gamma=0.6\) & \(\pm 10\%\) \\ Hospitalization probability of symptomatic cases & \(\eta=0.076\) & \(\pm 10\%\) \\ Probability of fatal outcome (if already hospitalized) & \(\mu=0.205\) & \(\pm 10\%\) \\ Effectiveness of vaccination & \(\
u=0.75\) & \(\pm 10\%\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Presumed model parameters with their short description and their estimated uncertainty.  
\[R_{0}=\beta\left(\frac{1}{\xi}+\frac{\gamma}{\rho_{I}}+\frac{\delta(1-\gamma)}{\rho_ {A}}\right), \tag{3}\]  
It must be noted that using \(R_{0}=2.2\), an early estimate for Hungary commonly used in the literature, and expressing the transmission rate \(\beta\) from the equation, a nominal value of \(\beta=1/3\) can be given. However, as this parameter is highly influenced by the circumstances varying in time (e.g., the stringency of restrictions, new variants of the virus, etc.), it is considered to be a time-dependent parameter and estimated as such (the corresponding \(R_{0}\) being calculated afterwards).  
The time-dependent effective reproduction number \(R_{t}\) shows the average number of infections caused by a single individual, given the state of the model at time \(t\) (thus, taking into account the time-varying nature of beta and the decrease of the susceptible population). This is calculated as follows,  
\[R_{t}=\beta_{t}\left(\frac{1}{\xi}+\frac{\gamma}{\rho_{I}}+\frac{\delta(1- \gamma)}{\rho_{A}}\right)\frac{\mathbf{S}_{t}}{\mathbf{N}}\quad\text{with} \quad t\in\left\{0,1,\dots,T\right\} \tag{4}\]  
It can be a base for comparison between different epidemic-handling strategies, incorporating the strictness of the restrictions as well. In accordance with the traditional notation \"\(R_{t}\)\", we used \(t\in\mathbb{I}_{0}^{T}\) (instead of \(k\)) as the time parameter of the time-dependent reproduction number.  
Inspired by [21], we considered the daily transmission rate \(\beta_{k}\) of the disease an unknown time-dependent parameter. The past values of \(\beta_{k}\) were computed such that the evolution of the epidemic spread matched the available observations.