# Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control
## 5 Results and Discussion
Plots 4, 5, and 6 of Figure 2 illustrate three derived probabilistic quantities \(\hat{z}_{k}=h(\hat{x}_{k},\hat{u}_{k},\hat{\theta})\), namely the time-dependent effective reproduction number (4) (\([\hat{z}_{k}]_{1}=R_{k}\)), the number of all infected people (\([\hat{z}_{k}]_{2}=\mathbf{\hat{L}}_{k}+\mathbf{\hat{P}}_{k}+\mathbf{\hat{1} }_{k}+\mathbf{\hat{A}}_{k}+\mathbf{\hat{H}}_{k}\)), and the number of daily new infections (\([\hat{z}_{k}]_{3}=\hat{\beta}_{k}\left(\mathbf{\hat{P}}_{k}+\mathbf{\hat{1} }_{k}+\mathbf{\hat{\delta}}\,\mathbf{\hat{A}}_{k}\right)\frac{\mathbf{\hat{S} }_{k}}{\mathbf{N}}\)). The first and third coordinates of \(\hat{z}_{k}\) are nonlinear functions of random variables. Therefore, the mean and the variance of \(\hat{z}_{k}\) were approximated by the first-order Taylor polynomial of function \(h\), namely,  
\[\hat{z}_{k}\ \approx\ \frac{\partial h}{\partial(x,u,\theta)}\left(u_{k} \right)\left( \begin{smallmatrix}\hat{x}_{k}\\ \hat{u}_{k}\\ \hat{\theta}_{k}\end{smallmatrix} \right)+h(\mu_{k})-\frac{\partial h}{ \partial(x,u,\theta)}(\mu_{k})\ \mu_{k}\ \sim\ \ \mathcal{N}\Big{(}h(\mu_{k})\,\ \frac{\partial h}{ \partial(x,u,\theta)}(\mu_{k})\ \Sigma_{k}\ \Big{(}\frac{\partial h}{ \partial(x,u,\theta)}(\mu_{k})\Big{)}^{\top}\Big{)}. \tag{25}\]  
The yellow curve in Plot 4 illustrates the estimated reproduction number published online by Atlo Team in [74]. \begin{table}
\begin{tabular}{l r r} \hline \hline
**Quantitative Properties of the Optimization** & **Problem 1** & **Problem 2** \\ \hline Total number of variables & 4869 & 70,614 \\ Number of variables with only lower bounds & 4383 & 4383 \\ Number of variables with lower and upper bounds & 486 & 486 \\ Total number of equality constraints & 4383 & 70,128 \\ Number of nonzeros in the Lagrangian Hessian & 10,826 & 386,575 \\ Number of iterations & 212 & 74 \\ Elapsed time (s) \({}^{*}\) & 8 & 1187 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational complexity of Problems 1 and 2 illustrated through the epidemiological data assimilation case study. In this comparison, the cumulative number of recovered people \(\mathbf{R}^{\text{(all)}}\) as an additional state variable is _not_ considered. Accordingly, the number of state variables is \(n=9\), the number of uncertain parameters is \(p=10\), and the number of inputs and the measured disturbances are \(m=1\) and \(q=1\), respectively. \({}^{*}\) The processing time was measured on a laptop PC with Intel Core i7-4710 MQ CPU at 2.50 GHz and 16 GB of RAM. Figure 2: Reconstructed epidemiological data computed for System (1) with Gaussian model parameters: expected value of states (Plot 1), transmission rate of the pathogen (Plot 2), time-dependent reproduction number (Plot 4), number of reconstructed hospitalized patients compared to the recorded data (Plot 3), sum of all infected compartments (Plot 5), and the daily new infections (Plot 6). The gray dotted vertical grid lines show the first days of the months. In Plot 12 of Figure 3, we present three uncertain time series, namely the number of recovered, but not yet vaccinated people (blue), the cumulative number of recovered people (red), and the cumulative number of immune people (green). Figure 3: Unknown epidemiological data computed for System (1) with Gaussian model parameters: number of people in the different phases of the disease (Plots 7–10), number of deceased people (Plot 11), and number of recovered and/or immune people (Plot 12). The gray dotted vertical grid lines emphasize the first days of the months. In Figures 2 and 3, we can observe the successfully suppressed first wave with a dramatic effect of the strict lockdown introduced in March 2020. As a result, the disease was mainly confined to closed institutions such as certain hospital wards and elderly homes. This policy could not be maintained in the autumn of 2020, and therefore, a substantially larger second wave occurred, causing a huge burden on the healthcare system.