# Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control
## 6 Conclusions  
In this paper, we proposed an optimization-based data assimilation approach to compute the unknown inputs and states of discrete-time compartmental epidemic models with uncertain normally distributed parameters. We started from the assumption that the joint state input parameter distribution is Gaussian. Then, a deterministic mean-variance recursion was developed, which made it possible to formulate the stochastic data assimilation problem as a single model predictive control design with a nonlinear mean-variance prediction model. We noted and demonstrated that the resulting \(\mu\Sigma\)-NMPC is computationally intensive, but its local optimum can be well approximated by a more efficient ordinary NMPC and further closed-form variance computations. We proposed simple heuristics to predict appropriate feedback gains, which realize state-dependent control actions along the prediction horizon. In this way, a trade-off can be made between the uncertainty of the computed states and inputs as the predicted control action is scheduled by the deviation between the actual realization of the state and its predicted expectation. As the approach does not make a difference between the unknown parameters and inputs, the joint state observation, change detection, and parameter estimation are also possible. Through the finite horizon predictive control computation, we estimated the unknown data of the past evolution of the COVID-19 epidemic spread within a fixed time window in Hungary. Among the unknown quantities, we considered the daily number of people in the different phases of the disease and the transmission rate of the pathogen, which highly depends on the actual social distancing rules, mobility restrictions, and virus mutations. The unknown time series were computed such that the expected value of the computed number of hospitalized patients fit the truly observed data as much as possible. Compared to our previous results [21; 22], we considered an augmented and uncertain compartmental epidemiological model with normally distributed random model parameters and a simple vaccination model as well. The main limitations of this study were the following. The length of hospital treatment was considered to be constant in the model, since no data have been published on the daily new hospital admissions with COVID-19, from which this parameter could be tracked efficiently. Moreover, no representative nationwide serological testing in Hungary has been organized since the summer of 2020. Such a result would definitely be helpful in making the estimate on the number of immune people more precise. Finally, the waning of immunity after infection or vaccination was also not taken into consideration in the model. However, such an extension does not affect the applicability of the proposed MPC-based estimation. With a few modifications, the approach can be applied to compute multiple uncertain possibly time-dependent parameters, but also for the prediction of the future behavior of the epidemic spread. The proposed methodology is able to extract and reconstruct detailed information from the whole time horizon of the epidemic process beyond giving estimates for the cumulative number of infected and recovered people. Future work will be focused on the analysis of other European countries. Conceptualization, G.S. and P.P. ; methodology, P.P. and G.S. ; formal analysis, P.P. and B.C. ; software, B.C. and P.P. ; visualization, B.C. and P.P. ; writing--review and editing, P.P. and B.C. ; supervision, G.S. ; funding acquisition, G.S. ; project administration, G.S. All authors have read and agreed to the published version of the manuscript. This work was supported by the grants \"OTKA\" SNN125739 and K131545 and the Thematic Excellence Program (TKP2020-NKA-11) of the National Research, Development, and Innovation (NRDI) Office.