# Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control
## 1 Introduction  
The recent and still ongoing COVID-19 pandemic has brought unprecedented challenges for most countries to protect human lives and operate the economy and society at an acceptable level at the same time [1, 2]. In supporting the related difficult decisions, dynamical modeling of the epidemic process has had a key importance in all developed societies [3]. Depending on the modeling goal, a wide range of computational techniques is available to describe, forecast [4], and even control the epidemic process [5]. Here, we can only mention a few selected references from the related extensive literature. The majority of the modeling solutions are based on deterministic compartmental description derived from susceptible-exposed-infected-recovered (SEIR)-type models [6, 7, 8]. Due to the rapidly increasing computing power, agent-based models have also become popular for modeling epidemics and studying control possibilities [9, 10]. Logistic wavelets were used in [11] tocompute the possible cumulative number of people infected with COVID. The application of artificial intelligence and machine learning has also been successful in COVID-related modeling and prediction [12; 13; 14]. The online tracking of informative epidemic parameters and variables proved to be essential for the continuous monitoring and evaluation of the situation, making predictions, or planning interventions. A key parameter in such analyses is the time-varying reproduction number (\(R_{t}\)) which is the population-level transmission potential of the disease at time \(t\), which is known to be non-trivial to estimate [15]. Therefore, several different computational approaches have been proposed to track this quantity for epidemic outbreaks [16; 17]. In [18], the authors fit a stochastic compartmental model defined by a random walk to estimate the time-dependent reproduction number of the COVID-19 epidemic in Wuhan from publicly available datasets. A discrete-time Hawkes process was used for the estimation of \(R_{t}\) in [19], which allowed the detection of events such as restrictions and their relaxation. The estimation of other non-measured variables such as the number of people in latent or asymptomatic stages is also a relevant problem in data reconstruction [20]. In [21], a state estimator with proven convergence was proposed to implement model predictive control (MPC) satisfying complex constraints for an eight-compartment model of the COVID-19 pandemic. Essentially the same model was used in [22] for an inversion-based estimation of \(R_{t}\) from Hungarian data. Similar to [21], an MPC approach was presented in [23] to determine optimal social distancing rules (and hence, \(R_{t}\)) to mitigate the epidemic. The majority of the data analysis approaches use primarily the daily infected data possibly together with the recovery statistics. However, it is well known that only a fraction of the true cases are detected, which also depends on the testing intensity [24; 25; 26]. Moreover, the recording of recoveries is also often not immediate and sometimes not precise enough as well. Therefore, similar to [22; 27], we used the official data on the daily number of hospitalized people in Hungary, assuming that current testing rules and protocols in hospitals give sufficiently reliable and timely information. The basic system theoretic idea behind our proposed solution is that the transmission parameter \(\beta\), which is closely related to \(R_{t}\), can be considered as an input of a nonlinear system describing epidemic spread, and its estimation can be traced back to a trajectory-tracking control problem, where the output to be tracked is the true reported number of hospitalized people. A straightforward choice for the solution is MPC, where we can computationally handle model nonlinearities and parametric uncertainties [28].