# Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control
## 1 Introduction
The theory of optimization- and prediction-based simultaneous state and parameter estimation is well founded in the literature [29; 30; 31; 32], and it is applied in a wide spectrum of sciences, e.g., in geosciences [30], in medicine [33], in agriculture [34; 35], in aerospace [36], or in meteorology [37]. The classical approaches [30; 37] use variational principles and consider continuous-time models. Due to its simplicity and transparency, the MPC approaches with discrete-time model descriptions are widely used to solve optimal filtering problems, e.g., [32; 33; 34; 35] addressed model predictive data assimilation, i.e., optimal state/parameter reconstruction to minimize the deviation between the measurement and model output. In [36], an optimal design parameter was computed through a constrained MPC for a small satellite system, which provides constraint satisfaction during operation time. Furthermore, optimal dosing of cancer chemotherapy was addressed in [33] by solving a predictive control problem with joint state and parameter estimation. In general, the nonlinear MPC (NMPC) approaches result in a cumbersome optimization problem, especially when the model equations are stochastic. However, the available sequential convex programing approaches [38; 39], the algorithmic differentiation techniques [40; 41], and the numerical software tools [42; 43] exploit the special structure of a typical MPC problem and provide an efficient toolkit to solve the nonlinear problems precisely in a reasonable time. Among the several approaches to cope with stochastic dynamics [44], we mention two groups of techniques, which are popular in control theory. First, the particle-based approaches [45; 46; 47; 48; 49] with scenario trees allow coping with general (not necessarily Gaussian) models. Secondly, the tube-based approaches [50; 51; 52; 53] approximate each predicted state and input by a Gaussian distribution. Therefore, the dynamic equations in these references are recast as a deterministic mean-variance recursion. In this paper, we propose a generic optimization-based approach to reconstruct epidemiological data through the approximation of unknown time-dependent quantities (such as the state, unknown input, or parameter) of a class of discrete-time nonlinear stochastic dynamical models by a sequence of Gaussian distributions. The problem was formulated as a single stochastic NMPC (SNMPC) computation. However, the solution of an SNMPC over a relatively large prediction horizon is challenging. Therefore, the problem was solved in multiple steps. First, we approximated the expected input and state trajectories through a nonlinear deterministic problem. If the expected values of the unknown quantities are fixed, the variance matrix of the joint distribution is well defined. An optional state feedback gain computation is proposed to reduce the solution's conservatism, i.e., the estimated standard deviation of the unknown quantities. Finally, the computed mean and variance values serve as an initial solution for the SNMPC problem, which ensures a fast convergence for the nonlinear optimization. The paper is organized as follows. First, we describe the applied compartmental model for the COVID-19 epidemic spread in Section 2. Then, we introduce our computational approach in two steps in Sections 3 and 4. The numerical results with a discussion are presented in Section 5.