Abstract
In this paper, we propose a model-based method for the reconstruction of not directly measured epidemiological data. To solve this task, we developed a generic optimization-based approach to compute unknown time-dependent quantities (such as states, inputs, and parameters) of discrete-time stochastic nonlinear models using a sequence of output measurements. The problem was reformulated as a stochastic nonlinear model predictive control computation, where the unknown inputs and parameters were searched as functions of the uncertain states, such that the model output followed the observations. The unknown data were approximated by Gaussian distributions. The predictive control problem was solved over a relatively long time window in three steps. First, we approximated the expected trajectories of the unknown quantities through a nonlinear deterministic problem. In the next step, we fixed the expected trajectories and computed the corresponding variances using closed-form expressions. Finally, the obtained mean and variance values were used as an initial guess to solve the stochastic problem. To reduce the estimated uncertainty of the computed states, a closed-loop input policy was considered during the optimization, where the state-dependent gain values were determined heuristically. The applicability of the approach is illustrated through the estimation of the epidemiological data of the COVID-19 pandemic in Hungary. To describe the epidemic spread, we used a slightly modified version of a previously published and validated compartmental model, in which the vaccination process was taken into account. The mean and the variance of the unknown data (e.g., the number of susceptible, infected, or recovered people) were estimated using only the daily number of hospitalized patients. The problem was reformulated as a finite-horizon predictive control problem, where the unknown time-dependent parameter, the daily transmission rate of the disease, was computed such that the expected value of the computed number of hospitalized patients fit the truly observed data as much as possible.
Highlights
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]
The basic system theoretic idea behind our proposed solution is that the transmission parameter β, which is closely related to reproduction number (Rt), 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
To capture the spread and the evolution of the COVID-19 epidemic, we considered a modified version of the compartmental model introduced in [21]
Summary
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]. The basic system theoretic idea behind our proposed solution is that the transmission parameter β, which is closely related to Rt , 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. 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. 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.
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