Data-driven optimization and parameter estimation for a metric graph epidemic model with applications to COVID-19 spread in Poland: A real-world example of optimization for a challenging Rosenbrock-type objective function

In this paper, we apply data-driven optimization to estimate key parameters in a metric graph-based epidemiological model, with the aim of analyzing the effect of road networks on the geographic spread of epidemics. As a case study, we fit our model to data from the COVID-19 pandemic in Poland during 2021. Our dataset integrates county-level daily case reports, national census information, and traffic flow studies. This framework allows us to examine the relative contribution of specific travel routes over time and infer unobserved transmission patterns in the presence of incomplete or unreliable case reporting. The optimization problem that arises from the model fitting yields an objective function resembling the Rosenbrock “banana” or “valley” function, a classical difficult benchmark for optimization algorithms. To our knowledge, this represents the first appearance of a Rosenbrock-type function in a real-world epidemiological context. We demonstrate that such a structure can emerge naturally from a simple uncoupled SIR model under specific conditions: a low initial incidence rate and a prolonged infectious period. This suggests that the Rosenbrock behavior is an intrinsic feature of fitting compartmental models to approximately Gaussian epidemiological data, providing a realistic yet simple scenario with which to test optimization algorithms. We explore optimization strategies suited to the Rosenbrock-type structure and identify a feasible parameter set for modeling the spread of COVID-19 in Poland. We use this set of parameters to identify discrepancies between the model and the data, explore how reducing traffic flow into urban areas can help flatten the infection curve, and identify some patterns in the distribution of intra- versus inter-city incidence rates. While recognizing the complex interplay of social and behavioral elements that cannot be fully captured in a high-level geographic model, our findings highlight the usefulness of metric graph-based models for understanding large-scale disease transmission in structured transportation networks.