Order reduction methods are important tools for systems engineering and can be used, for example, for parameter estimation of kinetic models for systems biology applications. In particular, the Proper Orthogonal Decomposition (POD) method produces a reduced-order model of a system that is used for solving inverse problems (parameter estimation). POD is an intrusive model order reduction method that is aimed to obtain a lower-dimensional system for a high-dimensional system while preserving the main features of the original system. We use a singular value decomposition (SVD) to compute a reduced basis as it is usually numerically more robust to compute the singular values of the snapshot matrix instead of the eigenvalues of the corresponding correlation matrix. The reduced basis functions are then used to construct a data-fitting function that fits a known experimental data set of system substance concentrations. The method is applied to calibrate a kinetic model of carbon catabolite repression (CCR) in Escherichia coli, where the regulatory mechanisms on the molecular side are well understood and experimental data for a number of state variables is available. In particular, we show that the method can be used to estimate the uptake rate constants and other kinetic parameters of the CCR model.

Mathematical modeling has become an indispensable tool for understanding, predicting, and controlling the spread of infectious diseases. Over the years, a wide variety of models have been developed to analyze disease dynamics and forecast epidemic trajectories. Deterministic and stochastic frameworks provide quantitative insights into transmission mechanisms and allow for rigorous evaluation of public health interventions such as quarantine, vaccination, and lockdown strategies. The integration of computational and data-driven methods has significantly advanced epidemic modeling. Techniques from network analysis, large-scale data processing, and artificial intelligence (AI) have improved both the accuracy and efficiency of model predictions. In particular, deep learning methods—most notably in medical imaging—enable fast and reliable automated diagnosis of disease. Moreover, the combination of mathematical modeling with AI facilitates real-time tracking and forecasting of outbreaks, thereby supporting public health authorities in optimizing resource allocation and ensuring timely responses. The increasing availability of open-source datasets, including case reports, demographic information, mobility patterns, and medical images, has further expanded the capabilities of data-driven epidemic models. Such approaches play a critical role in managing emerging infectious diseases, strengthening preparedness, and mitigating the societal impact of future outbreaks. This work provides a comprehensive overview of mathematical modeling approaches in infectious disease dynamics, emphasizing their relevance for public health emergency management and evidence-based intervention strategies.

Machine learning (ML) is a powerful tool in drug design, enabling the rapid analysis of large and complex molecular graphs that represent the structural and chemical properties of medications. It enhances the precision and speed with which molecular interactions are predicted, drug candidates are refined, and potential therapeutic targets are identified. When combined with graph theory, ML allows for the prediction of structural properties, molecular behaviour, and the performance of chemical compounds. This integration promotes drug development, reduces costs, and increases the likelihood of producing effective medicines. In this study, we focus on the efficacy of medications used in the treatment of coronary artery disease (CAD) using graph-theoretical methodologies, such as topological indices. We computed several degree-based topological descriptors from chemical graphs, capturing essential connectivity and structural properties. These variables were incorporated into a machine learning framework to develop predictive models that identify structural factors influencing medication performance. Our study explores a dataset of known CAD drugs using supervised learning techniques to estimate their potential efficacy and support improved molecular design. The findings highlight the utility of graph-theoretical descriptors in enhancing prediction accuracy and providing insights into fundamental structural elements related to drug efficacy. Furthermore, this work emphasises the synergy between chemical graph theory and machine learning in accelerating drug development for CAD, offering a scalable and interpretable framework for future pharmaceutical applications.