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.

Time series forecasting plays a critical role in domains such as energy, finance, and healthcare, where accurate predictions inform decision-making under uncertainty. Although Transformer-based models have demonstrated success in sequential modeling, their adoption for time series remains limited by challenges such as noise sensitivity, long-range dependencies, and a lack of inductive bias for temporal structure. In this work, we present a unified and principled framework for benchmarking three prominent Transformer forecasting architectures-Autoformer, Informer, and Patchtst-each evaluated through three architectural variants: Minimal, Standard, and Full, representing increasing levels of complexity and modeling capacity. We conduct over 1500 controlled experiments on a suite of ten synthetic signals, spanning five patch lengths and five forecast horizons under both clean and noisy conditions. Our analysis reveals consistent patterns across model families. To advance this landscape further, we introduce the Koopman-enhanced Transformer framework, Deep Koopformer, which integrates operator-theoretic latent state modeling to improve stability and interpretability. We demonstrate its efficacy on nonlinear and chaotic dynamical systems. Our results highlight Koopman based Transformer as a promising hybrid approach for robust, interpretable, and theoretically grounded time series forecasting in noisy and complex real-world conditions.

Time series forecasting plays a vital role across scientific, industrial, and environmental domains, especially when dealing with high-dimensional and nonlinear systems. While Transformer-based models have recently achieved state-of-the-art performance in long-range forecasting, they often suffer from interpretability issues and instability in the presence of noise or dynamical uncertainty. In this work, we propose DeepKoopFormer, a principled forecasting framework that combines the representational power of Transformers with the theoretical rigor of Koopman operator theory. Our model features a modular encoder-propagator-decoder structure, where temporal dynamics are learned via a spectrally constrained, linear Koopman operator in a latent space. We impose structural guarantees-such as bounded spectral radius, Lyapunov based energy regularization, and orthogonal parameterization to ensure stability and interpretability. Comprehensive evaluations are conducted on both synthetic dynamical systems, real-world climate dataset (wind speed and surface pressure), financial time series (cryptocurrency), and electricity generation dataset using the Python package that is prepared for this purpose. Across all experiments, DeepKoopFormer consistently outperforms standard LSTM and baseline Transformer models in terms of accuracy, robustness to noise, and long-term forecasting stability. These results establish DeepKoopFormer as a flexible, interpretable, and robust framework for forecasting in high dimensional and dynamical settings.