In this paper, we developed a nonlinear mathematical model for the transmission of the monkeypox virus among populations of humans and rodents under the fractal-fractional operators in the context of Atangana-Baleanu. For the theoretical analysis, the renowned theorems of fixed points, like Banach's and Krasnoselskii's types, were used to prove the existence and uniqueness of the solutions. Additionally, some results regarding the stability of the equilibrium points and the basic reproduction number were provided. In addition, the numerical schemes of the considered model were established using the Adams-Bashforth method. Our analytical findings were supported by the numerical simulations to explain the effects of changing a few sets of fractional orders and fractal dimensions. Some graphic simulations were displayed with some parameters calculated from real data to understand the behavior of the model.

The initial value problem in Cauchy-type under the $ (k, \psi) $-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the $ (k, \psi) $-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.

The ability to accurately predict urban traffic flows is crucial for optimising city operations. Consequently, various methods for forecasting urban traffic have been developed, focusing on analysing historical data to understand complex mobility patterns. Deep learning techniques, such as graph neural networks (GNNs), are popular for their ability to capture spatio-temporal dependencies. However, these models often become overly complex due to the large number of hyper-parameters involved. In this study, we introduce Dynamic Multi-Graph Spatial-Temporal Graph Neural Ordinary Differential Equation Networks (DMST-GNODE), a framework based on ordinary differential equations (ODEs) that autonomously discovers effective spatial-temporal graph neural network (ST-GNN) architectures for traffic prediction tasks. The comparative analysis of DMST-GNODE and baseline models indicates that DMST-GNODE model demonstrates superior performance across multiple datasets, consistently achieving the lowest Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) values, alongside the highest accuracy. On the BKK (Bangkok) dataset, it outperformed other models with an RMSE of 3.3165 and an accuracy of 0.9367 for a 20-min interval, maintaining this trend across 40 and 60 min. Similarly, on the PeMS08 dataset, DMST-GNODE achieved the best performance with an RMSE of 19.4863 and an accuracy of 0.9377 at 20 min, demonstrating its effectiveness over longer periods. The Los_Loop dataset results further emphasise this model’s advantage, with an RMSE of 3.3422 and an accuracy of 0.7643 at 20 min, consistently maintaining superiority across all time intervals. These numerical highlights indicate that DMST-GNODE not only outperforms baseline models but also achieves higher accuracy and lower errors across different time intervals and datasets.