This research presents a novel mathematical study for analyzing the chemo-immune dynamics of non-muscle-invasive bladder cancer (NMIBC). We generalize a foundational three-compartment model (Mitomycin-C, Tumor, Effector-cells) by employing the fractal-fractional (FF) differential operator in the Caputo sense, characterized by a fractional order $\alpha$ and a fractal dimension $\beta$. This advanced operator is uniquely suited to capture the non-local memory effects inherent in immune system activation and the fractal (non-Euclidean) nature of the tumor microenvironment. We first establish the model's mathematical integrity by rigorously proving the existence, uniqueness, positivity, and boundedness of its solutions. The long-term behavior of the system is then analyzed, centered on the derivation of the basic reproduction number ( ${\mathcal{R}}_0$). We demonstrate that the disease-free equilibrium is globally asymptotically stable if ${\mathcal{R}}_0\le 1$, while a unique endemic (tumor) equilibrium emerges and gains stability if ${\mathcal{R}}_0>1$, indicating a transcritical bifurcation. For the numerical solution, we develop a semi-analytical scheme using the fractal-fractional Adomian decomposition method (FF-ADM) and validate its high accuracy against established numerical methods. Extensive numerical simulations are presented, including 2D and 3D plots, which visualize the profound impact of the fractional parameters $\alpha$ and $\beta$ on the system's trajectory, revealing that they significantly alter the time to tumor clearance. A comprehensive sensitivity analysis identifies the most critical parameters for controlling the disease, and 3D bifurcation plots illustrate the thresholds between tumor elimination and persistence. This work provides a more realistic and flexible tool for understanding NMIBC, with direct implications for optimizing treatment strategies.

This article presents and analyzes a mathematical model for smoking-related cancer that involves fractional-order derivative with seven different compartments. The model uses the ABC fractional derivative to describe the transmission dynamics of cancer caused by the smoking habit. We employed the Adams-Bashforth-Moulton method to find the numerical and graphical results of the model and we achieved a good level of accuracy. The existence and uniqueness of the model solution were established using Banach's fixed-point theory. For stability, we investigated the steady state points and basic reproduction number of the system. Additionally, the model's stability was discussed using the Hyers-Ulam criterion. The two-dimensional (2D) and three-dimensional (3D) simulations were performed for the different compartments and for the various values of the fractional-order parameters.

This paper presents a comprehensive investigation into the transmission dynamics of Lassa fever through a sophisticated mathematical model formulated with a fractal-fractional operator in the Atangana-Baleanu sense. The model's architecture is an SEIQR (susceptible-exposed-infected-quarantined-recovered) framework that incorporates crucial real-world epidemiological features, including a saturated incidence rate to account for behavioral changes at high infection levels and a relapse mechanism for recovered individuals. A rigorous qualitative analysis is conducted to establish the fundamental properties of the model, wherein we prove the existence, uniqueness, positivity, and boundedness of the solutions, ensuring the biological viability of the system. The stability of the model's equilibria is thoroughly examined. We derive the basic reproduction number ( $R_0$) using a next-generation matrix method, which serves as the critical threshold for disease persistence. We prove that the disease-free equilibrium is both locally and globally asymptotically stable when $R_0<1$. For the numerical investigation, we implement and validate a robust predictor-corrector scheme tailored for fractional-order systems, comparing its accuracy and convergence against the fractional Euler method and the classical Runge-Kutta scheme. Extensive numerical simulations are performed to visualize the system's dynamics. The results demonstrate that the fractional order provides enhanced flexibility in capturing memory effects, often leading to a more realistic, delayed epidemic peak. Furthermore, a series of novel three-dimensional dynamic surface plots reveals the geometric sensitivity of the solution manifold to key infection parameters, visually confirming that the transmission rate ( $\beta$) and quarantine rate ( $\eta$) are the most powerful levers in shaping the epidemic landscape. These findings offer a deeper understanding of Lassa fever's dynamics and highlight critical points for effective public health interventions.