This research introduces a new method to control pneumonia spread through a fractal-fractional mathematical model, which combines quantum-inspired neural control (QINN) systems. The model uses Caputo fractional differential equations to simulate pneumonia dynamics through self-similar memory-based systems. The system achieves stable operation through standard rigorous analysis together with a quantum-inspired neural controller, which maintains system stability during tracking and disturbance handling. The control system uses sliding mode control (SMC) combined with adaptive quantum-inspired neural networks, which update their weights through fractional-order gradient descent algorithms. Lyapunov stability analysis is employed to prove the global asymptotic stability of the closed-loop system. Numerical simulations demonstrate the effectiveness of the proposed approach in achieving accurate trajectory tracking and robust performance under varying conditions. The quantum-inspired controller provides three main benefits through superposition, which allows simultaneous control action assessment for fast response and entanglement-based synchronization that unifies all disease control sectors and quantum measurement systems, which safeguard against uncertain data inputs. The research establishes a connection between fractal-fractional calculus and quantum-inspired computing and control theory to develop a novel approach for modeling and controlling complex biological systems.

Time-fractional diffusion-wave equations (TFDWEs) effectively model various phenomena, including heat transfer, chemical and disease diffusion, reactive substance propagation, seismic wave dynamics, and signal transmission in electrical systems. This study presents an enhanced operational collocation method to solve TFDWEs with variable order. The proposed approach integrates the collocation technique with eighth-kind Chebyshev polynomials (EKCPs) as basis functions. To improve computational accuracy, new operational matrices for integer-order integrals in time and space are constructed using an innovative matrix framework. Additionally, a pseudo-operational matrix is developed to approximate variable-order integrals of the basis vector. The novel operational matrices comprise two components: an enhanced operational matrix and a supplementary vector. An error bound for the residual function is derived in a Chebyshev-weighted space. Comparative analysis with a second-order Chebyshev wavelets method demonstrates the efficacy of the proposed scheme, yielding small absolute errors.

The study develops a new approach that combines machine learning (ML) with fractional-order differential equations (FDE) to study tumor development in mammograms and evaluate cancer risk. XGBoost delivered its peak performance results through a 961-case dataset (46.3

Accurate prediction of chemical toxicity to aquatic organisms remains a critical challenge in environmental risk assessment. Traditional machine-learning (ML) models capture static correlations but cannot represent time-dependent, memory-driven processes such as bioaccumulation and cumulative physiological damage. This study presents a hybrid framework that integrates an ensemble ML classifier with a system of fractional-order differential equations (FDEs) for both toxicity prediction and dynamic modeling. Using the QSAR Fish Toxicity dataset (908 compounds, six molecular descriptors), a LogitBoost classifier (MATLAB implementation of an XGBoost approximation) was trained after MRMR feature selection and SMOTE oversampling, achieving an AUC of 0.9015. The continuous probability output of this classifier serves as the external forcing function for six coupled Caputo-type FDEs that evolve the normalized molecular descriptors over a biologically motivated pseudo-time axis. The resulting hybrid model is validated by an existence-uniqueness theorem, Adams–Bashforth–Moulton numerical integration, and constrained nonlinear optimization. Risk assessment identifies compounds that exceed regulatory thresholds. The bidirectional ML–FDE architecture supplies both high predictive accuracy and explicit memory-aware mechanistic trajectories, thereby complementing purely data-driven or purely mechanistic approaches. Further experimental validation on independent datasets is warranted.

Background: Scabies continues to affect around 300 million individuals each year and persists as a major neglected tropical disease. Problem statement: Current mathematical models generally fail to incorporate gender- and age-specific population structure (particularly the vulnerable 15–24 age group), comprehensive mite population dynamics, real clinical datasets, and the memory-dependent behavior captured by fractional calculus. Methods: This study proposes a novel ten-compartment Caputo fractional-order model that distinguishes four susceptible subgroups according to gender and age (15–24 years versus all other ages), while also tracking exposed, symptomatic, asymptomatic, and recovered individuals together with an explicit mite population compartment. The model is parameterized using real hospital records collected between 2017 and 2023. Model properties are examined analytically, and numerical simulations are performed. The chosen age grouping is motivated by hospital data revealing markedly higher incidence rates among 15–24-year-olds, attributable to increased social and sexual contact as well as shared living environments. Results and findings: Sensitivity analysis identifies the transmission rate as the most influential parameter. The susceptible population in the 15–24 age bracket decreases at a markedly faster rate than other groups. Raising the fractional order accelerates early epidemic growth yet enhances the apparent effectiveness of control measures and produces stronger reduction in the mite population. The fractional formulation accounts for memory effects and delivers improved long-term forecasting performance compared with conventional integer-order models. Conclusion: The proposed framework identifies clear intervention priorities focused on young adults and mite control, thereby supporting global efforts to meet the WHO 2030 targets for scabies elimination.

This study proposes a matrix-based collocation method employing sixth-kind Chebyshev polynomials (SKCPs) to explore the dynamics of a fractional-order compartmental model for Glaucoma characterized by progressive damage to the optic nerve and subsequent loss of vision. The pseudo-operational matrices for integration are formulated based on shifted SKCPs over an extended interval [0, 𝒯] , enabling efficient numerical approximation of the model’s solutions. The approximations provide insights into the behavior and interdependencies of the model’s compartmental variables. The theoretical analysis confirms the existence and uniqueness of the solution set via Schauder’s fixed-point theorem. Furthermore, the Ulam–Hyers stability of the approximate solutions is established, ensuring the robustness of the numerical scheme under perturbations. To evaluate model responsiveness, a sensitivity analysis is conducted by varying key parameters, identifying critical factors influencing disease dynamics. A feedback control strategy is integrated into the model to enhance its predictive and corrective capabilities, aiming to inform treatment strategies for improving eye health. This control mechanism is derived from the pseudo-operational collocation method combined with the Lagrange multiplier technique. Lastly, error bounds for the residual functions are rigorously estimated in a Chebyshev-weighted norm, verifying the accuracy and convergence properties of the proposed numerical method.

This study introduces a hybrid approach combining fractional-order dynamical modeling with machine learning (ML) to predict Parkinson’s disease (PD) using vocal biomarkers. A fractional-order system captures memory-dependent vocal impairment dynamics, with proven stability and uniqueness. Calibrated with clinical data, it achieves 97.44 % accuracy, while a Deep Neural Network (DNN) attains 91.8 % accuracy (AUC: 95.8 %). Key biomarker Pitch Period Entropy (PPE) (importance: 0.9014) emerges as most discriminative. The fusion of fractional calculus and ML enhances interpretability and predictive power, enabling early, non-invasive PD detection. This framework advances computational neurology, offering potential for personalized monitoring and improved diagnostic tools.

A fractional-order mathematical model of lung cancer is used to describe the dynamics of tumor growth and the interactions between cancer cells and immune cells. To obtain approximate solutions and better understand the behavior of the state functions, a pseudo-operational collocation scheme employing shifted Jacobi polynomials as basis functions is introduced. Initially, the existence and uniqueness of solutions to the model are established using the Leray-Schauder fixed-point theorem. Error bounds for the residual functions are estimated within a Jacobi-weighted L2-space. To enhance the accuracy and reliability of the results, two distinct strategies are implemented: sensitivity analysis and feedback control. The feedback control of the proposed pseudo-operational spectral method is performed using the method of Lagrange multipliers, marking its first application in this context. Spectral solutions are derived by applying the pseudo-operational scheme to both the original model and the model with control functions. Improved performance and outputs are anticipated following the application of the feedback control strategy. Finally, comprehensive biological interpretations of the results are provided, offering insights into the practical implications of the model.

Spinal disorders, such as disk hernia and spondylolisthesis, affect millions worldwide, leading to chronic pain and reduced quality of life due to disruptions in biomechanical alignment. Traditional diagnostic methods often overlook the viscoelastic memory effects in spinal tissues, necessitating advanced models that integrate machine learning with fractional calculus for improved accuracy and interpretability. The research introduces a new fractional-order machine learning system that analyzes vertebral column abnormalities through biomechanical motion analysis by using the University of California, Irvine (UCI) vertebral column dataset. The system selects the best machine learning model from Random Forest (RF), Gradient Boost (GB), XGBoost, Deep Neural Network (DNN), and Voting Ensemble (VE) models to work with Caputo fractional-order differential equations, which simulate spinal tissue viscoelasticity memory effects through pseudo-time analysis of pelvic incidence data. The study demonstrates that GB achieves its highest accuracy at 0.937, and RF attains its top Area Under the Curve (AUC) at 0.949, and the fractional model achieves a weighted Mean Squared Error (MSE) of 0.0017. The optimized parameters showed that the growth rates and coupling coefficients worked in an inhibitory manner. The biological findings demonstrated that patients with higher pelvic incidence and lumbar lordosis variability experienced greater spinal stress, which led to compensatory curvature patterns that: disk hernia and spondylolisthesis development. The results showed that pelvic incidence and sacral slope had the strongest correlation at 0.87, and pelvic incidence and lumbar lordosis angle showed a correlation of 0.74, which informs the fractional-order growth rates where higher-correlated features evolve faster under pathological stress due to their mechanical interdependence. Theoretical proofs establish solution existence and uniqueness and boundedness, and stability, and numerical efficiency emerges from the Adams-Bashforth-Moulton method, while a diagnostic Graphic User Interface (GUI) enables clinical application. The framework enables better spinal disease detection through fractional derivatives, which replicate real biomechanical operations.

The research introduces a new system that combines machine learning with fractional-order differential equations to forecast cervical cancer risk and track its progression through vital risk elements. The Hospital Universitario de Caracas in Venezuela provided a dataset of 858 records with 36 features, which Random Forest, XGBoost, Support Vector Machines, and Gradient Boosting models used to achieve an XGBoost AUC of 0.992 for binary classification despite class imbalance, which was handled using Synthetic Minority Over-sampling Technique (SMOTE). The Caputo FDE system with fractional-order 0.6085 received the ML-derived probabilities to simulate five normalized risk factors, which include first sexual intercourse and age, intrauterine device years, sexually transmitted diseases, and hormonal contraceptives during normalized pseudo-time. The optimization process using sequential quadratic programming achieved a weighted mean squared error of 0.0042, which demonstrates precise matching between predictions and actual data. The main results show that ML-FDE probability surfaces improve risk prediction by demonstrating higher cancer risks for women who use hormonal contraceptives for extended periods and have sexually transmitted diseases. The method provides accurate results for both high and low probability cases. The system integrated the predictions through its real-time diagnostic system.

This research investigates glaucoma, a leading cause of blindness marked by progressive optic nerve damage due to uncontrolled Intraocular Pressure (IOP). We propose a novel fractional-order model capturing the dynamic relationship between aqueous humor production, drainage efficiency, and optic nerve health. Through sensitivity analysis, three critical intervention points emerge: the aqueous humor inflow rate, baseline secretion levels, and the optic nerve's vulnerability to pressure-induced damage. Our findings suggest that prolonged IOP elevation triggers irreversible nerve degeneration, particularly when outflow pathways are compromised, whereas stable aqueous humor circulation helps preserve vision. To counteract disease progression, we design an Adaptive Fractional-order Fuzzy Sliding Mode Controller (AFFSMC) that precisely adjusts IOP by modulating aqueous humor dynamics and protecting neural tissue. Simulation results demonstrate that the AFFSMC reduces IOP fluctuation variance by 25% and achieves convergence within 3 seconds, while sensitivity analysis reveals that the aqueous humor inflow rate has the highest sensitivity index, identifying it as a primary therapeutic target.

This work presents a new and efficient numerical framework for solving three-dimensional time-fractional diffusion and mobile–immobile equations in the Caputo sense. The method is formulated using four-variable Jacobi polynomials, constructed systematically via the Kronecker product of one-dimensional Jacobi bases to accurately represent the multidimensional nature of the governing equations. Within a pseudo-operational collocation formulation, these polynomials enable a highly accurate and computationally efficient approximation of the fractional operators in both temporal and spatial directions. From the theoretical standpoint, the existence and uniqueness of the approximate solution are rigorously established through Schauder’s fixed-point theorem. Furthermore, the Ulam–Hyers stability of the numerical solution is verified, demonstrating the robustness of the method with respect to perturbations in the input data. To reinforce the reliability of the approach, an explicit error bound for the residual function is derived in a Jacobi-weighted Sobolev space, offering a firm analytical basis for assessing convergence. Numerical experiments confirm that the proposed approach achieves superior accuracy and efficiency, highlighting its potential as a powerful tool for high-dimensional fractional partial differential equations.

This research develops a mathematical model to explore the complex dynamics linking healthcare, environmental sustainability, and Sustainable Development Goals (SDGs). Using fractional calculus with Caputo derivatives, the model captures the intertwined nature of these systems. It analyzes relationships among health status, healthcare economics, public awareness, and SDG progress over time, offering a comprehensive framework for these global challenges. Through detailed simulations, the study reveals dynamic feedback loops and time-dependent effects, highlighting system complexity. Calibrating fractional order parameters clarifies each variable's influence, enhancing understanding of overall dynamics. The results provide valuable insights for policymakers and researchers, supporting efforts toward a healthier, sustainable future. This work advances mathematical modeling in interdisciplinary fields and lays a foundation for future studies addressing global sustainability challenges.

This study introduces a novel hybrid framework that integrates machine learning (ML) with fractional-order differential equations (FDE) to enhance the prediction and clinical management of psoriasis, leveraging real-world data from the UCI Dermatology Dataset. By optimizing ML models, particularly the Voting Ensemble, to inform FDE parameters, and developing a user-friendly graphical user interface (GUI) for real-time diagnostics, the approach bridges computational efficiency with physiological realism, capturing memory-dependent disease progression beyond traditional integer-order models. Key findings reveal that the Voting Ensemble achieves a precision of 0.986 ± 0.007 and an AUC of 0.992 ± 0.005. At the same time, the fractional-order model, with an optimized order of 0.6781 and a mean square error (MSE) of 0.0031, accurately simulates disease trajectories, closely aligning with empirical trends for features such as Age and SawToothRete. The GUI effectively translates these insights into clinical tools, demonstrating probabilities ranging from 0% to 100% based on input features, supporting early detection and personalized planning. The framework’s robustness and potential for broader application to chronic conditions highlight its significance in advancing healthcare.

BACKGROUND AND OBJECTIVE:Malignant melanoma, an aggressive form of skin cancer, poses significant challenges due to its rapid progression, metastatic potential, and resistance to therapies. This study aims to develop a fractional-order mathematical model capturing melanoma dynamics (tumor-immune interactions, extracellular matrix remodeling, nutrient dynamics) and introduce a Stabilized Fuzzy Sliding Mode Control (SFSMC) strategy to suppress tumor growth and restore microenvironmental homeostasis. METHODS:A fractional-order model was derived using Caputo derivatives to incorporate memory effects and long-range dependencies. The SFSMC combines sliding mode control with fuzzy logic to manage uncertainties. Theoretical analysis included well-posedness, stability (via Lyapunov functions), and computation of the reproduction number R0. Numerical simulations were performed using a predictor-corrector method with parameters calibrated from clinical data. RESULTS:The model demonstrated stability when R0<1, indicating tumor suppression. SFSMC reduced tumor cell populations by 78% and circulating tumor cells by 65% while improving immune response (45% increase in immune cells) and nutrient availability (30% recovery). Sensitivity analysis revealed R0 is mostly influenced by tumor growth rate, natural degradation rate of extracellular matrix (ECM), rate of ECM degradation by tumor cells, and ECM production rate, suggesting their potential role in suppressing tumor growth. CONCLUSIONS:The fractional-order framework and SFSMC offer a robust approach to modeling and controlling melanoma, with potential clinical implications for adaptive therapy.

Two-dimensional multi-term time-fractional diffusion-wave equations (TFDWEs) have numerous applications in fields such as material science, complex media, thermodynamics, heat conduction, quantum systems, finance, and economics. This study focuses on solving a specific type of 2D TFDWE by developing a pseudo-operational collocation scheme that utilizes three-variable Jacobi polynomials over the domain [-1,1] x [-1,1] x [0,1]. To achieve this, we construct three-variable Jacobi polynomials using classical orthogonal Jacobi polynomials P-i(sigma,sigma)(x) and P-j(theta,& vartheta;)(y) for x, y is an element of [-1,1], where the parameters sigma,sigma,theta, and & vartheta; are greater than -1. Additionally, we incorporate shifted orthogonal Jacobi polynomials P-k(& varepsilon;,epsilon)(t) for t is an element of [0,1] with & varepsilon;,epsilon>-1. This represents the first instance of such a construction. Variations in the parameters sigma,sigma,theta,& vartheta;,& varepsilon;, and epsilon lead to the generation of different basis functions, each with distinct root distributions that serve as collocation points. Specifically, if sigma > sigma, the roots are more densely concentrated around x=-1; if sigma< sigma, the roots cluster around x=1; and if sigma=sigma, the roots are uniformly distributed across both ends. The next step involves deriving pseudo-operational matrices of the integration for both fractional and integer orders corresponding to the three-variable basis vector. To accomplish this, the Kronecker product of integral pseudo-operational matrices connected to one-variable basis vectors is utilized. A simpler system of algebraic equations is derived by substituting matrix relations and approximations into the given equation and collocating the resulting algebraic equation at the collocation nodes. This simplification facilitates finding an approximate result compared to directly solving the original equation. Before tackling the given problem, the existence and uniqueness of the 2D multi-term time-fractional diffusion-wave equations (MTFDWEs) are examined. Error bounds for the approximate solutions derived from the proposed method and their derivatives concerning the independent variables are evaluated in a Jacobi-weighted Sobolev space. This assessment confirms that selecting an appropriate number of basis functions yields an approximate solution with satisfactory accuracy. To exhibit the efficiency and effectiveness of the method, several 2D TFDWEs are solved that incorporate a diverse range of fractional-order terms.

Glioblastoma Multiforme (GBM) is a highly aggressive and lethal form of brain tumor, presenting significant challenges in treatment and management. This study proposes an advanced control model for the management of GBM, focusing on the optimization of drug therapies and adaptive regulation of tumor volume. Leveraging fractional-order mathematical modeling, the approach integrates mathematical modeling techniques with control theory to develop comprehensive strategies for GBM management. It introduces a novel fractional-order mathematical model that uniquely integrates nutrient dynamics and blood flow into the analysis of Glioblastoma Multiforme (GBM) progression. By combining fractional-order calculus with adaptive PID control, the approach provides a more accurate representation of tumor behavior and enables personalized, real-time treatment optimization. The proposed control models aim to optimize drug therapies by dynamically adjusting treatment parameters based on real-time tumor volume monitoring. This adaptive approach allows for personalized treatment regimens tailored to the specific characteristics of the tumor, enhancing therapeutic efficacy while minimizing adverse effects. The proposed framework advances current methodologies by addressing memory effects and long-range dependencies in tumor growth, offering new insights for precision medicine. By harnessing the capabilities of fractional-order modeling, our research provides a novel framework for advancing the management of GBM, offering new insights and strategies for combating this formidable disease.

This paper introduces a comprehensive modeling framework for in vitro fertilization (IVF) by combining two distinct yet complementary modeling approaches. First, an artificial neural network (ANN) model is developed to capture intricate relationships within the IVF system, leveraging its ability to learn complex patterns from empirical data. The ANN model is followed by a fractional-order model (FOM) to capture enhanced temporal dynamics for the process under consideration, and it represents a new approach towards systems characterized by a fractional derivative. The proposed ANN model learns nonlinear relationships between different inputs, including sperm morphology and motility, oocyte quality, and other relevant biological variables that affect the outcome of IVF. Furthermore, the FOM expands traditional differential equations into fractional-order derivatives, which provide more accurate representation for the intricate fractional dynamics inherited by reproductive processes. Combining the ANN and FOM frameworks leads to a synergistic model, which improves our understanding in terms of fractional-order dynamics that regulate IVF outcomes and improves predictive capabilities. The experimental results show the effectiveness of the integrated approach in modeling the complexities of IVF dynamics. The integrated ANN and FOM approach gives a more complete view of the underlying biological processes involved in IVF, making better predictions of the success rate of IVF. This integrated modeling paradigm, by incorporating artificial intelligence into the fractional order dynamics, is furthering the science of reproductive biology while enhancing the forecasting capacity of the success of IVF. The results of the simulations suggest that the framework can improve awareness of the complex dynamics in assisted reproductive technologies and is a useful tool not only for academics but also for medical professionals.

This paper proposes a sophisticated mathematical model for the dynamics of lung cancer, involving Caputo-type fractional derivatives and interaction-specific time delays. The model characterizes the populations of lung epithelial cells, immune cells, and mutated cells, elucidating their nonlinear interactions and external influences. Fractional derivatives introduce memory effects, while time delays capture the biological latency in cellular interactions. This refined approach allows a deeper understanding of lung cancer progression and offers a robust framework to investigate therapeutic strategies. Hyers-Ulam stability and delay sensitivity were performed alongside other analytic computations. An advanced numerical analysis using the Adams-Bashforth-Moulton method was used to simulate the model. The model demonstrates the competitive balance between cellular growth and decay, which are interdependent and serve as the basis for further applications in cancer studies. The results show that the delays account for the latency of the immune response and the time required for cellular interactions, leading to a reduced growth rate compared to the basic reproduction number. These new perspectives highlight areas and directions to improve medical interventions.

This study presents a novel dual approach for diabetes risk prediction in women, combining machine learning classification with fractional-order physiological modeling. We employ seven machine learning algorithms: Decision Tree, Logistic Regression, Support Vector Machine (SVM), Random Forest, Bagged Trees, Naive Bayes, and XGBoost, to identify key risk factors, with XGBoost demonstrating higher performance. Glucose levels, BMI, blood pressure, and Diabetes Pedigree Function emerged as the most significant predictors across all models. Complementing these data-driven insights, we develop a Caputo fractional-order model that captures the temporal dynamics of glucose-insulin regulation, BMI, and blood pressure. Through fixed-point theorem analysis, we prove the existence and uniqueness of solutions, while numerical implementations using Lagrange polynomial interpolation reveal how varying fractional orders affect metabolic response patterns. This mathematical framework provides unique insights into the progression of diabetes, particularly through its ability to model memory effects and long-term physiological changes. The practical implementation of our research features an intuitive graphical user interface (GUI) that integrates both approaches, enabling real-time risk assessment with dynamic feedback. Our analysis of the Pima Indians dataset confirms important physiological relationships, including age-pregnancy and BMI-skin thickness correlations. This dual-method framework offers clinicians a comprehensive tool for diabetes management, combining the immediate predictive power of machine learning with the longitudinal perspective of fractional-order modeling. The machine learning component provides accurate short-term risk stratification, while the fractional-order model enhances understanding of long-term disease progression. Together, they enable more personalized and proactive care strategies, advancing both the theory and practice of diabetes risk assessment.