This paper investigates a four-dimensional predator-prey model with cross-species infection and Holling type II functional response. The model incorporates logistic growth for susceptible prey, susceptible-infected (SI)-type disease transmission with mass-action incidence, and a biologically realistic mechanism by which predators become infected through consumption of infected prey. We derive five key ecological and epidemiological threshold parameters governing predator persistence and disease invasion in different ecological scenarios. Analytical results establish positivity, boundedness, and conditions for the existence, feasibility, and stability of all equilibria. These include disease-free coexistence, endemic prey-only, and full endemic coexistence states. Global stability of the disease-free coexistence equilibrium is obtained using the Lyapunov method. Bifurcation analyses reveal transcritical, Hopf, and saddle-node bifurcations, explaining the transitions between extinction, stable coexistence, oscillatory dynamics, and bistability. Co-dimension-two analysis identifies organizing centers that structure the parameter space and clarifies mechanisms underlying regime shifts. Numerical simulations using MATLAB and MatCont confirm theoretical findings and illustrate diverse dynamical behaviors. These behaviors include predator extinction driven by highly infectious diseases, predation-mediated disease control, sustained oscillations, and multiplicity. Spatial extension via a reaction-diffusion framework demonstrates diffusion-driven Turing instability and pattern formation. The results provide integrated ecological and epidemiological insights into cross-infection dynamics and predator-prey coexistence.

Thisarticle develops and analyzes a fractional-order Leslie-Gower prey-predator-parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model's central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order alpha is an element of(0,1], (ii) two distinct biological delays-an infection transmission delay tau 1 and a predator handling delay tau 2-and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators (-Delta)gamma/2. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (L & eacute;vy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions-a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order alpha modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent gamma controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that beta, alpha, and tau 1 may be priority targets for parameter estimation.

This paper investigates a fractional-order mathematical model for the co-infection dynamics of pneumonia and typhoid fever using the Liouville-Caputo derivative. We establish the existence, uniqueness, non-negativity, and boundedness of solutions using Banach's fixed point theorem and fractional comparison principles. The Hyers-Ulam and generalized Ulam-Hyers-Rassias stability of the system are rigorously proved; this stability analysis is epidemiologically significant because it guarantees that small perturbations in initial conditions or model parameters-inevitable in real-world data collection-do not lead to unbounded deviations in disease trajectory predictions. To approximate solutions numerically, we develop a Laplace-Based Optimized Decomposition Method (LODM) and validate its convergence against a modified predictor-corrector scheme. The LODM provides a semi-analytical series solution, while the predictor-corrector method serves as a numerical benchmark; this dual approach ensures reliability of simulations. Numerical simulations illustrate the influence of the fractional order xi on system dynamics. Quantitative comparison between xi=1 (integer order) and xi<1 (fractional order) demonstrates that fractional modeling reduces peak infection by 12-18% and delays epidemic peaks by 15-30 days, confirming that memory effects capture long-term epidemiological dependencies that integer-order models fail to reproduce. A biological interpretation links the fractional order to immune memory, pathogen persistence, and intervention latency. This study provides both theoretical and numerical evidence supporting the use of fractional calculus in epidemiological modeling.

This paper develops a fractional-order mathematical framework to investigate the co-infection dynamics of Pneumonia and Typhoid fever using the Caputo-Fabrizio (CF) fractional derivative. The CF operator, characterized by a non-singular exponential kernel, captures the fading memory and hereditary effects inherent in biological processes, providing a more realistic representation of disease transmission. The total human population is divided into eight interacting compartments-susceptible, singly infected (with either Pneumonia or Typhoid), co-infected, recovered groups, and the environmental bacterial concentration. A system of CF-based fractional differential equations is formulated to describe the temporal evolution of each compartment. Theoretical analysis establishes the positivity, boundedness, and Hyers-Ulam stability of the system. Existence and uniqueness of solutions are rigorously proved using Lipschitz continuity and fixed-point theorems in Banach spaces. A modified fractional Euler method tailored for the CF operator is employed to obtain numerical solutions and assess convergence behavior. Numerical simulations under various fractional orders demonstrate that decreasing the fractional order enhances memory effects, leading to slower epidemic progression, attenuated infection peaks, and delayed recovery. The results confirm that memory-driven fractional dynamics significantly influence co-infection behavior and bacterial persistence. The proposed CF-Euler framework improves the biological realism of epidemic modeling and provides a foundation for developing memory-based control strategies in managing co-infectious diseases.

This paper investigates the stochastic dynamics of a fractional-order Leslie-Gower eco-epidemiological model under white noise perturbations. The fractional derivatives are interpreted in the Caputo sense, and the stochastic fractional differential equations are formulated in integral form, with dt^α denoting the fractional-order time increment. We prove the existence and uniqueness of global positive solutions and establish sufficient conditions for stochastic boundedness, extinction, and mean persistence using stochastic Lyapunov functionals, a fractional Itô formula, and comparison theorems. The stochastic stability of equilibrium points is analyzed, revealing noise-induced shifts between deterministic and stochastic dynamical regimes, including stochastic stabilization. An extended Euler-Maruyama scheme is developed for numerical simulation, and the results illustrate complex phenomena, such as noise-induced stabilization and stochastic bifurcations. Our findings highlight the interplay between environmental randomness and fractional memory in ecological systems. To the best of our knowledge, this work provides a systematic analytical and numerical framework for fractional-order stochastic Leslie-Gower systems under white noise.

This study develops a novel fractal-fractional (FF) model using the Caputo derivative to describe computer virus propagation dynamics in networked environments. The model incorporates memory effects and fractal geometry to accurately simulate virus transmission across complex networks. Fixed-point theory establishes solutions' existence and uniqueness, while stability analysis ensures robustness. Numerical simulations demonstrate the efficacy of targeted interventions, such as antivirus deployment in high-degree network hubs. The proposed framework highlights the potential of FF derivatives for enhancing our understanding of virus dynamics and informing control strategies. This work bridges a critical gap in the study of computer viruses using fractal and fractional calculus. It provides tools for robust analysis and mitigation in interconnected systems.

In this study, the Michaelis-Menten function was incorporated into a fractional stochastic model of glucose-insulin degradation to describe insulin degradation. The existence, uniqueness, boundedness, and nonnegativity properties of the model were derived through rigorous mathematical analysis. This paper presented an analysis of stability, including asymptotic and Hyers-Ulam stability. By integrating stochastic perturbations into fractional differential equations with Caputo derivatives, we enhanced the realism of diabetes modeling. A stochastic Runge-Kutta method of type 4 was employed to solve the system, ensuring improved accuracy and stability in simulating glucoseinsulin dynamics under random fluctuations. Through sensitivity analysis, four optimized control strategies, including insulin injection regimens and pharmaceutical interventions, were identified. An analysis of the dynamic behavior of the system under various physiological conditions was carried Moulton type and Runge-Kutta type 4. Based on the results, it appeared that fractional stochastic modeling could capture long-term memory effects and inherent randomness in glucose metabolism, thereby providing a more comprehensive framework for diabetes management and prediction.

Diabetes mellitus represents a growing global health crisis, marked by complex glucose–insulin regulatory dynamics that exhibit stability transitions and chaotic behaviors. This study employs fractional-order calculus to model these dynamics, capturing critical memory effects inherent in biological systems. The ARA-residual power series method (ARA-RPSM) is utilized to derive approximate analytical solutions for the fractional-order glucose–insulin system, which are validated through numerical simulations. The analysis reveals critical stability thresholds, bifurcations, and transitions to chaos, emphasizing the influence of fractional-order parameters and physiological factors. Stability and chaos analyses, supported by Lyapunov exponents and bifurcation diagrams, highlight the system’s sensitivity to parameter variations and initial conditions. These findings underscore the potential of fractional-order modeling in diabetes research, offering actionable insights for stabilizing glucose–insulin interactions and managing chaotic fluctuations. The study further demonstrates the computational efficiency of ARA-RPSM in exploring fractional-order systems, paving the way for advanced therapeutic strategies in diabetes management.

In this paper, we examine the pathological behavior of pneumonia transmission. The proposed model is examined using three fractional derivative operators. Caputo, Caputo-Fabrizio, and Atangana-Baleanu use an efficient numerical Euler method to approximate fractional-order systems. These three operators lead to several asymptomatic behaviors not found in integer-order derivative models. The paper aims to generalize four ODEs with four unknowns (susceptible, carrier, infected, and recovered). Hyers-Ulam stability, local and global stability are examined. If R-0 is less than one, the free equilibrium point is local, asymptotic, or otherwise global. We provide simulation results to confirm the theory. Compared to the theoretical results, the simulations were well-approximated.

This paper presents a mathematical model for zoonotic disease transmission between baboons and humans in the Al-Baha region using a fractal-fractional derivative approach. The model incorporates compartments for susceptible, infected, and recovered populations of both species. The Atangana-Baleanu fractal-fractional derivative captures memory effects and interactions between humans and baboons. The existence and uniqueness of the solution are demonstrated through fixed-point theorems, and Hyers-Ulam stability analysis is applied to assess the robustness of the model under small perturbations. Numerical simulations evaluate various control strategies, including sterilization, food restrictions, and limiting human interaction. They reveal the critical role of managing wildlife-human contact in reducing disease transmission. The Hyers-Ulam stability confirms that small deviations in initial conditions or parameter values do not significantly alter the long-term behavior of the system, ensuring the reliability of the model's predictions. This research not only models zoonotic disease transmission, but also explores the broader implications of fractal-fractional derivatives in dynamics and transport processes, offering insights into memory-driven phenomena across various fields. Results highlight the importance of integrated control measures in mitigating zoonotic disease risks.

This paper introduces a novel fractional-order model based on the Caputo-Fabrizio (CF) derivative for analyzing computer virus propagation in networked environments. The model partitions the computer population into four compartments: susceptible, latently infected, breaking-out, and antivirus-capable systems. By employing the CF derivative-which uses a nonsingular exponential kernel-the framework effectively captures memory-dependent and nonlocal characteristics intrinsic to cyber systems, aspects inadequately represented by traditional integer-order models. Under Lipschitz continuity and boundedness assumptions, the existence and uniqueness of solutions are rigorously established via fixed-point theory. We develop a tailored two-step Adams-Bashforth numerical scheme for the CF framework and prove its second-order accuracy. Extensive numerical simulations across various fractional orders reveal that memory effects significantly influence virus transmission and control dynamics; smaller fractional orders produce more pronounced memory effects, delaying both infection spread and antivirus activation. Further theoretical analysis, including Hyers-Ulam stability and sensitivity assessments, reinforces the model's robustness and identifies key parameters governing virus dynamics. The study also extends the framework to incorporate stochastic effects through a stochastic CF formulation. These results underscore fractional-order modeling as a powerful analytical tool for developing robust and effective cybersecurity strategies.

This paper presents a new fractional-order model that captures the propagation dynamics of computer viruses within digital networks by incorporating memory effects and nonlocal interactions via the Caputo derivative. The network is partitioned into four functional compartments: susceptible, latently infected, breaking-out (actively infectious), and recovered computers. The analytical framework establishes the fundamental mathematical properties of the model-existence, uniqueness, nonnegativity, and boundedness of solutions-ensuring its well-posedness. The stability of the infection-free and endemic equilibria is investigated using Matignon's theorem for fractional-order systems, Lyapunov functionals, and the fractional LaSalle invariance principle. The basic reproduction number R0 is derived via the next-generation matrix approach to determine threshold conditions governing the transition between virus eradication and persistence. To approximate the fractional dynamics, two robust numerical algorithms are implemented and compared: the multistep generalized differential transform method (MSGDTM) and the Jacobi spectral collocation scheme (JSCS). The model is extended to a stochastic framework to account for random fluctuations inherent in real-world network environments, with both deterministic and stochastic solvers evaluated through Monte Carlo simulations. The stochastic analysis reveals key phenomena including noise-induced transitions, enhanced extinction probability, and memory-noise interplay that are not captured by the deterministic model. Numerical experiments reveal that the fractional order strongly influences infection intensity and system memory, with smaller orders yielding delayed peaks and prolonged transients. While MSGDTM provides computational simplicity and flexibility for nonlinear problems, JSCS attains superior spectral accuracy and convergence efficiency. The comparative results highlight the role of fractional modeling in representing long-memory effects in cyber-epidemics and offer quantitative guidance for designing antivirus strategies in memory-dependent network environments.

This study investigates a fractional-order computer virus model using the Caputo derivative to capture memory effects in digital networks. The model classifies nodes into four compartments: susceptible (S), latent (L), actively infected (B), and recovered (R). Analytical results establish the well-posedness, positivity, boundedness, and stability of solutions, with the basic reproduction number R0 determining the threshold between virus extinction and persistence. Sensitivity analysis identifies the transmission rate as the most influential parameter, and a transcritical bifurcation at R0=1 separates disease-free and endemic equilibria. An optimal control framework based on Pontryagin’s Maximum Principle is designed to balance infection reduction with intervention costs. Stochastic perturbations are incorporated via the Milstein scheme to validate theoretical predictions and illustrate the impact of randomness. By integrating fractional calculus, stochastic modeling, and optimal control, this work provides a mathematical foundation for adaptive cybersecurity strategies in interconnected networks.

This study investigates the reliable numerical analysis of chaotic dynamics in the Glukhovsky-Dolzhansky system, which models convective fluid motion in a rotating ellipsoidal cavity. Hidden and self-excited attractors are localized using the numerical continuation method (NCM), Pyragas time-delayed feedback control, and Leonov's analytical dimension formula following global stability loss. A critical assessment of Lyapunov exponents and Lyapunov dimensions in a finite-time setting shows that positive values over long but finite intervals may incorrectly indicate sustained chaos due to transient effects and shadowing breakdown. Furthermore, we demonstrate that the fractional order gamma plays a bidirectional control role: it induces chaotic behavior at rho=5 for gamma<0.94 and suppresses chaos at rho=15 for gamma<0.93. The multifractal spectrum and correlation dimension are used to quantify attractor complexity, where transient chaos exhibits a broader spectrum (Delta alpha approximate to 0.67) compared to sustained chaos (Delta alpha approximate to 0.48). Monte Carlo simulations, Sobol sensitivity analysis, Kaplan-Meier survival analysis, and bootstrap-based hypothesis testing confirm the robustness of the results. Overall, the findings provide a unified framework for analyzing hidden attractors, transient chaos, and fractional-order effects in nonlinear fluid dynamical systems.

This study presents a novel application of fractal-fractional differential equations (FFDEs) to model heat and mass transfer phenomena in porous media, particularly in the context of chaotic systems. We introduce a new formulation of the fractal-fractional heat and advection-diffusion equations, incorporating Caputo-Fabrizio operators to capture the complexities of anomalous diffusion in heterogeneous porous structures. The existence and uniqueness of solutions are rigorously established using the Banach contraction principle, while stability analyses ensure the robustness of the proposed model. Furthermore, numerical simulations illustrate the model's capability to describe intricate transport dynamics, including the emergence of chaotic behavior arising from fluid instabilities in porous channels. By extending classical chaotic models such as the Lorenz-L & uuml;-Chen system to the fractal-fractional framework, this research provides deeper insights into nonlinear transport mechanisms in porous media. The application of the Adomian decomposition method enables highly accurate solutions, demonstrating its effectiveness in solving nonlinear FFDEs. These findings have significant implications across various scientific and engineering disciplines, including energy systems, environmental science, and biomedical engineering, by offering a refined approach to modeling complex dynamical processes with memory and hereditary effects.

This paper introduces novel numerical schemes based on Newton's interpolation polynomial for solving a fractal-fractional glucose-insulin regulatory system modeled using the Lotka-Volterra framework. The model incorporates fractal-fractional derivatives with a power-law kernel, extending classical glucose-insulin dynamics. Fixed-point theory establishes solutions' existence and uniqueness under the Caputo fractal-fractional operator. Hyers-Ulams stability is investigated. By using linear controllers, the Caputo fractal-fractional-order glucose-insulin system can be controlled to equilibrium. This control system has the potential to improve the management of diabetes by providing precise regulation of blood glucose levels. Numerical examples illustrate the effectiveness of the proposed method. To solve the system, the Atangana-Seda numerical scheme, incorporating Newton's interpolation polynomials, is applied. Simulated results are compared with analytical findings, confirming the accuracy and applicability of the approach to real-world biological systems.

This study presents a novel fractional-order mathematical model to investigate zoonotic disease transmission between humans and baboons, incorporating the Generalized Euler Method and highlighting key control strategies such as sterilization, restricted food access, and reduced human–baboon interaction. The model’s structure exhibits an inherent symmetry in the transmission dynamics between baboon and human populations, reflecting balanced interaction patterns. This symmetry is further analyzed through the stability of infection-free and endemic equilibrium points, guided by the basic reproduction number R0. Theoretical analyses confirmed the existence, uniqueness, and boundedness of solutions, while sensitivity analysis identified critical parameters influencing disease spread. Numerical simulations validated the effectiveness of intervention strategies, demonstrating the impact of symmetrical measures on minimizing zoonotic disease risks and promoting balanced population health outcomes. This work contributes to epidemiological modeling by illustrating how symmetry in control interventions can optimize zoonotic disease management.

This study proposes a generalized numerical scheme for simulating nonlinear fractal-fractional glucose-insulin systems with a Mittag-Leffler kernel. The model takes into account both the acute and chronic effects of beta-cell kinetic changes, and can be used to predict therapeutic interventions' effects. The model can also be used to identify potential targets for diabetes treatments. Solutions are examined for existence, uniqueness, non-negativity, and boundness. System stability is analyzed locally and globally, including Hyers-Ulam stability. Simulation results are compared to analytical solutions to validate the model. Finally, the effects of different parameters on the model are studied. We apply the Adomian decomposition method to approximate solutions to nonlinear glucose-insulin systems of fractional order when solving for fractional glucose levels. Based on this method, the solution can be represented as a series of easily computed convergent components. Numerical examples illustrate the method's effectiveness, showing that it accurately captures the system's dynamic behavior with precision. The generalized numerical scheme represents a significant advancement in glucose-insulin system simulation, providing a powerful and practical tool for researchers and clinicians. This tool can enhance the understanding and management of metabolic disorders, offering new insights into glucose and insulin interactions.

This study explores the glucose-insulin regulatory system using a fractal-fractional framework based on the Atangana-Baleanu derivative. By formulating a set of differential equations incorporating the Atangana- Baleanu fractal-fractional derivative, we capture the intricate, nonlinear dynamics of glucose and insulin. This is with enhanced accuracy compared to traditional models. We establish the existence and uniqueness of solutions through fixed-point theory and analyze Hyers-Ulam stability. Moreover, we employ a linear controller to stabilize chaotic behavior in the system, mitigating fluctuations that can lead to erratic physiological responses. Both analytical and numerical results validate the model's robustness to representing physiological processes. Our findings demonstrate that the fractal-fractional model significantly improves glucose-insulin dynamics modeling, highlighting its potential as a powerful tool for diabetes management and prediction of complex biological behaviors.

This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate long-term physiological processes. Key aspects include the derivation of Lyapunov exponents, bifurcation diagrams, and phase diagrams to explore system stability and chaotic behavior. A novel control strategy using simple linear controllers is introduced to stabilize chaotic oscillations. The effectiveness of this approach is validated through numerical simulations, where Lyapunov exponents are reduced from positive values (λ1=0.123) in the uncontrolled system to negative values (λ1=−0.045) post-control application, indicating successful stabilization. Additionally, bifurcation analysis demonstrates a shift from chaotic to periodic behavior when control is applied, and time-series plots confirm a significant reduction in glucose–insulin fluctuations. These findings underscore the importance of fractional calculus in accurately modeling nonlinear and memory-dependent glucose–insulin dynamics, paving the way for improved predictive models and therapeutic strategies. The proposed framework provides a foundation for personalized diabetes management, real-time glucose monitoring, and intelligent insulin delivery systems.