Dynamical analysis of a fractional-order influenza model with stability and control perspectives

Purpose This study aims to develop and analyze a four-dimensional influenza model using fractional calculus, specifically the Caputo derivative. It investigates solution existence and uniqueness, examines fixed points and Lyapunov stability and explores dynamic behavior through time series and phase portraits. The model's response to varying fractional orders and parameters is numerically simulated using modified Adams-Bashforth-Moulton and Runge-Kutta methods. The work also compares system dynamics with and without control measures and evaluates how model parameters affect the reproduction number, offering insights into the role of fractional operators in disease modeling. Design/methodology/approach The study constructs a four-dimensional influenza model using the Caputo fractional derivative to capture memory effects in disease dynamics. Existence and uniqueness of solutions are established analytically, followed by stability analysis through fixed points and Lyapunov methods. Numerical simulations are conducted using the Adams-Bashforth-Moulton and Runge-Kutta schemes adapted for fractional-order systems. Time series plots and phase portraits are generated by varying fractional orders and parameters. Comparative analysis is performed to assess the impact of control measures and parameter variations on the reproduction number, providing a comprehensive understanding of the model's behavior under different epidemiological scenarios. Findings The study reveals that fractional-order dynamics significantly influence the behavior of the influenza model, with varying fractional orders altering system stability and trajectories. The existence and uniqueness of solutions are confirmed under specific conditions, and Lyapunov analysis validates system stability. Numerical simulations demonstrate that fractional operators provide richer dynamics compared to classical models. The reproduction number is shown to be sensitive to parameter changes, and control measures effectively modify disease spread. Overall, the findings highlight the importance of fractional calculus in capturing complex epidemiological patterns and improving predictive accuracy in infectious disease modeling. Originality/value This work introduces a novel four-dimensional influenza model grounded in fractional calculus, offering a fresh perspective on disease dynamics through the Caputo derivative. Unlike classical models, it captures memory effects and complex temporal behaviors, enhancing realism in epidemiological simulations. The dual use of Adams-Bashforth-Moulton and Runge-Kutta methods tailored for fractional systems adds computational depth. By analyzing control measures and reproduction number sensitivity, the study provides valuable insights for public health strategies. Its originality lies in integrating fractional-order analysis with stability theory and numerical experimentation, contributing meaningfully to the advancement of mathematical epidemiology and fractional modeling frameworks.