Dynamical analysis and neural network approximation of a fractional-order predator–prey model with disease transmission

Abstract This paper presents a fractional-order S–I–P eco-epidemiological model that incorporates fear effects, refuge mechanisms, and nonlinear transmission to capture memory-dependent interactions in biological systems. The model is formulated using Caputo fractional derivatives, and the existence and uniqueness of solutions are established via fixed point theory. Stability of equilibrium points is analyzed using eigenvalue conditions, and sensitivity analysis is performed to identify key parameters influencing the basic reproduction number. Numerical simulations based on the Grünwald–Letnikov scheme illustrate the impact of fractional order on system dynamics, showing smoother and more stable behavior compared to the integer-order case. In addition, a feedforward artificial neural network trained with the Levenberg–Marquardt algorithm is employed to approximate the numerical solutions, achieving high accuracy with low mean square error and strong regression performance. The results indicate that the proposed hybrid framework provides an effective and reliable approach for analyzing complex eco-epidemiological systems with memory and behavioral effects.