Fractal fractional modelling of fruit disease dynamics with memory and heterogeneity

Fruit diseases pose a significant threat to agricultural productivity and food security, and their complex dynamics often involve memory effects, latent infection stages, and spatial heterogeneity that classical integer-order models poorly capture. In this study, we develop and analyse a fractal–fractional compartmental model for fruit disease dynamics that extends the standard SEIR-type framework by incorporating both the fractional order $$\:\alpha\:$$ (memory) and the fractal dimension $$\:\beta\:$$ (heterogeneity). We rigorously establish the existence and uniqueness of solutions for the proposed model in the Caputo sense and prove the positivity and boundedness of all state variables. The basic reproduction number $$\:{\mathcal{R}}_{0}$$ is derived as a threshold parameter, and a stability theorem for the disease-free equilibrium is presented. In addition, we investigate the Ulam–Hyers stability, demonstrating robustness with respect to small perturbations in initial data. Comprehensive numerical simulations are performed to assess the impact of varying $$\:\alpha\:$$ and $$\:\beta\:$$ on disease progression in susceptible, exposed, infected, and recovered compartments. The results show that decreasing $$\:\alpha\:$$ delays and reduces epidemic peaks, while varying $$\:\beta\:$$ spreads infection over time and modifies peak magnitudes, thereby reproducing a wide range of outbreak patterns observed in real orchards. These findings highlight the capacity of fractal–fractional models to provide a more realistic description of fruit disease dynamics and to serve as a decision-support tool for designing targeted intervention strategies. This work contributes both a novel mathematical framework and a comprehensive analytical and numerical analysis of its properties, bridging the gap between theoretical modelling and practical disease management. The approach opens new directions for incorporating optimal control, stochastic effects, and data-driven parameter estimation into plant epidemiology in future research.