Qualitative analysis of fractal–fractional order Coffee Berry epidemic model on an agricultural farm

Abstract

Coffee Berry Disease (CBD) poses a serious threat to global coffee production by reducing both yield and bean quality. Existing mathematical models based on classical integer-order calculus often fail to reflect the true dynamics of the disease, as they overlook memory effects and spatial variations commonly observed in agricultural systems. In this study, we introduce a new mathematical model called the Fractal–Fractional Coffee Berry Disease Infestation (FFCBDI) model. It is the first of its kind to apply Caputo-type fractal–fractional derivatives to model CBD progression. A key innovation in this model is the inclusion of a recovered berry compartment, which accounts for the possibility of re-infection due to environmental stress or pathogen persistence. The model incorporates two important parameters-fractional order and fractal dimension-to capture time delays and spatial heterogeneity more accurately. We analyze essential mathematical properties such as positivity, boundedness, existence, uniqueness, and stability. A bifurcation analysis is also performed to identify threshold values that determine whether the disease will spread or die out. Numerical simulations show how different parameter values influence disease dynamics. The proposed FFCBDI model offers a more realistic and flexible framework for understanding CBD transmission, which can support better prediction and control strategies in real-world agricultural settings.