Deterministic and Fractional-Order Modeling of Potato Disease Transmission Dynamics

Potato plant diseases are a significant concern because they reduce crop yields and quality, putting the global food supply at risk. Our goal in developing this deterministic, fractional-order epidemic model is to explain the transmission of these diseases by simulating the interactions between susceptible, exposed, infected, treated, and recovered potato plants. The deterministic model captures the mainstream dynamics of epidemics. At the same time, the fractional formulation based on the Atangana–Baleanu–Caputo (ABC) derivative represents the memory-dependent and nonlocal effects typical of interactions between plants and pathogens. We find that the fundamental reproduction number can be derived analytically, that there is an endemic and disease-free equilibrium, that solutions are positive and bounded, and that stability can be evaluated locally and globally using Jacobian eigenvalues and Lyapunov functions. Bifurcation theory reveals a forward bifurcation, and game theory is applied to identify a Nash equilibrium strategy for disease control. Transmission rates and disease progression are the two most important variables impacting epidemic outcomes, as shown by a sensitivity analysis. Bifurcation diagrams and Lyapunov exponents show that oscillatory or chaotic behavior can undergo transitions in numerical simulations of fractional dynamics. The fractional-order dynamics model outperforms the deterministic model in terms of persistence, decay rate, and richness of transient behavior. The results show that memory-driven fractional models can help develop agricultural intervention strategies by more accurately depicting the spread of potato diseases.