Dynamics and optimal control of a fractional-order plant disease model

A fractional-order model (FOM) was developed to investigate plant disease transmission (PDT) through a system of dimensionally consistent fractional differential equations (FDEs) with the Caputo derivative. The model's well-posedness was established by proving existence and uniqueness of solutions via a fixed-point theory and the contraction mapping principle. Positivity, boundedness, and the equilibrium points (EPs) of the system were then characterized, followed by an analysis of their local and global stability using the Routh-Hurwitz criteria and LaSalle's invariance principle. The control reproduction number ( $R_c$) was derived using the next-generation matrix method, and a sensitivity analysis highlighted the parameters most influential to disease spread. A fractional optimal control problem (FOCP) incorporating preventive and curative time-dependent interventions was formulated, and necessary optimality conditions (NOCs) were obtained through a kind of Pontryagin's maximum principle (PMP). The resulting optimality system was solved numerically using a forward-backward sweep method (FBSM) based on the fractional Euler scheme, enabling the evaluation of control strategies. Three optimal intervention strategies emerged, each shaping the epidemic trajectory differently depending on the distinguishing parameter $\epsilon$ in the two-stage transmission process. Numerical simulations depicted the behavior of $R_c$ across different $\epsilon$ and fractional order $\alpha$, while tabulated objective functional values exhibited the efficacy of the proposed controls. Overall, the framework offered practical insights for mitigating and potentially eliminating plant epidemics under diverse control strategies.