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.

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ϝ$. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.

In this work, we present a comprehensive analysis of the spatio-temporal $\mathrm{S}\mathrm{E}\mathrm{I}\mathrm{R}$ epidemic model of fractional order. The infection dynamics in the proposed fractional order model (FOM) are described by a system of partial differential equations (PDEs) within a time-fractional order and diffusion operator in one-dimensional space, considering that the total population is split into four compartments: Susceptible, exposed, infected, and recovered individuals denoted as $\mathrm{S}$, $\mathrm{E}$, $\mathrm{I}$ and $\mathrm{R}$, respectively. Our contributions commence by establishing the existence and uniqueness of positively bounded solutions for the proposed FOM. Moreover, we determined all equilibrium points (EPs) and investigated their local stability based on the basic reproduction number (BRN) ${\mathcal{R}}_0$, which is calculated by the next-generation matrix (NGM) method. Additionally, we demonstrated global stability using an appropriate Lyapunov function with fractional LaSalle's invariance principle (LIP). Sensitivity analysis of the FOM parameters was discussed to identify the most critical parameters by which the volume of disease propagation can be measured. The theoretical findings were corroborated by numerical simulations of solutions that are displayed in 3D and 2D graphs. Graphical simulations highlight the effect of vaccination on infection severity. Changing the fractional order $\alpha$ in the proposed FOM has an influence on the speed of convergence to the steady state as a result of the memory effect. Furthermore, vaccination emerges as an effective strategy for disease control.