A Multi-Pathogen Epidemiological Model: Analysis, Optimal Control, and a Deep Neural Network Approach for the Integer-Order System

Turmeric (Curcuma longa L.) is one of the most important spice crops and a valuable medicinal plant, but it is seriously affected by various types of diseases such as fungal, bacterial, nematode and viral diseases. In this paper, a complete mathematical model of the turmeric plant disease dynamics is developed under a fractal-fractional model in this context, encompassing all four types of pathogens and associated treatment classes. The fractal-fractional Caputo derivative operator captures memory effects and, through its fractal exponent, a genuine deformation of the classical memory kernel, allowing the underlying biological dynamics to be represented more flexibly than under the classical integer-order derivative; we do not, however, claim that this kernel deformation corresponds to demonstrated self-similarity or spatial heterogeneity in the turmeric plant–pathogen system. We show the positivity and boundedness of the solutions, calculate the next-generation matrix approach-based basic reproduction number R0 and investigate the local and global stability of both disease-free and endemic equilibria by Lyapunov functionals. A sensitivity analysis of R0 is conducted to determine the most important parameters influencing disease transmission and control. The existence and uniqueness of solutions and Ulam-Hyers stability of solutions are established by fixed point theory. For the associated integer-order system, we formulate an optimal control problem is formulated with three time-dependent controls: the prevention effort (u1), the enhancement of treatment (u2), and the care management (u3), and the optimality conditions are derived via Pontryagin’s maximum principle. Numerical simulations are conducted with three different fractal-fractional operators, namely Caputo, Caputo-Fabrizio and Atangana-Baleanu. A deep neural network is developed and trained to approximate the solution of the integer-order system. The third-layer deep neural network consists of neurons of sizes 80, 32, and 24, with activation functions of logistic sigmoid, radial basis and hyperbolic tangent, respectively, and is trained to approximate the system dynamics with the fourth-order Runge-Kutta method as a reference. The DNN is found to be very accurate in predicting the values with Nash-Sutcliffe Efficiency between 0.79 and 0.99 and Theil Inequality Coefficient around 10−2 in all 11 compartments, and hence proved capable of being a good surrogate modelling tool for the ODE systems. The present work contributes towards SDG 2 (Zero Hunger) and SDG 3 (Good Health and Well-being) by laying a mathematical basis for integrated disease management in turmeric cultivation for sustainable agriculture and food security.