Optimizing tomato plant and insect disease control using a fractional model and deep neural networks

Plants are vital as a primary source of food and provide essential contributions to the production of medicines, fibers, and the preservation of ecological equilibrium. Nevertheless, they are often endangered by viral infections, primarily transmitted by insect vectors that transmit the virus from host plants to susceptible plants. This work aims to investigate a biological control approach by integrating predators into a plant-virus propagation model controlled by delay differential equations. The model incorporates a representation of five separate populations. The tomato plant population is classified into three segments: susceptible, infected, and recovered. Similarly, the insect vectors are classified into two categories: susceptible insects and infected insects. This study investigates the integration of insect vector dynamics into a fractional-order system of tomato plant populations. Through the examination of the fundamental reproduction number (R₀) and the implementation of sensitivity analysis, this work highlights crucial factors that impact the rate of disease transmission. The study utilizes a new stochastic deep neural network technique to estimate the optimal solutions and compares them with the outcomes obtained by the Adams-Bashforth-Moulton method. To assess the accuracy and efficiency of the suggested method, we employed statistical metrics including mean square error, root mean square error, mean, variance, and regression. The MSE values range within 10^-02 to 10^-04 , while RMSE lies in the range 10^-01 to 10^-02. Furthermore, we integrate the absolute error of the most accurate predicted solution with the exact answer. The AE occurs within the range of 10^-02 to 10^-08.