The current study aims to design an integrated numerical computing-based scheme by applying the Levenberg–Marquardt backpropagation (LMB) neural network to solve the nonlinear susceptible (S), infected (I) and recovered (R) (SIR) system of differential equations, representing the spreading of infection along with its treatment. The solutions of both the categories of spreading infection and its treatment are presented by taking six different cases of SIR models using the designed LMB neural network. A reference dataset of the designed LMB neural network is established with the Adam numerical scheme for each case of the spreading infection and its treatment. The approximate outcomes of the SIR system based on the spreading infection and its treatment are presented in the training, authentication and testing procedures to adapt the neural network by reducing the mean square error (MSE) function using the LMB. Studies based on the proportional performance and inquiries based on correlation, error histograms, regression and MSE results establish the efficiency, correctness and effectiveness of the proposed LMB neural network scheme.