In this study, an advanced computational numerical scheme based on the Levenberg-Marquardt backpropagation (LMB) neural network (NN) process, i.e., LMB-NN is presented for solving the nonlinear mathematical influenza disease model. The nonlinear mathematical influenza disease model depends on four categories named susceptible S(t), infected I(t), recovered R(t) and cross-immune individuals proportion C(t). Six different cases of the nonlinear mathematical influenza disease model have been numerically treated using the LMB-NN process and the comparison of the results has been presented by using the reference data-based solutions designed based on the Adams results. The numerically obtained results of the nonlinear mathematical influenza disease model using the verification, testing, and training procedures are calculated using the LMB-NN process to reduce the functions of mean square error (MSE). For the correctness, competence, effectiveness, and efficiency of the LMB-NN process, the proportional and analysis methods are performed using the analysis of correlation, MSE results, error histograms and regression.