The aim of this study is to present the numerical simulations of the influenza disease nonlinear system (IDNS) using the stochastic artificial neural networks (ANNs) procedures supported with Levenberg-Marquardt backpropagation (LMB), i.e., ANNs-LMB. The IDNS is constructed with four classes, susceptible S(t), infected I(t), recovered R(t) and cross-immune people C(t), based stiff nonlinear ordinary differential system. The numerical computations have been performed through the stochastic ANNs-LMB for solving six different variations of the IDNS. The obtained numerical solutions through the stochastic ANNs-LMB for solving the IDNS have been presented using the training, verification and testing measures to reduce mean square error (MSE) from data-based reference solutions. To observed the correctness, efficiency, competence and proficiency of the designed computing paradigm ANNs-LMB, an exhaustive analysis is presented using the correlation studies, error histograms (EHs), mean squared error (MSE), regression and state transitions (STs) information. The worth and significance of ANNs-LMB is substantiated through comparisons of the outcomes admitted the good agreement from data derived results with 5–7 decimal places of accuracy for each scenario of IDNS.