<abstract> <p>In this study, a nonlinear mathematical SIR system is explored numerically based on the dynamics of the waterborne disease, e.g., cholera, that is used to incorporate the delay factor through the antiseptics for disease control. The nonlinear mathematical SIR system is divided into five dynamics, susceptible <italic>X</italic>(<italic>u</italic>), infective <italic>Y</italic>(<italic>u</italic>), recovered <italic>Z</italic>(<italic>u</italic>) along with the <italic>B</italic>(<italic>u</italic>) and <italic>C_h</italic>(<italic>u</italic>) be the contaminated water density. Three cases of the SIR system are observed using the artificial neural network (ANN) along with the computational Levenberg-Marquardt backpropagation (LMB) called ANNLMB. The statistical performances of the SIR model are provided by the selection of the data as 74% for authentication and 13% for both training and testing, together with 12 numbers of neurons. The exactness of the designed ANNLMB procedure is pragmatic through the comparison procedures of the proposed and reference results based on the Adam method. The substantiation, constancy, reliability, precision, and ability of the proposed ANNLMB technique are observed based on the state transitions measures, error histograms, regression, correlation performances, and mean square error values.</p> </abstract>

The current research aims to implement the numerical results for the Holling third kind of functional response delay differential model utilizing a stochastic framework based on Levenberg-Marquardt backpropagation neural networks (LVMBPNNs). The nonlinear model depends upon three dynamics, prey, predator, and the impact of the recent past. Three different cases based on the delay differential system with the Holling 3 rd type of the functional response have been used to solve through the proposed LVMBPNNs solver. The statistic computing framework is provided by selecting 12%, 11%, and 77% for training, testing, and verification. Thirteen numbers of neurons have been used based on the input, hidden, and output layers structure for solving the delay differential model with the Holling 3 rd type of functional response. The correctness of the proposed stochastic scheme is observed by using the comparison performances of the proposed and reference data-based Adam numerical results. The authentication and precision of the proposed solver are approved by analyzing the state transitions, regression performances, correlation actions, mean square error, and error histograms.

<abstract> <p>The purpose of this work is to present the stochastic computing study based on the artificial neural networks (ANNs) along with the scaled conjugate gradient (SCG), ANNs-SCG for solving the predator-prey delay differential system of Holling form-III. The mathematical form of the predator-prey delay differential system of Holling form-III is categorized into prey class, predator category and the recent past effects. Three variations of the predator-prey delay differential system of Holling form-III have been numerical stimulated by using the stochastic ANNs-SCG procedure. The selection of the data to solve the predator-prey delay differential system of Holling form-III is provided as 13%, 12% and 75% for testing, training, and substantiation together with 15 neurons. The correctness and exactness of the stochastic ANNs-SCG method is provided by using the comparison of the obtained and data-based reference solutions. The constancy, authentication, soundness, competence, and precision of the stochastic ANNs-SCG technique is performed through the analysis of the correlation measures, state transitions (STs), regression analysis, correlation, error histograms (EHs) and MSE.</p> </abstract>