Investigating the deterministic and stochastic dynamics of Monkeypox: a hybrid numerical approach using spectral method and feedforward neural networks

Monkeypox (Mpox) has recently emerged as a serious threat to human health. Multiple outbreaks of human Mpox have been reported in Africa and North America over the past few years. This study presents a comprehensive computational analysis of Mpox dynamics using a two-model approach. Firstly, a deterministic model is formulated to study the transmission dynamics of Mpox. A basic analysis of the model is conducted, and the corresponding threshold parameter is evaluated. The model’s sensitivity is discussed in detail to study the effect of different parameters on disease incidence. To capture the inherent randomness in disease dynamics, the deterministic model is extended to a stochastic case by incorporating the Brownian motion, which is solved numerically by the Legendre spectral collocation method (LSCM). The criteria for a global positive solution and disease extinction are presented. Comparison of the two frameworks show that while the deterministic model provides a clear baseline threshold for disease persistence, the stochastic model additionally reveals a noise-dependent extinction regime ( \(\mathcal {R}_0^{S}<1\) ) that has no deterministic counterpart, indicating that the stochastic approach yields more conservative and realistic short-term outbreak forecasts in the presence of environmental noise. The application of a Feedforward Neural Network (FFNN) further enriches the approach by simulating the complex interactions within Mpox dynamics, offering an advanced computational modeling framework. The simulation of FFNN-based predictions shows excellent convergence, with minimal gradient values, small learning rates \(\mu \) , and near-zero error distribution, demonstrating the significance of the proposed neuro-computing technique. A perfect correlation of ( \(R = 1\) ) is achieved for the deterministic model, while a correlation of ( \(R = 0.99\) ) is obtained for the stochastic model. The mean squared and mean absolute errors of FFNN approach remain extremely low, ranging from \(10^{-5}\) to \(10^{-9}\) for the deterministic model and from \(10^{-2}\) to \(10^{-7}\) for the stochastic model, confirming accuracy comparable to the LSCM while maintaining strong generalization across training, validation and test datasets.