Numerical computation of the stochastic hepatitis B model using feed forward neural network and real data.

Hepatitis B is a global health burden and can persist for years, with nearly two billion infections worldwide, where its spread is influenced by environmental heterogeneity, host-pathogen interactions, and vaccination-induced immune variability. Proper understanding and developing models with a suitable framework is essential to accurately capture the complexity of the hepatitis B virus (HBV) and its transmission. In this work, we present a novel framework of a stochastic model and a forward neural network that combines neural networks and stochastic differential equations to analyze the dynamics of hepatitis B virus transmission, as it is important to capture the inherent uncertainty of disease spread in heterogeneous environments. We formulate the stochastic model with a saturated incidence rate, incorporating the long-term persistence of the disease following key characteristics of the disease transmission. The theoretical analysis of the model is proven to ensure the well-posedness and to determine the conditions for extinction and persistence of the disease. Further, a set of real data of hepatitis B reported cases will be used to produce stochastic simulations, and to train a feed-forward neural network (FFNN), while approximating the model dynamics more effectively. To evaluate the efficacy of the hybrid framework, we demonstrate its performance by the presenting mean squared error (MSE), absolute error (AE), and regression analysis showing strong agreement between the stochastic simulations and neural network predictions.