Application of neural-network hybrid models in estimating the infection functions of nonlinear epidemic models

Hybrid neural network models are effective in analyzing time-series data by combining the strengths of neural networks and differential equation models. Although most studies have focused on linear hybrid models, few have examined nonlinear problems. This work explores the potential of a hybrid nonlinear epidemic neural network in predicting the correct infection function of an epidemic model. We design a novel loss function by combining bifurcation theory and mean-squared error loss to ensure the trainability of the hybrid model. Additionally, we identify unique existence conditions that support ordinary differential equations for estimating the correct infection function. Moreover, numerical experiments using the Runge-Kutta method confirm our proposed model's soundness both on our synthetic data and the real COVID-19 data.