In this talk, we explore the integration of data-driven techniques and deep neural networks into a Susceptible-Infected-Recovered (SIR) type model with social dynamics, including a saturated incidence rate, to enhance the prediction and forecasting of epidemic trends. Our approach emphasizes the use of robust data augmentation strategies through suitable data-driven models, which significantly improve the performance and reliability of Feed-Forward Neural Networks (FNNs) and Nonlinear Autoregressive Networks (NARs). This framework offers a valuable alternative and complement to Physics-Informed Neural Networks (PINNs), particularly in contexts where data derived from mechanistic models can be leveraged to strengthen the learning process. We demonstrate how incorporating synthetic and realistic data allows the neural networks to adapt to complex epidemic dynamics, even when observational data is limited or noisy. The methodology is applied to simulate various scenarios of the COVID-19 outbreak, including both lockdown and post-lockdown phases, with a focus on the cases of Italy and Spain. Our numerical experiments confirm the effectiveness of the proposed strategy in improving forecasting accuracy and capturing key features of epidemic evolution.