Early forecasting of influenza is an important task for public health to reduce losses due to influenza. Various deep learning-based models for multi-regional influenza forecasting have been proposed to forecast future influenza occurrences in multiple regions. While they only use historical data for forecasting, temporal and regional patterns need to be jointly considered for better accuracy. Basic deep learning models such as recurrent neural networks and graph neural networks have limited ability to model both patterns together. A more recent approach uses an attention mechanism or its variant, self-attention. Although these mechanisms can model regional interrelationships, in state-of-the-art models, they consider accumulated regional interrelationships based on attention values that are calculated only once for all of the input data. This limitation makes it difficult to effectively model the regional interrelationships that change dynamically during that period. Therefore, in this article, we propose a recurrent self-attention network (RESEAT) for various multi-regional forecasting tasks such as influenza and electrical load forecasting. The model can learn regional interrelationships over the entire period of the input data using self-attention, and it recurrently connects the attention weights using message passing. We demonstrate through extensive experiments that the proposed model outperforms other state-of-the-art forecasting models in terms of the forecasting accuracy for influenza and COVID-19. We also describe how to visualize regional interrelationships and analyze the sensitivity of hyperparameters to forecasting accuracy.