Traffic forecasting is crucial for transportation systems optimisation. Current models minimise the mean forecasting errors, often favouring periodic events prevalent in the training data, while overlooking critical aperiodic ones like traffic incidents. To address this, we propose DualCast, a dual-branch framework that disentangles traffic signals into intrinsic spatial-temporal patterns and external environmental contexts, including aperiodic events. DualCast also employs a cross-time attention mechanism to capture high-order spatial-temporal relationships from both periodic and aperiodic patterns. DualCast is versatile. We integrate it with recent traffic forecasting models, consistently reducing their forecasting errors by up to 9.6% on multiple real datasets.