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During the onset of a disaster event, filtering relevant information from the social web data is challenging due to its sparse availability and practical limitations in labeling datasets of an ongoing crisis. In this paper, we hypothesize that unsupervised domain adaptation through multi-task learning can be a useful framework to leverage data from past crisis events for training efficient information filtering models during the sudden onset of a new crisis. We present a novel method to classify relevant tweets during an ongoing crisis without seeing any new examples, using the publicly available dataset of TREC incident streams. Specifically, we construct a customized multi-task architecture with a multi-domain discriminator for crisis analytics: multi-task domain adversarial attention network. This model consists of dedicated attention layers for each task to provide model interpretability; critical for real-word applications. As deep networks struggle with sparse datasets, we show that this can be improved by sharing a base layer for multi-task learning and domain adversarial training. Evaluation of domain adaptation for crisis events is performed by choosing a target event as the test set and training on the rest. Our results show that the multi-task model outperformed its single task counterpart. For the qualitative evaluation of interpretability, we show that the attention layer can be used as a guide to explain the model predictions and empower emergency services for exploring accountability of the model, by showcasing the words in a tweet that are deemed important in the classification process. Finally, we show a practical implication of our work by providing a use-case for the COVID-19 pandemic.