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Coronavirus Disease 2019 (COVID-19) still presents a pandemic trend globally. Detecting infected individuals and analyzing their status can provide patients with proper healthcare while protecting the normal population. Chest CT (computed tomography) is an effective tool for screening of COVID-19. It displays detailed pathology-related information. To achieve automated COVID-19 diagnosis and lung CT image segmentation, convolutional neural networks (CNNs) have become mainstream methods. However, most of the previous works consider automated diagnosis and image segmentation as two independent tasks, in which some focus on lung fields segmentation and the others focus on single-lesion segmentation. Moreover, lack of clinical explainability is a common problem for CNN-based methods. In such context, we develop a multi-task learning framework in which the diagnosis of COVID-19 and multi-lesion recognition (segmentation of CT images) are achieved simultaneously. The core of the proposed framework is an explainable multi-instance multi-task network. The network learns task-related features adaptively with learnable weights, and gives explicable diagnosis results by suggesting local CT images with lesions as additional evidence. Then, severity assessment of COVID-19 and lesion quantification are performed to analyze patient status. Extensive experimental results on real-world datasets show that the proposed framework outperforms all the compared approaches for COVID-19 diagnosis and multi-lesion segmentation. [ABSTRACT FROM AUTHOR]