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In this paper, we investigate jointly sparse signal recovery and jointly sparse support recovery in Multiple Measurement Vector (MMV) models for complex signals, which arise in many applications in communications and signal processing. Recent key applications include channel estimation and device activity detection in MIMO-based grant-free random access which is proposed to support massive machine-type communications (mMTC) for Internet of Things (IoT). Utilizing techniques in compressive sensing, optimization and deep learning, we propose two model-driven approaches, based on the standard auto-encoder structure for real numbers. One is to jointly design the common measurement matrix and jointly sparse signal recovery method, and the other aims to jointly design the common measurement matrix and jointly sparse support recovery method. The proposed model-driven approaches can effectively utilize features of sparsity patterns in designing common measurement matrices and adjusting model-driven decoders, and can greatly benefit from the underlying state-of-the-art recovery methods with theoretical guarantee. Hence, the obtained common measurement matrices and recovery methods can significantly outperform the underlying advanced recovery methods. We conduct extensive numerical results on channel estimation and device activity detection in MIMO-based grant-free random access. The numerical results show that the proposed approaches provide pilot sequences and channel estimation or device activity detection methods which can achieve higher estimation or detection accuracy with shorter computation time than existing ones. Furthermore, the numerical results explain how such gains are achieved via the proposed approaches.