Medical images are used as an important basis for diagnosing diseases, among which CT images are seen as an important tool for diagnosing lung lesions. However, manual segmentation of infected areas in CT images is time-consuming and laborious. With its excellent feature extraction capabilities, a deep learning-based method has been widely used for automatic lesion segmentation of COVID-19 CT images. However, the segmentation accuracy of these methods is still limited. To effectively quantify the severity of lung infections, we propose a Sobel operator combined with multi-attention networks for COVID-19 lesion segmentation (SMA-Net). In our SMA-Net method, an edge feature fusion module uses the Sobel operator to add edge detail information to the input image. To guide the network to focus on key regions, SMA-Net introduces a self-attentive channel attention mechanism and a spatial linear attention mechanism. In addition, the Tversky loss function is adopted for the segmentation network for small lesions. Comparative experiments on COVID-19 public datasets show that the average Dice similarity coefficient (DSC) and joint intersection over union (IOU) of the proposed SMA-Net model are 86.1% and 77.8%, respectively, which are better than those in most existing segmentation networks.