Coronavirus (COVID-19), an air-borne disease, has affected the lifestyle of people all around the world. The World Health Organization (WHO) classified the disease as a pandemic due to its rapid spread of infection. Tracing patients infected with Coronavirus has become a steep uphill process supervened by the limited availability of tests based on reverse transcription-polymerase chain reaction (RT-PCR), which calls for efficient and highly responsive detection and diagnostic methods. Recently, methodologies based on image processing have been proposed by various researchers, especially using deep learning-based models. However, most models need millions of parameters to learn the complex input-output relationships and demand massive computational resources. This paper proposes the detection of COVID-19 from CT scan images using deep convolutional neural networks (CNN). A dynamic mode decomposition (DMD) based attention-driven image enhancement is proposed to extract localized enhanced features from CT scan images. Localized features can improve the model's performance by making inferences about the complete object. Pre-trained deep CNN models including VGGNet, ResNet50, and InceptionV3, are then transfer-learned on the DMD-enhanced CT scan images for COVID-19 detection. The paper proposes a custom shallow CNN architecture for detecting COVID-19 using DMD-enhanced CT scan images. The custom shallow CNN with significantly reduced learnable parameters improves the accuracy of the model and reduces the computational burden. The performance of the CNN architectures (custom shallow CNN and transfer learned deep CNNs) is evaluated using benchmark performance metrics, including accuracy, precision, recall, and F1 score. The experimental results demonstrate that the proposed shallow CNN network trained on DMD-enhanced images can better detect COVID-19 and outperform existing architectures in accuracy and computational complexity. The accuracy obtained for the proposed shallow CNN network trained on DMD-enhanced images is around 92.3%, with an F1-score of 0.918.