The COVID-19 pandemic has caused a devastating impact on the social activity, economy and politics worldwide. Techniques to diagnose COVID-19 cases by examining anomalies in chest X-ray images are urgently needed. Inspired by the success of deep learning in various tasks, this paper evaluates the performance of four deep neural networks in detecting COVID-19 patients from their chest radiographs. The deep neural networks studied include VGG16, MobileNet, ResNet50 and DenseNet201. Preliminary experiments show that all deep neural networks perform promisingly, while DenseNet201 outshines other models. Nevertheless, the sensitivity rates of the models are below expectations, which can be attributed to several factors: limited publicly available COVID-19 images, imbalanced sample size for the COVID-19 class and non-COVID-19 class, overfitting or underfitting of the deep neural networks and that the feature extraction of pre-trained models does not adapt well to the COVID-19 detection task. To address these factors, several enhancements are proposed, including data augmentation, adjusted class weights, early stopping and fine-tuning, to improve the performance. Empirical results on DenseNet201 with these enhancements demonstrate outstanding performance with an accuracy of 0.999%, precision of 0.9899%, sensitivity of 0.98%, specificity of 0.9997% and F1-score of 0.9849% on the COVID-Xray-5k dataset.