COVID-19 is a virus with high transmission rate that demands rapid identification of the infected patients to reduce the spread of the disease. The current gold-standard test, Reverse-Transcription Polymerase Chain Reaction (RT-PCR), has a high rate of false negatives. Diagnosing from CT-scan images as a more accurate alternative has the challenge of distinguishing COVID-19 from other pneumonia diseases. Artificial intelligence can help radiologists and physicians to accelerate the process of diagnosis, increase its accuracy, and measure the severity of the disease. We designed a new interpretable deep neural network to distinguish healthy people, patients with COVID-19, and patients with other pneumonia diseases from axial lung CT-scan images. Our model also detects the infected areas and calculates the percentage of the infected lung volume. We first preprocessed the images to eliminate the batch effects of different devices, and then adopted a weakly supervised method to train the model without having any tags for the infected parts. We trained and evaluated the model on a large dataset of 3359 samples from 6 different medical centers. The model reached sensitivities of 97.75% and 98.15%, and specificities of 87% and 81.03% in separating healthy people from the diseased and COVID-19 from other diseases, respectively. It also demonstrated similar performance for 1435 samples from 6 different medical centers which proves its generalizability. The performance of the model on a large diverse dataset, its generalizability, and interpretability makes it suitable to be used as a reliable diagnostic system.