Indoor air quality plays an essential role in the safety and well-being of occupants, especially in the context of airborne diseases. This paper introduces AeroSafe, a novel approach aimed at enhancing the efficacy of indoor air purification systems through a robotic cough emulator testbed and a digital-twins-based aerosol residence time analysis. Current portable air filters often overlook the concentrations of respiratory aerosols generated by coughs, posing a risk, particularly in high-exposure environments like healthcare facilities and public spaces. To address this gap, we present a robotic dual-agent physical emulator comprising a maneuverable mannequin simulating cough events and a portable air purifier autonomously responding to aerosols. The generated data from this emulator trains a digital twins model, combining a physics-based compartment model with a machine learning approach, using Long Short-Term Memory (LSTM) networks and graph convolution layers. Experimental results demonstrate the model's ability to predict aerosol concentration dynamics with a mean residence time prediction error within 35 seconds. The proposed system's real-time intervention strategies outperform static air filter placement, showcasing its potential in mitigating airborne pathogen risks.