Transmission of infectious respiratory diseases through airborne dispersion of viruses poses a great risk to public health. In several major diseases, one of the main modes of transmission is through respiratory droplets. Virus laden respiratory droplets and aerosols can be generated during coughing, sneezing and speaking. These droplets and aerosols can remain suspended in air and be transported by airflow posing a risk of infection in individuals who might come in contact with them. With this background, in this work, we present a numerical framework for simulation of dispersion of respiratory sputum droplets using implicit large-eddy simulations. A combination of discrete Lagrangian droplet model and fully compressible Navier-Stokes flow solver is employed in this study. The method is applied to analyze cases such as droplet dispersion during speech and cough under different environmental settings. Furthermore, the performance of the numerical framework is evaluated through strong and weak scaling analysis.