Violent respiratory events play critical roles in the transmission of respiratory diseases, such as coughing and sneezing, between infectious and susceptible individuals. In this work, large-scale multiphase flow large-eddy simulations have been performed to simulate the coughing jet from a human's mouth carrying pathogenic or virus-laden droplets by using a weakly compressible smoothed particle hydrodynamics method. We explicitly model the cough jet ejected from a human mouth in the form of a mixture of two-phase fluids based on the cough velocity profile of the exhalation flow obtained from experimental data and the statistics of the droplets’ sizes. The coupling and interaction between the two expiratory phases and ambient surrounding air are examined based on the interaction between the gas particles and droplet particles. First, the results reveal that the turbulence of the cough jet determines the dispersion of the virus-laden droplets, i.e. whether they fly up evolving into aerosols or fall down to the ground. Second, the droplet particles have significant effects on the evolution of the cough jet turbulence; for example, they increase the complexity and butterfly effect introduced by the turbulence disturbance. Our results show that the prediction of the spreading distance of droplet particles often goes beyond the social distancing rules recommended by the World Health Organization, which reminds us of the risks of exposure if we do not take any protecting protocol.