In the real world, pathogens do not exist in isolation. The transmission of one pathogen may be affected by the presence of other pathogens, and certain pathogens generate multiple strains with different spreading features. Hence, the behavior of multi-pathogen transmission has attracted much attention in epidemiological research. In this paper, we use the pairwise approximation method to formulate two-pathogen models capturing cross-immunity, super-infection, and co-infection phenomena, in which each pathogen follows a susceptible-infected-susceptible (SIS) mechanism. For each model, we calculate the basic reproduction number and analyze the stability of equilibria, and discuss the differences from the mean-field approach. We demonstrate that simulations are in good agreement with the analytical results.