Most developed models for solving problems in epidemiology use deterministic approach. To cover the lack of spatial sense in the method, one uses statistical modeling, reaction-diffusion in continuous medium, or multi-patch model to depict epidemic activities in several connected locations. Here, we show that an epidemic model that is set as an organized system on networks can yield Turing patterns and other interesting behaviors that are sensitive to the initial conditions. The formed patterns can be used to determine the epidemic arrival time, its first peak occurrence and the peak duration. These epidemic quantities are beneficial to identify contribution of a disease source node to the others. Using a real structure network, the system also exhibits a comparable disease spread pattern of Diarrhea in Jakarta.