In a simple susceptible-infected-recovered (SIR) model of how a disease spreads, it is assumed that all individuals in a community interact with each other. This ignores some of the actual community structure in the population, e.g. some individuals will interact with more people than you or me. We can model these community structures by a graph with a heavy-tailed degree distribution. Each day, every infected vertex in this graph transmits the disease to its susceptible neighbors, who become infected the next day, and then the infected vertex recovers. The vertices in the graph with very large degree are the super-spreaders. We show that for epidemics on certain critical random graphs with \$n\$ total vertices, the number of people infected on day \$t = 0,1,\dotsm\$ possesses a random scaling limit as \$n\to\infty\$ relating to stable processes. These limits are related to stable processes and we can still "see" the super-spreaders because these scaling limits possess positive jumps.