Consider a disease spreading from person-to-person through contact infection. Although nearly impossible to recover in practice, we can model this contact network by a graph generated randomly from all graphs with a pre-defined degree sequence. When the vertex degrees themselves are chosen at random from a heavy-tailed distribution, we provide a limiting description of the number of people infected at time t≥ 0 as a process over time. The limiting process contains strictly positive jumps corresponding to super-spreading individuals in the population. In this regime, the connected components of the graph viewed as metric spaces have scaling limits called continuum random graphs. Our results provide new insights for these objects.