The burning number of a graph models the rate at which a disease, information, or other externality can propagate across a network. The burning number is known to be NP-hard even for a tree. Herein, we define a relative of the burning number that we coin the contagion number (CN). We aver that the CN is a better metric to model disease spread than the burning number as it only counts first time infections (i.e., constrains a node from getting the same disease/same variant/same alarm more than once). This is important because the Centers for Disease Control and Prevention report that COVID-19 reinfections are rare. This paper delineates a method to solve for the contagion number of any tree, in polynomial time, which addresses how fast a disease could spread (i.e., a worst-cast analysis) and then employs simulation to determine the average contagion number (ACN) (i.e., a most-likely analysis) of how fast a disease would spread. The latter is analyzed on scale-free graphs, which are used to model human social networks generated through a preferential attachment mechanism. With CN differing across network structures and almost identical to ACN, our findings advance disease spread understanding and reveal the importance of network structure. In a borderless world without replete resources, understanding disease spread can do much to inform public policy and managerial decision makers’ allocation decisions. Furthermore, our direct interactions with supply chain executives at two COVID-19 vaccine developers provided practical grounding on what the results suggest for achieving social welfare objectives.