The spread of a disease, product or idea in a population is often hard to predict. We tend to observe one or few realizations of the contagion process and therefore limited information can be obtained for anticipating future similar events. The stochastic nature of contagion generates unpredictable outcomes throughout the whole course of the dynamics. This might lead to important inaccuracies in the predictions and to the over or under-reaction of policymakers, who tend to anticipate the average behavior. We analyze the unpredictability (or uncertainty) of a contagion process exploiting the functional nature of the data and define a novel non-parametric measure of variance based on a weighted version of the depth-based central region area. We apply this methodology to the susceptible-infected-susceptible epidemiological model and the small-world networks. We find that maximum uncertainty is attained at the “stable contagion threshold”, which represents the parameter conditions for which the steady state is reaching a plateau as a function of the contagion rate. The density of the network has a significant effect on the uncertainty of contagion, whereas only a mild effect is seen for the network’s structure.