The spread of disease can be slowed by certain aspects of real-world social networks, such as clustering and community structure, and of human behaviour, including social distancing and increased hygiene, many of which have already been studied. Here, we consider a model in which individuals with essential societal roles—be they teachers, first responders or health-care workers—fall ill, and are replaced with healthy individuals. We refer to this process as relational exchange, and incorporate it into a dynamic network model to demonstrate that replacing individuals can accelerate disease transmission. We find that the effects of this process are trivial in the context of a standard mass-action model, but dramatic when considering network structure, featuring accelerating spread, discontinuous transitions and hysteresis loops. This result highlights the inability of mass-action models to account for many behavioural processes. Using empirical data, we find that this mechanism parsimoniously explains observed patterns across influenza outbreaks in the USA at a national level, 25 years of influenza data at the state level, and 19 years of dengue virus data from Puerto Rico. We anticipate that our findings will advance the emerging field of disease forecasting and better inform public health decision making during outbreaks.