One of the most effective ways to improve quality of life in dementia is by exposing people to meaningful activities. The study of engagement is crucial to identify which activities are significant for persons with dementia and customize them. Previous work has mainly focused on developing assessment tools and the only available model of engagement for people with dementia focused on factors influencing engagement or influenced by engagement. This article focuses on the internal functioning of engagement and presents the development and testing of a model specifying the components of engagement, their measures, and the relationships they entertain. We collected behavioral and physiological data while participants with dementia (N = 14) were involved in six sessions of play, three of game-based cognitive stimulation and three of robot-based free play. We tested the concurrent validity of the measures employed to gauge engagement and ran factorial analysis and Structural Equation Modeling to determine whether the components of engagement and their relationships were those hypothesized. The model we constructed, which we call the ENGAGE-DEM, achieved excellent goodness of fit and can be considered a scaffold to the development of affective computing frameworks for measuring engagement online and offline, especially in HCI and HRI.