Impulsive differential equations are often used in mathematical modelling to simplify complicated hybrid models. We propose an inverse framework inspired by impulsive differential equations, called impulse extension equations, which can be used as a tool to determine when these impulsive models are accurate. The linear theory is the primary focus, for which theorems analoguous to ordinary and impulsive differential equations are derived. Results explicitly connecting the stability of impulsive differential equations to related impulse extension equations are proven in what we call time scale consistency theorems. Opportunities for future research in this direction are discussed. Following the work of Smith? and Hove-Musekwa on malaria vector control by impulsive insecticide spraying, we propose a novel autonomous vector control scheme based on human disease incidence. Existence and stability of periodic orbits is established. We compare the implementation cost of the incidence-based control to a fixed-time spraying schedule. Hybrid control strategies are discussed.