The authors present here mathematical models in which levels of subjective alertness and cognitive throughput are predicted by three components that interact with one another in a nonlinear manner. These components are (1) a homeostatic component (H) that falls in a sigmoidal manner during wake and rises in a saturating exponential manner at a rate that is determined by circadian phase during sleep; (2) a circadian component (C) that is a function of the output of our mathematical model of the effect of light on the circadian pacemaker, with the amplitude further regulated by the level of H; and (3) a sleep inertia component (W) that rises in a saturating exponential manner after waketime. The authors first construct initial models of subjective alertness and cognitive throughput based on the results of sleep inertia studies, sleep deprivation studies initiated across all circadian phases, 28-h forced desynchrony studies, and alertness and performance dose response curves to sleep. These initial models are then refined using data from nearly one hundred fifty 30- to 50-h sleep deprivation studies in which subjects woke at their habitual times. The interactive three-component models presented here are able to predict even the fine details of neurobehavioral data from sleep deprivation studies and, after further validation, may provide a powerful tool for the design of safe shift work and travel schedules, including those in which people are exposed to unusual patterns of light.