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Mathematics explains why babies sleep during the day, teenagers fall asleep late, and the elderly wake up early
Last reviewed: 27.07.2025
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Ever wondered why babies nap some days but not others? Or why older people wake up earlier? Mathematical modelling of sleep regulation provides unexpected answers to these and other questions, according to a new study from the University of Surrey.
In a paper published in npj Biological Timing and Sleep, the researchers analyzed the mathematical structure of the dual-process model (2PM) of sleep regulation, first proposed in the 1980s. 2PM explains how our sleep patterns are shaped by two factors: sleep pressure, which increases the longer we stay awake and decreases during sleep, and the circadian rhythm of our internal biological clock, which lasts for ~24 hours.
The Surrey team used mathematics to show how 2PM mirrors the brain's processes as it switches between sleep and wakefulness. They showed that the model helps explain why babies at certain stages of development only take naps on certain days - a phenomenon known as the "Devil's staircase" among oscillator scientists. The same model explains sleep patterns in animals.
The researchers also combined the mathematics of sleep-wake switches with the mathematics of how light affects the biological clock. This integrated model helps explain how many sleep phenomena are determined by a combination of internal physiological processes and the environment.
For example, the model explains why teenagers tend to fall asleep and wake up later than younger children. A slower increase in sleep pressure during wakefulness allows them to remain awake longer, and evening exposure to bright light further delays sleep.
The model also offers new insights into other common patterns. One surprising finding: Early awakening in old age may be driven not so much by changes in the biological clock, as is commonly thought, but by how the different systems that govern sleep interact, and how these interactions change with age, environment, and individual biology.
The team's work shows that the 2PM + light model provides insight into why some people have difficulty waking up early or going to bed at a "socially acceptable" time - not because their biological clock is "broken", but because their (light) environment or physiology pushes sleep later.
Professor Anne Skeldon, head of the School of Mathematics at the University of Surrey and lead author of the study, said:
“This model offers hope for better understanding and solutions to sleep problems. Using mathematics, we can see how small changes in lighting, routine, or biology shift sleep patterns and test practical ways to improve sleep for everyone. This is a step toward more personalized, effective solutions that can improve people’s quality of life.”
Using mathematics, the researchers showed that the 2PM + light model behaves like a system of nonlinear oscillators - a sleep-wake oscillator, oscillations of the biological clock, and a light/dark pattern that reaches the brain through the eyes.
Scientists explain that the sleep-wake oscillator does not typically follow a 24-hour rhythm, but it is the interaction with the biological clock and light cycles that helps us stay in sync with the day-night cycle through a process known as “entrainment.”
To further explore these oscillatory interactions, the scientists conducted mathematical simulations using the 2PM + light model. The simulations showed that being indoors for most of the day and having bright lights on in the evening disrupts the oscillator system and, with it, sleep. This allowed them to predict a number of behaviors, such as sleep shifts after evening light exposure or difficulty getting regular sleep.
Professor Derk-Jan Dijk, co-author of the study and director of the University of Surrey's Sleep Research Centre, added:
"This work shows how mathematics can shed light on complex and personal processes like sleep. With the right data and models, we can make better recommendations and develop new interventions to improve sleep in people whose rest is disrupted by modern routines, aging or disease."