Alertness Optimization for Shift Workers Using a Physiology-based Mathematical Model

Abstract

Sleep is vital for maintaining cognitive function, facilitating metabolic waste removal, and supporting memory consolidation. However, modern societal demands, particularly shift work, often disrupt natural sleep patterns. This can induce excessive sleepiness among shift workers in critical sectors such as healthcare and transportation and increase the risk of accidents. The primary contributors to this issue are misalignments of circadian rhythms and enforced sleep-wake schedules. Regulating circadian rhythms that are tied to alertness can be regarded as a control problem with control inputs in the form of light and sleep schedules. In this paper, we address the problem of optimizing alertness by optimizing light and sleep schedules to improve the cognitive performance of shift workers. A key tool in our approach is a mathematical model that relates the control input variables (sleep and lighting schedules) to the dynamics of the circadian clock and sleep. In the sleep and circadian modeling literature, the newer physiology-based model shows better accuracy in predicting the alertness of shift workers than the phenomenology-based model, but the dynamics of physiological-based model have differential equations with different time scales, which pose challenges in optimization. To overcome the challenge, we propose a hybrid version of the PR model by applying singular perturbation techniques to reduce the system to a non-stiff, differentiable hybrid system. This reformulation facilitates the application of the calculus of variation and the gradient descent method to find the optimal light and sleep schedules that maximize the subjective alertness of shift worker. Our approach is validated through numerical simulations, and the simulation results demonstrate improved alertness compared to other existing schedules.

Figures (9)

• Figure 1: Periodic solution of three-process model accommodates to the periodic light schedule of 16-hour light 8-hour dark. 48 hours of the state trajectories are plotted here.
• Figure 2: Periodic solution of PR model accommodates to the periodic light schedule in Eq.\ref{['eq:light']}. Fig. 2(a) includes the circadian states $x,x_c,n$ and Fig. 2(b) includes the neuron potentials $V_m,V_v$ and sleep homeostasis $H$.
• Figure 3: Bifurcation curves of PR model. The x-axis shows the sleep drive $D_v$ and y-axis shows the corresponding $V_m$ at equilibrium. In the wake and sleep region, the wake and sleep branch overlap because there is only one stable equilibrium. In the bistable branch, there are two stable equilibrium points for each $D_v$. When simulating sleep deprivation, we keep $V_m = 1.04 mV$.
• Figure 4: (a) Forced wake branch. (b) Comparison of the real sleep branch (blue) and the fitted sleep branch in hybrid PR model (red) (c) The filter function $g(D_v)$ between $0 \leq D_v \leq 0.2$. (d) Comparison of the real forced wake branch (red) and the fitted forced wake branch in hybrid PR model(blue).
• Figure 5: These are the comparisons of three models. The x axes show the time in hours. The starting times are 6 AM and the durations are 48 hours. (A) Circadian rhythm Process C which includes states $x,x_c$ and $n$. These three states have the identical dynamics in the full PR model and hybrid PR model. (b) Sleep state $\beta(t)$. 0 means the subject is awake and 1 means the subject is asleep. For the full PR model, the subject wakes up at 6:33 AM and sleeps at 10:25 PM. For the hybrid PR model, the subject wakes up at 6:27 AM and sleeps at 10:17 PM. (c) Comparison of the sleep homeostasis between the full PR model and hybrid PR model (d) Comparison of the $V_m$ between the full PR model and hybrid PR model.