Background

Clinical scores (DAS28, SLEDAI) fluctuate over time in ways that resist simple parametric modeling. Linear mixed models assume Gaussian residuals; time series models (ARIMA) assume stationarity. Neither captures the mean-reverting behavior with stochastic volatility observed in practice.

Hypothesis

We propose that disease activity X(t) follows a regime-switching Ornstein-Uhlenbeck (OU) stochastic differential equation:

dX(t) = θ_r(t) × (μ_r(t) - X(t)) dt + σ_r(t) dW(t)

where r(t) ∈ {quiescent, active, flaring} is a latent regime governed by a continuous-time Markov chain, and:

• θ_r: mean-reversion speed (how fast scores return to equilibrium) — differs by regime
• μ_r: equilibrium level — quiescent ≈ 1.5, active ≈ 3.5, flaring ≈ 5.5 (DAS28 scale)
• σ_r: volatility — highest in flaring regime (stochastic instability)
• W(t): standard Wiener process (Brownian motion)

Why OU With Regime-Switching

1. Mean reversion: Disease activity tends toward an equilibrium (treatment effect) — pure Brownian motion would be unbounded
2. Regime switching: The equilibrium itself shifts (remission vs flare) — single OU misses bimodality
3. Analytic tractability: OU has closed-form transition density → exact likelihood → efficient parameter estimation via EM algorithm
4. Prediction intervals: The SDE naturally produces calibrated prediction intervals, not just point forecasts

Testable Predictions

1. Regime-switching OU achieves CRPS (continuous ranked probability score) <0.8 on 12-week DAS28 forecasts, outperforming ARIMA (CRPS ~1.2) and random forest (no calibrated intervals)
2. Prediction interval coverage at 80% nominal level achieves 78-82% empirical coverage (calibrated)
3. The flaring regime has θ_flare < θ_quiescent (slower mean reversion — flares persist longer than recovery)
4. σ_flare > 2 × σ_quiescent (increased stochastic volatility during flares)
5. Regime transition rates correlate with medication changes: DMARD escalation increases P(active → quiescent) by >3×

Parameter Estimation

• EM algorithm: E-step = Kim filter for regime probabilities; M-step = MLE for OU parameters per regime
• Data: Monthly DAS28 from ≥200 RA patients, ≥2 years
• Implementation: Python (numpy + scipy.optimize), closed-form OU likelihood

Clinical Application

The SDE provides time-to-flare distributions (first passage time from quiescent to X > 3.2), enabling personalized visit scheduling: patients with high σ or low θ need more frequent monitoring.

References

• Uhlenbeck GE, Ornstein LS. On the theory of Brownian motion. Phys Rev. 1930.
• Hamilton JD. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica. 1989.
• Iacus SM. Simulation and Inference for SDEs. Springer, 2008.
• Prevoo MLL, et al. DAS28. Arthritis Rheum. 1995.