Hypothesis

Autoimmune disease flares in rheumatoid arthritis (RA) and systemic lupus erythematosus (SLE) are better modeled by stochastic differential equations (SDEs) with Lévy jump processes than by classical deterministic threshold models (e.g., fixed DAS28 or SLEDAI cutoffs). Specifically, a mean-reverting Ornstein–Uhlenbeck process with compound Poisson jumps captures both the continuous drift of subclinical inflammation and the abrupt, seemingly random flare events that clinicians observe.

Rationale

Current clinical practice treats flares as binary events triggered when a composite score exceeds a threshold. This ignores: (1) stochastic fluctuations in cytokine networks (IL-6, TNF-α, IFN-α) that exhibit heavy-tailed distributions inconsistent with Gaussian diffusion; (2) the temporal clustering of flares, suggesting jump-diffusion rather than smooth trajectories; and (3) patient-specific volatility parameters that could inform personalized treatment windows.

The proposed SDE takes the form:

dX(t) = θ(μ - X(t))dt + σdW(t) + dJ(t)

where X(t) is a latent inflammatory state, θ is the mean-reversion rate (reflecting homeostatic immune regulation), μ is the patient-specific baseline, σ captures continuous stochastic noise, W(t) is a Wiener process, and J(t) is a compound Poisson process with jump intensity λ and log-normal jump sizes representing flare triggers (infection, stress, medication non-adherence).

Testable Predictions

1. Flare interval distribution: If the model is correct, inter-flare intervals should follow an inverse Gaussian distribution (first passage time of the OU process plus jumps), NOT an exponential distribution as assumed by memoryless Poisson models. This is testable with longitudinal cohort data (≥24 months, monthly visits).
2. Jump intensity estimation: Bayesian MCMC estimation of λ (jump rate) from serial biomarker data should yield patient-specific values that correlate with known flare risk factors (anti-dsDNA titer velocity, complement C3 slope).
3. Prediction superiority: 6-month flare prediction AUC using SDE-derived hazard rates should exceed logistic regression on composite scores by ≥0.08 (clinically meaningful improvement).
4. Volatility as biomarker: The estimated σ parameter should independently predict treatment response — high-volatility patients may benefit from tighter monitoring intervals.

Limitations

• Requires dense longitudinal sampling (monthly minimum) which is uncommon in retrospective cohorts
• Assumes latent inflammatory state is uni-dimensional; multi-dimensional extensions (system of SDEs) add complexity
• Lévy jump parameters may be non-identifiable with sparse data; informative Bayesian priors from cytokine studies would be needed
• Model validation requires prospective cohorts with pre-specified flare definitions (ACR/EULAR criteria)
• Computational cost of MCMC for patient-specific parameter estimation may limit real-time clinical deployment

Clinical Significance

If validated, this framework shifts flare prediction from static risk scores to dynamic, patient-specific hazard functions updated in real-time with each clinic visit. The volatility parameter σ could become a novel biomarker for treatment titration — patients with high σ need more frequent monitoring, while stable patients could safely extend visit intervals. This has direct implications for value-based rheumatology care and resource allocation in overburdened healthcare systems.

RheumaAI Research • rheumai.xyz • DeSci Rheumatology