Hypothesis

The temporal sequence of autoantibody seroconversion events in systemic lupus erythematosus (SLE) follows a self-exciting Hawkes process, where each new seroconversion increases the instantaneous hazard of subsequent seroconversions and organ involvement events. Fitting a multivariate Hawkes process to longitudinal autoantibody panels (anti-dsDNA, anti-Sm, anti-RNP, anti-Ro, anti-La, anti-ribosomal P, aPL) will yield a triggering kernel that predicts cascading organ damage accrual with higher discrimination (C-statistic ≥ 0.78) than current static SLICC/ACR classification approaches.

Background and Rationale

SLE is characterized by progressive autoantibody diversification — epitope spreading generates new specificities over months to years. Current models treat each autoantibody as an independent risk factor, ignoring the temporal dependency structure between seroconversion events. Hawkes processes, originally developed for earthquake aftershock modeling, capture exactly this phenomenon: each event increases the probability of future events through a triggering kernel.

The biological basis is epitope spreading and immune network amplification. Anti-dsDNA seroconversion activates B-cell clones that cross-react with nucleosomal antigens, lowering the activation threshold for anti-Sm and anti-RNP specificities. This cascade is inherently self-exciting — the hallmark of Hawkes process dynamics.

Proposed Model

Let $N_k(t)$ denote the counting process for seroconversion of autoantibody species $k \in {1, ..., 7}$. The conditional intensity is:

$$\lambda_k(t) = \mu_k + \sum_{j=1}^{7} \sum_{t_i^j < t} \alpha_{jk} \cdot \omega_{jk} \cdot e^{-\omega_{jk}(t - t_i^j)}$$

where $\mu_k$ is baseline intensity, $\alpha_{jk}$ captures cross-excitation from species $j$ to $k$, and $\omega_{jk}$ governs temporal decay. The branching matrix $\alpha_{jk}/\omega_{jk}$ encodes the expected number of triggered seroconversions per event.

Organ involvement extension: Append organ damage events (renal, neuropsychiatric, hematologic, serositis) as additional process dimensions, yielding a 11-dimensional Hawkes model where autoantibody events trigger organ events through learned kernels.

Testable Predictions

1. The branching ratio (spectral radius of $\alpha_{jk}/\omega_{jk}$) will be subcritical (< 1.0) in mild SLE and approach criticality (> 0.85) in patients who develop severe organ involvement within 5 years.
2. Anti-dsDNA → anti-Sm and anti-dsDNA → aPL cross-excitation coefficients ($\alpha_{jk}$) will be the strongest off-diagonal elements, reflecting known epitope spreading pathways.
3. Time-varying Hawkes models fitted to the first 3 seroconversion events will predict nephritis onset timing with median absolute error < 8 months.
4. The model will outperform Cox proportional hazards with autoantibody count as a covariate (ΔC-statistic ≥ 0.08, p < 0.01 by DeLong test).

Study Design

Retrospective cohort from longitudinal SLE registries (e.g., Hopkins Lupus Cohort, Toronto Lupus Cohort) with ≥ 5 years follow-up and serial autoantibody panels at ≥ 6-month intervals. Minimum n = 400 patients. Parameter estimation via maximum likelihood with EM algorithm. Validation by temporal cross-validation (train on first 60% of follow-up, predict remaining 40%).

Limitations

• Hawkes processes assume exponential decay kernels; true immunological memory may follow power-law dynamics requiring Hawkes models with non-parametric kernels
• Interval-censored seroconversion data (tested every 3-6 months) introduces timing uncertainty — requires modified likelihood accounting for discrete observation windows
• The model assumes stationarity of baseline intensities $\mu_k$, which may not hold during immunosuppressive treatment changes
• Requires registries with consistent longitudinal autoantibody panels — many cohorts test selectively based on clinical suspicion, introducing informative observation bias
• Sample size requirements for 11-dimensional Hawkes estimation are substantial; regularization (L1 penalty on $\alpha_{jk}$) may be necessary

Clinical Significance

If validated, this framework transforms SLE monitoring from static risk stratification to dynamic cascade prediction. Clinicians could identify patients approaching criticality (branching ratio trending upward) and intensify treatment before irreversible organ damage. The Hawkes triggering kernel also reveals causal ordering of autoantibody cascades, informing targeted B-cell depletion timing. This represents a paradigm shift from counting autoantibodies to modeling their temporal interaction topology.

RheumaAI Research • rheumai.xyz • DeSci Rheumatology