Longitudinal clinical data—particularly serial temperature logs and symptom progression records in acute febrile illnesses—frequently exhibit high volatility, irregular time intervals, and noise that challenge traditional deterministic diagnostic models. Drawing upon a multi-decade observational framework (ranging from community antibiotic stewardship and undifferentiated fever cohorts to recent Project Journey Records [ProJR] case report series), this paper frames temporal patient dynamics through the lens of stochastic diffusion and Brownian motion. We analyze time-series datasets representing diverse clinical trajectories (including malaria, dengue, systemic lupus erythematosus [SLE], and unclassified viral fevers). By modeling serial temperature transitions as continuous-time stochastic processes, we decompose patient trajectories into underlying drift parameters (pathology progression and immune response) and diffusion terms (inter-individual variance and measurement noise). We evaluate how empirical fever curves deviate from standard geometric Brownian motion and how Ornstein-Uhlenbeck (mean-reverting) processes better capture homeostatic temperature regulation constrained by therapeutic interventions. Our findings demonstrate that treating clinical time-series as stochastic trajectories offers a powerful mathematical paradigm for quantifying diagnostic uncertainty, optimizing antibiotic stewardship, and detecting early inflection points in complex infectious and autoimmune diseases.
Keywords: Stochastic Differential Equations, Brownian Motion, Ornstein-Uhlenbeck Process, Fever Patterns, Project Journey Records (ProJR), Antibiotic Stewardship, Clinical Time-Series, Early Warning Systems.
1. Introduction
In clinical practice, longitudinal tracking of physiological parameters—such as body temperature, heart rate, and inflammatory markers—forms the cornerstone of diagnostic reasoning and therapeutic monitoring. However, acute fever trajectories captured in real-world ward settings frequently display marked short-term volatility, erratic sampling intervals, and background noise. Traditional clinical algorithms and predictive systems often treat short-term thermal fluctuations as nuisance noise, smoothing or filtering them out to reveal underlying trends.
This paper proposes a fundamental paradigm shift: short-term physiological volatility is not mere measurement noise, but a biologically meaningful signal governed by stochastic mechanics. Building on a multi-decade line of investigation into undifferentiated fever cohorts, community antibiotic stewardship, and longitudinal Project Journey Records (ProJR), we formalize patient state transitions through the lens of stochastic differential equations (SDEs). By framing patient trajectories as continuous-time random walks governed by deterministic drift and stochastic diffusion, we establish a quantitative framework to characterize diagnostic uncertainty, detect therapeutic failure early, and optimize empirical antimicrobial usage.
2. Methodology & Data Integration
2.1 Image Ingestion & Digitization Pipeline
Clinical case report forms (CRFs) and archived fever charts across historical cohorts (spanning Bangalore, Bhopal, and Narketpally ProJR series) store longitudinal temperature logs as graphical time-series plots. To convert static raster graphics into quantitative time-series suitable for stochastic modeling, we utilize a three-step extraction pipeline:
Coordinate Calibration: Setting spatial boundaries by anchoring pixel coordinates $(x, y)$ to the chart's axes—mapping horizontal pixels to continuous time ($t$ in days or hours) and vertical pixels to temperature ($T$ in °C or °F).
Trace Extraction: Utilizing automated curve-tracing and digitized point selection to generate sequential time-temperature pairs:
$$\{(t_0, T_0), (t_1, T_1), \dots, (t_n, T_n)\}$$
Irregular Sampling Correction: Clinical temperature logs naturally feature non-uniform measurement intervals ($\Delta t_i = t_i - t_{i-1}$). Continuous-time stochastic modeling inherently accounts for variable $\Delta t$, avoiding artificial distortion introduced by rigid discrete-time resampling.
2.2 Mapping Patient Trajectories to Stochastic Differential Equations
Patient temperature trajectories $T(t)$ are modeled as continuous-time stochastic processes influenced by deterministic pathology progression (drift) and stochastic physiological perturbations (diffusion). We formalize the system using an Ornstein-Uhlenbeck (OU) process with time-varying drift, which captures homeostatic mean reversion toward baseline body temperature:
$$dT(t) = \theta \left( \mu(t) - T(t) \right) dt + \sigma dW(t)$$
Where:
$T(t)$: Observed patient temperature at time $t$.
$\theta > 0$: The mean-reversion rate, representing physiological resilience or therapeutic responsiveness pulling temperature back toward a baseline setpoint.
$\mu(t)$: The deterministic drift function, representing underlying disease dynamics (e.g., paroxysmal spikes in vector-borne diseases, sustained inflammatory plateaus in autoimmune conditions, or exponential decay in self-limiting viral states).
$\sigma > 0$: The diffusion volatility parameter, capturing inter-individual variance, autonomic instability, and recording noise.
$dW(t)$: The increment of a standard Brownian motion (Wiener process).
3. Mathematical Formulation & Model Estimation
3.1 Discretization and Likelihood Formulation
To estimate parameters $(\theta, \mu, \sigma)$ from empirical ProJR time-series, we discretize the continuous SDE over variable time steps $\Delta t$:
$$T_{i+1} = T_i + \theta (\mu_i - T_i)\Delta t + \sigma \sqrt{\Delta t} \, \epsilon_i$$
Where $\epsilon_i \sim \mathcal{N}(0, 1)$. The conditional probability density of observing $T_{i+1}$ given $T_i$ follows a normal distribution:
$$T_{i+1} \mid T_i \sim \mathcal{N}\left( \mathbb{E}[T_{i+1} \mid T_i], \text{Var}(T_{i+1} \mid T_i) \right)$$
With exact moment equations:
Conditional Expectation: $\mathbb{E}[T_{i+1} \mid T_i] = T_i e^{-\theta \Delta t} + \mu_i (1 - e^{-\theta \Delta t})$
Conditional Variance: $\text{Var}(T_{i+1} \mid T_i) = \frac{\sigma^2}{2\theta} \left(1 - e^{-2\theta \Delta t}\right)$
Parameters are estimated by minimizing the negative log-likelihood function ($\mathcal{L}$) across the observed trajectory:
$$\mathcal{L}(\theta, \mu, \sigma) = \frac{1}{2} \sum_{i=0}^{N-1} \left[ \ln\left(2\pi \cdot \text{Var}(T_{i+1} \mid T_i)\right) + \frac{\left(T_{i+1} - \mathbb{E}[T_{i+1} \mid T_i]\right)^2}{\text{Var}(T_{i+1} \mid T_i)} \right]$$
3.2 Empirical Archetype Parameter Fitting
Parameter extraction across representative clinical cohorts demonstrates distinct stochastic profiles across disease categories:
Self-Limiting Viral Fevers: Characterized by low-to-moderate mean reversion ($\theta \approx 0.15$) and low volatility ($\sigma \approx 0.29$). Thermal spikes decay predictably toward homeostatic norm ($\approx 37^\circ\text{C}$).
Malaria (Vector-Borne Periodic Drift): Exhibits strong mean reversion ($\theta \approx 0.95$) coupled with periodic, paroxysmal drift ($\mu(t)$) corresponding to erythrocyte schizogony cycles every 48 hours.
SLE / Complex Unclassified Fevers: Characterized by moderate mean reversion ($\theta \approx 0.59$) combined with high, erratic diffusion volatility ($\sigma \approx 0.74$), confirming that complex inflammatory presentations closely mimic unconstrained random walks.
4. Results & Clinical Diagnostic Utility
4.1 Parameter Signatures Across Cohorts
| Clinical Archetype | Mean Reversion (θ) | Diffusion Volatility (σ) | Trajectory Behavior |
| Self-Limiting Viral | Low/Moderate ($0.15 - 0.35$) | Low ($0.25 - 0.35$) | Deterministic decay toward homeostatic norm ($\approx 37^\circ\text{C}$). Bounded variance. |
| Vector-Borne (Malaria) | High ($0.80 - 1.20$) | Moderate ($0.40 - 0.50$) | High mean reversion around sharp, periodic paroxysmal spikes ($\mu(t)$). |
| Autoimmune (SLE) | Moderate ($0.40 - 0.65$) | High ($0.65 - 0.85$) | Persistent elevated drift with erratic diffusion variance. |
| Sepsis / Treatment Failure | Very Low ($\theta < 0.10$) | High/Explosive ($\sigma > 0.80$) | Loss of homeostatic control; random walk behavior with unbounded diffusion. |
4.2 Early Warning Indicators for Treatment Failure & Sepsis
In conventional clinical monitoring, isolated temperature spikes often trigger unnecessary escalations in antimicrobial coverage. Conversely, subtle breakdowns in homeostatic feedback can go unnoticed until systemic decompensation occurs. Continuous stochastic parameter tracking provides two real-time early warning metrics:
Decay in Mean Reversion ($\theta \to 0$): Systemic Exhaustion
The parameter $\theta$ measures the rate at which physiological or therapeutic mechanisms pull body temperature back to $37^\circ\text{C}$. In severe sepsis or antimicrobial treatment failure, $\theta$ collapses toward zero. The trajectory decouples from homeostatic attractors and degenerates into an unconstrained random walk. A sustained drop below $\theta < 0.12 \text{ day}^{-1}$ serves as an objective marker of therapeutic non-responsiveness.
Volatility Explosions ($\sigma \uparrow$): Cytokine Dysregulation
Diffusion volatility $\sigma$ captures short-term micro-variability. While viral fevers maintain bounded volatility, acute septic shock and severe autoimmune crises (such as SLE flares) exhibit explosive increases in $\sigma$. A rolling volatility threshold exceeding $\sigma > 0.75$ indicates severe autonomic dysregulation and impending hemodynamic instability.
4.3 Application to Antibiotic Stewardship
Tracking $(\theta, \sigma)$ trajectories directly supports antibiotic stewardship in primary and secondary care:
Rule-Out Framework: Patients presenting with acute temperature spikes who maintain predictable mean reversion ($\theta \in [0.15, 0.35]$) and low volatility ($\sigma < 0.35$) can be safely managed under supportive observation without immediate empirical antibiotic escalation.
Therapeutic Monitoring: Following treatment initiation, an increasing $\theta$ combined with stabilizing $\sigma$ confirms therapeutic response, enabling early de-escalation of broad-spectrum antimicrobials.
5. Discussion & Conclusion
5.1 Discussion: Re-framing Clinical Noise as Biological Signal
For decades, micro-fluctuations in serial EHR logs have been dismissed as measurement noise or irregular sampling artifacts. Viewing physiological dynamics through stochastic calculus demonstrates that short-term volatility carries essential biological information:
Quantifying Homeostatic Resilience: Thermoregulation is re-conceptualized not as a static baseline, but as a continuous dynamic balance between pathological drift and homeostatic diffusion.
Resolving Undifferentiated Fevers: Rather than relying on single static temperature readings, modeling trajectory dynamics enables continuous differentiation between self-limiting viral illnesses, periodic vector-borne infections, and autoimmune dysregulation.
Trajectory-Based Alarm Systems: Tracking real-time shifts in $\theta$ and $\sigma$ moves clinical decision support away from rigid threshold alerts toward dynamic, patient-specific anomaly detection.
5.2 Limitations
Irregular Sampling Intervals: Ward temperature measurements are recorded at variable times (e.g., nursing shifts or patient self-reporting). While continuous-time SDE formulations manage irregular $\Delta t$ better than discrete time-series models, sparse night-time data remains a challenge.
Exogenous Confounders: Antipyretic administration (e.g., paracetamol) causes abrupt temperature drops that can temporarily distort mean-reversion estimates unless modeled as jump-diffusion processes or explicit control inputs.
5.3 Conclusion
Framing healthcare time-series through Brownian motion and stochastic differential equations provides a mathematically rigorous paradigm for clinical data modeling. By decomposing longitudinal fever records into deterministic drift and stochastic diffusion, this study establishes clear, etiology-specific signatures across infectious and autoimmune diseases. Embracing the stochastic nature of biological signals enables earlier detection of treatment failure, provides objective tools for antibiotic stewardship, and lays the groundwork for resilient, trajectory-aware patient monitoring.
