Incubation Curve Modeling Console (Log-normal)

📁 Data ingestion

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Drop an Excel file here, or click to browse

Supported: .xlsx, .xls, .csv · Max: 10MB

📈 Interactive curve editor

R² =1.0000

RMSE =0.0000

χ² =0.0000

n =51

Status:Original

🎬 Motion definition & clinical interpretation

Animated trajectory ("breathing" band): a visual surrogate for parametric uncertainty around the fitted incubation curve. The animation does not alter the underlying dataset; it illustrates plausible local perturbations that remain consistent with the current $(\mu,\sigma)$ and the displayed goodness-of-fit.

Deviation band (conceptual ±1σ envelope): an intuitive depiction of dispersion under a log-normal incubation model. Clinically, wider dispersion is compatible with heterogeneous inoculum, immune competence variation, age distribution, and comorbidity burden.

Inference note: interpret the oscillation as an uncertainty narrative, not as a time-varying hazard. Formal inference should rely on fitted parameters and the reported diagnostics.

📐 Log-normal probability density function

$$f(t;\mu,\sigma) = \frac{1}{t \cdot \sigma \cdot \sqrt{2\pi}} \cdot \exp\left(-\frac{(\ln t - \mu)^2}{2\sigma^2}\right)$$

μ (location):4.0943

σ (scale):0.3381

Median:60.0 h

Mode:53.5 h

Mean:63.6 h

Variance:—

📊 Analytical views

🧩 Layered analytical stage

Drag any analytical tile and drop it here to compose a multi-layer figure. Layers are clamped to data-consistent limits by default.

Axes time (hours) on X · cases / scaled density on Y · export is enabled only after any stage modification (including equation layers).

Active layers

order = rendering order

Axis configuration

X min: X max: Y max:

🧮 Equation → path workspace

Expression (use t as time in hours)

Variables:

Functions:

Spreadsheet bindings (scalars / ranges)

Bindings are injected into the evaluator as either scalars or numeric vectors (ranges). For vectors, use LIN_INTERP(t, xVec, yVec) or summary functions (SUM(), MEAN(), MIN(), MAX(), LEN()).

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exp

sin

cos

abs

Available symbols are translated into a safe subset of Math.*. For reproducibility, the exported figure encodes the exact expression in the footer and the stage export becomes available only after adding (or modifying) an equation layer.

Rendered as LaTeX (documentation)

$$y(t)=\text{(enter an expression to render)}$$

Note: LaTeX rendering is for manuscript readability. The numeric layer is computed from the expression (not from the LaTeX string).

💾 Export figures

📁 Reference assets (as-delivered)

☀️

Light PNG

curve_clean_light.png

🌙

Dark PNG

curve_clean_dark.png

🔲

Transparent PNG

curve_transparent.png

📐

Vector SVG

curve_vector.svg

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JSON dataset

curve_data.json

📖 Scientific documentation & references

📐 Log-normal model (mathematical basis)

The log-normal distribution is a continuous probability model for positive-valued variables. If $X \sim \mathrm{LogNormal}(\mu,\sigma)$, then $Y=\ln(X)$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. This model is commonly used for incubation periods due to multiplicative biological processes and right-skew.

🔬 Clinical/epidemiological rationale

Incubation periods often exhibit log-normal-like behavior because:

Pathogen replication kinetics: multiplicative growth with stochastic variation.
Host response thresholds: symptom onset frequently reflects crossing a biological threshold.
Host heterogeneity: age, baseline immune competence, and comorbidity profiles shift the distribution.
Exposure dose: inoculum size influences time-to-threshold and therefore onset timing.

📊 Fit diagnostics (reporting-ready)

$R^2$ (coefficient of determination): proportion of variance explained. In this implementation it is used as a pragmatic concordance indicator on the plotted scale.

RMSE: $\sqrt{\frac{1}{n}\sum (y_i-\hat{y}_i)^2}$ summarising average deviation.

$\chi^2$ (Pearson-type): $\sum\frac{(O_i-E_i)^2}{E_i}$ for distributional concordance when expected values are non-trivial.

💻 Software components (attribution)

SheetJS: client-side Excel parsing (MIT License).
WebAssembly (optional ingestion engine): performance-oriented Excel parsing backend (implementation-dependent; user-supplied module).
MathJax v3: LaTeX rendering (Apache 2.0).
SVG: resolution-independent vector graphics for scientific plots.
Canvas API: deterministic raster export (PNG) for manuscript submission.

🧾 Excel-reproducible formulas (implementation equivalence)

The console is deliberately aligned with functions that are natively supported by Excel to allow verification and independent reproduction. Below, TimeRange denotes the incubation time vector in hours (positive values).

Log-space parameters (natural log): μ = AVERAGE(LN(TimeRange)), σ = STDEV.S(LN(TimeRange)).
Log-normal PDF: =LOGNORM.DIST(t, mu, sigma, FALSE).
Log-normal CDF: =LOGNORM.DIST(t, mu, sigma, TRUE).
Hazard (fitted): =LOGNORM.DIST(t, mu, sigma, FALSE) / (1 - LOGNORM.DIST(t, mu, sigma, TRUE)).
Gaussian KDE (Excel, dynamic arrays): =LET(d,TimeRange,bw,8,x,t, SUM(NORM.DIST((x-d)/bw,0,1,FALSE))/(ROWS(d)*bw)). Multiply by the observed peak count if you want a count-scaled curve.
Empirical CDF (ECDF): =COUNTIF(TimeRange, "<="&t)/COUNT(TimeRange).

For aggregated datasets (time, count) use weighted log-mean and weighted log-standard-deviation. Excel supports this via SUMPRODUCT (weights) if required.

📚 Key references

The Lognormal Distribution. Aitchison, J. & Brown, J.A.C. Cambridge University Press, 1957. ISBN: 978-0-521-04011-2 (Classic statistical reference; DOI not available for 1957 edition)

Incubation periods of acute respiratory viral infections. Lessler, J. et al. The Lancet Infectious Diseases, 2009; 9(5): 291-300. DOI

Statistical distributions useful in epidemiology. Sartwell, P.E. American Journal of Hygiene, 1950; 51(3): 310-318.

NIST/SEMATECH e-Handbook of Statistical Methods. Chapter 1.3.6.6.9: Lognormal Distribution. NIST

Incubation Period Curve Modeling