The shedding curves fitted across the repository, and worked examples of fitting your own.
All three are fitted on the log10 scale by censored maximum likelihood.
c(t) = c₀ · e−a₀t
Pure decay from the reference event. It assumes shedding is already at its peak when the clock starts, which is often true when the clock starts at symptom onset and the study caught people late.
c(t) = c₀ · tb₀ · e−a₀t
A rise to a peak at t = b₀/a₀, then decay. It needs observations
after the reference event to see the rise, and its curve is undefined at
t ≤ 0, so those readings are discarded.
c(t) = c₀ · (t−t₀)b₀ · e−a₀(t−t₀)
The same shape with shedding starting at t₀ rather than at the
reference event. It exists because the reference event is not the same event
across studies — symptom onset, enrollment, confirmation, vaccination and
hospital admission all appear — and because the plain gamma was discarding
tens of thousands of detected measurements recorded at exactly t = 0.
Its t₀ is what makes those reference events commensurable.
Across all three, a₀ is the decay rate, giving a half-life of
ln(2)/a₀; b₀ governs the rise; t₀ is the
onset of shedding.
Choosing between the three is not an AIC comparison. They are
fitted to different observation sets — gamma drops every reading at
t ≤ 0 and gamma-shifted keeps the detected ones — and AIC only
compares models fitted to the same data. Check the measurement count before
reading anything into the AIC.
Roughly a third of the measurements in the repository are reported negative. They enter the likelihood as left-censored observations — carrying the information that the concentration was below the limit — rather than being dropped or replaced with a substituted value.
Dropping them biases decay rates slow and inflates simulated late-phase shedding, which is exactly the quantity a wastewater or transmission model is most sensitive to.
The fitted parameters ship with the Python package, so a transmission or wastewater model can consume them directly without refitting: load a study, take its curve, and simulate a cohort that carries inter-individual variability rather than tracing one average.
Both work through
woelfel2020virological,
so they can be read against each other.
Have a modelling notebook that uses Shedding Hub data? Send it to us and we will host it here alongside these.