Relatedness of the incidence decay with exponential adjustment (IDEA) model, “Farr's law” and SIR compartmental difference equation models
Bibliographic record
Abstract
Mathematical models are often regarded as recent innovations in the description and analysis of infectious disease outbreaks and epidemics, but simple mathematical expressions have been in use for projection of epidemic trajectories for more than a century. We recently introduced a single equation model (the incidence decay with exponential adjustment, or IDEA model) that can be used for short-term epidemiological forecasting. In the mid-19th century, Dr. William Farr made the observation that epidemic events rise and fall in a roughly symmetrical pattern that can be approximated by a bell-shaped curve. He noticed that this time-evolution behavior could be captured by a single mathematical formula (“Farr's law”) that could be used for epidemic forecasting. We show here that the IDEA model follows Farr's law, and show that for intuitive assumptions, Farr's Law can be derived from the IDEA model. Moreover, we show that both mathematical approaches, Farr's Law and the IDEA model, resemble solutions of a susceptible-infectious-removed (SIR) compartmental differential-equation model in an asymptotic limit, where the changes of disease transmission respond to control measures, and not only to the depletion of susceptible individuals. This suggests that the concept of the reproduction number (R0) was implicitly captured in Farr's (pre-microbial era) work, and also suggests that control of epidemics, whether via behavior change or intervention, is as integral to the natural history of epidemics as is the dynamics of disease transmission.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".