Farr's Laws

Farr's laws are a set of empirical epidemiological regularities formulated between 1840 and 1866 by William Farr (1807–1883), Compiler of Abstracts at the General Register Office of England and Wales and a pioneer of vital biostatistics and quantitative epidemiology. Most prominently, Farr's Law of Epidemics (1840) states that epidemic waves rise and fall in a symmetric, bell-shaped curve whose rate of acceleration decreases at a constant geometric ratio across successive time intervals—meaning that the second difference of the logarithms of incident cases or deaths is a negative constant. Formulated decades before the germ theory of disease was established, Farr's empirical curve-fitting anticipated the mechanistic dynamics of susceptible depletion and threshold turnover later formalized in twentieth-century compartmental models in epidemiology and the basic reproduction number.

Quarterly Smallpox Mortality and the Constant Deceleration Ratio

Upon assuming office at the newly created General Register Office in London in 1839, William Farr instituted the first systematic national classification of causes of death (the precursor to the modern International Classification of Diseases, ICD). In his Second Annual Report of the Registrar-General of Births, Deaths, and Marriages in England (1840), Farr analyzed the 1837–1839 English smallpox (Variola major) epidemic across ten quarterly intervals. Observing that smallpox deaths rose from 2,513 to a peak of 4,489 and then subsided systematically, Farr discovered that while the ratio of cases from one quarter to the next (I_t+1 / I_t) changed over time, the ratio of successive ratios remained nearly constant.

Let I_t denote incident epidemic cases or deaths in discrete time interval t. Defining the first difference of natural logarithms as the empirical growth rate Δ ln(I_t) = ln(I_t+1 / I_t), Farr's Law of Epidemics states that the second difference of the logarithms is a constant negative value (−K):

Equivalently, the ratio of ratios (sometimes termed Brownlee's ratio) satisfies:

Integrating this discrete second-order difference equation twice with respect to time t yields a quadratic polynomial in the exponent—that is, a Gaussian normal curve for epidemic incidence I(t):

Using this formula during the 1865–1866 British cattle plague (rinderpest) outbreak, Farr publicly refuted parliamentary panic that the epizootic would grow without bound, accurately forecasting both the week of peak mortality and the subsequent decline.

Across his four decades at the General Register Office, Farr articulated three quantitative laws linking epidemic trajectories, urban crowding, and topography:

Brownlee's Pathogen Attenuation Fallacy vs. Susceptible Depletion

In 1906, Scottish physician and statistician John Brownlee revived Farr's Law, fitting Pearson Type IV bell curves to dozens of historical plague, measles, and scarlet fever outbreaks. However, because the Gaussian curve is perfectly symmetric, Brownlee erroneously concluded that epidemics end not because susceptible hosts are exhausted, but because the infectious organism undergoes an intrinsic, clock-like loss of virulence ("infectivity attenuation") at rate exp(−Kt).

In 1927, William Kermack and Anderson McKendrick resolved this controversy through their nonlinear compartmental models in epidemiology (and discrete chain-binomial equivalents such as the Reed–Frost model). By expanding the conserved SIR quantity in a Taylor series when the basic reproduction number R₀ is moderately above unity, Kermack and McKendrick proved that susceptible depletion alone generates a bell-shaped incidence trajectory approximated by the squared hyperbolic secant function:

Near the epidemic peak, ln[sech²(u)] ≈ −u² + O(u⁴), which matches Farr's quadratic logarithm law Δ² ln(I_t) ≈ −K without invoking any change in intrinsic pathogen virulence.

Methodological Limitations in Heterogeneous Populations

Modern epidemiological analysis (Fine, 1979; Bregman & Langmuir, 1990) demonstrates why naive application of Farr's Law can fail when forecasting multi-subpopulation pandemics (such as the 1990 Bregman–Langmuir projection of US HIV/AIDS incidence):

1. Reporting Artifacts and Diagnostic Saturation: A decreasing second difference Δ² ln(I_t) < 0 can reflect saturating surveillance capacity or lengthening reporting delays across calendar periods (visualized on a Lexis diagram) rather than true biological deceleration.
2. Subpopulation Cascade Waves: When an infection shifts sequentially from a high-contact core group into lower-risk demographic strata, early deceleration in the core group creates a false Farr peak before the broader population wave accelerates.

How did William Farr's elevation data help confirm John Snow's waterborne cholera theory?

During the 1848–1849 London cholera epidemic, William Farr initially interpreted the inverse correlation between elevation above the Thames and cholera mortality as evidence for atmospheric miasma settling in low-lying river districts. However, Farr's meticulous household death registries provided John Snow with the natural cohort study data needed in 1854 to compare customers of the Southwark & Vauxhall Water Company (drawing sewage-contaminated water at Battersea) against the Lambeth Water Company (which had moved its intake upstream to Thames Ditton), convincing Farr to champion waterborne sanitation by the 1866 East London outbreak.

Why are real SIR epidemic curves slightly right-skewed rather than symmetric like Farr's Gaussian curve?

In an exact Kermack–McKendrick SIR model with R₀ > 2, the initial growth phase occurs at rate (R₀ − 1)γ when S / N ≈ 1, whereas the late decay phase after the wave overshoots the herd immunity threshold occurs at a slower asymptotic rate [1 − R₀ (S_∞ / N)]γ. This asymmetry makes the true epidemic curve climb faster than it descends, whereas Farr's Gaussian approximation treats the ascent and descent rates as equal.

1. Farr, W. (1840). "Letter to the Registrar-General on the causes of death in England." Second Annual Report of the Registrar-General of Births, Deaths, and Marriages in England, pp. 69–98. Her Majesty's Stationery Office.
2. Brownlee, J. (1906). "Statistical studies in immunity: the theory of an epidemic." Proceedings of the Royal Society of Edinburgh, 26, 484–521. https://doi.org/10.1017/S037016460002472X
3. Kermack, W. O., & McKendrick, A. G. (1927). "A contribution to the mathematical theory of epidemics." Proceedings of the Royal Society of London. Series A, 115(772), 700–721. https://doi.org/10.1098/rspa.1927.0118
4. Fine, P. E. M. (1979). "John Brownlee and the measurement of infectiousness: an historical study in epidemic theory." Journal of the Royal Statistical Society: Series A, 142(3), 347–362. https://doi.org/10.2307/2982487
5. Lilienfeld, D. E. (2007). "Celebration: William Farr (1807–1883)—an appreciation on the 200th anniversary of his birth." International Journal of Epidemiology, 36(5), 985–987. https://doi.org/10.1093/ije/dym132