lactationcurve (Python)

A package for fitting dairy animal lactation curves, evaluating lactation curve characteristics (LCCs) (time to peak, peak yield, cumulative yield, persistency), and computing 305-day milk yield using the ICAR guideline.

Contact: Meike van Leerdam, mbv32@cornell.edu

Authors: Meike van Leerdam, Douwe de Kok, Judith Osei-Tete, Lucia Trapanese

Initial authored: 2025-08-12

Updated: 2026-10-26

Lactation curves are widely used in dairy science to model milk production over time. They provide insights into the lactation performance of individual animals and herds, enabling better management decisions and genetic evaluations. The 305-day yield is a standard metric for assessing milk production, and accurate estimation of this yield is crucial for dairy farmers, researchers, and breeding programs.

The International Committee for Animal Recording (ICAR) provides guidelines outlining approved methods for calculating the 305-day yield of milk, fat, and protein. This package provides an open-source Python implementation of these methods, with lactation curve modelling at its core.

In addition to the methods described in the original guideline, the package incorporates 14 lactation curve models, including traditional and Bayesian fitting approaches as well as AI-based models. It also provides tools to derive biologically relevant characteristics such as time to peak, peak yield, cumulative yield, and persistency.

The package can be integrated into analytical workflows to calculate 305-day yields, fit and compare lactation curves, and derive key lactation characteristics through a single function call. These functionalities are supported through an interactive website and an openly available GitHub repository.

We encourage everyone to use, test, and contribute to the package, which is available under the AGPL-3.0-or-later license. For bug reports or feature requests, please submit them on the GitHub issues page or contact us directly via email.

What can this package do?

• Frequentist fitting of lactation curve models (numeric optimization):
  • Wood, Wilmink, Ali & Schaeffer, Fischer, MilkBot
• Frequentist fitting of lactation curve models (algebraic least squares):
  • MilkBot
• Bayesian fitting of the MilkBot lactation curve model via MilkBot API:
  • MilkBot
• Calculate Lactation Curve Characteristics — symbolic + numeric:
  • time_to_peak, peak_yield, cumulative_milk_yield, persistency
• Calculate cumulative milk yield over a lactation period:
  • Test Interval Method (also called the Fleischmann method)
  • Centering Date Method
  • Interpolation Standard Lactation Curve (ISLC) Method
  • Best Predict Method
• Data quality control for lactation curves:
  • Input validation/normalization via validate_and_prepare_inputs

Main Lactation curve models implemented

Fischer: A combination of an exponential and a linear component, characterized by an early exponential phase followed by an approximately linear decline.

Wood: An incomplete gamma-type function combining a power-law growth term t^b with an exponential decay term e^(-ct).

Ali & Schaeffer: A linear regression model based on quadratic polynomials in standardized time (t/305) and in the log-transformed inverse time log(305/t).

Wilmink: A combination of an exponential and a linear component. An additional parameter scaling the exponential term increases flexibility in describing early-lactation dynamics and peak formation.

MilkBot: An empirical, mechanistically motivated four-parameter model describing lactation as the development and decay of udder capacity. Both Bayesian and frequentist fitting approaches are implemented.

Additional models available for symbolic lactation curve characteristics: Brody, Sikka, Nelder, Dhanoa, Emmans, Hayashi, Rook, Dijkstra, Prasad.

Model formulas

• Wood: y(t) = a * t^b * exp(-c * t)
• Wilmink: y(t) = a + b * t + c * exp(k * t) with default k = -0.05
• Ali & Schaeffer: t_scaled = t / 305, L = ln(305 / t), y(t) = a + b*t_scaled + c*t_scaled^2 + d*L + k*L^2
• Fischer: y(t) = a - b*t - a*exp(-c*t)
• MilkBot: y(t) = a * (1 - exp((c - t)/b) / 2) * exp(-d*t)

Package structure

The package is organized into three main modules:

1. lactationcurve.fitting
2. lactationcurve.characteristics
3. lactationcurve.preprocessing

Generated API documentation is available at bovi-analytics.github.io/bovi/lactationcurve.html.

Core functions

The functions below are the main tools most users need. A lactation means the milk records belonging to one cow after one calving. A TestId is the label used to keep each lactation separate.

For the dataframe-based methods, the result usually has two columns: TestId identifies the lactation, and LactationMilkYield contains the estimated cumulative milk yield. The yield uses the same unit as the input records, such as kilograms or pounds.

The meaning of a TestId

The TestId is an identifier for a lactation, which can be used to group records belonging to the same lactation together. It is not the same as a cow ID, as a cow can have multiple lactations across different calvings.

If a TestId column is not provided, the package assumes all records belong to a single lactation and creates a TestId column with all values set to 0.

Minimum number of records per lactation needed per method

The minimum number of records needed to technically calculate the 305-day yield depends on the method:

• Test Interval Method: At least 2 records.
• Centering Date Method: At least 2 records.
• ISLC Method: Can be applied to lactations without measurements; the result is then the population mean from the standard curve.
• Best Predict Method: Can be applied to lactations without measurements; the result is then the population mean from the standard curve.
• Lactation curve models: In general, at least one record more than the number of model parameters is needed. Wood and Fischer (3 parameters) need at least 4 records, Wilmink and MilkBot (4 parameters) at least 5 records, and Ali & Schaeffer (5 parameters) at least 6 records.

In practice, more records are recommended to ensure a reliable fit and avoid overfitting. Ideally, records should be well distributed across the lactation period to capture the curve's shape accurately.

Often used abbreviations

• DIM: Days in Milk (the days since calving in the current lactation)
• LC: Lactation Curve
• LCC: Lactation Curve Characteristic
• ISLC: Interpolation using the Standard Lactation Curve
• ICAR: International Committee for Animal Recording
• API: Application Programming Interface

Bayesian fitting (MilkBot API)

• Set fitting="bayesian" and model="milkbot" in fit_lactation_curve or calculate_characteristic.
• Provide an API key via .env.
• Choose priors with custom_priors:
  • CHEN for published priors from Chen et al. (2023).
  • A dictionary containing custom MilkBot priors. Use the build_prior helper to create one. Each MilkBot parameter (scale, ramp, decay, onset) needs a mean and standard deviation, and seMilk specifies the expected noise in milk yield. The default seMilk is 4 kg.
• If no custom priors are provided, set custom_priors=None and specify the desired continent and breed:
  • continent="USA" (default) or continent="EU".
  • breed="H" for Holstein (default) or breed="J" for Jersey.
  • If parity is provided, continent-specific priors are further refined by parity-specific priors.
• The priors are sensitive to the unit. The default is kilograms; specify milk_unit="lb" when using pounds.
• The helper bayesian_fit_milkbot_single_lactation(...) normalizes differing API responses.
• Request an API key by emailing Jim Ehrlich.
• More information is available in the MilkBot API documentation and the corresponding paper.

Tutorials

Hands-on programming example notebooks are available in the GitHub tutorials and example notebooks. They demonstrate how to import the package, fit lactation curves, calculate 305-day yields, and derive lactation curve characteristics.

Citing the lactationcurve package

If you use the lactationcurve package in your research, please consider citing it as follows:

van Leerdam, M. B., de Kok, D., Osei-Tete, J. A., & Hostens, M. (2026). Bovi-analytics/bovi: v.1.1.6. Zenodo.

https://doi.org/10.5281/zenodo.18715145

If you use the Bayesian fitting functionality that relies on the MilkBot API, please also cite:

Ehrlich, J. L. (2013). Quantifying inter-group variability in lactation curve shape and magnitude with the MilkBot lactation model. PeerJ 1, e54.

https://doi.org/10.7717/peerj.54

If you use the 305-day yield calculation methods based on the ICAR guideline, please also cite:

Best Predict method: VanRaden, P. M. (1997). Lactation yields and accuracies computed from test day yields and (co)variances by best prediction. Journal of Dairy Science, 80(11), 3015-3022.

ISLC: Wilmink, J. B. M. (1987). Comparison of different methods of predicting 305-day milk yield using means calculated from within-herd lactation curves. Livestock Production Science, 17, 1-17.

Test Interval Method: Sargent, F. D., Lyton, V. H., and Wall, O. G. (1968). Test interval method of calculating Dairy Herd Improvement Association records. Journal of Dairy Science, 51, 170.

Centering Date Method: U.S. Department of Agriculture, Bureau of Dairying. (1925). Cow Testing Association Letter No. 1. Washington, DC.

License

GNU AGPLv3 License