**ORIGINAL ARTICLES**

**Using the distributed-delay model to predict egg production in
laying hens ^{¤}**

**Uso del modelo de distribución con retardo para predecir la producción de huevos en gallinas ponedoras**

**Uso do modelo de distribuição com atraso para predizer a produção de ovos de galinhas poedeiras **

**Luis Galeano-Vasco ^{1*}, MSc; Mario Cerón-Muñoz^{1}, PhD; Daniel Rodríguez^{2}, MSc; José M Cotes^{3}, PhD.**

* Correspondig author: Luis Fernando Galeano-Vasco. Grupo de investigación en Genética, Mejoramiento y Modelación Animal (GaMMA), Facultad de Ciencias Agrarias, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín, Colombia. Email: lf.galeano.vasco@gmail.com

1Grupo de investigación en Genética, Mejoramiento y Modelación Animal (GaMMA), Facultad de Ciencias Agrarias, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín, Colombia.

2Universidad Militar Nueva Granada, Facultad de Ciencias Básicas, Carrera 11 # 101-80, Bogotá.

3Universidad Nacional de Colombia, Departamento de Ciencias Agropecuarias, Calle 59^{a} # 63-20, Bloque 11, Oficina 101-7.

(Received: June 14, 2012; accepted: May 27, 2013)

**Summary**

**Background:** using mathematical models to characterize and estimate egg production curves is of great
importance for assessing the productive efficiency of hens. These models can be used in identifying and
modeling real-time factors affecting animal production and implementing corrective measures to minimize
its effect. **Objective:** we compared the ability to model and adjust the egg production curve in hens using
the distributed-Delay model versus the Adams-Bell and Lokhorst models. **Methods:** 225 records of weekly
production of Hy Line Brown (62 data), Lohmann LSL (54 data), Isa Brown (54 data), and Lohmann Brown
(55 data) were used. All analyzed flocks were raised at Hacienda La Montaña Farm, owned and managed by the
University of Antioquia (Colombia). Models used were Adams-Bell, Lokhorst and Delay; all were validated
and contrasted by Durbin-Watson statistic, MAD, determination (R^{2}) and correlation (r) coefficients. Results:
the Delay and Lokhorst models resulted in R^{2} values greater than 0.8 and r-values greater than 0.9 (p<0.01).
For the Lohmann Brown curve, the Adams-Bell model had the lowest R2 value (0.81), while the Lokhorst and
Delay models resulted in the highest R^{2} value for the Isa Brown curve (1.0). The Delay model fit the curve
(28 and 40 for the k parameter; 63 and 64 for the DEL parameter). The Hy Line Brown curve presented a high
number of irregularities, generating great difficulty for adjustment with the evaluated models. Conclusion:
Delay and Lokhorst models are efficient for predicting egg production curve of the bird strains tested. Unlike
the Adams-Bell and Lokhorst models, goodness of fit of the Delay model could be increased by including
physiological relationships and supply/demand of resources as input variables, which would allow the model
to fit the fluctuations observed in the production curves.

**Key words:** mathematical model, modeling, regression analysis.

**Resumen**

**Antecedentes:** los modelos matemáticos permiten caracterizar y estimar las curvas de producción de
huevos, siendo de gran importancia para la evaluación de la eficiencia productiva de las gallinas, posibilitando
identificar factores que afecten la producción animal y aplicar correctivos para minimizar su efecto. Objetivo:
se comparó la capacidad para ajustar la curva de producción de huevos utilizando el modelo de distribución
con retardo (Delay) y los modelos Adams-Bell y Lokhorst. **Métodos:** se utilizaron 225 datos de registros
semanales de producción de cuatrolíneas: Hy Line Brown (62 datos), Lohmann LSL (54 datos), Isa Brown
(54 datos), y Lohmann Brown (55 datos). Los lotes analizados pertenecieron a la Hacienda La Montaña, de la
Universidad de Antioquia (Colombia). Los modelos fueron validados y contrastados con MAD, el coeficiente
de determinación (R^{2}) y de correlación (r), y el estadístico Durbin-Watson. **Resultados:** los modelos Delay
y Lokhorst presentaron valores de R^{2} superiores a 0,8 y valores de r superiores a 0,9 (p<0,01). El modelo
Adams-Bell para la curva Lohmann Brown obtuvo el menor valor de r (0,81), mientras que los modelos Delay
y Lokhorst presentaron el valor más alto de R^{2} (1,0) para la curva de Isa Brown. El modelo Delay se ajustó
a la curva, con valores de 28 y 40 para el parámetro k, y de 63 y 64 para el parámetro DEL. La curva de la
línea Hy line Brown presentó gran cantidad de irregularidades (altibajos), generando mayor dificultad para ser
ajustada con los modelos evaluados. **Conclusión:** los modelos Delay y Lokhorst son eficientes para predecir
la curva de producción de huevos de aves de las estirpes probadas. La bondad de ajuste del modelo Delay
podría aumentarse mediante la inclusión de otras variables de entrada tales como las relaciones fisiológicas,
relaciones de oferta y demanda de recursos, y variables ambientales, posibilitando que el modelo Delay se
ajuste a las fluctuaciones de las curvas.

**Palabras clave:** análisis de regresión, modelación, modelo matemático.

**Resumo**

**Antecedentes:** os modelos matemáticos para caracterizar e estimar curvas de produção de ovos são de
grande importância para avaliar a eficiência produtiva de galinhas poedeiras. Estes possibilitam identificar os
fatores que afetam a produção animal e aplicar os corretivos para minimizar seus efeitos. **Objetivo:** comparar
a capacidade de ajustar a curva de produção de ovos utilizando o modelo de distribuição com atraso (Delay) e
os modelos Adams-Bell e Lokhorst. Métodos: foram utilizados 225 dados de registros de produção semanal
de quatro linhas de galinhas poedeiras: Hy Line Brown (62 dados), Lohmann LSL (54 dados), Isa Brown (54
dados) e Lohmann Brown (55 dados). Os lotes testados pertenceram à Fazenda La Montaña da Universidade
de Antioquia (Colômbia). Os modelos foram validados e comparados com MAD, coeficiente de determinação
(R^{2}) e de correlação (r), e estatística de Durbin-Watson. Resultados: os modelos Delay e Lokhorst tiveram
valores de R^{2} superiores a 0,8 e de r superiores a 0,9 (p<0,01). O modelo de Adams-Bell para a curva na
linha Lohmann Brown teve o menor valor de r (0,81), enquanto os modelos Delay e Lokhorst apresentaram
o maior valor de R^{2} (1,0) para a curva na linha Isa Brown. O modelo de atraso foi ajustado para a curva, com
valores de 28 e 40 para o parâmetro k, e 63 e 64 para o parâmetro DEL. A curva da linha Hy line Brown
apresentou muitas irregularidades (solavancos) gerando maior dificuldade para ser ajustada pelos modelos.
Conclusão: os modelos Delay e Lokhorst são eficientes na previsão de curvas produção de ovos de aves das
linhas testadas. A bondade de ajustar com o modelo de atraso pode ser melhorada com a inclusão de variáveis
de entrada adicionais, tais como relações fisiológicas, relações de oferta, demanda de recursos e as variáveis
ambientais. Permitindo que o modelo Delay ajuste as flutuações da curva.

**Palavras chave:** análise de regressão, modelação, modelo matemático.

**Introduction**

The onset of egg production is conditioned by
several factors, including sexual maturity, weight,
nutritional profile, and environmental conditions
such as luminosity (Abad, 2003). The egg-laying
curve begins at approximately 18 weeks of age,
followed by peak production eight to nine weeks
later, and subsequent persistence, defined as the
number of weeks when production is constant
post-peak. Next comes the declining phase, which
extends until the exit of the batch (Grossman *et al.*, 2000). Some of the factors influencing egg
production are body weight (Álvarez and Hocking,
2007), environmental conditions (i.e., temperature
and humidity) (Abiodun *et al.*, 2006; Hester, 2005) especially thermoneutral or comfort zone
for chickens (Rozenboim, 2007; Mashaly *et al.*,
2004), bacterial diseases (Peebles *et al.*, 2006), viral
diseases (Sun *et al.*, 2009), respiratory or intestinal
problems (Yegani *et al.*, 2008), and nutritional
balance (Safaa *et al.*, 2008; Jewers, 1990; Flores,
1994; Gerber, 2006).

The use of mathematical modeling in animal production has allowed farmers and researchers to describe and understand biological processes and prioritize the aims of production research from identifying the study's components to evaluating the response variable's effects.

This mathematical abstraction of biological events helps identify problems and generate solutions without incurring the costs of experimenting or animal manipulation, while decreasing the time to find solutions according to the production system. But the efficiency and accuracy of the simulation depends on the actual knowledge of the system's situations and quality of information incorporated into the model (Bindya, 2010; Spedding, 1988).

The purpose of modeling the production curve in
poultry eggs is to achieve a more detailed analysis
of the egg production cycle and describe the curve
phases and duration (Fialho, 2001). The curve also
facilitates the production prediction, the long-term
projection of eggs yield, and economic planning of
production and decision-making, among others (Yang
*et al.*, 1989; Groen *et al.*, 1998; Gavora *et al.*, 1982).

The egg production curve has been modeled using
weekly production information (Miyoshi *et al.*, 1996)
and logistic functions (Adams and Bell, 1980; Cason
and Britton, 1988), polynomial functions (Bell and
Adams, 1992), exponential functions (Foster *et al.*,
1987), segmented polynomials (Lokhorst, 1996;
Narushin and Takama, 2003), and nonlinear models
(Savegnago *et al.*, 2011).

The Adams-Bell and Lokhorst models have been compared to other models, such as the compartmental or McMillan model (McMillan, 1981), based on a logistic growth curve and modified compartmental models, by several researchers (Narushin and Takama, 2003; Lokhorst, 1996; Cason and Britton, 1988). These studies have concluded that both models have the best fit and suggest that the Adams- Bell and Lokhorst models can be used for describing and predicting the egg production curve (Lokhorst, 1996; Narushin and Takama, 2003; McMillan, 1981; Cason and Britton, 1988).

The Delay model has been successfully used
to simulate population dynamics in various living
organisms (Gutierrez *et al.*, 1984; Gutierrez *et al.*,
1988a; Gutierrez *et al.*, 1988b; Gutierrez *et al.*,
1991; Wermelinger *et al.*, 1992; D'Oultremont and
Gutierrez, 2002). The Delay model, developed by
Manetsch (1976) and modified by VanSickle (1977),
could be used as an alternative model to predict egg
production.

The Delay model can estimate the productive performance of a batch of birds by creating a population structure based on the number of individuals, age, or even changes in production, and by including therates of production decline (mortality) and increase (fecundity). Variations in production rates generate continuous entry and exit of individuals from each of the subsystem states. These variations are associated with the animal's physiological condition, changes in supply and demand of resources, and environmental factors. Therefore, the Delay model, structured by age, can be used to model egg production by considering eggs as the individuals in the system, and rate change in the number of eggs produced during a time period as the substates flow.

The aim of this study was to compare the Delay model with the Lokhorst (1996) and Adams-Bell (1980) mathematical models for their ability to model the egg production curve of Hy line Brown, Lohmann LSL, Isa Brown, and Lohmann Brownhen strains.

**Materials and methods**

*Data*

Weekly yield information, composed of 255 data (%), was used to fit the production curves. The weekly egg production rates (%) were calculated as the ratio between the number of eggs laid per week and the average number of hens per week. For comparing the capacity of model fit, four curves of egg production with different shapes such as age at start of egg production, increase rate of the curve, maximum number of eggs, length after the peak of production, and total weeks of production were selected. The strains selected were: Hy Line Brown (HB), Lohmann LSL (LSL), Isa Brown (IB), and Lohmann Brown (LB). All birds were raised in the University of Antioquia's (Colombia) Hacienda La Montaña Farm, located at 6°19'19''N and 1°37'40''W, at 2,350 m above sea level, with 15 °C average temperature (22 °C maximum and 7 °C minimum temperature).

During the production period hens were housed
in cages, ensuring 750 cm^{2} per bird. Hens were
fed according with the dietary recommendations
of each line. Water was supplied *ad libitum*, and
the environmental conditions (temperature and
humidity) were not controlled.

*Models*

The Delay and the Adams-Bell (1980) and Lokhorst (1996) models were used for modeling the laying behavior of hens. According to Narushin (2003), Adams-Bell and Lokhorst can be used to accurately describe daily egg production.

According to Gutierrez (1996), the Delay model can be stated as:

Where: r_{0} (t) is the number of eggs at the
beginning of the production phase of the flock at
time *t*. Now, r_{q} (t) is the number of eggs produced
at the end of the system at time *t*. The duration of
each substate is Δt. Variables r_{1} (t), r_{2} (t), ... , r_{q} (t)
are termed ''intermediate production rates'' of the
model, and refer to the increase in the number of
eggs, according to the specific loss rate μ_{i} (*t*), with
*i*= 1,2,3,.. ,*q*; where μ_{i} (*t*) takes values between 1<
μ_{i} <1 for time *t*. Exchange rates between states are
based on the number of laid eggs entering from
the previous intermediate flow, and the ones laid
towards the end that pass to the next state. The DEL
value is defined as the optimum length of the egg
production period estimated by the model. The k
parameter was obtained from the Erlang frequency
distribution (Van Sickle, 1977). The k and DEL
values allow the model to more accurately replicate
the properties of the process being modeled
(Manetsch, 1976).

The Adams-Bell model is shown below (Adams and Bell, 1980):

The Lokhorst model is shown below (Lokhorst, 1996):

Where: y_{i} is the production percentagefor the
*i ^{th}* week. Parameters

*a*and

*b*allow the model to adjust for initiation of production. The time period between the start of production and the peak of the curve is influenced by the

*r*parameter. The weekly post-peak production decline rate (%) is determined by the value of parameter

*c*. The slope of the final decrease is given by factor

*d*. Variable ti refers to the ith age of the flock (weeks), and ε

_{i}is the residual effect associated with the

*i*time.

^{th}For both models (Adams-Bell and Lokhorst), production was expressed in percentage terms, calculated as the ratio between the number of eggs laid per week and the average number of hens per week. The percentage of eggs was multiplied by a theoretical population of 1000 hens in order to compare the Delay with the mathematical models.

*Statistical analysis*

The accuracy of the models was determined by:

1. Spearman correlation coefficient (r), which measures the strength of the linear relationship between each model's actual and estimated values. The r coefficient has values between -1 and 1, and values approaching zero show no relationship of dependence between variables. The correlation between actual and predicted number of eggs was performed using the CORR procedure (SAS Institute Inc., Cary, NC, USA) (2004).

2. Determination coefficient (R^{2}), which describes how well a regression line fits a set of data. R^{2}, in percentage values (0-100), is interpreted as the percentage change in the dependent variable due to changes in the independent variable, valuing the causal relationship between the two (explained and explanatory). To calculate the R^{2}coefficient for each model a linear regression analysis was performed, with the number of eggs predicted as the dependent variable and the number of eggs observed as the independent variable.

3. Mean Absolute Deviation (MAD), which measures average absolute deviation of forecast from actual values:

Where y_{i} equals to the observed value at time i, ŷ_{i} equals the estimated value, and n equals the number
of observations.

In addition, the Durbin-Watson statistic (DW) was calculated to assess for auto-correlation, using the following equation:

Where *n* is the number of observations, and *e _{t}* is
the residual value for time

*t*; while

*e*is the residual value for time t-1 (Durbin and Watson, 1951). The SAS 9.1 (SAS Institute Inc., 2004) and Microsoft Excel Solver

_{t-1}^{®}(Microsoft, 2010) was used for fit and compared the models.

**Results**

Egg production parameters of the commercial hen strains used to evaluate model fit appear in the table 1.

For the Delay model, parameter k value for HB was 28, while it was 40 for the other hen strains. Parameter DEL, which expresses the estimated mean duration of the productive period, was between 63 and 64 weeks. The decrease factor allows quantifying the extent to which productivity is steadily reduced after reaching the maximum weekly production, before the fast-decreasing phase starts (towards the end of the flock). The LSL had the highest decrease rate of productivity, with a daily proportion of 0.00737, which is roughly equivalent to 83 eggs/day/flock, followed by HB, whose rate was 0.004835, corresponding to 45 eggs/day/flock. The lowest rates occurred in LB and IB, corresponding to reductions of 39 and 31 eggs/day/flock, respectively (Table 2).

For each hen line, the initial phase of rapid increase in productivity was modeled considering some of the initial compartments of the model as a pre-laying period (Table 2). This period was shorter for LSL, followed by LB, and finally by the HB. The longest period was for IB. Thispre-laying value generated by the model coincides with the time periods between the week when production started and the week of peak production, presented in table 1 for each hen line.

The values of all the coefficients of
determination (R^{2}) were greater than 0.8, and
Spearman correlation (r) values exceed 0.9 in eleven
of twelve results (Table 3). The Delay and Lokhorst
models had correlation values above 0.9, allowing
us to conclude that both are efficient to model all
bird strains tested. Delay and Lokhorst models had
the highest values of R2 and minimum values of
MAD, except for Isa Brown strain, where the Delay
model was exceeded by Lokhorst and Adams-Bell
in both criteria.

According to the Durbin Watson (DW) test results in table 4, all models showed positive autocorrelation of residuals with DW values below 2. The ideal DW value is 2 (Grossman, 2000). For the HB flock, the highest DW value was the Delay model (0.64), and for the LSL, IB and LB strains, the highest values of DW were for the Lokhorst model with values of 1.85, 0.84 and 1.47, respectively.

**Discussion**

All four flocks exceeded 90% production, an optimal response in commercial egg-laying hens. The production difference was related to the precocity of the LB flock, which reached full production before the other flocks. Another difference is associated with production length, which is determined by management decisions. Production cycle of LSL, IB, and LB lasted 53 weeks on average, while HB averaged 63 weeks (Table 1). Based on these indicators, it can be seen that production curves for the four commercial strains were different, allowing us to evaluate the ability of the models to fit the data.

Goodness of fit criteria indicated that the models
provided a better fit to egg production data; R^{2} and r
were higher, and MAD was lower, but DW was not
always closest to 2.

The DW criteria results can be explained by the time-series data often exhibiting positive autocorrelation, noncompliant with the independence of errors assumption. Therefore a positive error (negative) tends to be followed by another error positive (negative), generating a cyclical pattern.

The calculation of the parameters of Delay model is performed by linear and nonlinear optimization algorithms, from the change in the values of the parameters DEL and k, with the objective function of minimize the sum of the differences between observed and estimated values (error) using Microsoft Solver tool developed for mathematical simulation, optimization, and modeling. For this reason, estimation of the Delay model does not need to check the variance of the disturbance term assumptions (autocorrelated data), which is a functional advantage compared to Lokhorst and Adams-Bell models.

Due to age differences at production onset,
maximum number of eggs, and persistence in
decay phase of egg production curves, the models
showed different performances in each line of birds
evaluated. To show this, first the Lokhorst, Adams-
Bell, and Delay models were sorted according
to R2 values within each hen strain. The models
were sorted as follows: HB (Delay, Lokhorst, and
Adams-Bell), IB and LSL (Lokhorst, Delay, and
Adams-Bell), and LB (Lokhorst, Adams-Bell, and
Delay). After averaging R^{2} and r-values of the four
lines evaluated, Lokhorst ranked first (R^{2}= 0.95,
r = 0.96), followed by Delay (R^{2}= 0.93, r =0.96)
and finally Adams-Bell (R^{2}= 0.88, r = 0.89). In
both analyses, the Lokhorst and Delay models were
ranked first or second; these results showed the
Delay model performance was similar to Lokhorst
in the production curve estimation process of the
four lines evaluated, and evidence the effect of each
curve on prediction ability of all models.

The prediction of models is shown in figure 1, where the Delay model yielded higher estimated values compared to the recorded data, especially during the post-peak production phase. This model underestimated the starting point of the curve for LB, whereby the MAD value for Delay model was higher than the other models. The model had its best performance in IB and HB curves, especially the best fit in the final production phase of HB line (weeks 49 to 63), compared with the other two models.

The Lokhorst model showed the best fit for LB and LSL curves, accurately describing the increase of the curve, the peak of production, and the downward trend near the end of the curve. However, the model underestimated peak production and failed to predict the irregularities in the HB production curve.

The Adams-Bell model had the best fit in LB strain. However, in the other three lines of hens this model failed to fit the start, the raising phase or peak of egg production curve, and thus achieved the highest MAD values. Regarding the fit flaws of Adams-Bell model, Lokhorst (1996) claims they are due to the assumption of 100% maximum production along with the linear decrease in egg production.

Despite the good overall performance of all models in estimating number of eggs per week, none of the models were able to reproduce the fluctuations between the maximum and the final production stages in HB strain due to the lack of accurate production fluctuation cause records.

In conclusion, based on the values of r, R^{2} and
MAD, both Lokhorst and Delay model have similar
prediction ability. This indicates there are several
models that fit egg production curves of the four
lines of birds evaluated.

On this basis, Delay and Lokhorst are the models recommended to adjust the egg production curve, although the first model has advantages, such as biological interpretation of the parameters, in that it does not require assumption validation for parameter estimates. Most important is the possibility of including other biological characteristic variables, such as environmental factors (temperature, relative humidity, etc.), or the relationship of supply/demand for resources in the system, allowing the model can estimate the fluctuations and variability in the egg production curve.

¤ To cite this article: Galeano-Vasco, L. Cerón-Muñoz M, Rodríguez D, Cotes JM. Using the distributed-delay model to predict egg production in laying hens. Rev Colomb Cienc Pecu 2013; 26:270-279.

**Acknowledgments
**

The authors wish to thank Departamento de Formación Académica de Haciendas de la Universidad de Antioquia for data collection. This research was funded by Universidad de Antioquia (CODI Sostenibilidad 2013 Código E01727, and Mediana cuantía Código E01533) and Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS) (Convocatoria Nacional para Estudios de Doctorados en Colombia año 2011).

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