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Design and Validation of a Dynamic Discrete Event Stochastic Simulation Model of Mastitis Control in Dairy Herds1 H. G. ALLORE,*,2 L. W. SCHRUBEN,† H. N. ERB,‡ and P. A. OLTENACU* †Department

ABSTRACT A dynamic stochastic simulation model for discrete events, SIMMAST, was developed to simulate the effect of mastitis on the composition of the bulk tank milk of dairy herds. Intramammary infections caused by Streptococcus agalactiae, Streptococcus spp. other than Strep. agalactiae, Staphylococcus aureus, and coagulase-negative staphylococci were modeled as were the milk, fat, and protein test day solutions for individual cows, which accounted for the fixed effects of days in milk, age at calving, season of calving, somatic cell count (SCC), and random effects of test day, cow yield differences from herdmates, and autocorrelated errors. Probabilities for the transitions among various states of udder health (uninfected or subclinically or clinically infected) were calculated to account for exposure, heifer infection, spontaneous recovery, lactation cure, infection or cure during the dry period, month of lactation, parity, within-herd yields, and the number of quarters with clinical intramammary infection in the previous and current lactations. The stochastic simulation model was constructed using estimates from the literature and also using data from 164 herds enrolled with Quality Milk Promotion Services that each had bulk tank SCC between 500,000 and 750,000/ml. Model parameters and outputs were validated against a separate data file of 69 herds from the Northeast Dairy Herd Improvement Association, each with a bulk tank SCC

Received February 10, 1997. Accepted October 2, 1997. 1This research is supported by National Research Initiative Competitive Grants Program and USDA Award 96-35204-3441; is a component of NC-119 Regional Project; and is, in part, supported by the USDA and by the Agricultural Experiment Station at Cornell University, Regional Hatch Project Number 476. Some data are from a dissertation submitted to Cornell University by H. G. Allore in partial fulfillment of requirements for the Ph.D. degree. 2Present address: Department of Clinical Sciences, Cornell University, Ithaca, NY 14853. 1998 J Dairy Sci 81:703–717

*Department of Animal Science, of Operations Research and Industrial Engineering, and ‡Department of Clinical Sciences, Cornell University, Ithaca, NY 14853

that was ≥500,000/ml. Sensitivity analysis was performed on all input parameters for control herds. Using the validated stochastic simulation model, the control herds had a stable time average bulk tank SCC between 500,000 and 750,000/ml. ( Key words: simulation, mastitis, milk yield, milk composition) Abbreviation key: BTLS = bulk tank LS, BTSCC = bulk tank SCC, CNS = coagulase-negative staphylococci, LS = linear score, NEDHI = Northeast DHI, PJ305 = projected 305-d milk yield, QMPS = Quality Milk Promotion Services, RHA = rolling herd average. INTRODUCTION Currently, the Pasteurized Milk Ordinance requires that Grade A raw milk have 1,000,000 somatic cells/ml (6.4 LS). The daily LS of an individual cow that stays in the same event from a given day to the next day is calculated as LSij = PLSi( j–1) + errorij

[10]

where LSij = LS of given pathogen i at j DIM, PLSi( j–1) = LS of a given pathogen i at ( j – 1 ) DIM, i = element of the pathogen set [Strep. agalactiae, Streptococcus spp. (excluding Strep. agalactiae) , Staph. aureus, CNS], and errorij = random dependent residual. As with all of the discussed correlated measures, there were two components to the dependent error term: the LS error of the previous day and a white noise process (independently and identically distributed from N(0, 0.005) truncated at 2 ± SD). The errors of the current and previous day were correlated ( r = 0.9858 on a daily basis; R. W. Everett, 1995, unpublished data) so that errorij = 0.9858 × errori( j–1) + WNj

y = LS × 0.016

[13]

for second and greater parity cows where y = reduction of fat yield (kilograms) per day. Protein percentage losses (12, 35) were linearly related to LS: y = LS × 0.0115

[14]

where y = reduction in protein percentage per day. Input parameters. The input values of the stochastic simulation model may be entered either directly or from a data file. This feature allows SIMMAST to be flexible to the needs of sequential experiments and to run designed experiments in batch mode. Input parameters (Table 4 ) and their initial values were chosen to describe herds without a trend in the time average BTLS between 5.3 and 5.9. Trends for the time average were measured using standardized time series. The time average accounts for how much simulated time a variable spends at a particular value. The probability of exposure on the first simulated day was set at 1.0 to avoid an initialization bias. The risk of infection of the heifer at parturition was set at 0.45 ( 3 0 ) to model the introduction of infectious replacements into the herd at parturition. The proportion of prevalent clinical IMI that was predicted to regress to subclinical IMI within 30 d was set at 0.95. The ratio of Strep. agalactiae to Streptococcus spp. to Staph. aureus to CNS was based on the ratio of these pathogens in the QMPS

[11]

where WN = white noise. Therefore, the standard deviation of the dependent errorij is 0.04 (19). Losses of milk and milk components related to LS. The stochastic simulation model used the SCC transformation adopted by the National Cooperative DHI Program and its linear relationship to milk loss (18). When LS was 3, milk loss was 0.3 and 0.6 kg/d for first lactation and for cows in later lactations, respectively. Each increment of 1 LS above 3 was estimated to be equivalent to a milk loss of 90 and 180 kg per lactation for first lactation and mature cows, respectively (0.3 and 0.6 kg/d, respectively) (17). Fat yield losses for first and for second and greater parity cows ( 1 8 ) were calculated as y = LS × 0.00418

for first parity cows and

[12]

TABLE 4. Input parameters related to mastitis for the control herd (i.e., baseline scenario for SIMMAST, the discrete event stochastic simulation model). Description Probability of exposure on first simulated day Risk of infection for a heifer at parturition Proportion of prevalent clinical IMI that regressed to subclinical IMI within 30 d Number of clinical IMI this lactation for use with the culling rule Ratio of IMI caused by Streptococcus agalactiae, Streptococcus spp. (excluding Strep. agalactiae) , Staphylococcus aureus, or coagulase-negative staphylococci Lactation therapy ( 1 = yes; 0 = no) Mastitis prevention1 ( 1 = yes; 0 = no) Decrease in risk of new infection from prevention of mastitis Dry cow therapy ( 1 = yes; 0 = no)

1.0 0.45 0.95 3

33:19:29:19 0 0 0 0

1Preventive strategies to control mastitis might include milking hygiene (predipping all teats, cleaning and drying with a single-use paper towel, and postdipping), the maintenance of milking equipment, types of housing and bedding, genetics, and nutrition.

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data file for herds with a BTLS that was ≥5.3. All strategies used to control mastitis and the decrease in the risk of new infection (not applicable to the control herd) were set to 0. Pathogen-specific cure rates. All pathogenspecific cure rates, whether attributable to spontaneous recovery or therapy, were read from data files and are easily modified. For this experiment, cure rates by spontaneous recovery from subclinical IMI were 15, 59, 34, and 76% for infections caused by Strep. agalactiae, Streptococcus spp., Staph. aureus, and CNS, respectively (45). Bacteriologic cure rates after lactation therapy (intramammary antibiotic therapy) that were pathogenspecific were taken from the 10-yr retrospective study of Wilson ( 4 5 ) using the QMPS data file for data collected between 1985 and 1995 of which our data file is a subset. The pathogen-specific cure rates for clinical IMI were 75, 82, 56, and 62% for IMI caused by Strep. agalactiae, Streptococcus spp., Staph. aureus, and CNS, respectively; these percentages represent the overall treatment cure rates. Changes in infection status between dry-off and parturition for uninfected cows or for cows with IMI caused by Strep. agalactiae, Streptococcus spp., and Staph. aureus that were or were not treated with dry cow antibiotic therapy (penicillin dihydrostreptomycin) were from Eberhart and Buckalew (10), and changes in infection status for infections caused by CNS were from Harmon et al. (13). Initial prevalences of pathogen-specific IMI. The initial prevalences of IMI caused by Strep. agalactiae, Streptococcus spp., Staph. aureus, and CNS were input variables. The proportion of IMI caused by each pathogen in the control herd was 33% caused by Strep. agalactiae, 19% caused by Streptococcus spp., 29% caused by Staph. aureus, and 19% caused by CNS. These four pathogen groups (in these proportions) caused 82% of the recorded IMI in herds in the QMPS data file for herds with a BTSCC that was ≥500,000 cells/ml. Calculating Transition Probabilities Between Udder Health Events Transitions between udder health events have been described in detail elsewhere and have been compared with six published models ( 2 ) . The transition probabilities between udder health states proposed by Houben et al. ( 1 4 ) were modified according to the results of a survey among mastitis experts ( 2 ) . Udder health states were identified as uninfected, subclinical IMI, and clinical IMI. The addition of Journal of Dairy Science Vol. 81, No. 3, 1998

subclinical mastitis, the probability of milking time exposure, the risk of infection of the heifer at parturition, and the probability of recovery from IMI (both spontaneous and therapeutic) to the transition probabilities of Houben et al. ( 1 4 ) provided the closest fit to the observed QMPS data file. Transitions between udder health events depended on the month of lactation, parity, within-herd yields, the number of quarters with clinical IMI during the previous lactation, the number of quarters with clinical IMI in the current lactation, and whether any quarters were currently clinically infected. Additionally, the probability of exposure at milking was included when the probability was calculated for the transition from uninfected to either subclinical or clinical IMI and was calculated as the number of lactating cows that were currently infected divided by the number of lactating cows. As the probability of exposure approached 1.0, the probability of a cow having a subclinical or clinical IMI within 1 mo approached the estimated transition probabilities of Houben et al. (14), which represents an upper bound. Conversely, as the probability of exposure approached 0.0, the probability of an IMI approached 0.0. Random mixing of cows was assumed, and clinical cows were assumed to be milked with the same unit as all other cows, although their milk would be diverted from the bulk tank if cows were receiving lactation therapy. The probability of exposure applied to uninfected cows because our experimental unit was one cow, and we assumed no IMI were caused by more than one pathogen. Equations for transition probabilities are summarized in Table 5. Udder Health Events If the cow remains in the same udder health event, then the current LS is calculated as the sum of the LS of the previous day and the autocorrelated error (Equation [10]). Next, the current milk (Equation [2]), fat (Equation [5]), and protein (Equation [8]) yields are calculated. The original model ( 1 4 ) that formed the basis for our transition probabilities in the lactating period is used to calculate monthly transitions. Because our model runs on daily time intervals, the day of a future udder health event is drawn from a uniform distribution between 1 and 29 d. If the cow has one or more additional days in the current udder health event or if the cow has the same udder health event scheduled as the next event, the cow goes to the same udder health event on the next day. Otherwise, the cow goes to the next udder health event on the next day. Conversely,

DESIGN AND VALIDATION OF A SIMULATION MODEL

if it is the end of the lactation or if the cow meets the culling criteria, then the cow is scheduled to go to the dry event on the next day. Uninfected event. In addition to the description of the generic udder health event presented previously, when a cow newly arrives at the uninfected event, the cow is assigned a LS from a beta distribution ( 5 ) . Further modifications include the probability of exposure adjusted for any decrease in the risk of new infection from a preventive strategy to control mastitis. Subclinical event. In addition to the description of the generic udder health event presented previously, when a cow newly arrives at the subclinical event, the cow is assigned a pathogen based on the current proportion of pathogens. At the start of the simulation, the proportion of pathogens is from the user input. If the cow arrives at the subclinical event from a clinical event, the same pathogen is kept. The LS is determined from pathogen-specific distributions representing subclinical symptoms ( 5 ) . Transitions from subclinical IMI are not directly affected either by prevention, because the cow is already infected, or

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by lactation therapy, which is only applied to clinical IMI. The input file of spontaneous recovery rates from subclinical IMI, which is supplied by the user, is used to estimate the transitions from subclinical IMI to an uninfected udder health state. Clinical event. In addition to the description of the generic udder health event presented previously, when a cow newly arrives at the clinical event from the uninfected event, the cow is assigned a pathogen based on the current proportion of pathogens. At the start of the simulation, the proportion of pathogens is from the user input. If the cow arrives at the clinical event from the subclinical event, the same pathogen is kept. The LS is determined from pathogen-specific distributions representing clinical symptoms (32). The base transition probability between clinical IMI and an uninfected udder health state accounts for spontaneous recovery; therefore, bacteriological cure rates are in addition to the base rate. If lactation therapy is selected at the start of the simulation, then the input file with the bacteriological cure rates and the simulated time to start the strategy must also be

TABLE 5. Equations for the probabilities of udder health states for cows in uninfected, subclinical, or clinical udder health states in the discrete event stochastic simulation model. Current event and probability

Equation

Uninfected P(Clinical) 1 P(Subclinical) P(Uninfected) Clinical P(Clinical) P(Subclinical) P(Uninfected) Subclinical P(Clinical) P(Subclinical) P(Uninfected)

PEXPOS2 ×

⎛ eb0

⎜⎝1

+ b1 + b2 + b3 + b4 + b5 + b6

⎞

⎟⎠

× ( 1 – DRNI)

b0 + b1 + b2 + b3 + b4 + b5 + b6

+ e

PEXPOS × [1 – P(Clinical)] × ( 1 – DRNI) 1 – P(Clinical) – P(subclinical) ⎛ eb0

⎜⎝1

+ b1 + b2 + b3 + b4 + b5 + b6

⎞

⎟⎠

b0 + b1 + b2 + b3 + b4 + b5 + b6

+ e

× ( 1 – LT1)

[1 – P(Clinical)] × PROPSUB1 1 – P(Clinical) – P(subclinical)

⎛ eb0

⎜⎝1

+ b1 + b2 + b3 + b4 + b5 + b6

⎞

⎟

b0 + b1 + b2 + b3 + b4 + b5 + b6

⎠ + e ( 1 – P(Clinical) × P(SPON 1) 1 – P(Clinical) – P(subclinical)

1P(Clinical) = Probability that the next udder health event of a cow will be clinical, P(subclinical) = probability that the next udder health event of a cow will be subclinical, and P(uninfected) = probability that the next udder health event of a cow will be uninfected. 2PEXPOS = Probability of exposure to mastitis pathogens on a given day. The following variables were taken from Houben et al. (14): intercept ( b0) , effect of month of lactation ( b1) , effect of parity ( b2) , effect of yield level ( b3) , effect of number of quarters clinically infected in previous lactation ( b4) , effect of number of quarters infected up to and including previous month of current lactation ( b5) , and effect of current clinically infected quarters ( b6) . DNRI = Decrease in risk of new infection from prevention of mastitis, and LT1, SPON1, and PROPSUB1 = bacteriological cure rate of lactation therapy, spontaneous recovery rates, and proportion of prevalent clinical IMI that regresses to subclinical IMI in 30 d for pathogen 1, respectively, where pathogen 1 is an element of the following set: Streptococcus agalactiae; Streptococcus spp., Staphylococcus aureus, and coagulase-negative staphylococci.

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ALLORE ET AL.

provided. When lactation therapy is invoked, the input value of the proportion of prevalent clinical IMI that regress to subclinical IMI in 30 d should be adjusted if there is reason to think that the lactation therapy product may also reduce symptoms in cases it does not cure. This information is provided by the user. If lactation therapy is used, there is a 6-d withholding time (the current day and the next 5 d). Outcome Variables Daily sampling allowed uninfected cows to be exposed to infected cows; therefore, changes among udder health events are more realistic than the monthly transitions in the models of Sorarrain et al. ( 4 0 ) and Houben et al. ( 1 4 ) or the yearly transitions in the model of Carpenter ( 8 ) . Bulk tank LS was recorded daily to validate that the control herd had a stable BTLS and to perform sensitivity analysis.

proportion of prevalent clinical IMI that regresses to subclinical IMI in 30 d; and culling rule that specifies the maximum number of clinical IMI in this lactation and five levels of the ratio of Strep. agalactiae to Streptococcus spp. to Staph. aureus to CNS for the control herd. All strategies used to control mastitis and the decrease in the risk of new IMI (not applicable to the control herd) were set to 0. Ten replicates using commonly seeded random numbers were run for each level of each input variable; all other variables remained at their original values (Table 4). The random number seeds were between 0 and 65,534 from Snedecor and Cochran (39). The number of replicate runs was based on the variance of the BTLS (0.0044) estimated in a pilot project ( 2 ) . Differences to be detected between treatment means were of a LS difference of 0.1 using Type I and Type II error rates of 0.05 (two-tailed) and 0.20, respectively. Using Table 3 of Kastenbaum et al. (20), this process resulted in a sample size of about 10 per group.

Validation of SIMMAST The model output from 10 replicate runs for the control herd was validated against the NEDHI data file for the milk yield of individual cows, bulk tank fat and protein percentages, age at first calving, estimated prevalence of IMI, and culling rates. Because the test day milk yields assume a base age of calving, although adjustments are made for ages older than the base age, we compared the base calving ages for first parity cows. The NEDHI data file uses the LS of individual cows to estimate new and chronic IMI. The number of estimated new IMI was calculated as the number of cows that had a LS that was 750,000/ml or might have been in violation of the Pasteurized Milk Ordinance), and we restricted our control herd to be stable between 500,000 and 750,000 somatic cells/ml. The base age at first calving in the model was 28 mo, which was within one standard deviation of the age at first calving of the herds in the NEDHI data file (29.1 ± 3.8 mo). The NEDHI data file recorded 14.9 ± 7.3 estimated new IMI per month and 33.0 ± 11.9 chronic IMI per month. These estimates are lower than the estimates for the prevalence of IMI for the control herd, but calculation methods for these statistics differed. Cows included in the NEDHI data file might not have been sampled if they had had a clinical IMI or were removed prior to milking for health reasons on a sample day. Also, some subclinical IMI could have a LS of