Estimating herd prevalence of bovine brucellosis in 46 U.S.A. states using slaughter surveillance

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Preventive Veterinary Medicine 85 (2008) 295–316 www.elsevier.com/locate/prevetmed

Estimating herd prevalence of bovine brucellosis in 46 U.S.A. states using slaughter surveillance Eric D. Ebel 1, Michael S. Williams *, Sarah M. Tomlinson National Surveillance Unit, Veterinary Services, Animal and Plant Health Inspection Service, USDA, 2150 Centre Avenue, Building B, Fort Collins, CO 80526, United States Received 29 August 2007; received in revised form 11 February 2008; accepted 12 February 2008

Abstract Making valid inferences about herd prevalence from data collected at slaughter is difficult because the observed sample is dependent on the number of animals sampled from each herd, which varies with herd size and culling practices, and the probability of a positive test result, which depends on variable within-herd prevalence levels as well as test sensitivity and specificity. In this study, brucellosis herd prevalence among beef cow–calf operations is estimated from slaughter surveillance data using a method that combines process modeling with Bayesian inference. Inferences are made for two populations; the first population comprises cow–calf beef herds in a typical U.S. state. The second population represents all beef herds in a collection of 46 low-risk states. The Bayesian Monte Carlo method used in this study links process model inputs to observed surveillance results via Bayes Theorem. The surveillance evidence across multiple years is accumulated at a discounted rate based on the probability of introducing new infection into an area. The process model’s inputs include herd size, culling rate per herd, within-herd prevalence, serologic test performance, and the probability of successfully investigating positive results. The surveillance results comprise the number of cows and bulls tested at slaughter and the number of affected herds detected each year. The results find at least 95% confidence that brucellosis herd prevalence among beef cow–calf herds is less than 0.014% (3 per 21,500 herds) and 0.00081% (5 per 6,15,770) after 5 years of slaughter surveillance (with no detections of affected herds) in a typical U.S. state and across 46 low-risk U.S. states, respectively. These results were based on conservative modeling assumptions, but sensitivity analysis suggests only slight changes in the results from changing the assumed process model input values. The most influential analytic

* Corresponding author. Present address: Risk Analysis Division, Office of Public Health Sciences, Food Safety and Inspection Services, USDA, 2150 Centre Avenue, Building D, Fort Collins, CO 80526, United States. Tel.: +1 970 492 7189. E-mail address: [email protected] (M.S. Williams). 1 Present address: Risk Analysis Division, Office of Public Health Sciences, Food Safety and Inspection Services, USDA, 2150 Centre Avenue, Building D, Fort Collins, CO 80526, United States. 0167-5877/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.prevetmed.2008.02.005

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input was the probability of introducing new infection into a putatively brucellosis-free state or group of states. Published by Elsevier B.V. Keywords: Bayesian Monte Carlo; Herd-level prevalence; Disease surveillance

1. Introduction The effort to eradicate bovine brucellosis from the United States began in 1934 (Ragan, 2002). The current U.S. eradication program establishes classification levels for U.S. states based largely on the reported herd prevalence within the state (USDA-APHIS, 2003). Class-free states are those with a reported herd prevalence of 0% for at least 12 consecutive months. Class A states are those with a reported herd prevalence not exceeding 0.1%; while class B and C states are allowed progressively higher herd prevalence levels. As of 2007, 49 States were listed as classfree in accordance with the brucellosis regulations. Thirty-four of those States were class-free for 10 or more years. On-going surveillance of beef cow–calf herds in class-free states primarily consists of serologic testing of cows and bulls (2 years of age or older) at commercial slaughtering establishments. The program requires samples to be collected from at least 95% of all such cattle at the time of slaughter. Because the U.S. state classifications are based on herd prevalence, it is useful to assess the slaughter surveillance system’s effectiveness in detecting affected herds. Herd prevalence is the proportion of affected herds among the population of all herds. Affected herds are defined as herds in which any animal has been determined to be infected with Brucella abortus; a herd is considered a commingled group of animals under common ownership on a single premises. Slaughter surveillance mimics a two-stage sample, where the first stage consists of herds in which some cattle are culled in a given year. In this context, culling refers to marketing of cows and/or bulls to slaughter for any reason (e.g., age, reproductive performance, economics, etc.) (USDA-APHIS, 1997). The second stage consists of drawing a sub-sample of animals from each herd; this sub-sample represents the animals that were culled from the herd. Because virtually all herds periodically cull cows and bulls, the difference between the target and sampled populations is small. Nevertheless, making valid herd-level inferences with data collected at slaughter is difficult because the sampling results depend on the number of animals sampled from each herd, which varies with herd size and culling practices, and the probability of a positive test result, which depends on variable within-herd prevalence levels as well as test sensitivity and specificity. Furthermore, selection of culled animals may be poorly approximated by simple random sampling. Slaughter surveillance only detects an affected herd if an infected animal is culled from that herd, if that animal is test-positive at slaughter, and if it is successfully traced back to its affected herd of origin and testing demonstrates B. abortus infection among one or more cattle within that herd. Previous approaches to assessing the effectiveness of surveillance systems have focused on the statistical confidence about an assumed minimum expected prevalence (i.e., design prevalence) implied by the data collected (Amosson and Dietrich, 1984; Cannon, 2002; Martin et al., 2007a). These approaches can generate a quantitative probability of ‘‘freedom from disease’’ or infection only if the design prevalence is

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appropriately established as a minimum threshold below which a population can effectively be considered free. Nevertheless, these approaches are not designed to generate a probability distribution that reflects the relative confidence about different levels of occurrence of infection other than the design prevalence. If the design prevalence is economically or epidemiologically meaningful, then the confidence inferred from surveillance evidence about prevalence levels above and below the threshold should also be meaningful. Furthermore, because these approaches only measure confidence that prevalence is below the design prevalence, and are conditional on negative surveillance findings, they are not suitable for use if infection is detected via surveillance. Bayesian methods can be used to estimate the uncertainty about herd prevalence implied by slaughter surveillance. Advantages of a Bayesian approach include the ability to incorporate prior information about various model parameters and the ability to simultaneously estimate multiple parameters. For example, others have used Bayesian models to simultaneously estimate prevalence and the performance of diagnostic tests (sensitivity and specificity and correlation) in the absence of a ‘‘gold standard’’ test (Hanson et al., 2003; Mintiens et al., 2005). Use of Bayesian techniques has also been facilitated by the introduction of the WINBUGS software package, which has simplified the fitting of Bayesian models. While these Bayesian methods have advanced the field of animal surveillance, they do not address the underlying problem of describing how animals were selected from each herd. For example, the WINBUGS software does not easily accommodate the hypergeometric distribution, which is often essential for modeling the sampling of the proportion of herds that cull animals, as well as sampling of animals within small herds (i.e., sampling without replacement). One method for describing the dynamics of slaughter surveillance is to use a simulation or process model. Such a model can incorporate information regarding population demographics, culling practices, disease dynamics and testing methods. For a given set of input parameters, a process model simulates the introduction and spread of the disease in the population, draws samples from the population, and assesses the probability that the disease is detected. For example, Chriel et al. (2005) use a process model to assess the sampling and testing of Danish beef and dairy herds for three different surveillance systems. Their results were used to compare the probability of detecting the introduction of infectious bovine rhinotracheitis using the three systems. Process models are effective tools for this application because the goal is to compare the relative performance between competing surveillance systems. Process models do not directly support inferences about populations, but these models can be connected to the outputs of surveillance systems (i.e., the number of tests and the number of infected animals or herds found) to estimate characteristics of the underlying population that generated the process model outputs. The connection occurs via Bayes Theorem. Such an analysis answers the question: ‘‘What were the input parameters of the process model that generated the observed sample?’’ In this paper, brucellosis herd prevalence among U.S. beef cow–calf operations is inferred from multiple years of slaughter surveillance data using a method that combines process modeling with Bayesian inference. Inferences are made for two populations; the first population comprises approximately 21,500 beef herds in a single U.S. state that has not found an affected herd after more than 5 years of surveillance and is designated ‘‘class-free’’ for brucellosis. The second population represents all beef herds in a collection of 46 states considered free of brucellosis in 2007 (i.e., lowrisk states). This collection excludes Texas because it was not class-free at the time of the analysis. It

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also excludes Idaho, Montana and Wyoming because these states border the Greater Yellowstone Area and, therefore, are dissimilar to the majority of states considered class-free for brucellosis. The Greater Yellowstone Area is known to harbor brucellosis-infected wildlife that have previously transmitted infection to cattle (Godfroid, 2002). 2. Methods The method used here has been previously described as Bayesian Monte Carlo (Patwardhan and Small, 1992) and Bayesian synthesis (Givens et al., 1993; Raftery et al., 1995; Green et al., 2000). Calculations were completed using the R software package (www.r-project.org). We begin with an outline of the methods for estimating herd prevalence from observed data and a process model. This is followed by descriptions of how inferences are drawn from a single year of brucellosis testing data and how these data are combined across multiple years of surveillance. Using these methods, we assume slaughter surveillance data consist of the number of positive tests, the number of positive herds found through successful tracing of positive samples back to their herds of origin (i.e., traceback), and the total number of tested samples collected at slaughter. 2.1. Review of Bayesian Monte Carlo Consider a process model denoted by M and a vector of model inputs, denoted by u. The model mathematically transforms the set of possible inputs to the set of model outputs, denoted by M(u). Many of the input parameters are not fixed values, so the uncertainty is represented by the joint distribution f u ðuÞ. For a randomly sampled ui from f u ðuÞ, the observed output from the process model is Mðui Þ. Monte Carlo methods randomly draw from f u ðuÞ and generate a probability mass function that approximates the original distribution when a sufficiently large number of samples are simulated. In Monte Carlo sampling, these samples are drawn with equal probability and each sample contributes equally to the numerical approximation of the distribution. Thus, if K samples are drawn from f u ðuÞ, the true output distribution is approximated by a discrete probability mass function with mass 1/K corresponding to each sample outcome. In the Bayesian Monte Carlo approach, all available empiric or expert evidence on the process model’s inputs and outputs is combined with the process model to produce posterior probability distributions for those inputs and outputs. In this application, the information on the process model outputs is the observed surveillance results (i.e., the total number of samples collected and tested; and the number of truly affected herds found after an epidemiologic investigation of positive samples). Information concerning model inputs includes cattle population demographics, culling practices, disease dynamics and testing methods; this information comes from a variety of sources, including published research, government statistics and expert opinion. In this process, pðMðui ÞÞ is the prior probability mass function for the model output of replication i and E denotes the empiric evidence regarding the model output MðuÞ. A likelihood function is defined, pðEjMðui ÞÞ, which describes the error structure of the empiric evidence. If the form of the model and its parameterization are highly accurate, the empiric evidence should agree well with the distribution of model outputs and the likelihood pðEjMðui ÞÞ, value will be high. A poor agreement between the empiric evidence and the model output will result in small values of pðEjMðui ÞÞ.

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The Bayesian posterior probability that evaluates the empiric evidence with respect to the model outputs is given by pðMðuÞjEÞ ¼

pðEjMðuÞÞ pðMðuÞÞ : pðEÞ

Because the probability mass for each Monte Carlo observation is pðMðui ÞÞ ¼ 1=K, the Monte Carlo posterior probability reduces to pðEjMðui ÞÞ pðMðui ÞjEÞ ¼ PK : i¼1 pðEjMðu i ÞÞ This posterior probability applies to both the model’s output (M(u)) and the vector of model inputs (u) (Patwardhan and Small, 1992). The element of greatest interest in this study is uprev; the number of affected herds in the population. 2.2. Single year inference The primary functions of the process model are to simulate the occurrence of brucellosis among herds, its spread within affected herds, the culling of adult cattle from each herd, the testing of blood samples collected from the culled animals at slaughter and the investigation of test-positive animals back to their herds of origin. The model requires the input, u, which contains parameters that describe various processes such as herd prevalence, within-herd prevalence, culling rates, test sensitivity and specificity, and the probability of successfully investigating positive samples. For each vector of input values, the process model simulates the number of animals sampled at slaughter, denoted nanimals, and the number of test-positive animals detected at slaughter that originated from affected herds; true-positive animals are denoted ntrue.pos and false-positives are denoted sfalse.pos. Because the U.S. brucellosis eradication program intends to sample nearly all adult cattle presented for slaughter, nanimals is essentially equivalent to the observed number of animals slaughtered in 1 year. The process model propagates test-positive results to further determine the number of affected herds found as a result of tracing test-positive animals to their herds of origin and confirming infection. If an affected herd is successfully identified via tracing, confirmation of affected herd status is accomplished via a whole-herd investigation (e.g., blood testing followed by slaughter and culture of test-positive cattle) that is assumed to be 100% sensitive and specific. The process model’s predictions for affected herds found (xherds.found) account for the probability that true-positive animals are successfully traced to their herds of origin and allow for the rare occurrence that false-positives would be traced to an affected herd. It is assumed that only one test-positive animal, successfully traced to an affected herd, is required to detect an affected herd in 1 year. The surveillance program provides the observed number of affected herds found (Epositive.herd) during 1 year of surveillance. This information is combined with the process model output using pðEjMðui ÞÞ ¼ BinomialðEpositive:herds ; nanimals ; xherds:found =nanimals Þ: This function calculates the binomial probability that Epositive.herd occurs among nanimals sampled given the process model’s predicted prevalence of affected herds detected among all animals sampled. A particular likelihood value ultimately pertains to the specific number of affected herds in the population that generated the particular value of xherds.found; numbers of affected herds with larger likelihood values are more consistent with the empiric evidence.

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Suppose uprev is the number of affected herds in the population and u ¼ ðuprev ; u2 ; . . . ; ut ÞT is the vector of input parameters for the model. For example, assume the prior information about the number of brucellosis-affected herds is described by the distribution uprev  Discrete(0, B). This distribution function explains that the number of affected herds can only be integer values less than or equal to B. Information available prior to collection of the surveillance data is used to inform the exact parameterization of this distribution. For each sample ui ¼ ðuprev;i ; u2;i ; . . . ; ut;i ÞT of input parameters, the process model generates the output Mðui Þ. If K samples are drawn from f u ðuÞ; the expected value and variance of the estimator for the number of affected herds are ˆ uˆ prev ¼ m

K X

pðEjMðui ÞÞ uprev;i PK i¼1 j¼1 pðEjMðu j ÞÞ

and sˆ 2 ¼

K X

pðEjMðui ÞÞ ˆ 2uˆ : u2prev;i PK m prev pðEjMðu ÞÞ j i¼1 j¼1

Other statistics, such as the upper 95th percentile of the number of affected herds, are estimated by summing the probability mass for each outcome until the percentile of interest is exceeded. The Bayesian calculations require that annual observed sampling data are provided. Cattle demographics data for one average-sized, class-free, and predominantly beef-producing state (State Z) were summarized. For State Z, with 21,500 beef herds, nanimals averaged 57,250 per year and Epositive.herd was known for the past several years. To extrapolate the analysis to a collection of 46 low-risk states (including State Z), the model was parameterized based on the estimated total number of beef herds in these states. Serologic testing was assumed to follow the same protocol used in State Z. The low-risk states comprise 615,770 beef herds, nanimals  2.5 million and Epositive.herd = 0 for the past several years based on available testing data. 2.3. Combining surveillance information across time The Bayesian Monte Carlo method estimates herd prevalence using data from a single time period. Slaughter surveillance data are collected continuously but collation and analysis usually occurs annually. While it is possible to make inferences using only data from the most recent year, such an estimation strategy would ignore the substantial amount of data collected in previous years. Thus, it is often argued that an estimator of the current prevalence should ‘‘borrow strength’’ from data collected in the previous years. The origins of this concept date back to the rolling sample designs in the late 1950s (Alexander, 2002). The term time value of information has been used in veterinary epidemiology to describe the same concept (Schlosser and Ebel, 2001). The time value of information explicitly links infection dynamics to depreciation of historic surveillance evidence such that the value of historic data rapidly depreciates for rapidly transmitted infections, while data pertaining to slowly spreading infections maintain their relevance longer and, therefore, depreciate at a slower rate. This study uses a Bayesian approach for combining data across time, where the posterior from each year is adjusted to become the prior for the next year. For populations in which the disease

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has not been detected for a number of years, the greatest hazard is the introduction of new infection. Therefore, the posterior prevalence distribution is adjusted by the probability that brucellosis will be introduced. Suppose the probability of introducing infection is t and p(uprev = rjE) is the posterior probability that uprev = r, based on the surveillance information collected during the year. Given a value for t, the probability that there were 0 affected herds is reduced using pðuadjusted ¼ 0jEÞ ¼ pðuprev ¼ 0jEÞð1  tÞ and the probability that there were r affected herds is modified using pðuadjusted ¼ rjEÞ ¼ pðuprev ¼ rjEÞð1  tÞ þ pðuprev ¼ r  1jEÞt, for r = 1 to B. This approach effectively shifts the distribution for uprev to the right (Fig. 1). These equations only approximate P the adjusted probabilities because the adjusted distribution needs to be normalized so that Br¼0 pðuadjusted ¼ rjEÞ ¼ 1. The probability of introducing infection depends on population size, cattle imports, wildlife reservoirs and prevalence of infection outside the population (Dalrymple, 1993). For low-risk states, it was conservatively assumed that transmission from one of the possible sources of infection (e.g., international, domestic or wildlife) was likely to occur approximately once every 3 years (t = 0.33). This parameter agrees with other risk-of-importation estimates; for example, Jones et al. (2004) estimated that importation of infected cattle into Great Britain occurred once every 2.6–3.2 years. Nevertheless, the U.S. brucellosis program has not demonstrated a single occurrence of importation of brucellosis into a class-free state via a cattle source. Because this analysis only pertains to states that do not adjoin the Greater Yellowstone Area (where infected bison and elk may transmit infection to cattle herds), the assumption that introduction could occur once every 3 years is likely to be very conservative.

Fig. 1. An illustration of the brucellosis herd prevalence distributions following simulation of 1 year of slaughter surveillance. The flat prior assumes that it is equally likely for 0–25 affected herds to exist in the population before collection of surveillance evidence. The posterior distribution reflects the inclusion of 1 year of surveillance evidence; this distribution suggests a much higher probability of small numbers of affected herds relative to the prior distribution. The adjustment for introduction of new infection reduces the value of the surveillance evidence by shifting the posterior distribution to the right and increasing its variance.

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The probability of introducing brucellosis into a single state is assumed to be lower than this same probability for a collection of states. Therefore, the probability of introduction for a single state was conservatively set at once every 10 years (t = 0.10). Two different approaches support this value. First, those states that were class-free in 2007 were in that status for an average of 12– 16 years. There has not been a reported case of a class-free state becoming re-infected via importation of an infected animal during the period considered. These results were inserted into a Beta(s + 1, n  s + 1) distribution where s equaled zero instances of importation and n equaled 12 or 16 years of observation (Vose, 2000). It was determined from the Beta distribution that the average chance of importing a new case was between 5% and 7%. Second, the apparent animallevel prevalence across the United States is much less than one per million (e.g., fewer than six infected cows or bulls are detected among the approximately 5–6 million dairy and beef cows and bulls sampled each year). If a typical state (e.g., State Z) imports 10,000 cows and bulls per year, then the chance of importing one or more infected cows per year is about 1%. Our choice of t = 0.10 exceeds both of these empirically based estimates. 3. Description of process model inputs Each of the process model’s iterations generates a value for xherds.found. This value is a random sum across the number of affected herds uperv.i selected for that iteration. Pz¼u For a particular iteration, xherds:found ¼ z¼0prev:i Binomialðstrue:pos:z þ sfalse:pos:z ; vÞ, where Binomial(n, p) represents a random draw from a binomially distributed random variable with parameters n and p. In this case, strue.pos.z and sfalse.pos.z are the random numbers of true-positive and false-positive cattle detected from affected herd z (i.e., strue.pos.z  Binomial(jz, SeSeriesi) and sfalse.pos.z  Binomial(hz, SpSeriesi) where SeSeriesi and SpSeriesi are the overall sensitivity and specificity of a series of tests for iteration (i) and v is the probability of successfully investigating positive cattle to locate their affected herds of origin. The number of adult cattle sold for slaughter from an affected herd, gz, is a random value (i.e., gz  Binomial(Nz, cz), where Nz is the herd size and cz is the culling probability for affected herd z) from which jz  Binomial(gz, pz) (where pz is the within-herd prevalence of brucellosis in affected herd z) and hz = gz  jz are derived. 3.1. Herd size, N Published data for State Z provides numbers of herds and cows for seven herd-size strata (USDA-NASS, 2002). None of the commonly available parametric distributions adequately fit these data because those models are unable to accommodate the large proportion of small herds while still accounting for the small number of very large herds in the upper tail of the distribution. Therefore, the population was distributed into smaller strata using a piece-wise linear distribution function. The stratum level information was fit to a distribution function to closely match (within 1%) the reported total herds and animals for that state (Fig. 2). Once the distribution of herds was derived for a single state, each herd was replicated to expand these estimates to a national population of 615,770 herds in low-risk states. 3.2. Culling rate per herd, c Culling rate was modeled as the product of the annual fraction of herds that cull and the fraction of animals culled given that the herd culls (Fig. 3). Data from a national study found that only 43.8%

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Fig. 2. Distribution of herd sizes for a single U.S. state. The largest herd contains in excess of 2600 animals while the vast majority of herds have fewer than 100. The vertical lines at the base of the graph represent individual herds.

Fig. 3. Box and whiskers plot for the simulated number of cows and bulls culled per year for different herd sizes. Variability in numbers culled is influenced by the probability that a particular herd culls any animals in a given year and, if any culling occurs, the fraction of the herd culled. Although discrete size classes are illustrated here, herd size is in integer units between 1 and 2600 in the process model.

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of beef herds with 50 or fewer animals sell any adult cattle in a given year (USDA-APHIS, 1998). In contrast, 94.1% of operations with 300 or more cows sell one or more adult cattle in a given year. These data were fit to a logistic regression model to predict the probability of culling as a function of herd size (i.e., pðculledjx ¼ herdsizeÞ ¼ expða þ bxÞ=ð1 þ expða þ bxÞÞ, where a = 0.0243 and b = 0.0065 for beef herds). It was previously reported that 14% of a herd is sold when culling takes place (Amosson and Dietrich, 1984), but no information was provided on the variability of this estimate. To model this variability, a Beta(a = 5.2, b = 31.7) distribution was employed to obtain a 90% confidence interval between 6% and 24% of the herd’s cows and bulls were sold, given that cows were culled from the herd. To assess the appropriateness of this culling model, its output was compared to published statistics. The average cull rate for the model was 9.5% across all herds, which is slightly higher than published cull rates of approximately 9% for beef herds (Olson et al., 2004). 3.3. Within-herd prevalence, p Previously reported data were used to describe within-herd prevalence as a function of herd size (Amosson and Dietrich, 1984). These data suggest that within-herd prevalence generally decreases with increasing herd size. Such a pattern implies that cattle-to-cattle spread of brucellosis within a herd is mostly independent of cattle density. To account for variation in within-herd prevalence levels between herds of similar size, the published proportion of infected animals was treated as the mean for a beta distribution (Fig. 4).

Fig. 4. The variability in within-herd prevalence is illustrated for different herd size classes. Central tendency for each size class is based on Amosson and Dietrich (1984) and variability about the central tendency is modeled using beta distributions.

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Some overlap in adjoining beta distributions was modeled because we assumed that, in addition to herd size, variable management practices would influence within-herd prevalence. 3.4. Test sensitivity and specificity, SeSeries and SpSeries Several years of testing data from State Z were analyzed to determine the number of cattle tested from the state and the number of cattle determined to be test-positive on both screening and confirmatory tests. Data from State Z were used because it was one of a few states that entered complete data on every animal tested at its laboratories. Therefore, we were able to estimate the state of origin of cattle tested within State Z based on cattle identification information entered into the database. Because we knew that all cattle in the testing data subset were from State Z, we could assume each sample had the same probability of being from an infected animal and each positive result was similarly investigated. Using the 2004 data from State Z (a typical year of testing), the number of State Z’s cows and bulls tested within State Z was 45,969. The testing regime in State Z combined a buffered acidified plate antigen (BAPA) test with a brucellosis card test. The testing regime runs the tests in series, with the BAPA test run first. An animal is considered test-positive only when both tests are positive. The estimated sensitivity and specificity of this combination (SeSeries and SpSeries) is required in the process model. Among the 45,969 tests in 2004, State Z found 204 samples that were BAPA-positive and 51 samples that were BAPA- and card-positive. The SeSeries and SpSeries parameters were estimated using a hierarchical Bayesian model for assessing testing accuracy in the absence of a gold standard test (Hanson et al., 2003) (see Appendix A). Prior distributions for the sensitivity and specificity of both tests were taken from Gall and Nielson (2004). A prior distribution for cattle prevalence conservatively assumes a herd prevalence of 0.12% and an average within-herd prevalence of 12%. Point estimates from this model were SeSeries = 0.8826 and SpSeries = 0.999. 3.5. Probability of successfully investigating a herd of origin, v In the U.S. bovine brucellosis program, a test-positive result automatically provokes a full investigation of the herd of origin to determine its brucellosis status. Each investigation of a positive sample is begun by collecting available information to trace the animal back to its herd of origin. This information may consist of an official (e.g., brucellosis vaccination) or unofficial ear tag, a back tag or identification information from other animals that entered the slaughter facility in the same consignment. Not all animals can be successfully traced back to their herds of origin, but little information exists to describe the probability of a successful trace. Many factors contribute to the difficulty of tracing animals, with common issues being missing ear- or back tags and incomplete or inaccurate records at markets. One of the less obvious causes for an unsuccessful trace is a mismatch between the sample and the tag information, which was found to be 3% in one study (Heaton et al., 2005). The U.S. brucellosis eradication program requires that a minimum of 90% of those samples that are positive to a series of tests are successfully traced back to their herds of origin in a given year (USDA-APHIS, 2003). Investigation of a minimum of 95% of successful traces must also be satisfactorily completed. Satisfactory completion of an investigation typically requires testing the whole herd and finding no serologic evidence of brucellosis. Nevertheless, it is possible that program epidemiologists may claim satisfactory resolution of an investigation based on sufficient

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ancillary evidence of absence of brucellosis in the herd (e.g., whole herd vaccination status, closed herd without any new introductions, absence of clinical signs, etc.). These minimum standards suggest that each state must thoroughly investigate 85.5% (0.90  0.95) of positive samples. For the purposes of this analysis, it was conservatively assumed that positive samples would be successfully investigated 75% of the time. This probability, denoted by v, is conservative because it is likely that v is greater than 75%; for example, program records suggest that 97–98% of positive samples are successfully traced and satisfactorily resolved (Donch et al., 2006). Nevertheless, an assumed lower value for v generates higher, as opposed to lower, prevalence estimates. Higher prevalence estimates result because the reduced detection probability is consistent with the existence of non-detected affected herds in the population. 3.6. Prior distributions for uprev For the State Z model, the prior distribution for uprev at time zero (i.e., just after the state was declared class-free) was assumed to be a discrete distribution between 0 and 25 affected herds with each value having equal probability. This range of affected herds implies that herd prevalence could initially be from 0% to 0.12%. Such a range is conservative for a class-free state, given that the maximum allowable herd prevalence for the less stringent categorization of class A status is 0.1%; also, a class-free state must not report any affected herds for at least 12 months. For the low-risk states model, this same prior distribution for uprev was assumed. This range of affected herds implies herd prevalence between 0% and 0.004%. The effect of this assumed prior distribution on the model’s results was examined to determine if a wider range was needed. 4. Sensitivity analysis The process model’s inputs were derived from small studies, expert opinion, or a combination of the two. To address concerns that potentially biased inputs could substantially alter beef herd prevalence estimates, a sensitivity analysis was performed on the low-risk states’ estimates. The effect of altering the empiric evidence used in the Bayesian calculations (i.e., nanimals and Epositive.herd) was similarly assessed. To examine the effect of changes in v, t, and p on herd prevalence estimates, the sensitivity analysis chose values for these parameters considered to be substantially above or below their original values. Only lower values were assessed for t because the original value for the risk of importation (0.33) was considered extremely conservative. One-way analysis was completed by changing one parameter value and generating herd prevalence estimates across time with all other parameters remaining at their original values; this process was repeated for each alternative parameter value. Two statistics, the 95th percentile of the uprev distribution (at its minimum) and the number of years of surveillance before the estimated prevalence stabilized, are reported for each analysis. 5. Results 5.1. Prevalence estimation The Bayesian Monte Carlo analysis estimates herd prevalence (uprev) as a function of the time since attaining class-free status. The first estimation is based on actual testing evidence from

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Fig. 5. Mean, 5th, and 95th percentiles of the brucellosis herd prevalence distribution for the beef industry in a single U.S. state (State Z) estimated across 16 years of negative slaughter surveillance evidence using a Bayesian Monte Carlo analysis. The probability of introducing infection into this state (t = 0.10) influences the rate at which surveillance evidence depreciates such that, after 8 years of slaughter surveillance, the 95th percentile of this distribution stabilizes at three affected herds.

State Z that has conducted surveillance for many years without detecting an affected herd. During each of the process model’s iterations, the number of affected herds was randomly selected from the prior distribution. Each simulated affected herd’s size was also randomly determined; as a result, most affected herds were smaller herds. Following 5 years of slaughter surveillance of beef cattle without detecting any infection, the model estimates a 95% probability that three or fewer affected herds exist in this class-free state (Fig. 5). The accumulating surveillance evidence progressively suggests an increasing statistical confidence about lower numbers of possibly undetected affected herds. These results, however, do not imply an epidemiologic trend in the number of affected herds. It should be noted that even if State Z had zero affected herds throughout the surveillance period, these results would not change because the analysis is restricted to the statistical inferences from the surveillance data. The expected value (i.e., mean) for the number of affected herds decreases from approximately six herds after 1 year of surveillance to about one herd after 5 years of surveillance. The 5th percentile of prevalence remains at zero affected herds throughout the 16-year time span that is modeled. After 8 years of slaughter surveillance, the model predicts a 95% probability that two or fewer affected herds might exist, but beyond this point the estimate remains constant. Estimated prevalence tends to stabilize with more years of surveillance because accumulated surveillance evidence eventually depreciates at a steady rate that offsets the value of additional data. If surveillance evidence did not depreciate, then each year of additional surveillance evidence would continue accumulating and estimated prevalence would continue to decrease.

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Fig. 6. Mean, 5th, and 95th percentiles of the brucellosis herd prevalence distribution for the cow–calf beef industry in 46 low-risk U.S. states estimated across 16 years of negative slaughter surveillance evidence using a Bayesian Monte Carlo analysis. The accumulation of information improves confidence across the first 5 years. The probability of introducing infection into these states (t = 0.33) influences the rate at which surveillance evidence depreciates such that, after 5 years of surveillance, the 95th of this distribution percentile stabilizes at five affected herds.

Depreciation of slaughter surveillance information is influenced by the probability of introducing new infection into the state; a higher probability depreciates information faster and a lower probability depreciates information slower. The relatively low probability of introducing infection into a single state (t = 0.10) has the effect of increasing the ‘‘value’’ of older data. If the probability of importing infection was assumed to be 1% per year, then the estimated prevalence would stabilize at a slower rate and at a lower prevalence than illustrated in Fig. 5. The second estimation of beef herd prevalence is based on slaughter surveillance evidence collected across all 46 low-risk states. After 5 years of slaughter surveillance within low-risk states, we estimate a 95% probability that five or fewer affected herds could exist (Fig. 6). Beyond this point, however, prevalence of affected herds stabilizes and additional surveillance simply maintains 95% confidence of five or fewer affected herds. The mean number of affected herds predicted by the model is approximately two; this value stabilizes after 4 years of slaughter surveillance. As in the single state model, the 5th percentile for prevalence is zero throughout the 16 years modeled. The time-series depictions of estimated herd prevalence (e.g., Fig. 6) only provide percentile and mean values for numbers of affected herds, but the method actually generates probability distributions for each surveillance year. For example, the probability of zero affected herds is approximately 10% while the probability of two affected herds is about 28% after 6 years of surveillance (Fig. 7). Despite the different population sizes in the single and low-risk states, inferences about uperv are similar. For example, the predicted distribution of affected herds after 1 year of surveillance is essentially the same for both analyses, but slight differences in estimated uperv develop gradually

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Fig. 7. A probability mass function of the number of brucellosis-affected beef herds among 46 low-risk U.S. states after 6 years of negative slaughter surveillance evidence is shown. The distribution was estimated using Bayesian Monte Carlo analysis.

across time. This similarity in estimates suggests that the probability of detecting affected herds using slaughter surveillance is mostly independent of scale. Whether the population is 21,500 herds or 615,770 herds, slaughter surveillance is capable of detecting five or more affected herds with a high degree of confidence. Such a result is consistent with proportional sampling from a hypergeometric distribution; sampling the same fraction of herds without replacement essentially implies the same confidence of detecting affected herds regardless of the population size. The similarity of results between the two estimates also suggests their insensitivity to the initial prior distribution for number of affected herds. The initial range of affected herds considered (i.e., 0–25) is conservative for the single-state model, but this range is not necessarily conservative for the much larger population of low-risk states. The analysis for low-risk states estimates numbers of affected herds much less than the maximum of this range; therefore, the initial range of affected herds is also conservative for this larger population size. The differences in these estimates primarily reflect the different probability of introduction assumptions. Conservative values are used in both models but the 33% used in the low-risk states model is substantially larger than the 10% used in the single-state model. The larger probability of introduction causes the low-risk states’ model to predict that prevalence stabilizes sooner and at a higher level than the single-state model. 5.2. Sensitivity analysis Sensitivity analysis shows that alternative parameterizations of the process model change the 95th percentile for herd prevalence by no more than four affected herds (Table 1). The largest change occurs when t is reduced from 0.33 to 0.01; this reduces the estimated 95th percentile of uperv by four herds (i.e., from five herds to one herd) and increase the number of years until the

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Table 1 Summary of the values used in the sensitivity analyses of process model inputs for estimating brucellosis herd prevalence in 46 low-risk U.S. states Parameters

Values

95th percentile of uprev distribution

Number of years until uprev stabilized

Probability of introducing infection, t

0.01 0.10 0.33*

1 herd 2 herds 5 herds

11 9 5

Within-herd prevalence, p

2 infected animals per herd (regardless of herd size) Amosson and Dietrich (1984) values for 1st year of infection* Amosson and Dietrich (1984) values for 3rd year of infection

7 herds

7

5 herds

5

4 herds

3

0.50 0.75* 0.95

6 herds 5 herds 4 herds

6 5 6

Probability of successful tracing, v

The last two columns illustrate that the estimated 95th percentile value for number of affected herds, and the number of years required for the estimates to stabilize, are not greatly influenced by substantial changes in the input parameters (* indicates original values used to estimate herd prevalence).

estimate stabilizes by an additional 6 years (i.e., from 5 to 11 years). The increase in number of years to stability occurs because each year’s surveillance information is not depreciated as quickly when the probability of introducing infection is low. Adjustments to p were based on within-herd prevalence values reported by Amosson and Dietrich (1984). That study provides within-herd prevalence estimates for different herd-size strata when the disease had been undetected for 1–3 years. The original parameterization of the model assumed average within-herd prevalence values derived for 1 year post-infection. For example, the original average within-herd prevalence for herds with 19 or fewer cows was 27%. This value was originally 1% for herds with 1000 or more cows. As high alternative values, it was assumed p values that were consistent with the disease being undetected for 3 years were used. For comparison, these values were 39.4% and 11.6% for herds with fewer than 19 cows and more than 1000 cows, respectively. Similar adjustments were made for intervening herd sizes. These changes reduced the estimated 95th percentile of uperv by one herd and reduced the time until uperv stabilized by 2 years. Increased within-herd prevalence improves detection of affected herds, thereby improving the inferential power of slaughter surveillance. In contrast, increased p diminishes the time value of information by implying that more infection is transmitted across time; therefore, historic information is less relevant to future estimates of herd prevalence. The low alternative values for p were that each infected herd contained only two infected animals. This resulted in similar magnitudes of change, but in the opposite direction relative to the original estimates. Herd prevalence estimates were somewhat insensitive to v. Increasing the probability of successfully investigating positive animals from 0.75 to 0.95 reduced the estimated 95th percentile of uperv by one herd. A nearly equivalent reduction in v increased the estimated 95th percentile of uperv by one herd.

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Fig. 8. The 95th percentile of the brucellosis herd prevalence distribution in 46 low-risk U.S. states, after 5 years of slaughter surveillance, decreases as a function of the fraction of cows and bulls sampled. In this case, a sampling fraction of 1 equals annually sampling 2.5 million cows and bulls culled and slaughtered from 615,770 beef cow–calf herds in lowrisk states. This graph illustrates that the 95th percentile of uprev is relatively stable once the sampling fraction is greater than approximately 0.7 or 1.75 million cattle sampled.

Estimated herd prevalence was insensitive to reductions in the number of slaughter samples collected each year (nanimals) until less than 70% of the current sampling level was collected (Fig. 8). To illustrate the effect of nanimals on estimated herd prevalence, the number of slaughter cattle sampled each year was proportionally reduced from 100% to 1% of current

Fig. 9. Mean, 5th, and 95th percentiles of the brucellosis herd prevalence distribution for the beef cow–calf industry in 46 low-risk U.S. states estimated across 10 years of slaughter surveillance using a Bayesian Monte Carlo analysis. In this illustrative example, two affected herds were assumed to be detected in the 6th year of surveillance. The estimated herd prevalence distribution reflects these detections during the 6th and 7th years of surveillance, but it returns to its previous values by the 8th year if no additional affected herds are discovered after the 6th year.

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levels. At approximately 70% of current sampling levels (i.e., nanimals  1.75 million), the 95th percentile of uperv increases from five to six affected herds. Such a result suggests that current slaughter sampling intensity might be reduced without substantially changing surveillance effectiveness. A final perturbation to this analysis considered the effect of non-zero values for Epositive.herd. Such a scenario could occur if one or more affected herds were detected in a given year of slaughter surveillance. To illustrate this phenomenon, the evidence for the 6th year was modified so that two infected animals were traced to affected herds but no new infection was discovered in subsequent years (Fig. 9). In this illustration, the estimated herd prevalence increases in the 6th year but then returns to its stabilized level across the next 2–3 years. If the acceptable threshold for declaration of freedom was five affected herds or less with 95% confidence, then only 1 year might be needed to regain this classification (in this scenario) instead of the 5 years of negative surveillance evidence it took to initially stabilize estimated herd prevalence below the threshold. Nevertheless, the mean and 5th percentile values of the herd prevalence distribution do not return to their stabilized values until an additional 2 or 3 years of negative surveillance evidence are accumulated. 6. Discussion In the U.S. bovine brucellosis eradication program, slaughter surveillance is the only activity required for detecting affected beef herds once a state achieves class-free status. In low-risk states we estimate that, after 5 years of negative surveillance, we are at least 95% confident that there are five or fewer affected herds. This equates to being 95% confident in a herd prevalence of less than 0.00081% (5/615,770), which is well below the World Organization for Animal Health requirement that herd-level prevalence not exceed 0.2% to claim bovine brucellosis freedom (OIE, 2003). Several assumptions included in this analysis likely inflate the estimated herd prevalence. For example, the process model assumes that animal cull rates are independent of brucellosis infection status. If the likelihood of culling was increased for infected cattle in affected herds, then the effectiveness of slaughter sampling would increase and the resulting estimates would suggest a lower herd prevalence (given evidence that no affected herds were detected). We also assumed conservative values for the probability of successful investigation and probability of introduction. Both of these assumptions likely bias herd prevalence upward relative to less conservative, but possibly more realistic, assumptions. The process model also conservatively assigns affected herds randomly among all herds and lacks any consideration of spatial clustering of affected herds. Because larger herds will import cattle more frequently than smaller herds, it could be argued that larger herds should be more likely to become brucellosis-affected. If larger herds were disproportionately represented among affected herds, the probability of detecting herds via slaughter surveillance would improve relative to our process model’s assumptions of randomly distributed affected herds. Similarly, geographically clustered infection is more likely to be detected than randomly distributed infection once an index herd is identified via slaughter surveillance. Follow-up investigation of herds clustered about a detected herd would likely detect other affected herds. Estimated herd prevalence among low-risk states stabilizes after 5 years of accumulated slaughter surveillance evidence only if the probability of introduction remains constant. Yet, most class-free states have maintained this status for longer than 5 years. The longer class-

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free states remain in this status, the smaller their probability of introduction seemingly becomes. In fact, once all states have achieved class-free status for at least 5 years, the probability of introducing new infection should reflect the probability that the United States imports brucellosis from outside the country. This probability will likely be small, but it will require some amount of maintenance surveillance to perpetuate our confidence in the United States’ freedom. The surveillance requirements of the U.S. brucellosis program, combined with the low-risk states analysis reported here, imply that the objective of slaughter surveillance is to detect a herd prevalence of about 1 per 100,000 with 95% confidence. Nevertheless, slaughter surveillance is imperfect. Affected herds of small size are unlikely to be detected via slaughter surveillance in any given year because such herds cull fewer cows and bulls. Furthermore, this analysis implies that some small affected herds can avoid detection indefinitely if slaughter surveillance is the only activity employed for detecting such herds; such a finding is consistent with some small producers who might avoid brucellosis surveillance altogether by only culling their cattle to noncommercial slaughter facilities (e.g., custom slaughter). Bayesian Monte Carlo is a flexible tool for making herd-level inference when sampling is performed at the animal level and in situations where little animal identification information is available to directly link animals to their herds of origin during analysis. The primary advantage of this approach is that modeling the uncertainty distribution of disease prevalence provides a basis for estimating prevalence as a function of time; such an advantage is important in government disease control programs that extend across many years. Admittedly, this approach can be more complex than other methods for demonstrating freedom from disease and any inferences depend on a number of assumptions and models built from various data sources. Ideally, a surveillance system should provide virtually all of the data required to make valid inferences and the results would not heavily depend on modeling and external data sources. Alternative methods such as stochastic decisions trees (Martin et al., 2007b), however, still require the same collection and synthesis of data regarding the population demographics, disease characteristics, and management practices. Personal experience suggests that the collection of these data constitutes the most substantial proportion of the analysis effort. Rather than only estimating statistical confidence about a design prevalence value, the Bayesian Monte Carlo method used in this analysis allows inference about herd prevalence regardless of the number of positive samples or whether positive samples are successfully traced to their herd of origin. A limitation of previously proposed methods is that once a positive animal is found, those methods are no longer applicable; such methods can only determine the confidence that prevalence is less than some level given negative surveillance evidence. 7. Conclusions Based on slaughter surveillance evidence and conservative assumptions in a Bayesian Monte Carlo model, this analysis demonstrates at least 95% confidence that bovine brucellosis herd prevalence among beef cow–calf herds is less than 0.014% (3 per 21,500 herds) and 0.00081% (5 per 615,770) in a typical U.S. class-free state (State Z) and across 46 class-free (low-risk) U.S. states, respectively. These estimates are most sensitive to the probability of introduction of new infection assumed in the analysis. Because conservative values are assumed, the true herd prevalence is likely less than the values estimated.

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