Two approaches to the analysis of registry data for bovine diseases with regard to the relationship between disease incidence and cattle trade are proposed. Firstly, a parameter-driven spatio-temporal disease mapping model

formulated in a hierarchical Bayesian framework is used. Various cattle movement parameters, e.g. the number and proportion of in-movements from infected regions, can be included as potential covariates. Within this context problems of such an endogenous covariate are discussed. Since a purely parameter-driven approach is often not adequate to depict local epidemics, a so-called observationdriven infectious disease model is proposed as a second possibility. It includes an autoregressive part for counts in the region of interest in the past. Additionally,

the sum of previous cases in other regions weighted by cattle movements is added to assess the spread of the disease by trading. Both models are applied to cases

of Coxiellosis in Switzerland, 2005 to 2009

Held, L

Paul, M

Schrödle, B

ZORA

University of ZurichZurich Open Repository and ArchiveWinterthurerstr. 190CH-8057 Zurichhttp://www.zora.uzh.chYear: 2010Analyzing veterinary surveillance data: Approaches to model therelationship between disease incidence and cattle tradeSchrödle, B; Held, L; Paul, MSchrödle, B; Held, L; Paul, M (2010). Analyzing veterinary surveillance data: Approaches to model the relationshipbetween disease incidence and cattle trade. In: 25th International Workshop on Statistical Modelling, Glasgow, 05July 2010 - 09 July 2010, 1-6.Postprint available at:http://www.zora.uzh.chPosted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.chOriginally published at:25th International Workshop on Statistical Modelling, Glasgow, 05 July 2010 - 09 July 2010, 1-6.Schrödle, B; Held, L; Paul, M (2010). Analyzing veterinary surveillance data: Approaches to model the relationshipbetween disease incidence and cattle trade. In: 25th International Workshop on Statistical Modelling, Glasgow, 05July 2010 - 09 July 2010, 1-6.Postprint available at:http://www.zora.uzh.chPosted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.chOriginally published at:25th International Workshop on Statistical Modelling, Glasgow, 05 July 2010 - 09 July 2010, 1-6.Analyzing veterinary surveillance data: Approaches to model therelationship between disease incidence and cattle tradeAbstractTwo approaches to the analysis of registry data for bovine diseases with regard to the relationshipbetween disease incidence and cattle trade are proposed. Firstly, a parameter-driven spatio-temporaldisease mapping model formulated in a hierarchical Bayesian framework is used. Various cattlemovement parameters, e.g. the number and proportion of in-movements from infected regions, can beincluded as potential covariates. Within this context problems of such an endogenous covariate arediscussed. Since a purely parameter-driven approach is often not adequate to depict local epidemics, aso-called observationdriven infectious disease model is proposed as a second possibility. It includes anautoregressive part for counts in the region of interest in the past. Additionally, the sum of previouscases in other regions weighted by cattle movements is added to assess the spread of the disease bytrading. Both models are applied to cases of Coxiellosis in Switzerland, 2005 to 2009.Analyzing veterinary surveillance data:Approaches to model the relationshipbetween disease incidence and cattle tradeBirgit Schro¨dle1, Leonhard Held1 and Michaela Paul11 University of Zurich, Institute for Social and Preventive Medicine, Hirschen-graben 84, 8001 Zurich, Switzerland, Email:birgit.schroedle@ifspm.uzh.chAbstract: Two approaches to the analysis of registry data for bovine diseaseswith regard to the relationship between disease incidence and cattle trade areproposed. Firstly, a parameter-driven spatio-temporal disease mapping modelformulated in a hierarchical Bayesian framework is used. Various cattle move-ment parameters, e.g. the number and proportion of in-movements from infectedregions, can be included as potential covariates. Within this context problemsof such an endogenous covariate are discussed. Since a purely parameter-drivenapproach is often not adequate to depict local epidemics, a so-called observation-driven infectious disease model is proposed as a second possibility. It includes anautoregressive part for counts in the region of interest in the past. Additionally,the sum of previous cases in other regions weighted by cattle movements is addedto assess the spread of the disease by trading. Both models are applied to casesof Coxiellosis in Switzerland, 2005 to 2009.Keywords: Cattle trade; Spatio-temporal disease mapping; Infectious disease;INLA; Observation-driven1 IntroductionThe spread of a bovine disease can take place over short distances betweenadjacent or nearby farms borne by wind or insects, which typically resultsin local clustering of cases. However, disease dispersal also takes place overlong distances caused by trade of infectious animals (Gilbert et al., 2005).Hence, the inclusion of cattle trade in an analysis of veterinary surveillancedata might give hints towards the association of animal movement and dis-ease presence.As a first approach disease counts can be analyzed using a disease map-ping model that considers spatial, temporal, and spatio-temporal trends(Knorr-Held, 2000; Schro¨dle and Held, 2009). Additionally, various cat-tle movement parameters can be included in the model using ecologicalregression (Clayton et al., 1993). Bayesian inference is conducted usingintegrated nested Laplace approximations (INLA), which was recently pro-posed in Rue et al. (2009).2 Analyzing veterinary surveillance dataAs an alternative to this purely parameter-driven approach the spread ofa bovine disease by cattle trade can be modelled within a likelihood-basedinfectious disease framework (Paul et al., 2008). Here, an autoregressiveterm for past counts in the region of interest and the sum of past cases inother regions weighted by cattle movements are part of the model formu-lation. The advantage of this so-called observation-driven model is that itis able to describe local epidemics.Coxiellosis in cattle is an infectious, bacterial disease among ruminant ani-mals, which can be spread by airborne infection. It can be the reason for anabortion, even in a late phase of the pregnancy. Cases of Coxiellosis werereported to the Swiss Federal Veterinary Office (BVET) between 2005 and2009 and are available aggregated for 184 Swiss regions and the Principalityof Liechtenstein on a yearly basis. The number of herds mi in each regioni is known.Since 2008 it has been mandatory for Swiss stock-keepers to notify theBVET of all cattle movements. As the spatial pattern of movements is sim-ilar for 2008 and 2009 the considered movement parameters are assumedto be consistent from year to year.2 Spatio-temporal disease mappingFor modelling spatio-temporal disease counts yit ∼ Po(exp(ηit)) a nonpara-metric hierarchical Bayesian setting as proposed in Knorr-Held (2000) isused. The respective linear predictor can be written asηit = log(mi) + ξ + νi + ψi + γt + φt + δit, (1)where ξ is an intercept, νi and ψi are spatially unstructured and structuredeffects and γt and φt are temporal main effects, specified as an i.i.d. termand a random walk of first order, respectively. The term δit accounts forspatio-temporal interaction and can be specified assuming four differenttypes of interaction between time and space (Knorr-Held, 2000). To ac-count for covariates xit as introduced in the following paragraph (1) canbe extended toηit = log(mi) + ξ + νi + ψi + γt + φt + δit + β · xit. (2)The total number of in-movements (model acronym: TOT) and the ab-solute number (A) and proportion (P) of in-movements from regions withelevated risk are considered as potentially associated with disease presence.Since we assume that the movement pattern does not change from year toyear the total number of in-movements is constant over time. The move-ments from infected regions are time-varying and defined using a two-stageprocess: A separate spatial disease mapping model is fitted for each year,including only νi and ψi from (1) (Besag et al., 1991). Regions are indicatedSchro¨dle et al. 3as infected when exceeding two different thresholds, namely an estimatedrelative risk larger than 2 and 3, respectively. At the second stage the mod-els are fitted using a time lag of one and two years, respectively, to detectthe incubation period of the disease (TL1 and TL2). As the time-varyingcovariate is derived using previous observations it is a so-called endogenousor feedback variable. This issue will be discussed briefly in Section 4.All models are fit using integrated nested Laplace approximations (INLA).This approach for approximate Bayesian inference was recently proposedby Rue et al. (2009) as an alternative to Markov chain Monte Carlo mech-anisms. The advantage of INLA is that it runs in remarkably fast compu-tational time and returns accurate parameter estimates for a wide range ofmodels. Additionally, INLA computes the deviance information criterion(DIC) as tool for Bayesian model choice. All analyses in this paper wereconducted using the R INLA package build on the 1st of February 2010,version 1.668.3 Infectious disease modelA purely parameter-driven model as proposed in Section 2 might not beable to describe localized epidemics which can often be found in veterinarydisease surveillance data (Held et al., 2005). Hence, a so-called observation-driven model is built including the number of cases yi,t−1 in the past. Inits simplest formulation the observations yit are Poisson distributed withmeanµit = λ · yi,t−1 +mi · exp(α) (3)and λ > 0 (Held et al., 2005). The parameter α accounts for all residualvariation. Cases at times t− k, k > 1, could be considered as well.As an addition, the sum of counts in all other regions j weighted by a factorwji can be added to model the spatial spread of the disease over time. InPaul et al. (2008) the respective mean is specified asµit = λ · yi,t−1 + ρ ·∑j 6=iwji · yj,t−1 +mi · exp(α) (4)with λ, ρ > 0. To assess the association between cattle movement anddisease presence the square root of the absolute number of cattle movements(CM) between regions j and i are used as weights wji in this application.Other weights wji can also be considered (Paul et al., 2008). Here, modelswith wji = 1 for all j and for all j ∼ i, respectively, are fit as alternatives.The term j ∼ i denotes all regions j which are neighbours of region i.The parameter α in (4) can be split into an intercept and a linear timetrendµit = λ · yi,t−1 + ρ ·∑j 6=iwji · yj,t−1 +mi · exp(α+ ζ · t). (5)4 Analyzing veterinary surveillance dataTABLE 1. Spatio-temporal disease mapping (see Section 2): The DIC andthe posterior mean of the respective cattle trade parameter along with its95%-credible interval are shown for each model. For the model without covariate(1) the DIC is shown only.RR> 2 RR> 3DIC βˆ DIC βˆTL1 A 939.3 0.017 [0.008; 0.027] 942.7 0.012 [0.004; 0.020]P 947.0 1.29 [0.39; 2.18] 950.0 0.96 [−0.02; 1.94]TL2 A 945.2 0.009 [−0.000; 0.018] 951.0 0.007 [−0.003; 0.016]P 948.0 0.83 [−0.038; 1.69] 951.1 0.85 [−0.20; 1.90]TOT 955.8 0.005 [−0.006; 0.017](1) 954.6Extensions for a region-specific random effect are also possible (Paul etal., 2009), but the computation of the results might suffer from numericalproblems if the number of regions is large. Hence, they are not consideredhere.Maximum likelihood inference is performed using iterative algorithms asdescribed in Held et al. (2005) and Paul et al. (2008). The AIC is calculatedfor model choice.4 ResultsRegarding spatio-temporal disease mapping, model (1) was run withoutcovariate for all four possible types of space-time interaction. The modelincluding an interaction term of Type II had the lowest DIC and was chosenas basis model for the ecological regression including cattle trade quantities.All results are summarized in Table 1. The DIC is lowest for the modelincluding the absolute number of in-movements from infected regions withrelative risk larger than 2 and a time lag of one year. This is plausibleconsidering the nature of the disease. Models including the absolute numberof in-movements are generally preferred compared to models involving theproportion of cattle trade from infected regions. A positive association isobtained for all covariates. For three of the models with a time lag of oneyear the 95%-credible interval includes only positive values.As noted in Section 2 the number of in-movements from infected regions isan endogenous covariate. If the pattern of the disease exhibits local clusters,the infected areas chosen by the two-stage process typically are a few groupsof neighboring regions. As cattle trade is much larger between neighboringregions, the respective parameters might just explain parts of the localspatial clustering of cases in the data. One hint pointing in this direction isthe fact that the estimated variance of the spatially structured effect ψi inSchro¨dle et al. 5TABLE 2. Infectious disease model (see Section 3): The AIC and the estimatedcoefficients along with their standard errors are shown for each model.wji AIC λˆ ρˆ ζˆ(3) − 1092.8 0.44 (0.05)(4) 1 1094.8 0.44 (0.05) 0.0000 (0.0000)(4) 1, if j ∼ i 1062.7 0.43 (0.05) 0.0491 (0.0103)(4)√CM 1082.0 0.44 (0.05) 0.0005 (0.0002)(5)√CM 1083.5 0.44 (0.05) 0.0005 (0.0002) 0.08 (0.11)(1) drops after inclusion of cattle movement in the model. Unfortunately,it cannot be quantified to what extent such confounding is present.Results for the infectious disease models are shown in Table 2. With regardto AIC model (4) using√CM as weights performs better than model (3).Hence, the autoregressive inclusion of counts from other regions weightedby cattle trade provides a better fit. The respective parameter estimate ρˆis positive (0.0005) with a small standard error (0.0002). In contrast, thealternative model with wji = 1 for all j is not better than (3). If the countsin neighbouring regions j ∼ i are considered as additional explanatoryvariables the AIC is even smaller than for the model including cattle tradeand the estimated coefficient is significantly positive (0.0491). Hence, a highlocal clustering of cases is present. For (5) a positive linear time trend isestimated (0.08), but it is not significantly different from zero.5 DiscussionTwo very different approaches were applied to data on Coxiellosis in cattleto assess the spread of the disease by cattle trade. In both cases modelchoice criteria and estimated coefficients indicate a positive associationbetween animal movement and disease presence. Nevertheless, both ap-proaches are not without problems. The disease mapping approach makesuse of an endogenous covariate which might result in a confounding prob-lem. With regard to the infectious disease model first steps in the directionof a mixture of the parameter- and observation-driven approach are takenwhen fitting (5). Nevertheless, it would be desirable to include spatial effectsas well, especially a spatially structured effect to account for local cluster-ing of the disease. Hence, it must be explored if both approaches could becombined by substituting α in (3) by (1) (without (logmi)). In this newsetting parameter estimation might be possible in a Bayesian framework asalgorithms used for maximum likelihood inference (Paul et al., 2008) willpossibly suffer from numerical problems. Furthermore, the two approachesare not comparable at the moment as different model choice criteria arederived and different components are included.6 Analyzing veterinary surveillance dataAcknowledgments: Financial support by the Swiss Federal VeterinaryOffice (BVET) is gratefully acknowledged.ReferencesBesag, J., York, J. and Mollie, A. (1991). Bayesian image restoration withtwo applications in spatial statistics. Annals of the Institute of Sta-tistical Mathematics, 43, 1-59.Clayton, D., Bernardinelli, L. and Montomoli, C. (1993). Spatial correla-tion in ecological analysis. International Journal of Epidemiology, 22,1193-1202.Gilbert, M., Mitchell, A., Bourn, D., Mawdsley, J., Clifton-Hadley, R. andWint, W. (2005). Cattle movement and bovine tuberculosis in GreatBritain. Nature, 435, 491-496.Held, L., Ho¨hle, M. and Hofmann, M. (2005). A statistical framework forthe analysis of multivariate infectious disease surveillance counts. Sta-tistical Modelling, 5, 187-199.Knorr-Held, L. (2000). Bayesian modelling of inseparable space-time vari-ation in disease risk. Statistics in Medicine, 19, 2555-2568.Paul, M., Held, L. and Toschke, A. (2008). Multivariate modelling of in-fectious disease surveillance data. Statistics in Medicine, 27, 6250-6267.Paul, M. and Held, L. (2009). Predictive assessment of a non-linear ran-dom effects model for multivariate time series of infectious diseasecounts. Technical Report, University of Zurich.Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian infer-ence for hierarchical Gaussian Markov random field models. Journalof the Royal Statistical Society, Series B, 71, 319-392.Schro¨dle, B. and Held, L. (2009). Spatio-temporal disease mapping usingINLA. Technical Report, University of Zurich.

Analyzing veterinary surveillance data: Approaches to model the relationship between disease incidence and cattle trade

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