Epidemic modelling of bovine tuberculosis in cattle herds and badgers in Ireland
EEpidemic Modelling of Bovine Tuberculosis in CattleHerds and Badgers in Ireland
L.M. White* † G.E. KellyVisiting Nurse Service of New York School of Mathematics and Statistics(VNSNY) University College DublinIreland*Work undertaken as an M.Sc. student at University College Dublin. The paperdoes not represent the views of VNSNY † corresponding author: [email protected] Abstract
Bovine tuberculosis, a disease that affects cattle and badgers in Ireland, was studiedvia stochastic epidemic modeling using incidence data from the Four Area Project(Griffin et al., 2005). The Four Area Project was a large scale field trial conducted infour diverse farming regions of Ireland over a five-year period (1997-2002) to evaluatethe impact of badger culling on bovine tuberculosis incidence in cattle herds.Based on the comparison of several models, the model with no between-herdtransmission and badger-to-herd transmission proportional to the total number ofinfected badgers culled was best supported by the data.Detailed model validation was conducted via model prediction, identifiabilitychecks and sensitivity analysis. 1 a r X i v : . [ q - b i o . P E ] J u l he results suggest that badger-to-cattle transmission is of more importance thanbetween-herd transmission and that if there was no badger-to-herd transmission, lev-els of bovine tuberculosis in cattle herds in Ireland could decrease considerably.Keywords: Mycobacterium bovis, Tuberculosis, Epidemic model, Ireland. A bovine tuberculosis (bTB) (causative agents are any of the disease-causing my-cobacterial species within the M. tuberculosis-complex) eradication scheme was ini-tiated in Ireland in 1954. Although initial progress was good, the programme sub-sequently stalled. Presently bTB incidence in cattle herds in Ireland is roughly 4%and approximately 18,500 infected cattle were slaughtered in 2011, with costs bothto the farmer and the exchequer. BTB also infects wild badgers (
Meles meles ), aprotected species under the Wildlife Act 1964, and they have been implicated inthe transmission of the disease to cattle (Griffin et al., 2005, Kelly et al., 2008). Itis also possible for cattle to infect badgers and the relative contribution of the twospecies to the persistence of the disease in cattle is difficult to quantify from fieldexperiments.Epidemic models can play a role in this quantification and here we present astochastic model of transmission dynamics of bTB in cattle herds and wild bad-gers in Ireland. In contrast to deterministic models that describe average effects,stochastic models contain and produce variability. For example, the time when aherd becomes infected and the daily rate of infection is largely unpredictable. Adeterministic model might define the rate of infection as 0.25/herds/day but in astochastic model it may be defined as 0.14-0.40/herds/day, i.e. daily rates vary. Theprobability of each of these rates can be modeled to form a distribution that peaksat 0.25/herds/day. This variability provides a range of effects i.e. a confidence in-2erval within which the likely course of an epidemic will probably lie. To paraphraseDe Jong (1995), the gain of such modeling is not the resulting model, but insteadthe insight into the population dynamics of infectious agents that is obtained inthe process of model building and model analysis on the one hand, and interpretingexperimental and observational data on the other.Donnelly and Hone (2010) presented epidemic models for bTB corresponding toareas of the Randomised Badger Culling Trial (RBCT) in S-W England. However,transmission dynamics between badgers and cattle herds may be inherently differentin Ireland than in Britain and require separate modeling. There is a substantialgrowing body of evidence that badgers in Great Britain are different in many respectsto badgers in Ireland including genetic origins, diet and territoriality (O’Meara etal., 2012; Sleeman et al., 2008), with Irish badgers exhibiting more wide rangingbehaviour (Judge et al., 2009; Kelly et al., 2010).In this study we describe epidemiological models for the transmission of bTBboth from herd-to-herd and from badger-to-herd and apply them to data from theIrish Four Area Project (FAP), the purpose being to determine the relative contri-bution of the two infectious pathways. Five subsets of the FAP data are analysedseparately.
We consider extensions of the classical susceptible-infective-removed (SIR) epidemicmodel. One extension was formulated by Barlow et al. (1998) to explain the trans-mission of bTB between cattle herds and brush-tail possums in a region of NewZealand. The model described the transition of herds between three states: U (uninfected and susceptible to infection), I (infected and thus infectious, but undi-agnosed) and M (under movement control (MC), thus not infectious to other herds).3n analogous model was subsequently formulated by Donnelly and Hone (2010)for a single area with both risk of infection from wildlife and density-dependentbetween-herd risk of infection (i.e. a model which assumes that the rate of contactof one herd with another increases in proportion to the total number of herds in thepopulation).The model is described by the following differential equations: dUdt = Mp − U ( βI + k ) (1) dIdt = U ( βI + k ) − Ic (2) dMdt = Ic − Mp (3)with constant population size N = U + I + M (4)where c is the overall rate at which infected herds go on MC (year − ), p is theaverage length of time a herd spends on MC in years, hence, 1 /p is the rate at whichherds come off MC, k is the rate of infection from wildlife to cattle (year − ), and β is the between-herd transmission coefficient which represents the herd-to-herd riskof infection per year. A visual representation of the model is given in Figure 1.Setting Equations (1) and (3) to 0 to find equilibrium values, we obtain thefollowing quadratic equation for the equilibrium value of I , I ∗ . I ∗ ( β + pcβ ) + I ∗ ( − N β + k + pck + c ) − N k = 0 (5) I ∗ will later be used in the formulation of a probabilistic model to explain theproportion of herds with newly detected bTB infection in a year.In the specific case of no risk of infection from wildlife ( k = 0) and β >
0, the4quilibrium value of I is: I ∗ = (cid:18) N − cβ (cid:19) (cid:18)
11 + pc (cid:19) (6)Similarly, when there is no herd-to-herd transmission of infection ( β = 0) and k >
0, the equilibrium value of I is: I ∗ = N pc + ck (7)We also consider a second model which assumes herd-to-herd transmission tobe frequency-dependent i.e. the rate of herd-to-herd transmission is completelyindependent of the total number of herds, N . The model and associated equilibriumvalues are described in Donnelly and Hone (2010).As in Donnelly and Hone (2010), we consider four alternative values of k (therate of infection from wildlife to cattle per year):1. k = 0: there is no transmission from wildlife.2. k = αN w : k is proportional to the total number of badgers culled in the regionin question, N w .3. k = αI w : k is proportional to the total number of infected badgers culled inthe region in question, I w .4. k = αI w /N w : k is proportional to the proportion of infected badgers culledrelative to the total number of badgers culled in the region in question,where α is a proportionality constant assumed to be non-negative. The three typesof between-herd transmission (no transmission, density-dependent and frequency-dependent) combined with the four types of transmission from wildlife, gives elevendifferent alternatives for I ∗ (omitting the alternative that has neither between-herdtransmission nor transmission from wildlife, β = k = 0).5 .2 The Data Data were taken from the FAP, a large scale field project undertaken in matchedremoval and reference areas (each approximately 245 km ) in four counties in Ire-land: Cork, Donegal, Kilkenny and Monaghan over a five-year period (September1997-August 2002). The project was carried out to assess the impact of badgerremoval on bTB incidence in cattle herds. Badger removal was proactive in removalareas while minimal culling took place in reference areas. The badger-removal pro-cedures are described in detail in the Badger Manual prepared by the Departmentof Agriculture, Food and Forestry (DAFF, 1996). In summary, badgers were killedbe a member of the FAP team after being captured during an 11-night removaloperation in which restraints were placed at active setts for 11 nights and wereinspected each morning. All euthanased badgers went through gross post-morteminvestigation. If evidence of tuberculosis was detected, all affected tissues were sentfor histopathological examination and for culture. If no evidence of tuberculosiswas detected, bacteriological culture was conducted on multiple tissues, includingthe lymph nodes, kidney and lung tissue. A badger was diagnosed as positive fortuberculosis if it was positive at histopathological examination and/or culture. Thestudy is described in further detail in Griffin et al. (2005).We consider data on badgers and cattle from the removal areas of the FAPduring the study period (1997-2002) and the 5-year ’pre-study period’ (September1992-August 1997). Data on herds were collected routinely by all local DistrictVeterinary Offices and data relating to badger removal and infection status werecollected throughout the field trial by FAP staff.Every animal in every herd in Ireland is tested annually for bTB by the SingleIntradermal Comparative Tuberculin Test (SICTT) and a herd is considered positiveif any cattle test positive. Herds that test positive are placed under restriction -MC. There is, however, an incubation period for the disease in cattle; therefore, five6ubsets of the cattle and badger data set were considered for analysis, numbered 1-5as follows:1. Data on badgers were taken from the year of the initial badger cull (1997/1998)while cattle herd data were taken from the year prior to that. These data werechosen because the majority of badgers were culled in the first year of thestudy and data on cattle from the year previous to the initial badger cull hadnot been affected by the badger culling and to allow for an incubation periodfor the disease in cattle.2. Data on both badgers and cattle were taken from the year of the initial badgercull (1997/1998).3. Data on badgers were obtained by summing over the five years in the studyperiod for each area. Data on the number of restricted herds, B , were alsosummed over the five years in the study period for each area. However, dataon the total number of herds, N , were not summed as this would result incounting the same cattle herds more than once. Therefore, for each area, thetotal number of herds is taken to be the maximum number of herds in any oneyear over the five years.4. Data on badgers and data on the number of herds restricted were obtained bysumming over the ten years in the combined study and pre-study period ineach of the four areas. As above, for each area, the total number of herds, N ,over the ten-year period is taken to be the maximum number of herds in anyone year over the ten years.5. Similar to data set 3, except the pre-study period data (1992-1997) were usedinstead of the study period data.All five data sets are displayed in Table 1. Complete data sets can be foundin Griffin et al. (2005) and Corner et al. (2008). A recent study suggests a high7pecificity of between 99 . − .
8% for the SICTT in Irish settings (Clegg et al., 2011).For all data sets above, false positive misclassification was further minimised bylimiting positive infection status to those SICTT-positive reactors in herds restrictedfrom trading following disclosure of two or more such animals.
An epidemic model was employed above to formulate an expression for the equilib-rium value of I , the number of infected herds, in terms of the unknown parameters β , α , c and p . We estimated these unknown parameters by setting up a binomiallog-likelihood for the proportion of herds B out of the total number of herds N , thatexperience a bTB herd breakdown and become restricted in a year. Assuming B isapproximately Binomial( N , q = I ∗ c/N ) at equilibrium, the log-likelihood is: l = log( L ( q | N, B )) ∝ B log(( q )) + ( N − B ) log((1 − q )) (8)The model with the value of I ∗ c closest to B is the ’best’ fitting model. Thereare four sets of ( B, N ), one set for each of the four counties, hence, the log-likelihoodis: l = (cid:88) j =1 log( L ( q j | N j , B j )) ∝ (cid:88) j =1 B j log(( q j )) + ( N j − B j ) log((1 − q j )) (9)where q j = I ∗ j c/N j (10)For each of the eleven alternatives for I ∗ there is a separate likelihood.8 .3.2 Parameter Estimation There are four unknown parameters in each of the eleven likelihoods (via I ∗ ), α , β , c and p . These four parameters are assumed to be the same for each of the fourcounties. The method of maximum likelihood is used to estimate α and β and asso-ciated standard errors (s.e.) for each of the eleven likelihoods (using Equation (9)).Wald’s method is then used to construct confidence intervals (C.I.) for the unknownparameters.Empirical values for the other two parameters, c and p , which do not vary withthe model, are assumed. The parameter c, defined as the rate at which infectedherds are detected and put under MC, incorporates two rates of detection, 1 /µ , therate at which infected herds are detected via the annual SICTT, and a , the rate atwhich infected herds are detected through slaughterhouse surveillance. As in Coxet al. (2005), we define µ by: µ = b (cid:18)
12 + (1 − s ) s (cid:19) (11)where s is the herd test sensitivity (in per cent) and b is the number of years betweenroutine herd tests. The first term comes from the assumption that herds becomeinfected at random times between tests. The second term, (1 − s )/ s , is the number ofretests required when the test has less than 100% sensitivity, assuming a geometricdistribution. Now, c = a + 1 µ = a + 2 sb ( s + 2(1 − s )) (12)Values for the parameters s , b and a/c (which is the proportion of infectedherds detected through slaughterhouse surveillance) are obtained from the literature.Frankena et al. (2007) estimated that in recent years between 27% and 46% of allnew herd breakdowns in any year have been detected by slaughterhouse surveillancein Ireland. Taking the mid-point we approximate a/c by 36.5%. However, other9alues of a/c between 27% and 46% were also considered. The parameter b is equalto 1 since all herds in Ireland are subject to annual routine herd testing.Clegg et al. (2011) suggest a sensitivity of 52 . − .
8% for the SICTT andbased on this the value for the sensitivity, s , is taken as the mid-point of this interval,56.85%.Letting a = 36 . c , b = 1 and s = 0 . c in Equation (12) to obtain c = 1 .
25. Since c is the rate at which infected herds are detected, 1/ c (0.8 years) isthe average length of time in years that a herd is infected before it is detected.We assume that p , which is the average length of time a herd spends on MC,is equal to 0.5 years. When a herd is put on MC it must pass two consecutiveSICTT tests before it is taken off MC and the length of time between retests isapproximately 60 days. Hence, the minimum length of time a herd spends on MC isapproximately 120 days (0.329 of a year) if it passes its first two consecutive tests.It has been noted that the average restriction period is not very much longer than120 days (M. Good, Department of Agriculture, Food and the Marine, personalcommunication), while in the study of Donnelly and Hone (2010) a value of p = 0 . p =0.5 a point midway between these two values. The relative support for each of the eleven models, i = 1 , . . . ,
11, (for the elevenalternative I ∗ ) was calculated using Akaike weights (Burnham and Anderson, 2002).The Akaike weight of a model can be interpreted as the probability that the modelis the ’best’ model among a set of R models. For the i th model, the Akaike weightis given by: w i = exp( − ∆ i ) (cid:80) Rr =1 exp( − ∆ r ) (13)where ∆ i is defined by ∆ i = AICc i − min( AICc ) (14)10nd AICc is a modified version of Akaike’s Information Criterion (AIC) that correctsfor small sample sizes relative to the number of parameters. Small values of AICcare preferred. The AIC approach is a method for comparing the goodness-of-fit ofnested and non-nested models and discourages the use of models with too manyparameters that overfit the data (Burnham and Anderson, 2002). AICc is definedas
AICc = − L + 2 k nn − k − n is the sample size. AICc i is the AICc value for the i th model. Basic reproduction numbers, R , were also calculated to evaluate the invasive po-tential of the disease, based on the model. The basic reproduction number is theaverage number of secondary infections produced when one infected individual is in-troduced into a host population where everyone is susceptible (Anderson and May,1991). Here, the individual will refer to a herd. When R <
1, the infective maynot transmit the disease during the infectious period and so the disease will die outin the long run (no epidemic). However, if R >
1, the disease will spread and therewill be an epidemic. An epidemic will occur if dI/dt >
0. The derivation of ananalytical expression for R depends on the epidemiological model. Hence, for themodel with density-dependent between-herd transmission we have: dIdt = U ( βI + k ) − Ic > ⇒ U ( βI + k ) c > I ⇒ U βc + kIc > R is defined as R = U βc + kIc (17)11imilarly, for the model with frequency-dependent between-herd transmissionthe basic reproduction number is R = U βN c + kIc (18)For the model with no between-herd transmission (i.e. β = 0) the basic repro-duction number is R = kIc (19)Equation (19) holds for both density- and frequency-dependent transmission.The reproduction numbers for the four counties j = 1 , . . . ,
4, were calculated bysubstituting the parameter estimates of the best fitting model and associated valuesof U j and N j into the above equations. Green and Medley (2002) argue that proof of the accuracy or validity of any modelshould be required before it is used to influence policy. The most important aspect ofthis is that a valid model should be true for data not used in the modeling process.Therefore, the parameter estimates of the best fitting model to data set 1 wereapplied to the next year of data (i.e. badger data from 1998/1999 and cattle datafrom 1997/1998) to get estimates of the number of restricted herds, which were thencompared to the observed numbers of restricted herds.Models also need to be checked for identifiability, as was done in modeling mas-titis in dairy cows by White et al. (2001). In the models presented here, we notethat if there exist two pairs, ( α , β ) and ( α , β ) that give the same I ∗ value thenthe maximum likelihood estimates of α and β will not be unique. There were fourcounties so it would be extremely unlikely to get two α and β pairs that give thesame I ∗ j , for all j = 1 , . . . ,
4. Numerical checks were undertaken, however, to ensurethis had not occurred. The log-likelihood of each model, given by equation (9), was12omputed for a grid of α and β pairs and distinct values were obtained over the gridto indicate that identifiability was not a problem.Another issue in model validity is that the model should not be considered com-plete until sensitivity analysis is used to identify the rates that have a large impacton the modeling process. It is important that sensitive rates are estimated correctly;if the data for these rates are poor, then more data are required.A sensitivity analysis similar to that described in Cross and Getz (2006) wascarried out to determine the relative importance of several parameters of interest.Eight random values for six parameters of interest were chosen from uniform distri-butions bounded by the minimum and maximum values, as listed in Table 2. Over250,000 parameter sets were created from all possible combinations of the eight ran-dom values for the six parameters. For each parameter set, we calculated the totalpredicted infected herds ( I ∗ c ) based on the best fitting model to data set 1.Each of the six parameters were standardised by transforming them into thepercentage difference from the mean (i.e. ( x i − ¯ x ) / ¯ x , where x i is the value of theparameter on run i and ¯ x is the mean from all runs). A linear regression with thetotal predicted infected herds as the dependent variable and the six standardisedmodel parameters as the independent variables was carried out to determine whichparameters were of significant importance.In addition, another sensitivity analysis was conducted where parameter setswere created with p (the length of time on MC), s (SICTT sensitivity) and a/c (the proportion of infected herds detected via slaughterhouse surveillance) set totheir minimum and maximum likely values as indicated in Table 2. Model fitting ofmodels 1 to 11 was repeated to determine if model selection and parameter estimates( ˆ α and ˆ β ) remain stable across varying values of the other parameters.Data analyses were performed in R version 2.12.1.13 Results
From the analysis of the proportion of cattle herds with newly detected bTB infectionin a year, the model that best fit data set 1 was model 2, with the highest Akaikeweight of 0.520 (see Table 4). Model 2 assumes no between-herd transmission ( β = 0)and assumes badger-to-herd transmission is proportional to the total number ofinfected badgers culled ( ˆ α = 0 . × q j = I ∗ j c / N j by equation (10) are 12.4%, 5.5%, 4.5% and 5.8%respectively.Substituting the maximum likelihood estimates of the unknown parameters frommodel 2 into I ∗ j c/N j , j =1,2,3,4 (Equation (10)) (where I ∗ for model 2 is defined inEquation (7) with k = αI w ) and then into equation (19), gives the reproductionnumber for the jth county, j = 1 , . . . , R = kIc = αI wj I ∗ j c = (0 . I wj I ∗ j (1 .
25) (20)For Cork, Donegal, Kilkenny and Monaghan the reproduction numbers are 0.004,0.003, 0.005, and 0.002 respectively. All are very similar, and below 1, hence signi-fying that bTB will die out in the long-run.Models 6 and 10 both have Akaike weights of 0.229, signifying that there issubstantial support for them also. However, they are essentially the same modelsas model 2 since they both have badger-to-herd transmission proportional to the14otal number of infected badgers culled ( ˆ α = 0 . I w ) is shown in Figure 2 (based on dataset 1). Since ˆ α >
0, Figure 2 shows apositive relationship between the proportion of newly infected cattle herds in a yearand the number of infected badgers culled in a year. It also shows the implications,based on the model, that a highly infected badger population would have on theinfection rates of the cattle population, assuming that infection rates of the culledbadgers is indicative of the infection rates of the whole badger population. As theinfected badger population grows, so too will the proportion of infected herds in thepopulation. However, the fitted model is based on only four empirical observationsand that for county Cork is highly influential in the fitted model.Based on the best model fit to data set 1, model 2, a 50% reduction in the pro-portion of infected badgers culled would result in a 46% reduction of bTb incidencein herds in Cork and a 48% reduction in the other three counties- Donegal, Kilkennyand Monaghan.The results for data sets 2, 3 and 4 are very similar to those of data set 1.The best fitting model in each case is model 2, where β = 0 and the estimatedvalues for α differ only slightly from that of data set 1 (data set 2: ˆ α = 0 . , C.I. : 0 . − . α = 0 . C.I. : 0 . − . α = 0 . C.I. : 0 . − . . The results for data set 5 from thepre-study period only are completely different from those obtained for the otherfour data sets, with model 8 as the best fitting model, which estimates a non-zerofrequency-dependent between-herd transmission parameter ( ˆ β = 2 . α = 0 . .2 Model Validity The parameter estimates of the best fitting model to data set 1 (i.e. ˆ β = 0 andˆ α = 0 . I ∗ j , the number of restricted herds.Values of I ∗ j c =16, 6, 6, 21 for the four counties respectively were obtained comparedto the observed numbers of restricted herds 29 , , ,
19. The lack of agreementreflects the yearly variation in herd bTB incidence in these data and large changesin the badger population due to proactive badger culling.A question also arises as to how critical the assumed badger-to-herd infectionrates are in these models, as model 8 is best for the pre-study period while model 2 isbest for the study period. The fifth data set used data on badgers from the pre-studyperiod only, when few badgers were culled. In all models, it is assumed that thenumber of badgers culled is equal to the badger population size and thus the badgerpopulation size is greatly underestimated for data set 5. Models were re-fitted withadjustments to the population size, to examine how results changed. Assuming N w and I w as in the first year of the study period for the data of pre-study, all modelswere re-fitted to this data set and the best one found i.e. N and B from data set5 were used with the N w and I w from data set 1. The results for this adjusteddata set 5 were similar to those for data set 1. Model 2 (and 6, 10) was the bestfit with no between-herd transmission ( β = 0) and the badger-to-herd transmissionrate is proportional to the total number of infected badgers culled, estimated as( ˆ α = 0 . β =0, ˆ α =0.0022, 95% C.I.: 0.0018-0.0026).16 .2.2 Identifiability To check the identifiability of the models, the log-likelihood of each model, givenby equation (9), was computed for a grid of α and β pairs and distinct values wereobtained over the grid indicating identifiability was not a problem. Table 3 shows the results of the sensitivity analysis. The table of results ranksparameters in decreasing order of standardised coefficients (i.e. b/ S.E.) size. Thestandardised coefficients allow us to compare the relative importance of the sixparameters. The results indicate that all six parameters are important in the model, N , the total number of herds, being the most important and herd test sensitivity, s ,being the least. It can also be seen that the greater the rate of disease transmissionfrom badgers, the greater the predicted number of infected cattle. The average lengthof time a herd spends on MC in years also influences the course of the epidemic. Thelonger the average length of time a herd spends on MC, the smaller the predictednumber of infected cattle. Also, the greater the sensitivity of the SICTT herd test,the greater the predicted number of infected cattle, as to be expected.In the second sensitivity analysis several parameter sets were created with p (thelength of time on MC), s (SICTT sensitivity) and a/c (the proportion of infectedherds detected via slaughterhouse surveillance) set to their minimum and maximumlikely values as indicated in Table 2 and models 1 to 11 were re-ran. For all parametersets, the model selection and parameter estimates ( ˆ α and ˆ β ) remained unchangedthus indicating the robustness of the results to changes in values of the parameters p , s and a/c . 17 Discussion
The study has been performed on a small set of observational data and there is verylittle repetition in the data since there were only four counties, thus we cannot drawconclusions from the results with great certainty. However, the herd populationsize in each of the four counties studied was large. Thus, within the limitationsabove and other study limitations described below, estimated model parameters areprecise.Five data sets were taken from the cattle and badger data from the FAP in Irelandand for four of them, the model that best fit assumed transmission from badgersto cattle was proportional to the total number of infected badgers culled and hadno herd-to-herd-transmission. The assumption regarding the badger population sizewas critical in the differing model selection for the fifth data set. When this wasadjusted and a more realistic badger population size assumed, results similar to theother data sets were obtained. These findings suggest herd-to-herd transmission isof much lesser importance for these areas than badger-to-herd transmission.The results indicate that there is a significant association between levels of bTB inbadgers and cattle. This does not prove causality but is, however, in agreement withresults from the FAP and RBCT where some measure of causality was established(Griffin et al., 2005; Bourne et al., 2007). Reductions in cattle bTB incidence, due toproactive culling of badgers, ranged from 51% to 68% over a five-year culling periodin the FAP and 23% in the RBCT. However, since herd bTB was not eliminated,these results also indicated there are sources of infection other than the badger.In this model, it was assumed that there were only two possible sources of infec-tion of cattle herds, between-herd infection and infection from wildlife, namely wildbadgers, to cattle. Infection from any other wildlife, such as deer, was ignored, aswas within-herd infection (i.e. cattle-to-cattle infection in a single herd). As in Bar-low et al. (1998), re-infection of wildlife, in our case wild badgers, from cattle herds18as also considered to be negligible and the inferences made here were conditionalon any such reinfections being negligible. In addition, it was assumed that cattleinfection, as shown by reaction on a skin test, was equivalent to the animal beinginfectious and that there is no carrier state in cattle or badgers. Issues in relationto animal testing are discussed in Clegg et al. (2011).In human populations, one-third of the world’s population is infected, eitherlatently or actively, with tuberculosis (Ozcaglar et al., 2012). The rate of latentinfection of cattle herds in Ireland has yet to be established. Ozcaglar et al. (2012)in their simulation study, showed that a human tuberculosis epidemic can be viewedas a series of linked subepidemics: a fast tuberculosis subepidemic driven by directprogression, a slow tuberculosis subepidemic driven by endogenous reactivation, anda relapse tuberculosis subepidemic driven by relapse cases; thus, proving that youngand mature tuberculosis epidemics behave differently and suggested that differentcontrol strategies may be necessary for controlling each subepidemic. Thus, the issueof latent infection of a herd is an important aspect of bTB epidemiology as this willperhaps drive a subepidemic by endogenous reactivation. A similar statement can bemade in relation to wild badgers and has important implications for vaccine testingalso, now underway (Aznar et al., 2011). Previous history of infection in a herd hasbeen shown to be a risk factor for bTB in many studies (Griffin et al., 2005; Bourneet al., 2007) and this may be related to latent infection. Thus, the important aspectof epidemic modelling of latent infection with bTB needs study.Epidemic models are particularly suitable for investigating the likelihood of per-sistence versus fade-out of infection. Blower et al. (1995) demonstrated that it takesseveral hundred years for a tuberculosis epidemic in humans to rise, fall, and reachan endemic state. Our models estimated approximate reproduction numbers of lessthan 1, indicating the epidemic would ultimately fade-out, although slowly. Thiswas an asymptotic result and assumed a constant herd population size, while infact, the herd population size in Ireland is continually changing (Kelly et al., 2008).19ore accurate estimates of reproduction numbers and epidemic length would re-quire a separate simulation study. For example, Vynnycky and Fine (1998) includeestimates of infection and re-infection rates over time for different ages, and includerates at which individuals who have been infected or reinfected for a time withoutdeveloping disease move into the ‘latent’ class, in their models.The models used here assumed the population of herds was closed - no immi-gration i.e. no recruitment. However, from the sensitivity analysis, we can concludethat bTB transmission is influenced by the herd population size i.e. the numberof herds. This suggests that increases in population size as a result of economic orother policies may impact strongly on the course of the disease. The assumption ofa closed herd population also implies there are no introduced cattle into any herdover the study period and no long range cattle movements. Both of these assump-tions are unrealistic, particularly in the Irish context and both factors are knownrisk factors in herd bTB (Sheridan, 2011).Our results are quite different to those of Donnelly and Hone (2010) that werebased on data from the RBCT. They found the model that best fit the RBCT data,using a data set comparable to data set 1 here, was model 11, which has frequency-dependent between-herd transmission and badger-to-herd transmission proportionalto the proportion of infected badgers culled. They found much stronger support forfrequency-dependent badger-to-cattle transmission than density-dependent. Our re-sults, based on the FAP data, suggest that there is stronger support for density-dependent badger-to-cattle transmission, with the wildlife transmission variable be-ing the number of infected badgers culled. In a simple comparison we found thatthe biggest difference between the RBCT data used by Donnelly and Hone (2010)and the FAP data used here, was that the proportion of infected badgers is muchhigher in each of the four removal areas in the FAP than in the majority, thoughnot all, badger proactive culling areas of the RBCT. The FAP and RBCT are alikein terms of numbers of herds per km and infection rates in herds.20t should also be noted that the models used here are idealised in form and inaddition to the limitations described above, they do not take into account otherfeatures of the epidemic in question. For example, it is assumed the rate of badger-to-cattle herd transmission is independent of herd size and this may not be true.Herd size has been shown in many studies to be one of the most important riskfactors for herd bTB (Griffin et al., 2005; Bourne et al., 2007; Kelly et al., 2008)and the sensitivity analysis above showed changes in rates of badger-to-cattle herdtransmission has a dramatic effect on model results. Thus, results should be inter-preted as representing transmission dynamics with transmission rates averaged overherd sizes.Farm management practice changes, control policy changes, economic changesand climate changes may all affect the course of an epidemic and these factors requireseparate study. In addition spatial correlation structures related to transmission mayalso be important. Cowled et al. (2012) found that epidemic models for swine feverin wild pigs in Australia that do not take realistic spatial structures of the wildlifepopulation into account may overestimate the rate at which a disease will spreadand overestimate the size of an outbreak. A more detailed and complete study couldbe considered for the future.However, this study does provide further evidence for the importance of the roleof wild badgers on bTB levels in cattle herds in Ireland. This study also demon-strates that, unlike in Great Britain (based on the RBCT), herd-to-herd transmissionof bTB is of much lesser importance. References
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See Section 2.2 for a full descriptionof each data set.Area Total Herds(N) Total HerdsRestricted(B) Total Bad-gers Culled(Nw) Total In-fectedBadgersCulled (Iw)Data set 1Cork 292 48 235 68Donegal 379 1 190 27Kilkenny 229 21 188 22Monaghan 680 36 165 29Data set 2Cork 288 29 235 68Donegal 375 3 190 27Kilkenny 230 14 188 22Monaghan 687 19 165 29Data set 3Cork 288 67 446 115Donegal 375 14 273 40Kilkenny 230 34 409 56Monaghan 701 112 414 66Data set 4Cork 294 230 467 116Donegal 396 87 282 44Kilkenny 233 119 618 107Monaghan 701 338 627 88Data set 5Cork 294 163 21 1Donegal 396 73 9 4Kilkenny 233 85 209 51Monaghan 680 226 213 2226able 2: Sensitivity Analysis - Parameter BoundariesParameter Symbol Minimum Maximum SourceTotal number of herds N
200 1,000 From data in thispaperAverage length of time a herdspends on MC in years p s a α I w
10 100 From data in thispaper27able 3: Linear regression sensitivity analysis using over 250,000 runs of model 2 withparameter values chosen from uniform distributions and total predicted infected herdsas the dependent variable.Parameter Symbol Coefficient( b ) StandardError (S.E.) b/ S.E.Total number of herds N α a I w p -143.01 1.49 -95.75Herd test sensitivity rate inper cent s All parameters were transformed to percentage difference from the mean (i.e. ( x i − ¯ x ) / ¯ x , where x i is the value of the parameter on run i and ¯ x is the mean from all runs). All coefficients have a p-value less than 0.001. ∝ : proportional to. AICc: Akaike Information Criterion corrected for small sample sizes relative to the number of parameters. In the case when we assume one of the transmission parameters, β or α , is zero, we omit the parameters estimate and no p-value is calculable. When we assume one of the transmission parameters, β or α , is zero, the calculation of the p-value of the other transmission parameter is notapplicable (N/A) since the null hypothesis would state that both parameters were zero and hence, there would be no transmission of tuberculosis tocattle. able 4: Parameter estimates and log-likelihood values from the eleven varying models fitted to data set 1 (see Table 1) onbovine tuberculosis in cattle and badgers from the Four Area Project. Each of the models represents some combinationof the two fitted parameters β and α , where β is a measure of herd-to-herd transmission per annum (Density Dependent(DD) or Frequency Dependent (FD)) and k is a measure of badger-to-herd transmission per annum, such that k = αN w , k = αI w or k = α ( I w /N w ), where N w equals the number of badgers culled in an area and I w equals the number ofinfected badgers culled in an area. Model 2, highlighted in dark grey, has the most support from the data with an Akaikeweight of 0.52.Model Between-herdtransmission Transmissionfromwildlife Numberofparameters β p-value H : β = 0 α p-value H : α = 0 Log-likelihood AICc Akaikeweight1 None ∝ N w - - -382.4703 766.94 0.0002 None ∝ I w - - -372.4763 746.96 0.5203 None ∝ I w /N w - - -376.2648 754.53 0.0124 DD None 1 0.0006 N/A - - - - -752.6828 1507.39 0.0005 DD ∝ N w < ∝ I w < ∝ I w /N w < - - - - -388.7466 779.50 0.0009 FD ∝ N w < ∝ I w < ∝ I w /N w < igure 1: Representation of the transfer of cattle herds between states: U (unin-fected), I (Infected) and M (on Movement Control). Parameters are as follows: k isthe rate of infection from badgers to cattle per year, β is the between-herd trans-mission coefficient, c is the rate at which infected herds go on movement controlper annum and p is the average length of time in years a herd spends on movementcontrol. 31igure 2: The four circles represent the observed proportion of herds which experi-ence a bTB herd breakdown in a single year in the four counties in the analysis andthe solid line represents the fitted model of the proportion of herds that experience abTB herd breakdown in a year ( I ∗ jj
10 100 From data in thispaper27able 3: Linear regression sensitivity analysis using over 250,000 runs of model 2 withparameter values chosen from uniform distributions and total predicted infected herdsas the dependent variable.Parameter Symbol Coefficient( b ) StandardError (S.E.) b/ S.E.Total number of herds N α a I w p -143.01 1.49 -95.75Herd test sensitivity rate inper cent s All parameters were transformed to percentage difference from the mean (i.e. ( x i − ¯ x ) / ¯ x , where x i is the value of the parameter on run i and ¯ x is the mean from all runs). All coefficients have a p-value less than 0.001. ∝ : proportional to. AICc: Akaike Information Criterion corrected for small sample sizes relative to the number of parameters. In the case when we assume one of the transmission parameters, β or α , is zero, we omit the parameters estimate and no p-value is calculable. When we assume one of the transmission parameters, β or α , is zero, the calculation of the p-value of the other transmission parameter is notapplicable (N/A) since the null hypothesis would state that both parameters were zero and hence, there would be no transmission of tuberculosis tocattle. able 4: Parameter estimates and log-likelihood values from the eleven varying models fitted to data set 1 (see Table 1) onbovine tuberculosis in cattle and badgers from the Four Area Project. Each of the models represents some combinationof the two fitted parameters β and α , where β is a measure of herd-to-herd transmission per annum (Density Dependent(DD) or Frequency Dependent (FD)) and k is a measure of badger-to-herd transmission per annum, such that k = αN w , k = αI w or k = α ( I w /N w ), where N w equals the number of badgers culled in an area and I w equals the number ofinfected badgers culled in an area. Model 2, highlighted in dark grey, has the most support from the data with an Akaikeweight of 0.52.Model Between-herdtransmission Transmissionfromwildlife Numberofparameters β p-value H : β = 0 α p-value H : α = 0 Log-likelihood AICc Akaikeweight1 None ∝ N w - - -382.4703 766.94 0.0002 None ∝ I w - - -372.4763 746.96 0.5203 None ∝ I w /N w - - -376.2648 754.53 0.0124 DD None 1 0.0006 N/A - - - - -752.6828 1507.39 0.0005 DD ∝ N w < ∝ I w < ∝ I w /N w < - - - - -388.7466 779.50 0.0009 FD ∝ N w < ∝ I w < ∝ I w /N w < igure 1: Representation of the transfer of cattle herds between states: U (unin-fected), I (Infected) and M (on Movement Control). Parameters are as follows: k isthe rate of infection from badgers to cattle per year, β is the between-herd trans-mission coefficient, c is the rate at which infected herds go on movement controlper annum and p is the average length of time in years a herd spends on movementcontrol. 31igure 2: The four circles represent the observed proportion of herds which experi-ence a bTB herd breakdown in a single year in the four counties in the analysis andthe solid line represents the fitted model of the proportion of herds that experience abTB herd breakdown in a year ( I ∗ jj c / N jj