Inferring bovine tuberculosis transmission between cattle and badgers via the environment and risk mapping

Bovine tuberculosis (bTB), caused by Mycobacterium bovis, is one of the most challenging and persistent health issues in many countries worldwide. In several countries, bTB control is complicated due to the presence of wildlife reservoirs of infection, i.e. European badger (Meles meles) in Ireland and the UK, which can transmit infection to cattle. However, a quantitative understanding of the role of cattle and badgers in bTB transmission is elusive, especially where there is spatial variation in relative density between badgers and cattle. Moreover, as these two species have infrequent direct contact, environmental transmission is likely to play a role, but the quantitative importance of the environment has not been assessed. Therefore, the objective of this study is to better understand bTB transmission between cattle and badgers via the environment in a spatially explicit context and to identify high-risk areas. We developed an environmental transmission model that incorporates both within-herd/territory transmission and between-species transmission, with the latter facilitated by badger territories overlapping with herd areas. Model parameters such as transmission rate parameters and the decay rate parameter of M. bovis were estimated by maximum likelihood estimation using infection data from badgers and cattle collected during a 4-year badger vaccination trial. Our estimation showed that the environment can play an important role in the transmission of bTB, with a half-life of M. bovis in the environment of around 177 days. Based on the estimated transmission rate parameters, we calculate the basic reproduction ratio (R) within a herd, which reveals how relative badger density dictates transmission. In addition, we simulated transmission in each small local area to generate a first between-herd R map that identifies high-risk areas.


.1 Infectious period
The infectious period of cattle in ROI depends on the test frequency.The onset of the infectious period of a bovine is usually not exactly known.Therefore, we assume that cattle get infected at the middle point of two tests.We extract cattle testing data results in Kilkenny trial area during the vaccination study and calculate the average duration infectious period.Details on test scheme are explained below in 2.4.2 cattle data.
Once badgers get infected, they seem to have a life-long infectiousness.However, the excretion may be intermittent and vary greatly throughout time.The effective duration of infection period can be approximately derived as the average life span of infected animals (excluding the latent period) (Anderson & Trewhella, 1985).The life expectancy after bTB infection varies from 35 days to 3.5 years in in laboratory studies, with the most of badgers survive between one to two years (Little et al., 1982;Cheeseman et al., 1985).Another study also found that badgers with bite infection usually have more acute disease progress and the survival of those badgers with bite infection was estimated as 117 days (CI 0 to 341 days) (Clifton-Hadley et al., 1993).However, badgers who have apparent respiratory origin infection have a mean survival time of 491 days with CI 253 to 729 days (Clifton-Hadley et al., 1993).Based on all those information, we assume a one year infection period for badgers.

Latent period
The latent period of bTB in badgers is not well understood due to a lack of effective method of detecting M. bovis in live animals.Culture and examination methods have been used in some old experimental studies.However, those methods are labour intensive and prone to error, and should be interpreted with caution (Anderson & Trewhella, 1985).Little et al. (1982) undertook an animal experiment with a small sample size and used the duration between the first exposure to the first time of recording M. bovis excretion.This study found the range of latent period is from 95 days to 158 days, with a comment that the excretion is intermittent.Since the previous badger transmission model assumed a 3-month latent period based on those estimation, we used the same assumption (Anderson & Trewhella, 1985).
In cattle, animal challenge study has found a short latent period ~30 days (Kao et al., 2007), while other models have assumed a lengthy latent period of 6 to 20 months (Barlow et al., 1997;Fischer et al., 2005).A more recent modelling study compared two models with long or short latency but could not distinguish the two assumption (SOR and SORI) from within herd transmission data (Conlan et al., 2012).In SOR model, cattle are infectious once infected, but cattle cannot be detected in occult stage.In this assumption, occult cattle are estimated to become responsive to test cattle in 1.8 days.In the other assumption, infected cattle go through occult, responsive to test and then infectious.In this assumption, cattle's latent period is estimated to be 406 days or 28 days based on different prior information with the similar model fit.As the animal experiments also suggest short latent period, we adopt a short latent period model assuming the latent period as 1.8 days (Conlan et al., 2012).

Background death rate
The natural death rate for badgers is calculated from the capture-mark-recapture study conducted in an undisturbed wild population in the west of England (Rogers et al., 1997).A survival probability each year was reported in the study.We assume an exponential distribution of death and therefore can derive an average lifespan of 1330 days with a natural death rate of 7.52e-4 per day.
The lifespan of cattle varies between farms, herd types and so on.In the Irish farming system, the average lifespan for beef breeds is almost 3 years, being slightly lower than for dairy breeds (M.H. Poola, 2005).Another study suggests that the annual culling rate is about 20% (Maher et al., 2008).Therefore, we assume the background death rate for cattle is about 1/3 per year (9.13e-4 per day).

Confidence bounds for partial reproduction ratio
We calculate the confidence bounds for partial reproduction ratios based on the confidence bounds of the transmission rates and the decay rate parameters estimation.The   and   have narrow confidence bounds than other partial R because   and   are estimated from the cattle infection data, which are more abundant than badger infection data.

Smoothing badger prevalence data
Badger infection data were extracted from the badger vaccination trial in Kilkenny.Despite of the intensive data collection, the badger prevalence data are scarce at the resolution in this spatial model.Therefore, we use statistical learning to investigate the association between the spatial location of badger territories, the time and the badger prevalence.With the best-fitted relationship, we can predict the badger in a finer resolution.
Badger annual prevalence at territory level has been calculated and fitted into several spline smooth models.The badger annual prevalence is the response of the model.The coordinates of the centroid of badger territories (x,y) and the time (day) are the three predictors in statistical model.The predictor (prevalence) has the value from 0 to 1, hence a binomial distribution with a logit link relationship between predictor and response is assumed.We fit predictors to the observed prevalence data using smoothing spines.The goal was to find a function s() that fits the observed data well, while not overfitting the data.This means to find a function that minimizes ∑ (  − (  )) 2 +  ∫  ′′ () 2   =1 , where  is a tunning parameter and the function s() is the smoothing spline.The functions for different predictors are additive, hence this statistical learning method is also called generalised additive models (GAM).Five statistical models with different predictors were tested: Model 1: only time  = () Model 2: time and the vaccination zone  = () + () Model 3: coordinates (x,y)  = s(, ) Model 4: coordinates (x,y) and time  = (, ) + () Model 5: coordinates (x,y) and time with interaction  = (, , ) From the AIC of these 5 models, the best fit smoothing spline is the Model 5 where the coordinates (x, y) and t are the three predictors with interaction between coordinates and time (x*y*t);(Table 3).The predicted prevalence over space is visualized in Figure 8.The predicted prevalence at each territory varies from 0 to 0.7 with the mean prevalence 0.28 (Figure 1B).Temporal prevalence changes in a few example badger territories are presented in Figure 9.

Model selection on different denominator in infection force
In this paper, we use the total number of cattle to represent a local.Badgers act as vectors that lives in farms but does not determine the area.The model structure is similar to vector-borne disease where the number of the host in an area is the denominator (Hartemink et al., 2009;Cecilia et al., 2020) The statistic model descripted in 2.3 is used to fit these three models to estimate the parameters.The goodness of fit for these three models was used to select the best fit model ( Despite the changes in transmission rate parameters, the decay rate parameter has limited impact on the partial R value.From the point estimation, higher decay rate ( =0.02) resulted in higher  , and  , .However, the changes in the partial R does not influence much on the R for the system (Figure 10).

Table 2
Confidence bound for partial reproduction ratios

Table 3
AICfor smoothing models Nb determines the areaCompared to Model 1, all the denominator in ODE is changed to Nb. Nb +Nc determines the area In the ODE version of transmission model, the denominator is the total number of cattle and badgers.

Table 5 .
Goodness of fit for three transmission model structures Therefore, we conduct a sensitivity analysis to investigate how would decay rate parameter influence the result of this study in terms of estimation on transmission rate parameter, R, and badger-cattle ratio threshold.

Table 7 . Sensitivity test of decay rate on NGM
] Supplementary Figure 3. Within-herd R in an isolated herd with different relative badger density.A) = .; B)  = .