Modelling the spread of bovine viral diarrhea virus (BVDV) in a beef cattle herd and its impact on herd productivity
 Alix Damman^{1, 2},
 AnneFrance Viet^{1, 2},
 Sandie Arnoux^{1, 2},
 MarieClaude GuerrierChatellet^{3},
 Etienne Petit^{3} and
 Pauline Ezanno^{1, 2}Email author
DOI: 10.1186/s1356701501458
© Damman et al.; licensee BioMed Central. 2015
Received: 1 April 2014
Accepted: 6 January 2015
Published: 24 February 2015
Abstract
Bovine viral diarrhea virus (BVDV) is a common pathogen of cattle herds that causes economic losses due to reproductive disorders in breeding cattle and increased morbidity and mortality amongst infected calves. Our objective was to evaluate the impact of BVDV spread on the productivity of a beef cowcalf herd using a stochastic model in discrete time that accounted for (1) the difference in transmission rates when animals are housed indoors versus grazing on pasture, (2) the external risk of disease introductions through fenceline contact with neighboring herds and the purchase of infected cattle, and (3) the risk of individual pregnant cattle generating persistently infected (PI) calves based on their stage in gestation. The model predicted the highest losses from BVDV during the first 3 years after disease was introduced into a naive herd. During the endemic phase, the impact of BVDV on the yearly herd productivity was much lower due to herd immunity. However, cumulative losses over 10 years in an endemic situation greatly surpassed the losses that occurred during the acute phase. A sensitivity analysis of key model parameters revealed that herd size, the duration of breeding, grazing, and selling periods, renewal rate of breeding females, and the level of numerical productivity expected by the farmer had a significant influence on the predicted losses. This model provides a valuable framework for evaluating the impact of BVDV and the efficacy of different control strategies in beef cowcalf herds.
Introduction
Bovine viral diarrhea virus (BVDV) affects most industrialized cattle farming systems by inducing reproductive disorders (abortion, delayed calving, reduced fertility) in breeding cattle and by lowering herd productivity through increased culling, morbidity, and mortality [1]. Introductions may occur through the direct purchase of infected animals when cattle are housed indoors as well as through fenceline contact with infected animals in neighboring herds when cattle are grazed outdoors on pasture. The likelihood of these introductions depends on the control measures that are implemented by individual farms. In some areas (e.g. Brittany, France: [2]), purchased animals are guaranteed not to be persistently infected based on knowledge of their dam status, previous diagnostic testing, or their source herd status. In other areas, although there are prepurchase diagnostic tests available for BVDV, most farmers tend not to use them [3,4]. It is also difficult to determine whether pregnant females on pasture are at subsequent risk of delivering persistently infected (PI) calves since there are few reliable prenatal tests for BVDV. The severity of production losses following disease introduction is also related to several additional management factors, including (1) the level of herd immunity from previous natural exposure [5] or preventative vaccination [6], (2) the percentage of dams that are at risk for generating PI calves through vertical transmission, and (3) the ability for BVDV to spread within and between different production subgroups within a herd [7]. In the absence of a calf surveillance scheme, it may be difficult for farmers to detect the presence of BVDV in the herd leading to the establishment of an endemic disease state and long term production losses.
Modelling is a pertinent approach to predict pathogen spread and persistence in a herd and to evaluate its impact on herd dynamics and productivity for a large range of management scenarios [8]. Many of the modelling studies conducted to date have concerned dairy cattle herds, both at the herd [7,9,10] and regional scales [1113]. However, beef cowcalf herds have a unique demographic structure that has not been captured by previously published models. First, beef cattle are frequently grazed outside for long periods, particularly at the time when pregnant dams have the greatest risk of generating PI calves following exposure to BVDV through fenceline contacts. Second, the calving period is concentrated over a few months and calves are raised with cows until weaning. This increases the duration and intensity of exposure to BVDV as PI animals mainly are observed in young stock due to a shortened life expectancy [14]. Therefore, conclusions drawn for dairy herds cannot be directly transferred to beef farming systems. Models of BVDV spread in a beef herd have been proposed to evaluate the costs associated with epidemics in naive herds [15] or to compare control [16] and testing [17] strategies. Two recent models account for BVDV introduction due to animal purchases or fenceline contacts [5,18], representing an endemic situation. However, none of these models simultaneously account for the withinherd contact structure, the difference between the indoor and outdoor periods in withinherd virus transmission, and the risk of continuous virus introduction due to the purchase of animals and contacts with neighboring infected herds. All of these processes are expected to greatly influence BVDV spread and persistence in a beef cowcalf herd and, consequently, impact the associated losses.
Our objective was to evaluate the impact of BVDV spread on the productivity of a beef cowcalf herd across a large range of management scenarios. A stochastic epidemiological model was proposed that accounted for different transmission rates between separately managed production groups during the outdoor grazing period versus a homogeneous population structure during the indoor period. The model also incorporated an external risk of BVDV introduction through fenceline contacts with neighboring herds as well as through animal purchases. A sensitivity analysis was conducted to determine the relative importance of management factors such as herd size, the length of breeding, grazing, and selling periods, the replacement rate of breeding females, and level of numerical productivity expected by the farmer. Comparisons of production losses in acute outbreaks and endemic situations were also performed.
Materials and methods
A stochastic compartmental model in discrete time was developed to simulate the spread of BVDV. A time interval of 7 days was chosen as the longest as possible to properly represent transientlyinfected animals in the infection process. The model was fully implemented in C++, allowing the model to be run rapidly.
Herd dynamics
Definitions and values of model parameters
Parameters  Values  Definitions  Sources 

ρ _{ sex }  0.5  Sex ratio  
a, b  8.7, 6  Parameters of the gamma distribution used to calculate the next start of pregnancy  
δ  0.035  Probability of twin birth  
τ _{ He }  0.02  Probability of infertility for heifers  
τ _{ Co }  0.08  Probability of infertility for cows  
breeding_start  15^{th} of March  Date of start of the breeding period  
breeding_dur  [16 18 20]  Duration of the breeding period (in weeks)  
weaning  1^{st} October  Date of weaning  
pasture_start  1^{st} April  Date of start of the pasture period  
pasture_dur  [29 32 35]  Duration of the pasture period (in weeks)  
renewal_rate  [0.286 0.317 0.349]  Ratio heifers/cows  
sell_period  [autumn winter spring]  Sell period of grassers^{1}  
size  [42 83 125]  Number of bred females^{2}  
μ _{ Ca,bi }  0.0225  Probability of mortality at birth of calves  
μ _{ Ca }  0.000333  Mortality rate of calves (d^{1})  
intro_week  [(20 25 30) 27 40]  Week of introduction of PI animal(s)^{3}.  
ε  [0.95 1 1.05]  Level of numerical productivity expected by the farmer  
μ _{ P,bi }  [0.06 0.0667 0.0733]  Probability of mortality at birth of PI calves  
μ _{ P }  [0.0017 0.0019 0.0021]  Mortity of PI animals per day  [20] 
ϕ _{ MS }  [0.006 0.00667 0.00733]  Trantion rate from state M to state S (d^{1})  [21] 
ϕ _{ TR }  [0.18 0.2 0.22]  Transition rate from state T to state R (d^{1})  [22] 
β ^{ T }  [0.027 0.03 0.033]  Daily transmission rate for T animals  
β ^{ P }  [0.45 0.5 0.55]  Daily transmission rate for PI animals  
\( {\beta}_b^P \)  [0.09 0.1 0.11]  Daily betweengroup transmission rate for PI animals  
α _{ Ra }  [0.72 0.8 0.88]  Abortion rate due to infection in early pregnancy  
α _{ Rb }  [0.18 0.2 0.22]  Abortion rate due to infection in midpregnancy  
η _{ X }  Probability of giving birth to a calf in state X if infection in midpregnancy and no abortion  
η _{ P }  [0.875 0.9375 1]  
η _{ M }  [0.0625 0.03125 0]  
η _{ R }  [0.0625 0.03125 0]  
K _{ ext }  0  Risk of virus introduction on pasture 
The herd was structured into 7 groups: calves from birth until weaning, male and female weaned calves for selling (grassers), heifers under the age of two kept for renewal, bred heifers, cows from the first pregnancy diagnosis until fattening decision, cows from fattening decision until culling, and bulls. The herd dynamics relies on specific dates when animals change groups (Figure 1; Table 1).
At weaning, some female calves were grouped with young heifers for renewal while others were fattened for sale during the year. The gender of calves was determined stochastically according to the sex ratio ρ _{ sex } (Table 1). The number of females selected for renewal was fixed. In case of unexpectedly high calf mortality, replacement calves could be purchased. The replacement of a dead calf was allowed from the beginning of the calving period until three months after its end. Purchase occurred if the number of calves present and to be born in the herd fell below the production objective.
At the beginning of the indoor period, all pregnant heifers and cows were merged to be raised together while nonpregnant ones were fattened for 100 days before being sold. The model determines the expected number of calvings according to the production objective. If the expected number of pregnant animals was below the target number of calvings to meet production objectives, pregnant females were purchased at the beginning of the indoor period to reach this number.
At the beginning of the breeding period, a fixed number of females was selected among the 2yearold heifers to form the group of bred heifers. The remaining 2yearold heifers were sold. Cows were split into two groups. A fixed number of cows formed the breeding stock while the others were fattened until their calves were weaned. Then, they were culled. The period began 2 weeks earlier for heifers than for cows, and ended when bulls were separated from breeding females. We integrated in the compartmental model an individualbased monitoring of pregnant females. From the start of pregnancy until calving, each female is represented individually to precisely predict her stage of pregnancy over time.
Calving occurred 285 days after the beginning of gestation and the mother was then not available for breeding for a period of 20 days. Twins were born with probability δ (Table 1). During the breeding period, the delay between the moment when a cow was available for breeding and the start of a new pregnancy was determined by a gamma distribution with parameters a and b. The values of these parameters (Table 1) were chosen to reproduce the observed calvingtocalving intervals as presented in [19]. The same gamma distribution with the same parameter values was used regardless of whether breeding females were indoors or at pasture. The date of a new pregnancy was calculated if the animal was not declared infertile. The probability of infertility of heifers and cows is given by parameters τ _{ He } and τ _{ Co }, respectively (Table 1). The calculated new pregnancy date must be before the end of the breeding period otherwise the animal was considered as nonpregnant.
The simulated average date of calving was 60 days after the beginning of the calving period. The indoor and breeding periods can be chosen within a certain range. It was assumed that both the calving and the breeding periods started during the indoor period. The date of weaning was chosen so that calves were weaned at the average age of 6 up to 8 months on the field.
All animals placed in the group of grassers after weaning were sold over the course of the year. The model simulates various periods for sales. These periods were specifically considered because they impact the duration that potentially infected animals are present in the herd. Indeed, animals can be sold at one, two or three periods in the year. Dates of selling were randomly chosen each year using triangular distributions for the three periods: (23/10, 15/11, 07/12), (23/01, 15/02, 07/03) and (23/05, 15/06, 07/07). The proportion of animals sold at each period was fixed at the beginning of each simulation.
The number of bulls present in the herd was assumed to be constant. The model assigns one bull per 20 bred heifers or cows. Each year, on November 1st, a bull was randomly selected for replacement.
Withinherd infection dynamics
where N ^{ P } and N ^{ T } are the total numbers of PI and T animals in the herd, respectively; N the herd size, and β ^{ P } and β ^{ T } the transmission rates per day associated with the PI and T animals, respectively (see Table 1).
with \( {N}_k^P \) and \( {N}_k^T \) the numbers of PI and T animals in pasture k, respectively, N _{ k } the number of animals in pasture k, k’ all the pastures in contact with pasture k, and \( {\beta}_b^P \) the betweengroup transmission rate per day associated with PI animals.
Immune cows (R) gave birth to calves protected by maternal antibodies (M) acquired via colostrum. Susceptible (S) or PI (P) cows gave birth to calves in the same state. If mothers were infected during pregnancy, consequences differed depending on the stage of the pregnancy at the time of infection. As females are individually monitored during pregnancy, their stages are known precisely. The pregnancy period was divided into three stages: early pregnancy (041 days; R _{ a }), mid pregnancy (42150 days; R _{ b }), and late pregnancy (151285 days; R _{ c }). Infection during the first stage led to either embryonic or fetal death with probability α _{ Ra } (Table 1), or the birth of a calf protected by maternal antibodies (M). Different consequences of infection during midpregnancy are possible [22,24,2730]. The pregnant female may abort with probability α _{ Rb } (Table 1). If not, the calf can be born in states M, R, or P with probabilities η _{ M }, η _{ R }, or η _{ P }, respectively. Finally, if infection occurs during late pregnancy, the mother gives birth to an immune calf (R). In case of abortion, a delay of 60 days is applied between the end of infection and abortion. The female is then unavailable for breeding for 20 days. If abortion occurs not too late, i.e. at least 20 days before the end of the breeding period, the female may return to the pregnancy state using the algorithm explained in the previous section.
where N _{ i } represents the number of individuals present in compartment i. In the case of multiple transfers, we used multinomials instead [32]. The transfers between groups were made by randomly selecting animals from all health states.
Females in the group of young heifers stayed two years in that group. For this group, all compartments were doubled to differentiate oneyear and twoyearold heifers. At the beginning of the reproduction period, only twoyearold heifers were either selected for breeding and transferred in the bred heifer group, or sold.
At birth, PI and nonPI calves had a probability of dying of μ _{ P,bi } and μ _{ Ca,bi }, respectively (Table 1). The proportion of deaths at birth for PI calves encompassed the deaths of abnormal calves. The model associates a mortality rate μ _{ Ca } with nonPI calves between calving and weaning and μ _{ P } with all PI animals (Table 1). Finally, 9% of all calves died before weaning and PI animals had a halflife of 1 year.
Purchased animals (calves, pregnant females, and bulls) can be of any health state in the model (S, T, P, RP).
Initial conditions, reference scenario, and simulations
A simulation year started after weaning (1^{st} October), corresponding to week 0 (Figure 1). In the reference scenario, the indoor period ranged from week 6 to 26 (midNovember to March). The breeding period started during the indoor period on week 23 (midMarch) for bred heifers and on week 25 (end of March) for cows. It finished on week 41 (midJuly) when bulls were separated from breeding females. The calving period ranged from week 12 to 29. It corresponds to the indoor period plus one week. The initial herd was obtained by running the model for 4 years without BVDV introduction. The fourth year was used to obtained mean reference values for purchases, sales, and number of weaned calves. Then, the birth of a PI calf was simulated at the beginning of the cow breeding period (week 27) in an average herd representative of herds in the Bourgogne region (France). The number of bred heifers and cows are 20 and 63, respectively. We assumed a basic level of numerical productivity expected by the farmer, i.e. the farmer does not expect losses to differ from usual infertility of breeding females and calf mortality (ε = 1). In such a situation, the production objective was equal to 73 weaned calves per year (for 83 bred females). After introducing a PI animal, the simulation continued for 15 years, the first three years being representative of an acute phase (infection arising in a naive herd), whereas years 6 to 15 were representative of an endemic phase (when infection persists in the herd). In the reference scenario, we assumed that all the purchased animals were susceptible and that no infection due to neighboring contacts occurred (K _{ ext } = 0). For each scenario considered thereafter, 3000 repetitions were performed.
Outputs
Outputs were selected to represent infection dynamics and the impact of BVDV on herd productivity. Outputs associated with infection dynamics are the probability of virus persistence in the herd (infected herds having ≥ 1 PI or T animal, or ≥ 1 immune dam carrying a PI fetus), and the prevalence of PI and T animals and of immune dams carrying a PI fetus (state RP). Prevalence of PI and T animals represents the proportion of PI and T animals in the whole herd while prevalence of immune dams carrying a PI fetus is restricted to breeding females only. Outputs related to herd productivity are the number of losses (abortions and deaths of PI animals), purchases (replacement calves, pregnant females and bulls), weaned calves, sales of grassers and young heifers, and sales of empty and fattened females. To evaluate the impact of BVDV on herd productivity, we subtracted the contribution of the reference year (the last year before the first BVDV introduction) from these last outputs, considering only the relative change with and without BVDV circulating. Losses and purchases were also evaluated per bred female to remove the direct impact of herd size on such outputs.
For all outputs except virus persistence, we calculated the annual median value with an 80% credible interval (P10P90) for each year after BVDV introduction by selecting only repetitions in which the virus was still present in the herd at the end of the year, i.e. at weaning. Virus persistence was calculated weekly.
Impact of the herd structure outdoors on BVDV spread
To test the effect of our assumptions regarding the structure of herds during the indoor and outdoor periods, two options were compared with the reference scenario: (1) no structure is considered, assuming all animals are homogeneously mixed as indoors; (2) three groups are considered as in the reference case but with no contact between them (β^{ P } _{ b } = 0).
Sensitivity analysis

parameters related to the herd management: level of numerical productivity expected by the farmer ε, duration of both the breeding (breeding_dur) and outdoor (pasture_dur) periods, renewal rate of breeding females (renewal_rate), herd size (size), and period of selling (sell_period);

parameters related to the infection dynamics: mortality at birth of PI animals (μ _{ P,bi }), mortality of PI animals (μ _{ P }), transition rates (ϕ _{ MS }) and (ϕ _{ TR }), transmission rates (β ^{ P }, β ^{ T }, \( {\beta}_b^P \)), abortion probabilities in early (α _{ Ra }) and midpregnancy (α _{ Rb }), probability of giving birth to a PI calf for a dam infected during midpregnancy (η _{ P }), and type of virus introduction (intro_week).
Three values were tested per parameter (Table 1). For continuous parameters (rates and proportions), we tested variations of 90%, 100%, and 110% of their nominal value (except for ε for which the variation was ± 5% to remain within a plausible range, and for η_{ P } which cannot be above 1). For other parameters (periods, herd size, and virus introduction), we tested for plausible values. Three selling periods are possible in the field, animals (including PI) being kept longer or shorter accordingly. We tested for selling all of the sold animals at each of these 3 periods. Durations of the breeding and the pasture periods varied by ± 2 and 3 weeks, respectively. To cover the variety of herd sizes in the Bourgogne region, we tested three numbers of females kept for breeding: 42 (10 heifers  32 cows), 83 (20 heifers  63 cows, reference scenario), and 125 (30 heifers  95 cows). Finally, BVDV introduction may occur through different ways in a naive beef cattle herd. In addition to the birth of a single PI (week 27, reference scenario), we tested the case of multiple births of PI calves (weeks 20, 25, and 30 successively, i.e. at the start, in mid, and at the end of the calving period), and the case where a PI replacement calf is purchased during the outdoor period (week 40). Other purchased animals could also be infected and therefore introduce BVDV in the herd. However, pregnant females are purchased at the start of the building period, thus when most of the females are in late gestation. Introducing a transiently infected female will barely have any effect. Introducing an immune dam carrying a PI fetus will have the same influence as introducing a PI calf at birth. Introducing a PI pregnant female is quite rare. Lastly introducing a PI bull could have a large effect but only if introduced directly in a group of bred females among which some are already pregnant, which also corresponds to the period of purchase of replacement calves. Therefore, we chose to present here the most probable cases that are the birth of PI calves and the purchase of a PI replacement calf.
Since herd size and the type of virus introduction in the herd were expected to largely impact model outputs, 9 (3 herd sizes × 3 types of introduction) sensitivity analyses were carried out to evaluate the effect of other model parameters. We used a fractional factorial design to sample parameter values [33]. A factorial design is appropriate when the levels of some input variables are discrete (such as periods). In such a design, all the combinations between variable levels are considered, leading to p ^{ n } scenarios when n parameters with p levels are considered. Using a fractional design (using the proc factex, SAS) enabled us to considerably reduce the number of scenarios and is appropriate when sensitivity indices for principal effects and firstorder interactions only are estimated. Two thousand onehundred and eightyseven scenarios were run for each analysis.
We analyzed aggregated outputs calculated as the mean values over the first 5 years after BVDV introduction of losses (mortality and abortion) and of the prevalence of T (prevT) and PI (prevP) animals, and of immune dams carrying a PI fetus (prevRP), in an infected herd.
For each output k, a linear regression model (ANOVA) was run with all model parameters: k _{ ij …} = μ + f(i, j, …) + ϵ, with μ a constant, f the relation between factors (i, j, …), and ϵ the residual. The total sum of squares then writes: \( S{S}_{tot}^k={\displaystyle \sum_{i,j,\dots }}{\left({k}_{ij\dots }k'\dots \right)}^2=S{S}_i^k+S{S}_j^k+S{S}_{i:j}^k+S{S}_{\in}^k \) (here for two factors i and j), with \( S{S}_i^k \) and \( S{S}_{i:j}^k \) the sum of squares related to factor i and to the firstorder interaction between factors i and j for output k, respectively. The contribution of factor i to variations in output k is \( {C}_i^k=\frac{S{S}_i^k+\frac{1}{2}{\displaystyle {\sum}_{j\ne i}}S{S}_{i:j}^k}{S{S}_{tot}^k} \). The sum of the contributions was equal to model R ^{2}.
Impact of BVDV spread in an endemic situation
Five years after BVDV first introduction, if the virus is still present in the herd then an endemic state has been reached. To evaluate the impact of BVDV spread in such an endemic situation, cumulated outputs over 10 years (year 6 to 15 after virus introduction) were calculated to enable a comparison between the endemic situation and the acute one (based on the first 3 years). Only repetitions for which virus was present at least one week were included. Moreover, cumulated outputs were normalized to account for the proportion of time the herd truly was infected. Outputs thus were multiplied by the ratio of the number of weeks the virus was present in the herd over the total duration in weeks of the acute and the endemic periods, respectively. The comparison between the acute and the endemic periods was initially evaluated without allowing virus reintroduction through purchases of infected animals or fenceline contacts with infected neighboring herds during the outdoor period. As these factors can significantly influence disease persistence, we also simulated BVDV reintroduction in the herd accounting for a probability of purchasing infected animals (TI, P or immune dam carrying a PI fetus) and for a probability of fenceline contacts with neighboring infected herds during the outdoor period (K _{ ext }). As no information was available on observed withinherd prevalence of BVDV infection in infected herds, we assumed a risk of purchasing infected animals on our best knowledge (1% for TI, 1% for P, and 0.5% for immune dams carrying a PI fetus) and assuming two levels of regional prevalence of infected herds: weak (10%) and strong (50%). For fenceline contacts, we evaluated three levels of external risk: nil (K _{ ext } = 0), weak (K _{ ext } = 0.0025), and strong (K _{ ext } = 0.01). Each case was tested for each of the three herd sizes and type of initial virus introduction which is expected to impact the acute phase.
Results
BVDV spread in a naive cowcalf herd
Herd size and the type of initial BVDV introduction in the herd impacted the spread and persistence of BVDV in a naive cowcalf herd.
Regardless of herd size, the annual prevalence of PI animals and immune dams carrying a PI fetus in an infected herd reached a maximum the year after the year of virus introduction (i.e. year 2). The prevalence of transiently infected animals was the highest during the year of virus introduction (year 1) when 3 PI were successively introduced. In the other scenarios, it was the highest the second year. In medium herds, after the birth of a single PI calf, the prevalence on year 2 of T, PI and dams carrying a PI fetus in 80% of the repetitions ranged from 1.6 to 2.0%, 1.0 to 3.6%, and 2.8 to 6.6%, respectively (Figure 3B). The prevalences slightly decreased with herd size. In small herds only, the median prevalences were higher if a PI calf was introduced outdoors (1.8%, 4.7%, 9.0%, respectively) than during the calving period (1.6%, 3.0%, 5.6%, respectively). For other herd sizes, the prevalences were mostly similar among types of virus introduction. Three to four years after BVDV introduction, the prevalences tended to stable values, denoting that an endemic state had been reached which was not affected by the initial virus introduction. An endemically infected herd was predicted to have from 0.3 to 1.1% of T animals, from 0.3 to 2.4% of PI animals, and from none to 0.9% of its dams carrying a PI fetus.
Regardless of herd size and type of virus introduction, the highest impact of BVDV spread on herd productivity occurred during the second year (Figure 3C). Losses associated with abortions and PI mortality were the highest the first two years, ranging from 0.16 to 0.26 per bred female. Losses per bred female were slightly lower in large herds. Additional purchases and sales of empty and fattened females, and losses in weaned calves that were due to BVDV spread were the highest the second year and then rapidly decreased. Purchases and sales of empty and fattened females per bred female were closely related and ranged from 0.05 to 0.17, with no effect of herd size. In more than half the repetitions (median), less than 2 weaned calves were lost due to BVDV spread over 3 years. The number of grassers and heifers sold was not impacted the first two years, a decrease occurring only in the third and fourth years. If the virus was introduced outdoors, losses were later and lower. Without any subsequent virus introduction, the productivity outputs tended to be barely affected 5 years after the first BVDV introduction.
Impact on BVDV spread of the herd structure outdoors
Assuming a heterogeneous mixing outdoors modified the predicted model outputs compared with assuming a homogeneous mixing. On the contrary, in the case of a heterogeneous mixing outdoors, assuming no contact between groups did not change model predictions compared with assuming the occurrence of betweengroup contacts (reference scenario).
The predicted prevalences of PI animals and immune dams carrying a PI fetus in infected herds were higher during the acute phase when assuming a homogeneous vs. heterogeneous mixing outdoors, regardless of herd size and type of virus introduction. In a herd of 83 bred females, the predicted median prevalence of PI animals on year 2 varied with the type of BVDV introduction and ranged from 3.1 to 3.8% in the homogeneous mixing scenario compared with 2.3 to 2.8% in the heterogeneous mixing scenario. The median prevalence of dams carrying a PI fetus on year 2 ranged from 6.0 to 6.7% in the homogeneous mixing scenario compared with 4.6 to 5.7% in the heterogeneous mixing scenario. Herd structure on pasture had no effect of the prevalence of T animals.
Predicted losses (abortion and PI mortality), purchases, and sales of empty and fattened females were slightly higher during the acute phase when assuming a homogeneous mixing outdoors, regardless of herd size and type of virus introduction.
Impact of herd management and infection characteristics on BVDV spread
Losses varied among scenarios of the sensitivity analysis in the ranges 815, 1330, and 1842 animals in small, medium, and large herds, respectively. The prevalence of T animals varied among scenarios between 0.8 and 1.6%, the prevalence of PI animals varied between 0.9 and 3.0%, and the prevalence of dams carrying a PI fetus varied from 1.4 and 4.8%, irrespective of herd size.
Losses variations were mainly explained by pasture duration (pasture_dur) and the choice of the selling period (sell_period) for parameters related to herd management, and by the abortion rate due to infection in early pregnancy (α _{ Ra }) and the transmission rate by PI animals (\( \beta \) ^{ P }) for parameters related to infection characteristics (Figure 5A). Introducing a PI replacement calf (week 40) in small herds led to a decrease in the contributions of these key parameters except sell_period, and led to an additional contribution of the probability of given birth to a PI calf for dams infected in midpregnancy (η_{ P }) and of PI mortality (μ_{ P }).
The variations in the prevalence of T animals in an infected herd (Figure 5B) were mainly explained by the choice of the selling period (sell_period), the transient infection duration (ϕ_{ TR }), and slightly by pasture duration (pasture_dur), the transmission rate by PI animals (β^{ P }), and PI mortality (μ_{ P }). The variations in the prevalence of PI animals (Figure 5C) were mainly explained by sell_period and the renewal rate (renewal_rate), and in some cases pasture_dur, as well as by the probability of given birth to a PI calf for dams infected in midpregnancy (η_{ P }), μ_{ P,} and in some cases β^{ P }. The variations in the prevalence of dams carrying a PI fetus (Figure 5D) were mainly explained by the same parameters as the variations in the prevalence of PI animals, except sell_period which barely contributed.
BVDV spread in an endemic situation
The risk of buying infected animals was low. Even when half of the source herds for replacement animals were assumed to be infected with BVDV, the model predicted that BVDV would be reintroduced to medium herds only once every 20 years during the endemic period. Indeed, only 1 to 5 animals were purchased per year per herd irrespective of the BVDV herd status, with a low risk that these animals were infected due to the low withinherd prevalence of infection.
Comparison of the outputs of the model of BVDV spread in a beef cowcalf herd cumulated over the acute (years 1 to 3 after initial introduction) versus the endemic phase (years 6 to 15)
Output definition  Acute phase  Endemic phase 

Average probability of virus presence^{1}  0.750.760.76  0.140.170.25 
Average frequency of herd reinfection^{1} (yr^{1})  00.020.08  00.140.38 
Median prevT (when the virus is present) (%)  1.31.5  0.51.0 
Median prevP (when the virus is present) (%)  1.43.5  01.2 
Median prevRP (when the virus is present) (%)  2.15.5  01.3 
Median losses/100 bred females  2337  1253 
Median purchases/100 bred females  1326  34125 
Median sales of grassers & heifers/100 bred females  72  129 
Median sales of fattened females/100 bred females  1023  23108 
Median weaned calves/100 bred females  52  239 
Discussion
We propose a stochastic model of BVDV spread in a structured beef cowcalf cattle herd. Originally, we accounted for a variation in exposure of animals (especially adults) between the indoor period, during which a homogeneous contact structure is assumed, and the outdoor period, during which groups are formed and raised on different pastures. Moreover, we accounted for a risk of continuous virus introduction in the herd through animal purchases and contacts with neighboring infected herds. This enabled the model to represent a large range of possible situations, from a single introduction into a naive herd to endemic situations potentially maintained by an external risk of virus reintroduction. Lastly, the model was flexible enough to represent beef herds of different sizes and management types, as illustrated by the different types of herds observed in Bourgogne, one of the main beef cattle farming regions in France. Such a model was pertinent to investigate the impact of BVDV spread on the productivity of a herd for a large range of scenarios.
To precisely represent the infection process, several modelling choices had to be made. The model was stochastic to enable an estimation of the probability of virus persistence. The variability in model outputs if the virus persisted was not very large. However, the prevalence in PI animals and in immune dams carrying a PI fetus in infected herds was low and therefore better estimated using a stochastic model. A discrete time step of 1 week was used as the longest permitting to precisely estimate the morbidity related to transiently infected animals (transient infection being quite short). We chose to implement a combination of compartmental and individualbased models  instead of using a fully individualbased approach as used in [34,35] – with the anticipation of using this withinherd model as part of a more complex simulation framework where computational efficiency is paramount, to represent betweenherd BVDV spread and control at a regional scale. Indeed, representing individually each animal all its lifetime was not necessary, except during pregnancy to precisely predict when infection will occur relative to the stage of pregnancy of the dam, and which consequence it will have for the calf to be born. Hence, we integrated in the compartmental model an individualbased monitoring of pregnant females (from the start of pregnancy until calving). To account for the seasonality of breeding and outdoor contacts, discrete periods were designed for reproduction and grazing. Such an approach is suitable when the continuous epidemic process (infection occurring possibly each day) is influenced by seasonal population dynamics (and therefore seasonal contacts) occurring on a discrete basis (with for instance only two periods in a year with different types or levels of contact). All of the model parameters related to herd size (number of bred cows and heifers) and management (target number of weaned calves, occurrence and intensity of purchases and of neighboring relationships, within herd contact structure, breeding/calving and indoor/outdoor periods) are userdefined. Hence, our model is highly flexible and may also be used to represent beef cattle herds in other regions.
The simulation results show that failing to account for the separation of cattle into different management groups on pasture could lead to overestimations of the predicted disease prevalence and persistence in affected herds. In our study, we assumed that there were three management groups and that these management groups remained fixed over the entire grazing period. However, in the real world, herd structure is often dictated by complex management constraints such as herd size, pasture availability, and labor resources. Furthermore, the management groups of bred females with bulls may be reformed several times during the grazing period, which may lead towards more homogeneous mixing dynamics. There is a need for further research into the effects of herd structure on BVDV spread using empirically derived data.
When introduced into a naive herd, BVDV spread was shown to have a large impact on herd productivity, especially the first 3 years after the initial introduction of the virus, during which yearly losses may be up to 6 times higher than in subsequent years when herd immunity has developed. The impact expressed per bred female was slightly lower in larger herds. Moreover, in the absence of control measures, BVDV may persist for years, which is currently observed in some regions [3,4,36,37]. We found that the virus was more likely to persist over time in larger herds. Virus persistence may increase with herd size because selfclearance may be more frequent in small herds due to stochastic events [38,39]. In such an endemic situation, our model predicted that yearly losses will be limited and that the production objective in weaned calves will be reached most of the time, in good agreement with the results obtained for Scottish beef herds [5]. However, the low yearly losses have to be balanced by the duration of virus persistence. Cumulated over several years, losses occurring during the endemic phase cannot be neglected, especially when virus reintroduction is frequent.
The sensitivity analysis shows that herd management (through pasture duration, selling period, and renewal rate) and the infection dynamics (though the transmission rate from PI, the duration of transient infection, the vertical transmission, PI mortality, and abortion rates) were very influential processes on at least some of the model outputs (prevalence of T and PI animals and of immune dams carrying a PI fetus, losses), regardless of herd size and type of virus introduction. As for dairy herds [10], it suggests that control of PI animals is the key for preventing BVDV spread in beef herds. PI animals may be introduced via either purchases or births to immune dams infected during midpregnancy. In Bourgogne, the breeding period occurs essentially during the outdoor period. Hence, it is an atrisk period for breeding females since they can become infected from contact with neighboring herds.
Among the available strategies to control BVDV spread in beef herds, vaccination is one of the main current options. According to our findings, female vaccination before breeding seems to be a valuable strategy to limit losses due to BVDV spread and persistence [4043]. Vaccination has been used largely in the US [44], but much less in the EU [43], except in Germany where it has been used in combination with eradication [45]. Such a strategy could become efficient and should be further evaluated in a context where herds may be regularly reinfected through neighboring contact. In the present study, the impact of BVDV on herd productivity was evaluated by measuring biological outputs (losses, variations in weaned calves, etc.). For the model to be useful in evaluating control strategies, it would be important to assign economic values to variations in herd productivity [5,18] and to account for farmer's decisions [46,47]. Indeed, farmers may be unwilling to implement control measures if the economic impact of BVDV on their herd is perceived to be low. Farmers of endemically infected herds may also become unaware of their own BVDV status [3,4,36] and of the risk they pose of transmitting the virus through animal movements and neighboring contact with other herds [48]. Moreover, if the risk of disease reintroduction is high, farmers may not perceive the value of controlling it even if they are aware it is present. Our model then can be used to estimate the expected prevalence of PI animals as well as of immune dams carrying a PI fetus in such endemically infected herds, therefore providing a prior for the risk of infection for (potentially naive) contact herds. Several groups are identified in the modelled herd based on age/physiological stages and health statuses, enabling the model to be used in the future to evaluate targeted control strategies at the herd scale in either naive or endemic situations.
Declarations
Acknowledgements
This work was carried out with the financial support of the French Research Agency (ANR), Program Investments for the Future, project ANR10BINF07 (MIHMES), by the European fund for the regional development (FEDER PaysdelaLoire), and by INRA. The authors thank J. Devun (Institut de l’Elevage) and R. Vermesse (GDS35) for their help in defining the assumptions on the herd demography. The authors also thank two anonymous referees for their helpful comments on a previous version of the manuscript.
Authors’ Affiliations
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