Transmission dynamics of hepatitis E virus in pigs: Estimation from field data and effect of vaccination

Epidemics 4 (2012) 86–92

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Transmission dynamics of hepatitis E virus in pigs: Estimation from field data and effect of vaccination J.A. Backer a,∗ , A. Berto a,b , C. McCreary b , F. Martelli b , W.H.M. van der Poel a a b

Central Veterinary Institute of Wageningen UR, P.O. Box 65, 8200 AB Lelystad, The Netherlands Animal Health and Veterinary Laboratories Agency, New Haw, Addlestone, Surrey KT15 3NB, United Kingdom

a r t i c l e

i n f o

Article history: Received 16 September 2011 Received in revised form 12 February 2012 Accepted 23 February 2012 Available online 3 March 2012 Keywords: Hepatitis E virus Mathematical model Reproduction number Bayesian analysis MCMC sampling

a b s t r a c t Hepatitis E is a viral disease that causes serious concerns for public health. Hepatitis E virus (HEV) genotype 3 is endemic in commercial pig farms worldwide that act as a reservoir. Pig-to-human transmission may occur when infectious animals enter the food chain at slaughter, through consumption of contaminated meat, direct exposure or use of by-products. To reduce the fraction of infectious animals at slaughter age and thus the risk for public health, it is important to understand the transmission dynamics of HEV in pig populations. In this study, we estimate the transmission rate parameter and mean infectious period of HEV in pigs from field data, using a Bayesian analysis. The data were collected in ten commercial pig herds that are each divided into three different age groups. Two transmission models were compared, assuming that animals are infected either locally by their group mates or globally by any infectious animal regardless of its group. For local and global transmission, the transmission rate parameters were 0.11 (posterior median with 95% credible interval: 0.092–0.14 day−1 ) and 0.16 (0.082–0.29 day−1 ), the mean infectious periods were 24 (18–33) days and 27 (20–39) days and the reproduction numbers were 2.7 (2.2–3.6) and 4.3 (2.8–6.9). Based on these results, global transmission is considered to be the more conservative model. Three effects of vaccination were explored separately. When vaccination is not sufficient to eliminate the virus, a shorter mean infectious period decreases the fraction of infectious animals at slaughter age, whereas a reduced transmission rate parameter adversely increases it. With a reduced susceptibility, vaccination of animals at a later age can be a better strategy than early vaccination. These effects should be taken into account in vaccine development. © 2012 Elsevier B.V. All rights reserved.

Introduction In 1997, hepatitis E virus (HEV) genotype 3 was first discovered in pigs in the United States (Meng et al., 1997), followed by detections in other countries, such as The Netherlands (Van der Poel et al., 2001) and the UK (Banks et al., 2004). HEV is widespread in commercial pig farms, of which more than 50% may be infected (Rutjes et al., 2007; Seminati et al., 2008). Although HEV infected pigs do not show severe disease (Halbur et al., 2001), they are thought to be a reservoir from which humans can be infected (Meng, 2010). This is supported by similarities between swine and human strains (Meng et al., 1998; Van der Poel et al., 2001). In humans, hepatitis E can cause clinical disease with symptoms such as jaundice, anorexia, hepatomegaly, and mortality in pregnant women (Emerson and Purcell, 2003). Pig-to-human trans-

∗ Corresponding author. E-mail address: [email protected] (J.A. Backer). 1755-4365/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.epidem.2012.02.002

mission may occur when infectious animals enter the food chain at slaughter, putting slaughterhouse workers at risk (Vulcano et al., 2007), as well as public health through contaminated pig meat (Yazaki et al., 2003; Feagins et al., 2007; Colson et al., 2010). Minimizing the fraction of infectious animals at slaughter age would reduce these risks. To achieve this, the transmission dynamics of HEV in pig populations should be studied and understood. Field studies – both cross-sectional (Fernández-Barredo et al., 2006; Di Bartolo et al., 2008) and longitudinal (De Deus et al., 2008; Casas et al., 2010) – have shown a peak prevalence of HEV RNA in pigs of intermediate age, and a non-zero prevalence in finishing pigs at slaughter age. This prevalence pattern can be understood using a model that describes the transmission between pigs. The pattern is determined by how fast a susceptible animal can be infected (expressed by the transmission rate parameter) and how long an infectious animal sheds virus (expressed by the mean infectious period). The product of these two parameters is the reproduction number R0 that represents the number of infections one infectious animal can cause in a fully susceptible popula-

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Table 1 Number of positive samples/sample size of fecal samples tested for Hepatitis E virus in three different age groups on 10 pig herds. Herd no.

Herd 1 Herd 2 Herd 3 Herd 4 Herd 5 Herd 6 Herd 7 Herd 8 Herd 9 Herd 10

Sin 1

Pig group (age) Weaners (6–9 weeks)

Growers (10–12 weeks)

Fatteners (13–26 weeks)

2/5 1/5 0/5 0/5 1/5 2/5 0/5 2/5 4/5 1/5

1/5 4/5 2/5 1/5 2/5 2/5 2/5 1/5 3/5 4/5

0/5 1/5 1/5 0/5 0/4 0/5 1/5 1/5 0/5 0/5

tion. The reproduction number has been estimated from field data using the results of a serological survey (Satou and Nishiura, 2007). However, the fraction of infectious animals at slaughter age depends on all transmission parameters and these have been determined in an experimental setting only (Bouwknegt et al., 2008). Here we estimate all parameters that determine the transmission dynamics of HEV between pigs from field data. Using the test results of sampled animals in three different pig groups (weaners, growers and finishers) in ten different commercial pig herds, it is possible to estimate the transmission rate parameter and mean infectious period simultaneously. We make a distinction in transmission between group mates and herd mates, to estimate the transmission parameters using a Bayesian analysis. With the estimated parameters we evaluate the fraction of HEV RNA positivity at slaughter age, and explore the effect of vaccination on it. Data Data were taken from McCreary et al. (2008) who sampled and tested ten fattening herds in the United Kingdom for HEV in 2007. Pigs entered the herd at a weaning age of 6 weeks, after which they passed through three different stages: weaners (6–9 weeks of age), growers (10–12 weeks of age) and fatteners (13–26 weeks of age), until slaughter at 26 weeks of age. Out of each of these age groups, fecal samples from five animals were taken, i.e. 15 per herd and 150 in total. As no information was available on the herd sizes, we will assume that these samples represent 5% of the group sizes. The fecal samples were tested by RT-PCR to detect viral RNA (McCreary et al., 2008). The results of this cross-sectional study (Table 1) showed that all herds were positive for HEV. In general, the peak prevalence is observed in the middle group of growers, although this varies per herd. Similar age-dependent prevalence patterns have been observed in previous field studies (FernándezBarredo et al., 2006; De Deus et al., 2008; Casas et al., 2010). HEV transmission model The model to describe HEV transmission in a pig herd has the same age structure as the data (Fig. 1). Each age group is subdivided in three distinct compartments that consist of pigs that are susceptible (S), infectious (I) or recovered (R) (Keeling and Rohani, 2008). The system described by this SIR model is assumed to be in an endemic equilibrium, i.e. the virus is assumed not to be introduced by infected weaners or other external sources but the disease can sustain itself in the regenerating pig population. This endemic equilibrium can only exist when the virus is sufficiently transmissible. The transmissibility is expressed by the reproduction number R0 that represents the number of secondary infections caused by one infectious animal during its entire infectious period in a fully sus-

weaners

S1

I1

R1

growers

S2

I2

R2

fatteners

S3

I3

R3

Sout 3

Iout 3

Rout 3

Fig. 1. Schematic overview of age-structured HEV transmission model. The pig population is divided in weaners, growers and fatteners (subscripts 1, 2 and 3), further subdivided in the fractions of susceptible (S), infectious (I) and recovered (R) animals.

ceptible and infinite population (Keeling and Rohani, 2008). When this number is smaller than one, the outbreak will die out and cannot sustain itself. So, the endemic equilibrium assumption also contains the assumption that R0 > 1. The SIR model choice means that we ignore the latent period (i.e. an infected animal is immediately infectious) and that we assume the infected animals to be immune upon recovery. The latent period between infection and shedding of infectious virus, was observed to be 3 (2–7) days in intravenously inoculated pigs (Bouwknegt et al., 2008) and deduced to be less than 7.2 (4.8–9.6) days in contactinfected pigs (Bouwknegt et al., 2009). As the latent period is small compared to the residence times in the different groups we will ignore it to avoid computational complexity. After infection, antibody levels increase (Bouwknegt et al., 2009), leading to increasing seroprevalence with increasing age (Takahashi et al., 2003; De Deus et al., 2008; Casas et al., 2010). When immunity is completely developed, the infected animal stops shedding virus and is assumed to be insusceptible to the infection. The animals entering the pig herd are assumed to be fully susceptible to infection. At the entering age of 6 weeks, the level of maternal antibodies has sufficiently decreased (Kanai et al., 2010; Jiménez de Oya et al., 2011) to ignore passive immunity in the model. The infectious period is assumed to be gamma distributed. Using the shedding periods of contact infected animals in two independent blocks (Bouwknegt et al., 2008) we found the best fitting shape parameter to be 4 by comparing corrected AIC values (Hurvich and Tsai, 1989). The probability density function g(t|) of the resulting infectious period distribution with a mean of  is:

g(t|) =

128 exp(−4t/)t 3 34

(1)

and G(t|) is the corresponding cumulative distribution function. The gamma distributed infectious period with shape parameter 4 can be modeled by four infectious stages in series. Infectious animals move from one infectious stage to another at a constant rate of 4/, which means that the duration in each stage is exponentially distributed. The fraction of susceptible animals S(a, t) of age a at time t and the fractions of infectious animals Ij (a, t) of age a at

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time t in infectious stage j are described by the general time- and age-dependent PDE system:

∂S(a, t) ∂S(a, t) = −S(a, t)(a, t) − ∂t ∂a ∂I1 (a, t) ∂I1 (a, t) 4 = S(a, t)(a, t) − I1 (a, t) −  ∂t ∂a ∂Ij (a, t) ∂Ij (a, t) 4 4 = Ij−1 (a, t) − Ij (a, t) − for 2 ≤ j ≤ 4,   ∂t ∂a

(2)

averaged over all infection times  occurring before entering the group. The second part gives the probability that an animal that entered susceptible is infected at time  and is still infectious at in of susceptible anideparture time Ti . Similarly, the fraction Si+1 mals entering group (i + 1) is equal to the fraction Siout of susceptible animals that escaped infection in the preceding group i: in Si+1 = Siout = Siin exp(−i Ti )

with (a, t) the infection pressure experienced by animals of age a at time t and boundary conditions S(0, t) = 1 and Ij (0, t) = 0. The endemic equilibrium assumption sets the left-hand sides of Eq. (2) to zero and the structure of the data dictates that the animals are grouped in age groups i with residence times Ti of 28, 21 and 91 days for weaners, growers and finishers. For the parameter estimation we will define the probability of testing a random animal positive for each group, depending on the model parameters and the number of infectious animals that can be expected in each group according to the measured data.

(6)

Weaners entering the first group are assumed to be susceptible, setting the boundary conditions S1in = 1 and I1in = 0. An animal in group i is tested positive either because it entered the group infectious and still is, or because it entered susceptible pos and is infected since. The probability pi of a positive test is: pos

pi

=

1 Ti

 0



Ti

Iiin



+Siin

1 



0

(1 − G(t − |))d −∞



t

i exp(−i )(1 − G(t − |))d

dt,

(7)

0

Parameter estimation The number of infectious animals J in a group is not directly observed. Only a sample of n = 5 animals is taken out of a larger group of assumed size N = 100, which restricts the outcome of observations to k = 0, 1, 2, 3, 4, or 5 positive animals. Assuming a test sensitivity and specificity of 100%, these positively tested animals are taken to be infectious. The ‘true’ number of infectious animals J follows a distribution, similar to a hypergeometric distribution:

  J k

f (J|N, n, k) =

N−J n−k

  N n



L(k, n, N, T |ˇL , ˇG , , J) =

n+1 , N+1

(3)

where the fraction (n + 1)/(N + 1) normalizes the distribution. The number of infectious animals Ji in group i as well as the total number of infectious animals determine the infection pressure i in that group i (Becker and Dietz, 1995). Assuming frequency dependent mixing (Keeling and Rohani, 2008),



TJ J j j j i = ˇL i + ˇG  , Ni Tj Nj

(4)

j

where the   fraction Ji /Ni is the prevalence in group i and the fraction

j Tj Jj / j Tj Nj is the prevalence in the entire herd, weighted to the residence times T . The local transmission rate parameter ˇL describes the transmission when an animal can only be infected by its group mates and the global transmission rate parameter ˇG describes the transmission when an animal can be infected by an infectious animal from any group. The reproduction number in this model is the sum of the transmission rate parameters multiplied by the mean infectious period, R0 = (ˇL + ˇG ). The prevalence in a group is also dependent on infectious animals that are transferred in of to this group from the preceding age group. The fraction Ii+1 infectious animals entering group (i + 1) is equal to the fraction Iiout of infectious animals leaving the preceding group i:

in = Iiout = Iiin Ii+1

 +Siin

1 



This expression is similar to Eq. (5), but instead of calculating the probability of being infectious at departure, the outer integral takes the average over all times t during the residence time Ti . This gives the probability of being infectious for a random animal from group i. With this probability, we can now calculate how likely the observations are if there were Ji infectious animals and (Ni − Ji ) noninfectious animals (either susceptible or recovered) in each group i given the parameters ˇL , ˇG and , using the likelihood function:

0

(1 − G(Ti − |))d −∞

Ti

i exp(−i )(1 − G(Ti − |))d.

(5)

0

The first part of this expression gives the probability that an animal that entered infectious, still is at departure time Ti ,

3  



f (Ji |Ni , ni , ki )

Ni Ji

pos Ji ) (1

(pi

pos Ni −Ji

− pi

)



(8)

i=1

Maximizing this function would lead to a maximum likelihood estimate for , ˇL and ˇG and J, but this is a computationally intensive procedure. Instead, we explore the parameter space in a Bayesian framework. Because of the limited information available, we restrict the analyses to a strictly local or a strictly global model, i.e. analyzing L( k, n, N, T |ˇL , 0, , J) or L( k, n, N, T |0, ˇG , , J). All integrals in Eqs. (5)–(7) are analytically solvable as a function of , i and Ti , so only simple substitution is needed to calculate the likelihood function. The transmission parameters and mean infectious period are estimated for each farm individually and assuming common parameters for all farms. Uninformative prior distributions are presumed for the transmission rate parameter (exponential with mean 1), the mean infectious period (gamma with mean 1 and shape 0.01) and the number of infectious animals (uniform). These three (set of) parameters are initialized at their maximum likelihood estimate and updated separately using a Metropolis–Hastings algorithm (Hastings, 1970), with gamma proposal distributions for ˇL , ˇG and , and normal proposal distributions for J. Each Bayesian analysis is run for 500,000 steps, with a burn-in of 1000 steps and a thinning parameter of 10. Convergence of the chains was monitored by simple visual inspection. Of the resulting posterior distributions the median and the 95% credible interval are reported. Effect of vaccination When animals are infectious at slaughter age, possibly infected meat can enter the food chain and form a risk for public health. To reduce this risk vaccination could be a valuable option. Vaccination has three distinct effects: it can reduce the infectiousness ˇG of infectious animals, it can reduce the susceptibility  of susceptible animals and it can reduce the mean period  of virus shedding. The fraction of infectious animals at slaughter age is explored as a function of these three effects. Endemic equilibrium is found by setting

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Table 2 Median values and 95% credible interval (between brackets) of posterior distributions of transmission rate parameter ˇ, infectious period  and reproduction number R0 , when only local transmission (subscript L) or only global transmission (subscript G) is assumed, for ten sampled herds separately and combined. Herd no.

Local transmission (ˇG = 0)

Global transmission (ˇL = 0)

ˇL (day−1 )

L (days)

R0,L

ˇG (day−1 )

G (days)

R0,G

Herd 1 Herd 2 Herd 3 Herd 4 Herd 5 Herd 6 Herd 7 Herd 8 Herd 9 Herd 10

0.21 (0.085–1.8) 0.13 (0.076–0.40) 0.17 (0.074–0.61) 0.29 (0.082–1.4) 0.18 (0.077–0.67) 0.18 (0.082–1.8) 0.17 (0.075–0.61) 0.16 (0.068–1.4) 0.58 (0.13–3.5) 0.16 (0.082–0.54)

14 (3.7–27) 26 (9.9–53) 14 (3.5–36) 6.0 (1.1–20) 12 (3.4–29) 18 (6.5–33) 14 (3.5–37) 16 (4.5–35) 30 (18–48) 19 (7.0–42)

2.4 (1.0–39) 3.5 (1.7–9.7) 2.3 (1.1–5.8) 1.7 (0.63–3.9) 2.1 (1.0–5.2) 2.9 (1.3–43) 2.3 (1.1–5.7) 2.3 (1.1–30) 18 (3.0–112) 3.0 (1.5–8.7)

0.90 (0.18–3.1) 0.22 (0.077–0.72) 0.23 (0.056–1.0) 0.72 (0.093–3.2) 0.53 (0.11–2.0) 0.63 (0.16–2.4) 0.23 (0.055–1.0) 0.39 (0.091–1.6) 0.91 (0.26–3.6) 0.32 (0.099–1.1)

11 (3.3–26) 29 (13–54) 13 (3.1–36) 4.3 (0.48–18) 12 (3.3–29) 16 (5.6–32) 13 (3.1–37) 15 (4.7–33) 28 (16–48) 23 (9.8–45)

9.2 (2.1–45) 6.2 (2.3–21) 2.8 (1.0–8.5) 2.7 (0.38–12) 5.7 (1.6–24) 9.3 (2.5–47) 2.8 (1.0–8.7) 5.1 (1.6–33) 26 (6.5–117) 7.2 (2.5–27)

All herds

0.11 (0.092–0.14)

24 (18–33)

2.7 (2.2–3.6)

0.16 (0.082–0.29)

27 (20–39)

4.3 (2.8–6.9)

the left-hand sides of Eq. (2) to zero and discretizing the population into na = 100 age classes. Animals move from one age class to another with a rate of ı = na /140 day−1 , assuming a total residence time of 140 days, or 20 weeks. For the global HEV transmission model: 1  0 = −(a)Sa ˇG (a)Ij,a − ıSa + ıSa−1 , na a

for 1 ≤ a ≤ na

j

(9)

for 1 ≤ a ≤ na 4 4 0= I − I − ıIj,a + ıIj,a−1 , (a) j−1,a (a) j,a for 2 ≤ j ≤ 4, 1 ≤ a ≤ na , is solved with boundary conditions S0 = 1 and Ij,0 = 0. Sa and Ij,a are the fractions of susceptible and infectious animals in age class a and infectious stage j. The fraction of infectious animals at slaughter age  is I . j,n a j Two types of vaccination are studied: early vaccination where all animals are assumed to be fully vaccinated and delayed vaccination where animals are vaccinated at 10 weeks of age. In the first case, the transmission rate parameter, susceptibility and mean infectious period do not depend on the age of the animals. They are reduced by reduction factors ˇ ,  and  , i.e. ˇG = ˇG,0 (1 − ˇ ),  = (1 −  ) and  = 0 (1 −  ), where the subscripts 0 denote the parameter values without vaccination. For delayed vaccination, the parameters do depend on the age a of the animals. It is assumed that they decrease linearly from 10 to 12 weeks at which point they achieve their reduction factor:

ˇG (a) =

(a) =

(a) =

⎧ ⎨ ˇG,0 

ˇ

⎩

14ı

ˇG,0 1 + 2ˇ −

ˇG,0 (1 − ˇ )

⎧ ⎨ 1 ⎩

28ı ≤ a < 42ı

(10)

42ı ≤ a ≤ na 1 ≤ a < 28ı



 a 1 + 2 − 14ı (1 −  )

⎧ ⎨ 0  ⎩

1 ≤ a < 28ı

a

28ı ≤ a < 42ı

(11)

42ı ≤ a ≤ na



 a 0 1 + 2 − 14ı 0 (1 −  )

1 ≤ a < 28ı 28ı ≤ a < 42ı

Results Parameter estimation

j

4 1  0 = (a)Sa ˇG (a)Ij,a − I1,a − ıI1,a + ıI1,a−1 , na (a) a

parameter ˇG , susceptibility  and mean infectious period  are studied separately by varying the reduction factors ˇ ,  and  between 0 (no effect) to 1 (full effect).

(12)

42ı ≤ a ≤ na

We explore the effect of early and delayed vaccination on the fraction of animals that are virus positive at slaughter age, using the estimated parameters of the combined herds assuming global transmission. The effects of reducing the transmission rate

The analyses yielded posterior distributions for ˇL , L and R0,L = ˇL L for the model with only local transmission and posterior distributions for ˇG , G and R0,G = ˇG G for the model with only global transmission. The results are summarized by the median posterior values and the 95% credible interval (Table 2). Herds 3 and 7 are interesting to compare as the numbers of positive samples in each group are identical. The results of these herds are similar, indicating that the Markov chain of length 500,000 is sufficiently large. The median transmission rate parameters of the global transmission model ranging from 0.22 to 0.91 day−1 , are higher or comparable to the median transmission rate parameters of the local transmission model ranging from 0.13 to 0.58 day−1 . The median infectious periods however are comparable for both models, ranging from 4.3 to 29 days for global transmission and from 6.0 to 30 days for local transmission. As a result the median reproduction numbers of the global transmission model ranging from 2.7 to 26, are higher or comparable to the median reproduction numbers of the local transmission model ranging from 1.7 to 18. Furthermore, the credible intervals for the transmission rate parameter and reproduction number of the global transmission model are much larger than those of the local transmission model. This is also apparent from the prevalence profiles for the local transmission model (Fig. 2) that show a greater flexibility to adjust to the observed prevalence. Based on the DIC values (Spiegelhalter et al., 2002), no substantial distinction between the models can be made, i.e. the difference in DIC is less than 5. Considering the wide ranges of the credible intervals, it is justifiable to regard the ten herds as independent samples of the same underlying system. This means that factors that can affect the virus transmission, such as the herd structure or hygiene level, do not differ significantly between the herds. The product of the likelihood functions of all herds is now analyzed in a 32dimensional parameter space. The local transmission model yields a transmission rate parameter of 0.11 (0.092–0.14) day−1 , a mean infectious period of 24 (18–33) days and a reproduction number of 2.7 (2.2–3.6), whereas the global transmission model yields a transmission rate parameter of 0.16 (0.082–0.29) day−1 , a mean infectious period of 27 (20–39) days and a reproduction number of 4.3 (2.8–6.9).

90

J.A. Backer et al. / Epidemics 4 (2012) 86–92

herd 1

1.0

local transmission (βG = 0) 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0.0

herd 2

herd 3

20

25

0.8

0.6

0.6

0.4

0.4

0.2

0.2 10

15

20

25

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 10

15

20

25

1.0

0.8

0.8

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0.4

0.2

0.2 10

15

20

25

1.0

0.8

0.8

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20

25

0.8

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25

0.0

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1.0

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0.4

0.4

15

20

25

0.0 1.0

0.8

0.8

0.6

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0.4

0.2

0.2 10

15

20

25

0.0

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

10

15

20

25

0.0

1.0

1.0

0.8

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25

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25

10

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20

25

10

15

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25

10

15

20

25

0.2 10

1.0

0.0

15

0.2 10

0.2

herd 10

0.0 1.0

0.0

10

0.2 10

1.0

0.0

herd 9

0.0

1.0

0.2

herd 8

0.0

1.0

0.0

herd 7

0.0

1.0

0.2

herd 6

0.0

0.8

0.0

herd 5

15

1.0

0.0

herd 4

0.2 10

1.0

0.0

global transmission (βL = 0)

10

15

20

age (weeks)

25

0.0

10

10

15

20

25

15

20

25

age (weeks)

Fig. 2. Measured prevalence profiles (dashed line) and estimated prevalence profiles (median: solid line, and 95% interval: shaded area) assuming only local transmission (left panels) or only global transmission (right panels), using the posterior distributions, for ten sampled herds.

Effect of vaccination The global transmission model is considered the most conservative, as it yielded higher transmission rate parameters and reproduction numbers than the local transmission model. For this reason, we have used the parameters estimated for the global model for all herds to explore the infectious fraction at slaughter age. Without vaccination the infectious fraction at slaughter age is 0.025 (0.0040–0.081), as shown in all plots of Fig. 3 when the reduction factor is zero. With vaccination the transmission rate parameter ˇG , the susceptibility  and the mean infectious period  can be reduced. Here we will evaluate these three effects of vaccination separately. When applying early vaccination, all animals are vaccinated before entering the weaner group. When in this case the transmission rate parameter is reduced, the infectious fraction starts to increase (Fig. 3A). The reason for this opposite effect is that because of the lower infection pressure, animals are infected at a later age and the prevalence peak shifts towards slaughter age. Only when the transmission rate parameter is sufficiently reduced – with a factor of 0.74 (0.58–0.84) – virus circulation is effectively halted by bringing the reproduction number below unity. The effect of reducing the susceptibiliy is identical to the effect of reducing the transmission rate parameter (Fig. 3B). This is because in early vaccination all animals are vaccinated which means the two are equivalent. When early vaccination reduces the mean infectious period, the infectious fraction decreases (Fig. 3C). Despite the lower infection pressure and the consequential later infection age, the shorter infectious period prevents the prevalence peak from shifting towards slaughter age. The infectious fraction at slaughter age is effectively brought to zero when the mean infectious period is reduced by a factor 0.76 (0.62–0.86). Delayed vaccination is aimed at decreasing the infection age by vaccinating animals at a later age. Our choice of vaccinating animals at 10 weeks of age and reaching full effect at 12 weeks of age, means that 25% of the population is unprotected. When vaccination only reduces the transmission rate parameter, the infectious fraction at slaughter age increases similarly to early vaccination (Fig. 3D), whereas this undesirable effect is not as large when vaccination reduces the susceptibility (Fig. 3E). A shorter mean infectious period in delayed vaccination has a similar effect as in early vaccination (Fig. 3F). Virus elimination can be achieved when the transmission rate parameter or the mean infectious period is reduced, albeit at higher reduction factors than when using early vaccination. When vaccination reduces the susceptibility, virus elimination is only achieved at almost full immunity.

Discussion We have studied the transmission dynamics for HEV in commercial pig farms. Our estimated transmission parameters are in general smaller than those determined in transmission experiments by Bouwknegt et al. (2008). The point estimates they found in the experiments can be compared with our median of the posterior distribution, but it should be kept in mind that the approaches are fundamentally different. The experimentally determined withinpen transmission rate parameter is 0.66 (0.32–1.35, 95% confidence interval) day−1 , which is 4–6 times higher than our estimates. Two possible reasons could explain this difference. Firstly, animals in an experimental setting are in closer contact, compared to a full farm situation where animals are also separated in pens. Secondly, in the transmission experiments, susceptible animals are brought into contact with infectious animals in the early stages of virus shedding. At this point, they are likely to be more infectious than in later stages, whereas in the endemic field situation,

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Fig. 3. Fraction of infectious animals at slaughter age, using posterior distributions of parameters estimated for all herds for the global transmission model, as a function of reduction of transmission rate parameter ˇG (left panels), susceptibility  (middle panels) and mean infectious period  (right panels) for early vaccination (top panels) and delayed vaccination (bottom panels); median values (solid line) and 95% interval (shaded area).

susceptible animals will encounter infectious animals of varying infection ages. In the transmission experiments, the mean infectious period was estimated as 49 (17–141, 95% confidence interval) days and 13 (11–17, 95% confidence interval) days for two separate blocks, while our estimates are in between. As a consequence, the reported experimental reproduction numbers of 32 (11.2–92, 95% confidence interval) and 8.8 (4.2–18.8, 95% confidence interval) are much higher than we have estimated from the field data. However, our estimates are in good agreement with the reproduction numbers that were determined by Satou and Nishiura (2007) from seroprevalence data of three Japanese pig farms. We have compared two transmission models, assuming that animals are infected either locally by their group mates or globally by any infectious animal regardless of its group. The global transmission model yielded higher estimates and wider distributions to fit the observed prevalence patterns. As in practice the transmission will be a mixture of the local and global transmission models, we consider the global estimates to be the most conservative. If the herd would be subdivided in more than three groups then the local and global transmission parameters may be estimated simultaneously to determine their relative importance. To reduce the fraction of infectious animals at slaughter age, vaccination could be used. A vaccine can reduce the transmission rate parameter, the susceptibility, the mean infectious period or a combination. In a fully vaccinated population, reduction factors of around 75% should be attained to eliminate the virus from the herd, which agrees reasonably with the theoretical critical vaccination coverage pc = 1 − 1/R0 = 1 − 1/4.3 = 0.77 (Keeling and Rohani, 2008). For lower reduction factors, reducing the transmission rate parameters or the susceptibility has an opposite effect on the infectious fraction at slaughter age, whereas the reduction of the mean infectious period has not. Thus, in the development of a possible vaccine against HEV in pigs, it would be most advantageous to focus on shortening the infectious period. Lowering the transmission rate parameter would only be useful when high reduction factors can be achieved, aimed at virus elimination. When a vaccine mainly reduces the susceptibility, delayed vaccination of animals at a later age can be a better strategy than early vaccination. Studying the combination of these different effects, can provide insight in the

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