A simple logit model was applied to the cumulative distribution of the time required for seroconversion of anti-HEV antibody based on inoculation experiments using swine HEV 9. Figure 1 compares the observed and predicted proportion of seroconversion with time post-inoculation. The maximum likelihood estimate of the median time required for seroconversion was 25.0 (95% confidence interval (CI): 20.9, 31.3) days. The coefficient of the time after inoculation (for the logit model) was estimated as 0.19 (95% CI: 0.11, 0.30). The χ² goodness-of-fit test did not reveal significant deviations between observed and predicted frequencies (χ²₅ = 2.31, p = 0.81).

The total sample sizes of seroprevalence surveys were 1400, 400 and 700 for Hokkaido, Honshu and Kyushu, respectively 24. The maximum likelihood estimates of the force of infection, λ, (and the corresponding 95% CI) of swine HEV for the three different geographic locations in Japan were 3.45 (3.17, 3.75), 2.68 (2.28, 3.14) and 3.11 (2.76, 3.50) [×10^-2 per day]. Expected values of λ ranging from 2.68–3.45 ×10^-2 day^-1 indicate that the average time of infection ranged from 29.0–37.3 days after the age of 30 days when individuals were first exposed to the risk of infection (i.e. average age at infection was 59.0–67.3 days). Observed and predicted age-specific seroprevalence are compared in Figure 2, confirming good overall agreement of the model with the data. Although a significant deviation was seen in Hokkaido (χ²₄ = 16.71, p < 0.01), this was not the case for Honshu (χ²₄ = 1.49, p = 0.89) or Kyushu (χ²₄ = 6.34, p = 0.17).

The population structure on pig farms in Japan is specific in that individuals are slaughtered immediately after the age of 180 days (i.e., 150 days after pigs are first exposed to the risk of infection). This satisfied a reasonable approximation of a simple age-specific survivorship function, referred to as Type I survivorship 32, to the data:

μ ( a ) = 0 , f o r a < 180 μ ( a ) = ∞ , f o r a ≥ 180 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGGaciab=X7aTjabcIcaOiabdggaHjabcMcaPiabg2da9iabicdaWiabcYcaSGGabiab+bcaGiab+bcaGiabdAgaMjabd+gaVjabdkhaYjab+bcaGiab+bcaGiabdggaHjabgYda8iabigdaXiabiIda4iabicdaWaqaaiab=X7aTjabcIcaOiabdggaHjabcMcaPiabg2da9iabg6HiLkabcYcaSiab+bcaGiab+bcaGiabdAgaMjabd+gaVjabdkhaYjab+bcaGiab+bcaGiabdggaHjabgwMiZkabigdaXiabiIda4iabicdaWaaaaa@55DB@

where μ(a) denotes age-specific mortality rate at age a. The basic reproduction number, R₀, is approximated as the product of the force of infection, λ, and the time from first exposure to death, L (= 150 days) 32:

R₀ ≈ λL

Thus, R₀ (and the 95% CI) was estimated to be 5.17 (4.76, 5.62), 4.02 (3.43, 4.71) and 4.66 (4.13, 5.25) for Hokkaido, Honshu and Kyushu, respectively.

Figure 3A shows the cumulative proportion of infected individuals according to our simple assumptions with different forces of infection, λ, being 0.01, 0.03 and 0.05 (day^-1). Accordingly, the average ages at infection, A (= 30 + 1/λ), are 130, 63 and 50 days, respectively. Using this range of λ, 0.03 (0.01, 0.05), it is predicted that 83.5 % (45.1, 95.0) of the herd would be infected at the age of 90 days, 93.3 % (59.3, 98.9) at 120 days and 97.3 % (69.9, 99.8) at 150 days. Based on the same assumptions, Figure 3B shows the age-specific absolute incidence (i.e. the number of newly infected individuals) at each given age in a hypothetical herd with a population size of 1000. Despite a few rough assumptions, the figure indicates that average age at infection directly influences the age-specific patterns of HEV incidence, enabling the prediction of incidence shortly before the age of finishing (i.e. 180 days). According to the above range of λ, the model predicts that 2.1 (4.1, 0.6) individuals would be newly infected at the age of 120 days, 0.8 (3.0, 0.1) at 150 days and 0.3 (2.3, 0.0) at 180 days. In other words, the smaller the force of infection is, higher the probability that virus excretion would occur at the finishing stage.

To estimate the force of infection, this study used two published datasets: (1) an experimental inoculation study of swine HEV 9, and (2) a large-scale seroprevalence survey in Japan 24. The experimental study recorded the time required for seroconversion of anti-HEV antibody following inoculation. The seroepidemiologic study consisted of a survey of pig farms in three discrete geographic locations (Hokkaido, Honshu and Kyushu; see Figure 4A). Anti-HEV IgG antibodies of 2500 pigs were measured by age group. The original study examined the age-specific seroprevalence by month, i.e., pigs at 60, 90, 120, 150 and 180 days of age (allowing ± 5 days variation for each) were sampled 24. We used this data because the detailed breeding methods of the piggery in Japan were known and the sample size was sufficiently large to allow statistical analysis. On the farm, a proportion of the suckling herd has a very short-lived maternal antibody (Tsunemitsu H, personal communication) and is not exposed to infection due to separate housing during this stage. A seroepidemiologic study and an isolation study of HEV RNA during natural infection in other countries also roughly satisfied this assumption: the number of seropositive pigs was negligible at the age of 30 days on several farms 50 and HEV RNA started to be detected at the age of 30 days 51. Since the pigs thereafter entered into weaning, growing and fattening stages, we assume that the risk of exposure starts at the age of 30 days. The pigs are slaughtered immediately after the age of 180 days.

Cumulative distribution of the time required for seroconversion, G(t) at t days post-inoculation was approximated by a simple logit model:

G ( t ) = exp ⁡ ( l ) 1 + exp ⁡ ( l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrcqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiGbcwgaLjabcIha4jabcchaWjabcIcaOiabdYgaSjabcMcaPaqaaiabigdaXiabgUcaRiGbcwgaLjabcIha4jabcchaWjabcIcaOiabdYgaSjabcMcaPaaaaaa@425C@

where l is a linear predictor given by:

l = b(t - t_m)

where b is the coefficient of the time since inoculation and t_m is the median time required for seroconversion. In order to apply a logistic curve to the cumulative distribution, the model has to satisfy G(0) ≈ 0 and G(∞) = 1. The maximum likelihood estimates of b and t_m were obtained by minimizing the binomial deviance of the model from the observed data. The 95% CI were determined by using the profile likelihood.

For simplicity and due to limited data availability, we assumed that the force of infection, λ, was independent of time (i.e. endemic equilibrium). Furthermore, except for an assumption that no exposure occured during the suckling stage (i.e. until the age of 30 days), λ was also assumed to be age-independent. Figure 4B shows a schematic illustration of the model. We denote the proportions of susceptible and infected individuals at age a days (for a ≥ 30) by S(a) and I(a), respectively. With an initial condition, S(30) = 1 and I(30) = 0, the model is given by the following ordinal differential equations:

d S ( a ) d a = − λ S ( a ) d I ( a ) d a = λ S ( a ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaamaalaaabaGaemizaqMaem4uamLaeiikaGIaemyyaeMaeiykaKcabaGaemizaqMaemyyaegaaiabg2da9iabgkHiTGGaciab=T7aSjabdofatjabcIcaOiabdggaHjabcMcaPaqaamaalaaabaGaemizaqMaemysaKKaeiikaGIaemyyaeMaeiykaKcabaGaemizaqMaemyyaegaaiabg2da9iab=T7aSjabdofatjabcIcaOiabdggaHjabcMcaPaaaaa@4BAC@

for a ≥ 30. The analytical solution is given by:

I(a) = 1-exp{-λ(a-30)}

for a ≥ 30. Under the same assumption, age-specific incidence, C(a), at age a, is given by product of λ and S(a):

C ( a ) = 0 f o r a < 30 C ( a ) = λ exp ⁡ { − λ ( a − 30 ) } f o r a ≥ 30 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGacaaabaGaem4qamKaeiikaGIaemyyaeMaeiykaKIaeyypa0JaeGimaadabaGaemOzayMaem4Ba8MaemOCaihcceGae8hiaaIae8hiaaIaemyyaeMaeyipaWJaeG4mamJaeGimaadabaGaem4qamKaeiikaGIaemyyaeMaeiykaKIaeyypa0dcciGae43UdWMagiyzauMaeiiEaGNaeiiCaa3aaiWaaeaacqGHsislcqGF7oaBcqGGOaakcqWGHbqycqGHsislcqaIZaWmcqaIWaamcqGGPaqkaiaawUhacaGL9baaaeaacqWGMbGzcqWGVbWBcqWGYbGCcqWFGaaicqWFGaaicqWGHbqycqGHLjYScqaIZaWmcqaIWaamaaaaaa@5CE8@

In addition to the estimation of λ, we examined the sensitivity of I(a) and C(a) to the different values of λ to explore the age-specific patterns and clarify the practical implications of λ to farm management.

Since all individuals are slaughtered immediately after the age of 180 days, the time delay from infection to seroconversion was thought to be non-negligible. That is, the age-specific seroprevalence data (at age a days) does not directly reflect all of those who were infected until age a, I(a), but rather indicates those infected until the age which is smaller than a. Thus, to account for the delay, we used probability density of the time to seroconvert (Figure 4C) in addition to the model of the cumulative distribution of HEV infection (Eq. 6). Considering the time-scale of the distribution of time to seroconvert (Figure 1) and the cumulative incidence up to the age of 60 days (Figure 2), we assumed that those seropositive at the age of 60 days approximately reflected those who were infected until the age of 45 days. Under these assumptions, the age-specific incidence density, f_i, whose infection is reflected as seropositive at i months after starting to be exposed (i.e., a = 30(i + 1) days) is assumed to be:

f i = I ( 45 ) f o r i = 1 f i = I ( 30 i + 15 ) − I ( 30 i − 15 ) f o r 2 ≤ i ≤ 5 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGacaaabaGaemOzay2aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWGjbqscqGGOaakcqaI0aancqaI1aqncqGGPaqkaeaacqWGMbGzcqWGVbWBcqWGYbGCiiqacqWFGaaicqWFGaaicqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGMbGzdaWgaaWcbaGaemyAaKgabeaakiabg2da9iabdMeajjabcIcaOiabiodaZiabicdaWiabdMgaPjabgUcaRiabigdaXiabiwda1iabcMcaPiabgkHiTiabdMeajjabcIcaOiabiodaZiabicdaWiabdMgaPjabgkHiTiabigdaXiabiwda1iabcMcaPaqaaiabdAgaMjabd+gaVjabdkhaYjab=bcaGiab=bcaGiabikdaYiabgsMiJkabdMgaPjabgsMiJkabiwda1aaaaaa@615F@

Using the same mid-point of the time-interval, the probability density of time to seroconvert, g_j, at j months after infection was also approximated by using G(t):

g j = G ( 15 ) f o r j = 1 g j = G ( 30 j − 15 ) − G ( 30 j − 45 ) f o r 2 ≤ j ≤ 5

The density of those who newly showed seropositive results at the age of 60 days (i.e., at i = 1 month), k₁, is given by k₁ = f₁g₁. In the same way, the densities at the ages of 90 and 120 days, k₂ and k₃, are given by k₂ = f₂g₁ + f₁g₂ and k₃ = f₃g₁ + f₂g₂ + f₁g₃. This can be generalized by using the following convolution equation 525354:

k i = ∑ j = 1 i f i − j + 1 g j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGRbWAdaWgaaWcbaGaemyAaKgabeaakiabg2da9maaqahabaGaemOzay2aaSbaaSqaaiabdMgaPjabgkHiTiabdQgaQjabgUcaRiabigdaXaqabaGccqWGNbWzdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGPbqAa0GaeyyeIuoaaaa@4169@

Since the observed seroprevalence data shows the cumulative distribution of those seroconverted after infection until month i, K_i, which is given by Ki=∑m=1ikm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaWgaaWcbaGaemyAaKgabeaakiabg2da9maaqahabaGaem4AaS2aaSbaaSqaaiabd2gaTbqabaaabaGaemyBa0Maeyypa0JaeGymaedabaGaemyAaKganiabggHiLdaaaa@3A3B@, we get:

K i = ∑ j = 1 i f i − j + 1 G j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaWgaaWcbaGaemyAaKgabeaakiabg2da9maaqahabaGaemOzay2aaSbaaSqaaiabdMgaPjabgkHiTiabdQgaQjabgUcaRiabigdaXaqabaGccqWGhbWrdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGPbqAa0GaeyyeIuoaaaa@40E9@

where Gi=∑m=1igm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaWgaaWcbaGaemyAaKgabeaakiabg2da9maaqahabaGaem4zaC2aaSbaaSqaaiabd2gaTbqabaaabaGaemyBa0Maeyypa0JaeGymaedabaGaemyAaKganiabggHiLdaaaa@3A2B@. Suppose that there were N_i pigs who had been seropositive until month i (where i = a/30 - 1) and M_i who had not, the likelihood function is

L = ∏ h = 1 5 K h N h ( 1 − K h ) M h MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGmbatcqGH9aqpdaqeWbqaaiabdUealnaaDaaaleaacqWGObaAaeaacqWGobGtdaWgaaadbaGaemiAaGgabeaaaaGccqGGOaakcqaIXaqmcqGHsislcqWGlbWsdaWgaaWcbaGaemiAaGgabeaakiabcMcaPmaaCaaaleqabaGaemyta00aaSbaaWqaaiabdIgaObqabaaaaaWcbaGaemiAaGMaeyypa0JaeGymaedabaGaeGynaudaniabg+Givdaaaa@43B8@

The maximum likelihood estimate of λ is obtained by minimizing the negative logarithm of Eq. 12. The 95% CI were derived from the profile likelihood. All statistical data were analyzed using the statistical software JMP IN ver. 5.1 (SAS Institute Inc., Cary, NC).