Quantitative analysis of the dose – response of white spot syndrome virus in shrimp

White spot syndrome virus (WSSV) is an important cause of mortality and economic losses in shrimp farming. Although WSSV ‐ induced mortality is virus dose dependent and WSSV infection does not necessarily lead to mortality, the relationships between virus ‐ particle dose, infection and mortality have not been analysed quanti-tatively. Here, we explored WSSV dose – response by a combination of experiments, modelling and meta ‐ analysis. We performed dose – response experiments in Penaeus vannamei postlarvae, recorded host mortality and detected WSSV infection. When we fitted infection models to these data, two models — differing in whether they incorporated heterogeneous host susceptibility to the virus or not — were supported for two independent experiments. To determine the generality of these results, we reanalysed published data sets and then performed a meta ‐ analysis. We found that WSSV dose – response kinetics is indeed variable over experiments. We could not clearly identify which specific infection model has the most support by meta ‐ analysis, but we argue that these results also are most concordant with a model incorporating varying levels of heterogeneous host susceptibility to WSSV. We have identified suitable models for analysing WSSV dose – response, which can elucidate the most basic virus – host interactions and help to avoid underestimating WSSV infection at low virus doses.

One important factor in determining whether WSSV infection and virulence will occur is the dose of virus particles to which a host has been exposed. For many viruses, hosts must be exposed to a large number of virus particles to become infected, as the probability of infection per virus particle is very small (Zwart, Daròs, & Elena, 2011;Zwart & Elena, 2015). Whereas very high virus doses may infect all hosts, in practice exposure to a broad range of doses will lead to infection in only some hosts. A better understanding of the relationship between dose and response can be useful for identifying when host organisms will be at risk of disease and for understanding the effects of different interventions to mitigate the effects of disease outbreaks. When mechanistic models of the dose-response are fitted to experimental data, this approach also can be useful for better understanding of the infection process. For example, if virus particles cooperate with each other during the infection process, then increases in dose will have disproportionate effects on the rate of host infection, leading to a steep dose-response. Observation of a steep dose-response has been crucial to demonstrating the strong cooperation between virus particles that stem from packaging of different genome segments into separate virus particles in some plant (Fulton, 1962;Sánchez-Navarro, Zwart, & Elena, 2013) and animal viruses (Ladner et al., 2016).
Many studies have shown that the rate of WSSV infection depends on virus particle dose (Escobedo-Bonilla et al., 2005;Laramore et al., 2009;Marks et al., 2005;Prior, Browdy, Shepard, Laramore, & Parnell, 2003). It has also been suggested that WSSV virus particles might be acting independently during the infection process because dose-response is similar to predictions of the independent action hypothesis (IAH) model . This model states that each virus particle can be assigned a nonzero probability of infection, and that virus particles act independently, leading to clear predictions for the shape of the dose-response (Druett, 1952;Regoes, Hottinger, Sygnarski, & Ebert, 2003;Zwart et al., 2009). However, none of the WSSV studies rigorously compared experimental data to mechanistically interpretable models, nor was model selection performed to identify which model is best supported by the empirical data. Furthermore, not all shrimp that become infected with WSSV will die (Flegel, 2007;Johnson et al., 2008;Venegas et al., 2000). One study has shown systematically that sublethal infection is infrequent in laboratory challenge experiments (Escobedo-Bonilla et al., 2005), although it is quite common in the field (Hoa, Zwart, Phuong, Oanh, et al., 2011). The relationship between dose, infection and mortality has also not been explored systematically or quantitatively.
Here, we set out to provide a quantitative description of the WSSV dose-response relationship. By performing model selection on a set of mechanistically interpretable models, we wanted to identify the underlying mechanisms that may give rise to this relationship.
Our contribution to these issues is threefold. First, we performed dose-response experiments and determined the infection status of all shrimp in the experiment, as well as recording mortality. We could therefore simultaneously consider dose-response on two levels: systemic infection and host mortality. Furthermore, the set-up described here is different from previous reports because the communal housing of shrimp allowed waterborne transmission, and we needed to include these effects in our models. Second, we fitted a number of generally applicable and mechanistically interpretable models of dose-response to these data using a maximum likelihood approach, and then performed model selection. Third, to gauge the generality of our conclusions, we performed a meta-regression analysis on reanalysed published dose-response data for WSSV in shrimp.

| Shrimp and virus isolates
Specific pathogen-free Penaeus vannamei (Boone, 1931) postlarvae (PL) were purchased from Gold Coin Singapore, imported to the Netherlands and reared by the Aquatic Research Facility at Wageningen University. Orconectes limosus (Rafinesque, 1817) were obtained from local fishers from the Meuse River and used for virus amplification. Shrimp were screened for common viral diseases by polymerase chain reaction (PCR) or reverse transcription (RT) PCR (Witteveldt, Cifuentes, Vlak, & van Hulten, 2004). Shrimp were kept in tanks with heating (28°C), continuous aeration and individual filter systems (Eheim).
For all experiments, we used purified WSSV virus particles (Xie, Li, Xu, & Yang, 2005) from isolate VN-T (Dieu et al., 2004). For dose-response experiment #1, the virus was amplified in O. limosus prior to purification, and for experiment #2, P. vannamei was used for amplification.

| Dose-response experiments
For the dose-response experiments, individual shrimp were kept in cages within a 180 L aquarium. On each side, the 8 × 8 cm slots had a round opening with a 6 cm diameter covered with a fine metal mesh (1 mm), allowing water to flow freely between the slots. Shrimp in these cages and aquaria were randomly assigned to the different treatments (i.e., different virus doses or mock-infected controls). Shrimp PL10-12 were weighed and then intramuscularly injected with 10 μl of WSSV virus particles or phosphate-buffered saline (PBS) for controls. Injections were performed with a 1.5 ml BD Pen (Becton Dickinson) and 28G 1/2″ NovoFine needles (Novo Nordisk). A later developmental stage (PL of 1-3 g in both experiments) was chosen because these shrimp can be injected intramuscularly, and because at this stage disease and mortality occur quickly, making experiments more tractable. The appearance, behaviour and mortality of shrimp were observed daily, and dead shrimp were removed immediately and stored at −20°C. Hence, our set-up with the cages prevented cannibalism of dead shrimp, but not waterborne transmission of the virus. After 10 days, all surviving shrimp were collected and stored. For experiment #1, cohorts of 13 shrimp each were inoculated with a 10 2 , 10 3 , 10 4 , 10 5 or 10 6 fold dilution of purified WSSV virus particles amplified in O. limosus, or PBS only for mock-inoculated controls. For experiment #2, cohorts of 17 shrimp each were inoculated with a 10, 10 2 , 10 3 , 10 4 , 10 5 or 10 6 dilution of WSSV virus particles amplified in P. vannamei, or PBS only. Shrimp taken from the same cohort were used for experiments #1 and #2, although experiment #1 was performed first and the shrimp were smaller.

| PCR detection of WSSV
The infection status of all shrimp from the dose-response experiments was determined by PCR. To avoid environmental contamination with WSSV DNA, a sample of muscle tissue was used for DNA extraction. A crude extract of DNA was used as template for two separate Taq-based PCR reactions with host-and virus-specific primers (Witteveldt et al., 2004; host-specific primers for 16S ribosomal RNA: 5′-GTGCGAAGGTAGCATAATC and 5′-CTGCTGCAACATAA GGATAC, WSSV-specific primers for VP26: 5′-ATGGAATTTGGC AACCTAACAAACCTG and 5′-GGGCTGTGACGGTAGAGA). Hostspecific primers were included as a positive control for DNA extraction and PCR. PCR products were resolved on a 0.7% agarose gel prestained with ethidium bromide.

| Dose-response models
To model dose-response, we took the simplest mechanistic model of virus infection as a starting point: the IAH model (Druett, 1952;Regoes et al., 2003;Zwart et al., 2009). The dose-response prediction of this model is obtained from the zero term of the Poisson distribution for the number of infecting virus particles, which represents those hosts that have not been infected by the virus (Zwart et al., 2009). The rate of infection I is then I ¼ 1 À S ¼ 1 À e Àλ ¼ 1 À e Àρn , where S is survival, λ is the number of infecting virus particles, ρ is the probability of infection for each virus particle and n is the number of virus particles in the inoculum.
In this case, we do not know the actual virus particle dose, but we know the relative viral dose as the different doses were obtained from serial dilutions of a WSSV stock. We therefore set the dose of the virus stock to an arbitrary high value (10 9 virus particles). The same convention also was used for all analysis and presentation of data throughout this study.
To apply this model to our experimental data, we must make three considerations. First, in our experiments, shrimp can become infected due to the intramuscularly injected virus particles, or due to waterborne transmission of the virus from other shrimp that have become infected during the experiment, as the shrimp are housed communally. The waterborne WSSV dose to which a shrimp is exposed probably will increase over time, and the probability of infection for each virus particle in the water will probably be lower than for injected virus particles (Soto & Lotz, 2001). As the waterborne virus particle load will be approximately the same for all shrimp sharing the same water, and both dose and infection probability are unknown, we simply introduce a second infection term analogous to λ to represent waterborne transmission (ω): Next, we also want to model the rate of mortality, because we measured both infection and mortality for the same group of hosts.
If we assume that infection does not necessarily lead to host death, and that the IAH principle also applies to this level of infection, we can introduce a probability that an infecting virus particle causes host death (φ). We consider this probability the same for infection resulting from intramuscular infection or waterborne transmission, in which case host mortality (M) is: In reality, for infection resulting from virus particles transmitted by waterborne transmission, the probability of host death might be smaller: These secondary infections will have started later than infections caused by the injected virus particles, and therefore, there will be less time for the virus to kill the host. However, as a first approximation and to avoid overparameterising the model, we ignore this effect.
Finally, as in many cases, the data did not support the IAH | 3 an overparametrized model-we chose to add dose dependence at the infection level. These dose-dependent effects on infection will carryover on the rate of mortality, and as a result, the shape of both responses will be affected. Also, as the infection routes for injected and waterborne virus particles are different, and the waterborne virus particle dose is assumed to be constant, we chose to only have dose-dependent effects stemming from the dose of injected virus particles. Therefore, following (Regoes et al., 2003), the rate of infection under the DA model is: where κ is a constant that determines what type of dose-dependent effects will occur. When κ < 1, there are antagonistic interactions between virus particles; as the dose is increased, it becomes harder for each virus particle to infect, and hence, the dose-response becomes more gradual than for IAH. When κ > 1, there are synergistic interactions; as the dose is increased, it becomes easier for each virus particle to infect, and hence, the dose-response is steeper than for IAH. the occurrence of waterborne transmission in our set-up, the doseresponse is as follows: where ν is a constant that determines the variance of the distribution of host susceptibilities (ρν 2 ). Differences in host susceptibility can only result in a more gradual dose-response: For increasing values of ν, the dose-response becomes more gradual, whereas when ν approaches zero, the shape of the dose-response becomes identical to the IAH prediction. By including the probability that an infecting virus particle causes host death in the terms for waterborne transmission and the injected inoculum, the relationship between dose and mortality is M ¼ 1 À e Àφω 1 1 þ ρφνn 1=ν . are infected by WSSV. There were no shrimp that died but were negative for WSSV detection, a host state that is also not possible for our infection models (per definition the probability φ ≤ 1). The IAH and DA models can be used to predict the number of animals in each state as a function dose, given S ¼ 1 À I and E ¼ I À M. The likelihood of observing a given number of shrimp in the M, E or S states at the end of the experiment (X 1 , X 2 , X 3 ) follows a multinomial distribution with probabilities p 1 , p 2 and p 3 (∑ 3 i¼1 p i ¼ 1). A particular realization (x 1 , x 2 , x 3 ) has a multinomial probability:

| Model fitting and model selection
We used a stochastic hill-climbing algorithm to minimize the negative log likelihood (NLL), with 10 3 searches from randomly chosen points in a broad range of plausible parameter-value space. We then used the Akaike information criterion (AIC) for model selection, an approach that weighs both the fit and the number of model parameters in choosing which model is best supported by the data. Note that the DA and HHS models add one model parameter, and that the number of model parameters is similar for all models. From the difference in AIC scores between the models (ΔAIC), we determined the Akaike weight (AW), an indicator of the relative likelihood of a model given the set of models compared (Johnson & Omland, 2004).
We also performed model selection on the model fittings for the two experiments, to gauge overall support for the different models (Navakatikyan, 2007). All analyses were performed in R 3.3 (R Core Team, 2016; RRID:SCR_001905), with custom scripts.

| Survival analysis
The status of shrimp in the dose-response experiments was scored daily, making it possible to analyse whether there are any effects of dose on the time until death. We tested for significant differences in time until death between doses using the log rank test, pooled over strata and using only the data from those animals that die by the end of the experiment. As infection status was only determined at the end of the experiment due to the invasive sampling method, we can only perform survival analysis for mortality.

| Meta-analysis
We searched the scientific literature for dose-response data on shrimp and WSSV. Studies had to meet the following criteria to be included: (a) use WSSV as an inoculum, using any method of exposure; (b) use shrimp as the challenged host; (c) consider at least three different doses, with a dose range such that infection or mortality is not greater than or less than 0.5 for all doses (i.e., to exclude any data sets with only very high or very low levels of response, which would not allow discrimination between the infection models); (d) consider >5 shrimp per dose; (e) report the numbers of shrimp for the response; and (f) the response should be mortality, or the rate of infection measured in an unbiased manner (i.e., testing of all shrimp, and not a subsample of dead or alive shrimp). If both usable mortality and infection data were reported (two of 16 data sets), we used only the infection data in this analysis to avoid including the same data set twice. In one data set with both infection and mortality data, there are discrepancies between them: that is, not all dead shrimp are WSSV infected (Laramore et al., 2009). In this case, infection is clearly the better indicator of the presence of WSSV.
We then fitted the IAH, DA and HHS models to the data, but considering only a one-step model (i.e., in effect considering all data sets to be infection data and using the models to estimate I). For the reanalysis, we also left out the waterborne transmission term from the models as none of these experiments reported infection or mortality in the mock-inoculated controls. Thus, for the IAH model, only parameter ρ must be estimated, for the DA model, ρ and κ must be estimated, and for the HHS model, ρ and ν must be estimated. Given that these data were then reduced to two possible outcomes (S and I), we could use binomial likelihoods to determine the NLL and AIC. A null model with virus dose-independent host mortality (I = θ, where θ is a constant) was also fit to the data, to allow for estimation of McFadden's pseudo-R 2 (subsequently referred to as R 2 ) for each models as R 2 ¼ 1 À lnðLÞ= lnðL 0 Þ, were L is the likelihood of the model and L 0 the likelihood of the null model (McFadden, 1973).
Given that IAH model is highly constrained, negative R 2 values could be expected in some cases, and these values were reset to zero.
R 2 values are bounded at zero and one, but for meta-analysis, effect sizes need to unbounded. For meta-analysis, we logit transformed R 2 values, that is zero instances were adjusted by 1-maximum value of R 2 (Warton & Hui, 2011). The sampling variance (s 2 ) for each value of logit transformed R 2 was estimated following Equation 6: where n is the number of shrimp in the experiment (Mengersen & Gurevitch, 2013

| Experimental data
To investigate the dose-response of WSSV in shrimp, P. vannamei PL were infected by intramuscularly injecting 10-fold dilutions of WSSV isolate VN-T (Dieu et al., 2004). Two experiments were per-  Table S1). Some mock-inoculated controls were also found to be infected with WSSV and dead (Figure 1d). This observation confirmed the occurrence of waterborne transmission, which was expected as the shrimp were physically separated by a fine mesh, but water flowed freely between cubicles. We found that some infected shrimp had not died by 10 dpi (Figure 1a,d). The number of shrimp that were both infected and survived until the end of the experiment was low, reaching a maximum of 1/3 of infected shrimp (four of 12 shrimp for second to lowest dilution of the virus amplified in P. vannamei). Shrimp from the same cohort kept in separate aquaria experienced no WSSV infection or mortality, indicating that the shrimp population used for the experiment was not infected.

| Quantitative analysis of dose-response and survival analysis
The IAH and DA dose-response models were fitted to the data with a maximum likelihood method ( Figure 1,  (Figure 1a-c, Table 2). For experiment #2-with the virus amplified in P. vannamei-the HHS model was clearly the best fitting and best-supported model, followed by the DA model (Figure 1d-f, that the response was more gradual than IAH predictions in experiment #2. When the model fits for the two experiments were combined in a single model selection, the HHS model was again better supported than the DA model, whilst there was virtually no support for the IAH model (Table 2). Overall, our experimental data therefore provide most support for the HHS model.
When shrimp mortality over time was plotted, there appears to be an effect of dose on shrimp survival in both experiments. Animals exposed to a higher dose appear to die sooner (Figure 2), and there was a significant effect of dose on time until death in both experiments (Log rank test; experiment #1: χ 2 = 14.266, df = 5, p = 0.014; experiment #2: χ 2 = 25.570, df = 6, p < 0.001).

| Meta-analysis of WSSV dose-response
To determine the generality of the results based on our experimental data, a reanalysis and meta-analysis of WSSV Motamedi Sedeh et al., 2012;Thuong et al., 2016), that met all our criteria for inclusion in this analysis. The IAH, DA and HHS models were fitted to these data sets, and model selection was performed individually for each data set (Supporting Information Table S2; a summary of the results is given in Table 3). Upon reanalysis of the individual data sets, there were two clear parallels with our own experimental results. First, in some experiments, the IAH model was supported, whereas in others, the more complex DA or HHS models were better supported (Table 3). Second, the dose-response was either similar to IAH, as seen for our experiment #1, or when this model was not supported, more gradual than model predictions, with ν ≫ 0 for the HHS model and κ < 1 for the DA model, as seen for our experiment #2 (Table 3). F I G U R E 2 The relationship between white spot syndrome virus dose and time until death. For both panels, the ordinate is the time postinoculation, and the abscissae are the proportion of mortality. The response for different log 10 -transformed arbitrary doses is denoted by the legend, and MC indicates the mock-inoculated controls. In panel a, the results for experiment #1 are given, and note the highest dose (10 8  We also performed a reanalysis and meta-analysis of published data and found similar results. In some cases, the dose-response was similar to IAH, whereas in others, it was more gradual. Although the IAH model was supported for some individual data sets, overall, the more complex DA and HHS models were better supported. Thus, we can conclude that the dose-response kinetics of WSSV in shrimp appear to be variable over experiments, and that in many cases, the IAH model is not suitable for the analysis of WSSV dose-response. We do not think that virus amplification would have affected the model selection procedure (see Supporting Information Text S1).
Given that neither the DA nor the HHS models are consistently For instance, all nine data sets reported in one study (Laramore et al., 2009)  plastic and linked to environmental factors such as host density (Reeson, Wilson, Gunn, Hails, & Goulson, 1998). Although we cannot rule out antagonistic dose-dependent action, we therefore think that the most likely explanation for deviations from IAH model predictions is differences in the distribution of host susceptibility to WSSV.
The meta-analysis results also suggest that the IAH model fits better when shrimp with a higher weight are used, raising the possibility that heterogeneous susceptibility to WSSV might decrease as development progresses. This contrasts with observations for baculovirus infection of insect larvae, in which the IAH model was supported early and not late in larval development (Zwart et al., 2009), and we speculate these differences might arise due to differences in the timing of investment in viral defences.
Similar to previous but shorter-running experiments (Escobedo-Bonilla et al., 2005), infection was mainly PCR detected in shrimp that had died by the end of the experiment. Some of these surviving shrimp may represent true sublethal infections (Johnson et al., 2008;Venegas et al., 2000;Wu & Muroga, 2004). These shrimp could also have become infected later in the experiment due to waterborne transmission, in which case death could have been delayed simply due to later onset of infection. Our data suggest that-for the conditions we have studied-sublethal infections will not have a high prevalence at any virus-particle dose. For all models and data sets, we estimated large values for the probability of host death upon virus infection (φ > 0.5).
On the other hand, these results are also puzzling because WSSV infection is often detected in shrimp sampled from ponds in which there are no indications of a disease outbreak (Flegel et al., 2004;Hoa, Zwart, Phuong, Oanh, et al., 2011 of the dose-response will shift, as some virus genotypes will infect more easily (see also Supporting Information Figure S1), and some shrimp populations will be more susceptible. By contrast, we expect the degree of HHS within the shrimp population to be a more impor-  (Cornforth, Matthews, Brown, & Raymond, 2015), highlighting the importance of using empirically supported infection models for assessments. Our results also suggest that infection will be underestimated by the IAH model for low WSSV doses, given the amount of heterogeneity in the host population estimated for experiment #2 and an extreme example from the data reanalysis (Figure 4). Second, consideration of heterogeneity of host susceptibility is indispensable to model and understand infectious disease dynamics. Although individuals with a low susceptibility could facilitate between-pond or between-herd transmission by virtue of being susceptible to low pathogen doses often associated with long-range transmission, in general, heterogeneity makes it harder for a pathogen to spread within a population (Cook, Otten, Marion, Gibson, & Gilligan, 2007;Dwyer, Elkinton, & Buonaccorsi, 1997 The predicted effects of heterogeneity in host susceptibility on the rate of infection are shown. The ordinate is log 10 of virus dose in median lethal dose (LD 50 ) units, whilst the abscissae are the log 10 -transformed rate of infection. The vertical grey line indicates one LD 50 and using the LD 50 scale allows us to compare the predicted rate of infection for different models compared to a common, convenient measure of infectivity. The solid black line is the dose-response predicted by the independent action hypothesis (IAH) model. The dashed purple line represents the heterogeneous host susceptibility (HHS) model prediction with the same variation in host susceptibility as estimated for experiment #2 (ν = 1.905), whereas the dotted dark red line represents the HHS model prediction based on the second highest level of variation predicted for data sets in the reanalysis of published data (ν = 10.233, based on the seventh data set in Supporting Information Table S2; white spot syndrome virus CH-1995-2). The IAH model will underestimate the rate of infection for virus dose below the equivalent of an LD 50 , whilst overestimating infection for higher doses. The underestimation of the rate of infection for low doses is probably more relevant for risk assessment because the infection and death of one or more highly susceptible individuals will lead to all individuals in the pond being exposed to higher doses