The viral decay rate in the soil determined by Roberts et al. was adopted because the experimental temperature and soil type are more relevant to lettuce growing conditions compared to the other decay study . Decay rates in the root and shoot were used from the hydroponic system predictions.The transport model was fitted to log10 viral concentration data from DiCaprio et al. , extracted from graphs therein using WebPlotDigitizer . In these experiments, NoV of a known concentration was spiked in the feed water of hydroponic lettuce and was monitored in the feed water, the root and shoot over time. While fitting the model, an initial feed volume of 800 mL was adopted and parameters producing final volumes of b200 mL were rejected. To fit the model while accounting for uncertainty in the data, a Bayesian approach was used to maximize the likelihood of the data given the parameters. A posterior distribution of the parameters was obtained by the differential evolution Markov chain algorithm, which can be parallelized and can handle multi-modality of the posteriors distribution without fine tuning the jumping distribution.
The rationale behind the model fitting procedure and diagnostics are discussed in Supplementary section S1H.The initial viral concentration in the irrigation water was drawn from an empirical distribution reported previously by Lim et al. for NoV in activated sludge treated secondary effluent. As justi- fied by Lim et al. ,vertical grow tables the sum of the concentrations of two genotypes known to cause illness was used to construct the distribution. To estimate the risk from consumption of lettuce, the daily viral dose was computed using Eq. 10 for the kth day. The body weight was drawn from an empirical distribution for all ages and genders in the United States, from a report of the percentile data of body weight. The lettuce consumption rate was drawn from an empirical distribution constructed from data reported by the Continuing Survey of Food Intakes by Individuals . The ‘consumer only’ data for all ages and genders was used, and hence the reported risk is only for those who consume lettuce. It is important to note that the daily viral dose was computed in from the model output using the shoot density ρshoot to be consistent with the consumption rate reported in CSFII. Several different NoV dose-response models have been pro posed based on limited clinical data. The validity of these models is a matter of much debate , which is beyond the scope of this study.
These models differ in their assumptions resulting in large variability of predicted risk out comes. To cover the range of potential outcomes of human exposure to NoV, we estimated and compared risk outcomes using three models: 1) Approximate Beta-Poisson ; 2) Hypergeometric ; and 3) Fractional Poisson . In the risk estimation, we considered NoV in the lettuce tissue exists as individual viral particles and used the disaggregated NoV models. The model equations are given by Eq. 11–13, Table 1. Ten thousand samples of the daily infectious risks were calculated from BP and FP models using MATLAB R2016a. Wolfram Mathematica 11.1 was used for the model estimation as it was faster. Using a random set of 365 daily risk estimates of , the annual infection risk was calculated according to the Gold Standard estimator using Eq. 14, Table 1. While there appears to be some dose dependence for illness resulting from infection Pill∣inf , this has not been clearly elucidated for the different dose response models. Hence, we adopted the procedure used in Mara and Sleigh and calculated annual illness risk with Eq. 15.Under the assumption of first order viral decay, NoV loads in water at two time points did not fall in the credible region of model predictions, indicating that mere first order decay was unsuitable to capture the observed viral concentration data. The addition of the AD factor into the model ad dressed this inadequacy and importantly supported the curvature ob served in the experimental data.
This result indicates the AD of viruses to hydroponic tank wall is an important factor to include in predicting viral concentration in all three compartments.The adequacy of model fit was also revealed by the credible intervals of the predicted parameters for the model with AD . Four of the predicted parameters: at, bt, kdec, s and kp, were restricted to a smaller subset of the search bounds, indicating that they were identifiable. In contrast, the viral transfer efficiency η and the kinetic parameters spanned the entirety of their search space and were poorly identifiable. However, this does not suggest that each parameter can independently take any value in its range because the joint distributions of the parameters indicate how fixing one parameter influences the likelihood of another parameter . Hence, despite the large range of an individual pa rameter, the coordination between the parameters constrained the model predictions to produce reliable outcomes . Therefore, the performance of the model with AD was considered adequate for estimating parameters used for risk prediction.Risk estimates for lettuce grown in the hydroponic tank or soil are presented in Fig. 4. Across these systems,cultivo de frambuesas en maceta the FP model predicted the highest risk while the 1F1 model predicted the lowest risk. For a given risk model, higher risk was predicted in the hydroponic system than in the soil. This is a consequence of the very low detachment rates in soil compared to the attachment rates. Comparison of results from Sc1 and Sc2 of soil grown lettuce indicated lower risks and dis ease burdens under Sc1 . Comparing with the safety guidelines, the lowest risk predicted in the hydroponic system is higher than the U.S. EPA defined acceptable annual drinking water risk of 10−4 for each risk model. The annual burdens are also above the 10−6 benchmark recommended by the WHO . In the case of soil grown lettuce, neither Sc1 nor Sc2 met the U.S. EPA safety bench mark. Two risk models predicted borderline disease burden according to the WHO benchmark, for soil grown lettuce in Sc1, but under Sc2 the risk still did not meet the safety guideline. Neither increasing holding time of the lettuce to two days after harvesting nor using bigger tanks significantly altered the predicted risk . In comparison, the risk estimates of Sales-Ortells et al. are higher than range of soil grown lettuce outcomes presented here for 2 of 3 models. The SCSA sensitivity indices are presented in Fig. 5. For hydroponi cally grown lettuce, the top 3 factors influencing daily risk are amount of lettuce consumed, time since last irrigation and the term involving consumption and ρshoot. Also, the risk estimates are robust to the fitted parameters despite low identifiability of some model parameters . For soil grown lettuce, kp ap pears to be the major influential parameter, followed by the input viral concentration in irrigation water and the lettuce harvest time. Scorr is near zero, suggesting lesser influence of correlation in the input parameters.In this study, we modeled the internalization and transport of NoV from irrigation water to the lettuce using ordinary differential equations to capture the dynamic processes of viral transport in lettuce.
This first attempt is aimed at underscoring the importance of the effect of time in determining the final risk outcome. The modeling approach from this study may be customized for other scenarios for the management of water reuse practices and for developing new guidelines for food safety. Moreover, this study identifies critical gaps in the current knowledge of pathogen transport in plants and calls for further lab and field studies to better understand risk of water reuse.To develop an adequate model to predict viral transport in plant issue, it is necessary to couple mathematical assumptions with an under standing of the underlying biogeochemical processes governing virus removal, plant growth, growth conditions and virus-plant interactions. For example, although a simple transport model without AD could predict the viral load in the lettuce at harvest, it failed to capture the initial curvature in the viral load in the growth medium . An alternative to the AD hypothesis that could capture this curvature is the existence of two populations of viruses as used in Petterson et al. , one decaying slower than the other. However, a closer examination of the double exponential model revealed that it was not time invariant. This means that the time taken to decay from a concentration C1 to C2 is not unique and depends on the history of the events that occurred . Other viral models, such as the ones used in Peleg and Penchina faced the same issues. The incorporation of AD made the model time invariant and always provided the same time for decay between two given concentrations. This model fitting experience showcases how mathematics can guide the understanding of biological mechanisms.
The hypothesis of two different NoV populations is less plausible than that of viral attachment and detachment to the hydroponic tank. While it appears that incorporating the AD mechanism does not significantly improve viral load prediction in lettuce shoot at harvest, this is a consequence of force fitting the model to data under the given conditions. Changing the conditions, for example, by reducing viral attachment rate to the tank wall, will underestimate virus load in the lettuce shoot in the absence of AD . Through this model fitting exercise, we also acknowledge that the model can be significantly improved with new insights on virus plant interactions and more data on the viral transport through plants. A potential cause for concern in the model fit is the wide intervals. However, there is significant uncertainty in the data as well suggesting that the transport process itself is noise prone. Moreover, from the perspective of risk assessment, the variability between dose-response models is higher than the within dose-response model variability . Since within dose-response model variability stems from uncertainty in viral loads at harvest among other factors, the wide intervals do not exert a bigger effect than the discordance from different dose response models.Some parameters are identifiable to a good degree through model fitting, but there is a large degree of uncertainty in the viral transport efficiencies and the AD kinetic parameters. While this could be a consequence of fitting limited number of data points with several parameters, the viral load at harvest and risk estimates were well constrained. This large variation in parameters and ‘usefully tight quantitative predictions’ is termed the sloppiness of parameter sensitivities, and has been observed in physics and systems biology . Well designed experiments may simultaneously reduce uncertainty in the parameters as well as predictions , and therefore increasing confidence in predictions. One possible experiment to reduce parameter uncertainty is recording the transpiration and growth rate to fit Eq. 6 independently to acquire at and bt.An interesting outcome of our analysis is the strong association of risk with plant growth conditions. The health risks from consuming lettuce irrigated with recycled wastewater are highest in hydroponic grown lettuce, followed by soil grown lettuce under Sc2 and the least in soil grown lettuce under Sc1 . This difference in risk estimates stems to a large degree from the difference in AD kinetic constants . Increasing katt, s will decrease risk as more viruses will get attached to the growth medium, while increasing kdet, s will have the opposite effect , as more detached viruses are available for uptake by the plant. The combined effect of the AD parameters depends on their magnitudes and is portrayed in Supplementary Fig. S5. This result indicates that a better understanding of the virus interaction with the growth environment can lead to an improved understanding of risk. More importantly, this outcome indicates that soil plays an important role in the removal of vi ruses from irrigation water through adsorption of viral particles. An investigation focused on understanding the influence of soil composition on viral attachment will help refine the transport model. The risk predicted by this dynamic transport model is greater than the EPA annual infection risk as well as the WHO annual disease burden benchmark. The reasons for this outcome are many-fold. First, there is a significant variability in the reported internalization of viruses in plants.