# Predictive Water Virology: Hierarchical Bayesian Modeling for Estimating Virus Inactivation Curve

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Systematic Review

#### 2.2. Probability Distribution of Inactivation Model Parameters

_{t}/N

_{0}) were fitted to the efficiency factor Hom (EFH) model (Equations (1) and (2)) for free chlorine disinfection, which can take the decay of the disinfectant’s concentration into account [11,12]. The decay of disinfectants is expressed as the first-order rate equation (Equation (3)).

_{t}is the virus concentration (plaque forming unit, TCID

_{50}, genome copy number) at contact time t [min], N

_{0}is the initial concentration of virus, C

_{0}is the initial concentration of disinfectant [ppm], k is the inactivation rate constant (ppm

^{−n}min

^{−m}), m is the Hom component (-), n is the coefficient of dilution (-), η is the efficiency factor (Equation (2)), C

_{t}is the concentration of disinfectant at contact time t, and k’ is the decay constant (min

^{−1}). Parameters (k, m and n) were determined by the least squares method. The Hom component was set as more than 0.3 in the least squares method to ensure an exact effect of the efficiency factor [13]. The decay constants that were unavailable from several reports were estimated using pH, temperature, type of water and initial concentration of disinfectant as parameters by hierarchical Bayesian modeling (HBM) and used as an approximation for estimating EFH model parameters. We first selected several probability distributions for the EFH model based on Akaike’s information criterion (AIC) using the “fitdistrplus” package of R software [14]. Hierarchical Bayesian models were constructed using the candidate distributions, and then appropriate probability distributions were selected by comparing the root mean squared error (RMSE) and the widely applicable information criterion (WAIC) [15].

#### 2.3. Hierarchical Bayesian Modeling

_{genotype}

_{[i]}. The “i” is the index of viral genotypes, and the µ

_{genotype}

_{[i]}followed a normal distribution (Equation (7)).

_{common}is the mean value among all genotypes and σ

_{G}is standard deviation bearing the genotype-dependent differences. To evaluate the goodness of fit, the generalized linear model (GLM) and HBM were evaluated by RMSE and WAIC. Both model constructions were executed on statistical software R (version 3.5.0, R Foundation for Statistical Computing, Vienna, Autstria) by R and Stan codes. (Tables S1, Code S1 and S2).

#### 2.4. Model Validation

#### 2.4.1. Measurement of Norovirus Concentration in Treated Wastewater

^{−1}).

#### 2.4.2. Inactivation Experiment of Rotavirus by Sodium Hypochlorite

^{−1}). In this experiment, the initial concentration of free chlorine was set at 2.27 ppm.

#### 2.4.3. Predictive Inactivation Curve

## 3. Result

#### 3.1. Article Selection and Data Extraction

#### 3.2. Probability Distribution of the Inactivation Model Parameters

#### 3.3. Comparison of a Goodness of Fit between GLM and HBM

#### 3.4. Predictive Inactivation Curves Based on HBM

^{−1}), pH = 7.0, temperature = 26.5 °C, infectivity and purified water [47]), whereas the second dataset were in the range of prediction after 1.5 min of contact time (Figure 4b; 0.49 ppm of free chlorine, k’ = 0.001 (min

^{−1}), pH = 7.8, temperature = 5.0 °C, infectivity and purified water [28]). The test data of adenoviruses gathered near the 50% predictive curve (Figure 4c; 5.4 ppm of free chlorine, k’ = 3.13 (min

^{−1}), pH = 8.0, temperature = 25 °C, infectivity and purified water [24]). The predictive curve of hepatitis A virus converged to about −1.3, and test data were on the predictive curve except for a value of inactivation efficiency at 2 min (Figure 4d; 0.5 ppm of free chlorine, k’ = 1.17 (min

^{−1}), pH = 6.0, temperature = 5.0 °C, infectivity and purified water [37]). The predictive curves of coxsackievirus displayed a weaker tailing, so that the range of the predictive curve widened as time went on (Figure 4e; 0.25 ppm of free chlorine, k’ = 0.07 (min

^{−1}), pH = 10, temperature = 26.5 °C, infectivity and purified water [47]). The test data of coxsackievirus were placed from 25% to 50% predictive curves. The test data of echovirus were on the center of the predictive inactivation curve (Figure 4f; 0.2 ppm of free chlorine, k’ = 0.47 (min

^{−1}), pH = 7.0, temperature = 5.0 °C, infectivity and contaminated water [34]). The RMSEs of from (a) to (f) were 0.95, 0.58, 0.52, 0.61, 0.84 and 1.02, respectively.

#### 3.5. Effect of a Decay Constant and Efficiency Factor on the Shape of a Predictive Inactivation Curve

## 4. Discussion

^{−4}infection/person/year and 10

^{−6}(10

^{−4}and 10

^{−5}as less stringent setting) DALY loss per person per year (DALYpppy loss) [57,58]. In the case of applying DALY, according to values of tolerable DALY adopted at each WWTP, free chlorine concentration and its contact time as critical limits must be determined (Figure 6). Also, risks estimated by predictive inactivation models are different among virus species, so that operators at each WWTP should determine critical limits to enable the estimated risks of all viruses to go below the employed tolerable risk or disease burden (Figure 6).

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The flow of article screening. Twenty-nine full-text articles relating to free chlorine were selected.

**Figure 2.**Comparison of a goodness of fit between the generalized linear model (GLM) and hierarchical Bayesian model (HBM) for EFH model parameters (circle: purified water, triangle: contaminated water). Each color shows the type of viral genotypes, and the gray shade indicates the results of GLM.

**Figure 3.**Predictive inactivation curves of norovirus GII and rhesus rotavirus by free chlorine using real datasets (red: 2.5%, purple: 25%, green: 50%, orange: 75%, blue: 97.5% curves). Circle plots are test data of inactivation efficiency. (

**a**) A predictive curve for norovirus GII was generated under the following conditions: 2.0 ppm of free chlorine, k’ = 17.9 (min

^{−1}), pH = 6.74, temperature = 16.6 °C, genome copy number and contaminated water. (

**b**) A predictive curve for a rhesus rotavirus strain was generated under the following conditions: 2.27 ppm of free chlorine, k’ = 7.01 (min

^{−1}), pH = 7.0, temperature = 20 °C, infectivity and contaminated water.

**Figure 4.**Validation of predictive inactivation models for free chlorine using test datasets (red: 2.5%, purple: 25%, green: 50%, yellow: 75%, blue: 97.5% curve). (

**a**) Poliovirus (type 1 Mahoney) inactivation under the condition; 0.29 ppm of free chlorine, k’ = 0.001 (min

^{−1}), pH = 7.0, temperature = 26.5 °C, infectivity and purified water [47], (

**b**) Poliovirus (type 1 Mahoney) inactivation under the condition; 0.49 ppm of free chlorine, k’ = 0.001 (min

^{−1}), pH = 7.8, temperature = 5.0 °C, infectivity and purified water [28], (

**c**) Adenovirus (type 2) inactivation under the condition; 5.4 ppm of free chlorine, k’ = 3.13 (min

^{−1}), pH = 8.0, temperature = 25 °C, infectivity and purified water [24], (

**d**) Hepatitis A virus (HM175) inactivation under the condition; 0.5 ppm of free chlorine, k’ = 1.17 (min

^{−1}), pH = 6.0, temperature = 5.0 °C, infectivity and purified water [37], (

**e**) Coxsackievirus (B5) inactivation under the condition; 0.25 ppm of free chlorine, k’ = 0.07 (min

^{−1}), pH = 10, temperature = 26.5 °C, infectivity and purified water [47], (

**f**) Echovirus (type 1 Farouk) inactivation under the condition; 0.2 ppm of free chlorine, k’ = 0.47 (min

^{−1}), pH = 7.0, temperature = 5.0 °C, infectivity and contaminated water [34]. The dashed lines indicate predictive curves that are inappropriate to use because the m-values were less than 0.3.

**Figure 5.**Predicted inactivation curves constructed by different k’-values (norovirus inactivation at the condition: 2.0 ppm of free chlorine, Ph = 7, temperature = 20 °C, genome copy number and contaminated water). (

**a**) k’ = 10, (

**b**) k’ = 17.9, (

**c**) k’ = 25.

**Figure 6.**Example curves of DALYpppy loss for three virus species. Blue circles indicate the critical limits at each value of acceptable disease burden suggested by WHO [2]. Critical limits are the intersection point of the x-axis and DALYpppy loss curve for norovirus, which shows a higher burden of disease than other viruses.

k | m | n | |
---|---|---|---|

Norovirus | Lognormal | →* | → |

Rotavirus | Lognormal | → | → |

Poliovirus | Lognormal | Gamma | Lognormal |

Adenovirus | Lognormal | → | → |

Hepatitis A virus | Lognormal | → | → |

Coxsackievirus | Lognormal | → | → |

Echovirus | Exponential | Lognormal | Lognormal |

**Table 2.**Comparison of a goodness of fit between generalized linear model (GLM) and hierarchical Bayesian model (HBM).

k | m | n | |||||
---|---|---|---|---|---|---|---|

Statistics | GLM | HBM | GLM | HBM | GLM | HBM | |

Norovirus | WAIC | 23.0 | 14.4 | 6.81 | 7.54 | 28.9 | 17.4 |

RMSE | 0.34 | 0.23 | 0.03 | 0.03 | 0.43 | 0.27 | |

Rotavirus | WAIC | 54.6 | 45.8 | −7.90 | −11.2 | 43.9 | 35.6 |

RMSE | 0.45 | 0.41 | 0.16 | 0.14 | 0.20 | 0.14 | |

Poliovirus | WAIC | 69.5 | 68.8 | 37.9 | 36.3 | 63.9 | 63.3 |

RMSE | 0.54 | 0.53 | 0.51 | 0.49 | 0.50 | 0.49 | |

Adenovirus | WAIC | 11.0 | 1.77 | −3.12 | −5.92 | 17.9 | −2.19 |

RMSE | 0.05 | 0.01 | 0.02 | 0.02 | 0.08 | 0.01 | |

Hepatitis A virus | WAIC | 19.4 | 11.9 | 4.02 | 2.04 | 18.1 | 18.4 |

RMSE | 0.29 | 0.27 | 0.17 | 0.15 | 0.16 | 0.15 | |

Coxsackievirus | WAIC | 67.9 | 65.5 | 2.21 | 3.98 | 48.1 | 47.5 |

RMSE | 0.70 | 0.68 | 0.20 | 0.19 | 0.51 | 0.49 | |

Echovirus | WAIC | 50.6 | 48.1 | −9.30 | −8.13 | 18.1 | 13.5 |

RMSE | 0.24 | 0.23 | 0.10 | 0.09 | 0.46 | 0.41 |

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**MDPI and ACS Style**

Kadoya, S.-s.; Nishimura, O.; Kato, H.; Sano, D.
Predictive Water Virology: Hierarchical Bayesian Modeling for Estimating Virus Inactivation Curve. *Water* **2019**, *11*, 2187.
https://doi.org/10.3390/w11102187

**AMA Style**

Kadoya S-s, Nishimura O, Kato H, Sano D.
Predictive Water Virology: Hierarchical Bayesian Modeling for Estimating Virus Inactivation Curve. *Water*. 2019; 11(10):2187.
https://doi.org/10.3390/w11102187

**Chicago/Turabian Style**

Kadoya, Syun-suke, Osamu Nishimura, Hiroyuki Kato, and Daisuke Sano.
2019. "Predictive Water Virology: Hierarchical Bayesian Modeling for Estimating Virus Inactivation Curve" *Water* 11, no. 10: 2187.
https://doi.org/10.3390/w11102187