# Optimal Selection of Sampling Points within Sewer Networks for Wastewater-Based Epidemiology Applications

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data

#### 2.2.1. Sampling Data

#### 2.2.2. Geographic Data

#### 2.3. General Procedure

#### 2.4. System Sensitivity

#### 2.5. Optimization of Sampling Point Location

#### 2.5.1. Information Theory

_{1}, C

_{2}, …, C

_{N}) are considered, the total information content can be calculated by the joint entropy $H\left({C}_{1},{C}_{2},\dots ,{C}_{N}\right)$, which is defined as

_{1}, C

_{2}, …, C

_{N}), and $p\left({c}_{1}^{{j}_{1}},{c}_{2}^{{j}_{2}},\dots ,{c}_{N}^{{j}_{N}}\right)$ is the joint probability of events $\left({c}_{1}^{{j}_{1}},{c}_{2}^{{j}_{2}},\dots ,{c}_{N}^{{j}_{N}}\right)$.

#### 2.5.2. Probability Distribution

#### 2.5.3. Signal Matrix and Entropy

#### 2.5.4. Objective Function and Optimization Algorithm

Algorithm 1. MIMR-based greedy selection algorithm to find optimal sampling points. | |

1 | Procedure $\mathrm{FIND}\_\mathrm{OPTIMAL}\_\mathrm{S}(C,N,X,S,z$)Input: candidate set $C$ including all $N$ candidate nodes, catchment set $X$ including all sub-catchments, and desired number of candidate nodes $z$.Output: selection set $S$ including optimally selected candidate nodes. |

2 | Initialize maximum joint entropy ${JE}^{MAX}\leftarrow H\left(X\right)$ using Equation (3), selection set $S\leftarrow \varnothing $, temporary joint entropy ${JE}^{TEMP}\leftarrow NULL$. |

3 | for $c\in \mathrm{C}$ do |

4 | Calculate entropy $H\left(c\right)$ for each candidate node using Equation (6) |

5 | end for |

6 | Assign optimal candidate node $s\leftarrow \mathrm{arg}\underset{c}{\mathrm{max}}\left[H\left(c\right)\right]$ |

7 | Update $C\leftarrow C\backslash \left\{s\right\}$ |

8 | Update $S\leftarrow \mathrm{S}\bigcup \left\{s\right\}$ |

9 | while ${JE}^{TEMP}\ne {JE}^{MAX}$ and $\left|S\right|\le z$ do |

10 | for $c\in C$ do |

11 | Calculate ${MIMR}_{S\bigcup \left\{c\right\}}$ using Equation (7) |

12 | end for |

13 | Find local optimal candidate node $s\leftarrow \mathrm{arg}\underset{c}{\mathrm{max}}{[MIMR}_{S\bigcup \left\{c\right\}}]$ using Equation (8) |

14 | Update $C\leftarrow C\backslash \left\{s\right\}$ |

15 | Update $S\leftarrow S\bigcup \left\{s\right\}$ |

16 | Assign temporary joint entropy ${JE}^{TEMP}\leftarrow H\left(S\right)$ |

17 | end while |

18 | return $\mathrm{selection}\mathrm{set}S$ |

## 3. Experimental Results and Discussion

#### 3.1. Determination of System Sensibility

_{S}), as values from the literature vary exponentially. Focusing on the M

_{S}in Figure 7a, at candidate node “A” the number of infected individuals needed to detect a positive signal exceeded the total population. This means no positive signal would be detected even if the entire population was infected. Compared to the median values, their optimal values from Table 3 were utilized in the second method. The results are illustrated in Figure 7b. With the optimal parameter combination, circa 40 infected individuals are needed in the catchment (104,231 residents) to detect a positive signal. In other words, one infected individual out of 2641 noninfected individuals could be detected under the combination of optimal parameter values. This value is close to our experience with real-world data, where SARS-CoV-2 RNA was detected in wastewater samples from the WWTP for incidences of approximately 40 infected individuals per 100,000 residents (1 infected individual in 2500). Thus, the optimal parameter values were used for the following analysis in Section 3.2.

#### 3.2. Optimization of Sampling Points

## 4. Conclusions

- Virus specific parameter values of SARS-CoV-2 from the literature are currently not sufficient for parametrizing our model.
- Number and locations for the sampling points depends on the expected sensitivity of the system.
- Increasing the number of sampling points does not necessarily improve the information content.
- Virus-related uncertainties have an impact on the placement and number of sampling points, but this impact is offset by the expected sensitivity.
- For the case study of Hildesheim, only 8 sampling points and less than 10 infected individuals per sub-catchment were required to identify potentially infected sub-catchments.

- The probability distribution function is simply based on the assumption that all infected people come from the same sub-catchment. For a better representation, epidemiological data could be used to estimate real infection distributions, as shown in Figure 5b.
- The flow time used to calculate the system sensitivity simply uses a constant. For further studies, 1D sewer models can be applied to better estimate the flow time and also simulate RNA loss, which is another limitation of the current approach.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameter: Virus RNA Shedding Magnitude | ||

$\mathrm{Values}({log}_{10}$ copies/mL) | Literature | Comments |

2.9 | [31] | |

3.75 | [32] | Units adjusted |

4.55 | [16] | Units adjusted |

4.7 | [33] | |

5.8 | [32] | |

6.28 | [34] | Units adjusted |

7.1 | [31] | |

Parameter: virus RNA Shedding probability | ||

Values (%) | Literature | Comments |

10.1 | [35] | |

15.3 | [33] | |

29 | [36] | |

47.7 | [37] | |

48.1 | [33] | |

53.4 | [38] | |

54.5 | [16] | |

55 | [39] | |

83.3 | [32] | |

Parameter: virus RNA decay in wastewater | ||

Values (-) | Literature | Comments |

0.06 | [40] | |

0.084 | [41] | |

0.09 | [42] | |

0.183 | [43] | |

0.286 | [41] | |

0.67 | [42] | |

Parameter: critical detection limit | ||

Values (copies/mL) | Literature | Comments |

3.7 | [44] | |

9.2 | [44] | |

39.04 | [45] | |

59.4 | [45] | |

72.42 | [45] | |

78.96 | [45] | |

79.08 | [45] | |

98.42 | [45] | |

133.02 | [45] | |

159.08 | [45] | |

183.34 | [45] | |

301.22 | [45] | |

374.86 | [45] | |

533.78 | [45] |

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**Figure 1.**Study area: the main sewer network of Hildesheim with candidate nodes and sub-catchments (source: Landesamt für Geoinformation und Landesvermessung Niedersachsen/LGLN).

**Figure 2.**Photos from fieldwork: (

**a**,

**b**) installation of an autosampler in a manhole; (

**c**,

**d**) damaged autosampler due to surcharge; (

**e**) quality of two-hour composite samples over 24 h from one autosampler.

**Figure 3.**Mass balance model from RNA shedding to sample analysis, where ${\mathrm{C}}_{\mathrm{t}}$ and ${\mathrm{C}}_{0}$ are the concentrations of virus RNA in wastewater at time $\mathrm{t}$ and time $0$, ${L}_{RNA}$ (copies/day) is the detected RNA load, ${C}_{RNA}$ (copies/mL) is the detected RNA concentration in the lab, ${c}_{crit}$(copies/mL) is the critical detection limit, ${Q}_{DWF}$ (mL/day) refers to dry weather flow (only domestic wastewater ${Q}_{D}$ is considered), ${p}_{s}$ (-) is the virus RNA shedding probability in stool, ${q}_{s}$ (mL/(person*day)) is the volume of stool produced per individual and day, ${M}_{s}$ (copies/mL) is the virus RNA shedding magnitude in stool, $k$ (/day) is the first-order decay value of RNA in wastewater, and $t$ (day) is the flow time of wastewater in the sewer network from RNA input to sampling point.

**Figure 4.**Venn diagrams describing (

**a**) the total system, $\mathsf{\Omega}$, and two events (A and B), (

**b**) individual entropy, (

**c**) joint entropy, and (

**d**) total correlation [26].

**Figure 5.**Different definitions of the probability distribution using network topology and settlement structure: (

**a**) sub-catchment-dependent probability distribution, (

**b**) realistic probability distribution. Dots indicate potentially infected individuals (green dot: the probability that the individual is infected is zero, red dot: the lighter the red, the less likely the individual is to be infected).

**Figure 7.**Longitudinal section of candidate nodes (capital letter with number) and covered residents (number below nodes’ identifier). The minimal number of infected individuals to detect a positive signal by changing one parameter and holding (

**a**) all other parameters with their median values and (

**b**) all other parameters with their optimal values. Virus-dependent parameters: $k$—Virus RNA decay in wastewater, ${c}_{crit}$—Critical detection limit, ${M}_{S}$—Virus RNA shedding magnitude, ${p}_{S}$—Virus RNA shedding probability.

**Figure 8.**Color maps of entropy based on different ${E}_{inf}$ within the detection capability 1:2641. The darker the color of the node, the higher the entropy.

**Figure 9.**Entropy for each node and optimal sampling point covering the corresponding sub-catchments with eight sampling points (system sensitivity 1:2641, ${E}_{inf}=$ 10). The numbers of the lines indicate the length of the specific sewer.

Data Used | Data Source |
---|---|

Sewer network | Stadtentwässerung Hildesheim (SEHi) (2021) |

Population statistics | Stadt Hildesheim (2022), https://www.stadt-hildesheim.de/rathaus-verwaltung/buerger-und-ratsinfo/stadtteile/ (accessed on 1 December 2022) |

Land use map | Stadt Hildesheim (2015), https://www.stadt-hildesheim.de/wirtschaft-bauen/stadtplanung-und-stadtentwicklung/stadtentwicklung/flaechennutzungsplan/ (accessed on 1 December 2022) |

Digital orthophoto (DOP) | Landesamt für Geoinformation und Landesvermessung Niedersachsen (LGLN) 2022, https://opengeodata.lgln.niedersachsen.de/#dop (accessed on 1 December 2022) |

3D building model | LGLN (2022), https://opengeodata.lgln.niedersachsen.de/#lod2 (accessed on 1 December 2022) |

ALKIS-Dataset | LGLN (2021), provided by SEHi |

Parameter | Value | Comments | Source |
---|---|---|---|

Feces production rate | 128 g/(person*day) | Wet mass | [18] |

Feces density | 1.06 g/mL | [19] | |

Average water consumption | 128 L/(person*day) | The value from 2019 | [20] |

Factors | Labels | Unit | Min | 25% | 50% | 75% | Max |
---|---|---|---|---|---|---|---|

RNA shedding magnitude | ${M}_{s}$ | $({\mathit{log}}_{10}\mathrm{copies}$/mL) | 2.90 | 4.15 | 4.70 | 6.04 | 7.10 |

RNA shedding probability | ${p}_{s}$ | (%) | 10.1 | 29.0 | 48.1 | 54.5 | 83.3 |

RNA decay in wastewater | $k$ | (/day) | 0.06 | 0.09 | 0.14 | 0.26 | 0.67 |

RNA critical detection limit | ${c}_{crit}$ | (copies/mL) | 3.70 | 62.66 | 88.75 | 177.28 | 533.78 |

^{1}Numbers in bold indicate the best value in each row.

**Table 4.**An example of a potential signal matrix using the system sensitivity of 1:2200 based on the hypothetical model with the probability of the infected individual from a specific sub-catchment (+ positive: signal detected on candidate node, − negative: no signal detected).

$\mathbf{Source}\mathit{X}$ | $\mathbf{Candidate}\mathbf{Nodes}\mathit{C}$ | Probability | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | E | S1 | S2 | S3 | S4 | S5 | S6 | $\mathit{p}\left({\mathit{X}}_{\mathit{i}}\right)$ | |

X1 | − | − | − | − | − | + | − | − | − | − | − | 0.16 |

X2 | − | + | + | − | − | − | + | − | − | − | − | 0.08 |

X3 | − | + | − | + | − | − | − | + | − | − | − | 0.26 |

X4 | − | + | + | − | + | − | − | − | + | − | − | 0.14 |

X5 | − | + | + | − | + | − | − | − | − | + | − | 0.22 |

X6 | − | + | − | + | − | − | − | − | − | − | + | 0.14 |

**Table 5.**Relationship between the number of infected individuals in one sub-catchment and entropies

^{1}. Numbers in brackets indicate that one more sampling point is needed to cover the catchment. ${E}_{inf}$ is the number of infected individuals in the sub-catchment, ${N}_{S}$ is the smallest number of sampling points needed to reach the maximum joint entropy, ${JE}^{MAX}$, $theo.{JE}_{CA}^{MAX}$ is the theoretical maximum joint entropy of the covered areas, ${N}_{covered}$ is the maximum number of covered populations, and ${N}_{total}$ is the total number of populations.

${\mathit{E}}_{\mathit{i}\mathit{n}\mathit{f}}$ | ${\mathit{N}}_{\mathit{S}}$ | ${\mathit{J}\mathit{E}}^{\mathit{M}\mathit{A}\mathit{X}}$ | $\frac{{\mathit{J}\mathit{E}}^{\mathit{M}\mathit{A}\mathit{X}}}{\mathit{t}\mathit{h}\mathit{e}\mathit{o}.{\mathit{J}\mathit{E}}_{\mathit{C}\mathit{A}}^{\mathit{M}\mathit{A}\mathit{X}}}$ | ${\mathit{N}}_{\mathit{c}\mathit{o}\mathit{v}\mathit{e}\mathit{r}\mathit{e}\mathit{d}}$ | $\frac{{\mathit{N}}_{\mathit{c}\mathit{o}\mathit{v}\mathit{e}\mathit{r}\mathit{e}\mathit{d}}}{{\mathit{N}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}}$ |
---|---|---|---|---|---|

(-) | (-) | (bits) | (%) | (-) | (%) |

1 | 17 | 1.86 | 37.4 | 26,000 | 24.9 |

2 | 20 | 3.55 | 71.3 | 58,046 | 55.7 |

3 | 20 | 4.43 | 89.1 | 80,925 | 77.6 |

4 | 16 | 4.85 | 97.4 | 94,018 | 90.2 |

6 | 15 | 4.85 | 97.4 | 94,018 | 90.2 |

8 | 11 | 4.95 | 99.4 | 99,834 | 95.8 |

9 | 8 | 4.95 | 99.4 | 99,834 | 95.8 |

10 | 7 (8) | 4.98 | 100.0 | 99,076 (104,231) | 95.1 (100.0) |

12 | 6 (7) | 4.98 | 100.0 | 99,076 (104,231) | 95.1 (100.0) |

13 | 5 (6) | 4.98 | 100.0 | 99,076 (104,231) | 95.1 (100.0) |

23 | 4 (5) | 4.98 | 100.0 | 99,076 (104,231) | 95.1 (100.0) |

24 | 3 (4) | 4.98 | 100.0 | 99,076 (104,231) | 95.1 (100.0) |

26 | 3 (4) | 4.98 | 100.0 | 98,750 (104,231) | 94.7 (100.0) |

29 | 2 | 4.98 | 100.0 | 104,231 | 100.0 |

40 | 1 | 4.98 | 100.0 | 104,231 | 100.0 |

^{1}Bold and underlined numbers indicate the reached values for the optimal set of sampling points.

${\mathit{N}}_{\mathit{S}}$ | $\mathit{S}$ | $\mathit{J}\mathit{E}$ | $\mathit{T}\mathit{C}$ | $\mathit{M}\mathit{I}\mathit{M}{\mathit{R}}_{\mathit{S}}$ | ${\mathit{N}}_{\mathit{c}\mathit{o}\mathit{v}\mathit{e}\mathit{r}\mathit{e}\mathit{d}}$ |
---|---|---|---|---|---|

1 | G3 | 1.51 | 0.00 | 1.21 | 24,623 |

2 | F1 | 2.88 | 0.11 | 2.28 | 25,785 |

3 | C3 | 4.02 | 0.34 | 3.15 | 21,192 |

4 | G2 | 4.40 | 0.51 | 3.42 | 9657 |

5 | C4 | 4.72 | 0.71 | 3.64 | 8344 |

6 | C1 | 4.91 | 0.92 | 3.74 | 6891 |

7 | S23 | 4.98 | 1.02 | 3.78 | 2584 |

8 | S24 | 4.98 | - | - | 5155 |

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## Share and Cite

**MDPI and ACS Style**

Yao, Y.; Zhu, Y.; Nogueira, R.; Klawonn, F.; Wallner, M.
Optimal Selection of Sampling Points within Sewer Networks for Wastewater-Based Epidemiology Applications. *Methods Protoc.* **2024**, *7*, 6.
https://doi.org/10.3390/mps7010006

**AMA Style**

Yao Y, Zhu Y, Nogueira R, Klawonn F, Wallner M.
Optimal Selection of Sampling Points within Sewer Networks for Wastewater-Based Epidemiology Applications. *Methods and Protocols*. 2024; 7(1):6.
https://doi.org/10.3390/mps7010006

**Chicago/Turabian Style**

Yao, Yao, Yibo Zhu, Regina Nogueira, Frank Klawonn, and Markus Wallner.
2024. "Optimal Selection of Sampling Points within Sewer Networks for Wastewater-Based Epidemiology Applications" *Methods and Protocols* 7, no. 1: 6.
https://doi.org/10.3390/mps7010006