# Agglomerative Clustering of Enteric Infections and Weather Parameters to Identify Seasonal Outbreaks in Cold Climates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Health Outcomes

#### 2.2. Meteorological Records

#### 2.3. Agglomerative Clustering

_{1t}and z

_{2t}, for salmonellosis and enteritis, respectively, and a set of time series of weather parameters, w

_{mt}, where m = {1,…,4} indicate a parameter, t = {1,…,T} is a study day, and T is the effective length of time series, T = 2848 days. As illustrated in Figure 1 and Figure 2, z

_{1t}and z

_{2t}are assumed to be derived from a Poisson distribution with parameters λ

_{l}and λ

_{2}, respectively; and w

_{mt}, are assumed to be derived from a multivariate normal distribution, N(

**μ**, Σ).

^{T}= {x

_{1}, …, x

_{T}}, where an object x

_{i}is a certain day with the centered daily values of disease counts and meteorological parameters, say x

_{i}= {z

_{1i}, w

_{1i}, w

_{2i}, w

_{3i}, w

_{4i}}. To each day x

_{i}∈X

^{T}, we assign a label, or a cluster identifier, of class y

_{i}. Next, to describe and visualize the multidimensional data, we examine the clustering structure using the t-Distributed Stochastic Neighbor Embedding (t-SNE) method [19], which relies on the conditional probabilities, as follow (Equation (1)):

_{j}are to those of day x

_{i}, given a normal distribution centered on x

_{i}with variance σ

_{i}

^{2}[19]. The variance σ

_{i}

^{2}of each day was selected considering the perplexia estimate, such that the days with the most common values of disease counts and weather parameters, or in other words days with higher density, have a smaller variance [19]. For days with similar disease counts and meteorological parameters, ${p}_{j|i}$ should be relatively high, whereas for days with very different counts and meteorological parameters, ${p}_{j|i}$ should be almost infinitesimal.

^{2}is the Euclidian distance between the geometric centers G

^{C}and G

^{H}of the clusters C and H; and |C| and |H| are the numbers of objects (or days) in the clusters C and H, respectively. Ward’s distance was chosen because it has tensile properties and shows good results in preliminary experiments on clustering [20,21,22]. These properties were ensured by applying normalization and standardization to each variable before the clustering procedure. In addition, the standard procedure implemented in the adapted package used PCA to transform data (the forward transformation—before clustering and reverse transformation—after clustering) to minimize the correlation across the features. Next, to assess the quality of clustering and to choose the number of clusters, we maximized the silhouette metric [20] (Equation (3)):

_{i}of the sample (i.e., each day of the time series). In the context of a daily time series, “distances from one day to another” can be interpreted as similarities among days based on disease counts and the averages of meteorological factors. Thus, the silhouette metric for time series data can be interpreted as a similarity profile of time series segments (clusters).

#### 2.4. Log-Linear and Harmonic Regression Models

_{t}]) = β

_{0}+ β

_{s}sin(2πωt) + β

_{c}cos(2πωt)

_{t}is the daily counts for z

_{1t}and z

_{2t}, e.g., salmonellosis and enteritis, respectively; seasonality in disease cases was assessed based on the significance of the two harmonic terms with t as consecutive days of the study period, and ω = 1/365.25. Estimates obtained from fitting Model A can be used to compute phase shift $\varphi $ based on join signs of ${\widehat{\beta}}_{s}\mathrm{and}\text{}{\widehat{\beta}}_{c}$:

_{t}, and weather parameters, w

_{jt}(Model B):

_{t}]) = β

_{0}+ β

_{m}w

_{mt}

_{m}is the regression coefficient for the m-weather parameter. Since ambient temperature, pressure, and dew point values are highly correlated, we ran Model B with each weather parameter separately. Relative risks (RR) associated with a 10 unit of change in each weather parameter and their 95th confidence intervals (CI

_{95%}) are estimated as RR

_{j}= exp{10 β

_{m}} and CI

_{95%}= exp{10(β

_{m}± 1.96σ

_{βm})}, respectively. The Model B was then further explored with respect to its stability by adding sequentially the first order autocorrelation term, seasonal harmonics, and indicator variables for day of the week.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Daily time series and histograms for health outcomes: (

**a**) salmonellosis, A02.8, and (

**b**) enteritis, A04.9 in Barnaul, Russia in 2004–2011.

**Figure 2.**Daily time series and histograms for weather parameters: (

**a**) temperature (°C), (

**b**) dew point (°C), (

**c**) relative humidity (%), and (

**d**) atmospheric pressure (hPa) in Barnaul, Russia in 2004–2011.

**Figure 3.**Results of t-SNE analysis depicted by the clustering distances in the vertical and horizontal axes for: (

**a**) salmonellosis, A02.8, and (

**b**) enteritis, A04.9 in Barnaul, Russia in 2004–2011. Colors represent health outcome counts from 0 to 12 for salmonellosis and 0 to 27 for enteritis.

**Figure 4.**Silhouette metric values and number of clusters for: (

**a**) salmonellosis, A02.8, and (

**b**) enteritis, A04.9 in Barnaul, Russia in 2004–2011.

**Figure 5.**Histograms of salmonellosis daily counts and meteorological characteristics for: (

**a**) cluster 5 (Set 1), (

**b**) cluster 4 (Set 2), and (

**c**) cluster 10 (Set 2).

**Figure 6.**Histograms of enteritis daily counts and meteorological characteristics for: (

**a**) Cluster 2 (Set 1); (

**b**) Cluster 8, (Set 1); (

**c**) Cluster 7 (Set 2); (

**d**) Cluster 8 (Set 2); (

**e**) Cluster 10 (Set 2); (

**f**) Cluster 11 (Set 2).

**Figure 7.**Sampling distribution of days selected annually and the corresponding time series for Cluster 5 of salmonellosis, A02.8 (Set 1) in Barnaul, Russia in 2004–2011.

**Figure 8.**Sampling distribution of days selected annually and the corresponding time series for Cluster 8 of enteritis, A04.9 (Set 1) in Barnaul, Russia in 2004–2011.

**Figure 9.**A scatter-plot matrix for health outcomes and weather variables and pair-wise Spearman correlation. *** p < 0.001, * p < 0.1.

Parameter | Mean | SD | Median | Minimum | Maximum | IR | Kurtosis |
---|---|---|---|---|---|---|---|

Salmonellosis (A02.8) | 0.67 | 1.01 | 0.00 | 0.00 | 12.00 | 1.00 | 9.77 |

Enteritis (A04.9) | 7.38 | 3.80 | 7.00 | 0.00 | 27.00 | 5.00 | 0.85 |

Temperature, °C | 2.96 | 14.55 | 4.83 | −40.16 | 27.81 | 22.61 | −0.61 |

Dew point, °C | −2.74 | 12.88 | −0.97 | −44.16 | 20.69 | 18.60 | −0.32 |

Humidity, % | 68.56 | 12.82 | 70.00 | 20.00 | 98.00 | 18.00 | −0.09 |

Pressure, hPa | 997.02 | 10.44 | 996.25 | 958.11 | 1042.55 | 14.68 | −0.01 |

Cluster | Daily Number of Cases | Confidence Interval for Number of Cases | Temperature, °C | Dew Point, °C | Humidity, % | Pressure, hPa | Seasonal Distribution, Number of Days | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

Spring | Summer | Autumn | Winter | Total Number of Days | |||||||

Set 1 | |||||||||||

Cluster 1 | 0.72 | [0.66;0.77] | 16.51 | 9.14 | 64.70 | 988.94 | 19 | 606 | 209 | - | 834 |

Cluster 2 | 0.48 | [0.41;0.54] | −18.41 | −22.28 | 70.26 | 1010.65 | 122 | - | - | 407 | 529 |

Cluster 3 | 0.27 | [0.22;0.32] | 1.16 | −2.76 | 75.71 | 1000.56 | - | - | 489 | - | 489 |

Cluster 4 | 0.33 | [0.27;0.38] | −4.86 | −8.03 | 77.48 | 997.34 | 232 | - | - | 265 | 497 |

Cluster 5 | 3.10 | [2.91;3.28] | 16.09 | 8.87 | 65.82 | 989.65 | 50 | 127 | 12 | - | 189 |

Cluster 6 | 0.55 | [0.47;0.63] | 10.32 | 0.10 | 52.10 | 993.90 | 310 | - | - | 310 | |

Set 2 | |||||||||||

Cluster 1 | 0.27 | [0.22;0.32] | 1.17 | −2.77 | 75.71 | 1000.57 | - | - | 489 | - | 489 |

Cluster 2 | 0.33 | [0.27;0.38] | −4.86 | −8.03 | 77.48 | 997.34 | 232 | - | - | 265 | 497 |

Cluster 3 | 0.55 | [0.47;0.63] | 10.32 | 0.10 | 52.10 | 993.90 | 310 | - | - | - | 310 |

Cluster 4 | 3.14 | [2.88;3.41] | 16.43 | 11.66 | 75.21 | 987.59 | 7 | 109 | 7 | - | 123 |

Cluster 5 | 0.54 | [0.45;0.63] | −22.27 | −25.27 | 74.44 | 1012.24 | 12 | - | - | 352 | 364 |

Cluster 6 | 1.52 | [1.42;1.62] | 11.92 | 4.01 | 61.37 | 994.62 | - | 50 | 134 | - | 184 |

Cluster 7 | 0.36 | [0.30;0.43] | 15.95 | 11.76 | 77.69 | 987.23 | 9 | 207 | 6 | - | 222 |

Cluster 8 | 0.33 | [0.26;0.41] | −9.88 | −15.71 | 61.05 | 1007.13 | 110 | - | - | 55 | 165 |

Cluster 9 | 0.02 | [0.00;0.03] | 17.28 | 8.06 | 57.48 | 989.78 | - | 168 | 69 | - | 237 |

Cluster 10 | 3.00 | [2.81;3.19] | 15.47 | 3.69 | 48.32 | 993.51 | 43 | 18 | 5 | - | 66 |

Cluster 11 | 1.23 | [1.14;1.32] | 20.65 | 12.41 | 61.80 | 984.41 | 10 | 181 | - | - | 191 |

Set | Cluster | Disease Counts | Temperature | Dew Point | Humidity | Pressure | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

IR | K | IR | K | IR | K | IR | K | IR | K | ||

Set 1 | Cluster 5 | 2.00 | 12.22 | 6.02 | 1.97 | 6.25 | 0.94 | 21.0 | −0.44 | 9.04 | −0.50 |

Set 2 | Cluster 4 | 2.00 | 9.76 | 4.80 | 0.25 | 4.65 | 0.74 | 9.50 | 0.69 | 7.82 | −0.44 |

Cluster 10 | 2.00 | −0.86 | 8.70 | 0.23 | 10.73 | −0.60 | 13.00 | 0.03 | 9.98 | −0.31 |

Cluster | Daily Number of Cases | Confidence Interval for Number of Cases | Temperature, °C | Dew Point, °C | Humidity, % | Pressure, hPa | Seasonal Distribution, Number of Days | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

Spring | Summer | Autumn | Winter | Total Number of Days | |||||||

Set 1 | |||||||||||

Cluster 1 | 7.21 | [6.99;7.43] | 17.60 | 11.10 | 68.47 | 987.33 | 24 | 702 | 48 | - | 774 |

Cluster 2 | 12.26 | [11.82;12.70] | −2.11 | −5.79 | 75.62 | 993.98 | 137 | 17 | 4 | 128 | 286 |

Cluster 3 | 7.62 | [7.25;8.01] | 8.43 | 1.02 | 62.56 | 997.18 | - | - | 332 | - | 332 |

Cluster 4 | 7.12 | [6.82;7.44] | −21.02 | −24.10 | 74.10 | 1011.89 | 17 | - | - | 399 | 416 |

Cluster 5 | 6.99 | [6.67;7.32] | 11.49 | 1.05 | 52.10 | 992.94 | 324 | 14 | - | - | 338 |

Cluster 6 | 5.01 | [4.70;5.31] | −6.89 | −9.76 | 78.64 | 1000.65 | 94 | - | - | 120 | 214 |

Cluster 7 | 4.17 | [3.90;4.45] | −0.96 | −4.27 | 78.06 | 1001.58 | - | - | 326 | 326 | |

Cluster 8 | 10.08 | [9.53;10.63] | −4.60 | −11.86 | 56.44 | 1004.68 | 137 | - | - | 25 | 162 |

Set 2 | |||||||||||

Cluster 1 | 7.13 | [6.82;7.44] | −21.02 | −24.10 | 74.10 | 1011.89 | 17 | - | - | 399 | 416 |

Cluster 2 | 6.99 | [6.67;7.32] | 11.49 | 1.05 | 52.10 | 992.94 | 324 | 14 | - | - | 338 |

Cluster 3 | 4.17 | [3.90;4.45] | −0.96 | −4.27 | 78.06 | 1001.58 | - | - | 326 | - | 326 |

Cluster 4 | 5.77 | [5.51;6.02] | 15.80 | 10.93 | 74.84 | 986.96 | 24 | 369 | 33 | - | 426 |

Cluster 5 | 5.45 | [5.11;5.79] | 12.14 | 3.18 | 57.37 | 994.13 | - | - | 183 | - | 183 |

Cluster 6 | 5.01 | [4.70;5.31] | −6.89 | −9.76 | 78.64 | 1000.65 | 94 | - | - | 120 | 214 |

Cluster 7 | 11.05 | [10.72;11.38] | −4.52 | −7.79 | 76.84 | 995.25 | 114 | 1 | - | 124 | 239 |

Cluster 8 | 10.08 | [9.53;10.63] | −4.60 | −11.86 | 56.44 | 1004.68 | 137 | - | - | 25 | 162 |

Cluster 9 | 8.99 | [8.70;9.27] | 19.81 | 11.30 | 60.67 | 987.80 | - | 333 | 15 | - | 348 |

Cluster 10 | 18.43 | [17.58;19.27] | 10.17 | 4.35 | 69.44 | 987.50 | 23 | 16 | 4 | 4 | 47 |

Cluster 11 | 10.30 | [9.83;10.77] | 3.87 | −1.63 | 68.93 | 1000.93 | - | - | - | 149 | 149 |

Set | Cluster | Disease Counts | Temperature | Dew Point | Humidity | Pressure | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

IR | K | IR | K | IR | K | IR | K | IR | K | ||

Set 1 | Cluster 2 | 4.00 | 0.74 | 10.48 | 0.35 | 10.13 | 0.11 | 11.00 | 0.78 | 9.07 | 1.43 |

Cluster 8 | 4.00 | 0.68 | 9.84 | 0.51 | 9.45 | 0.80 | 11.00 | −0.63 | 8.06 | −0.41 | |

Set 2 | Cluster 7 | 4.00 | −0.38 | 9.79 | −0.36 | 9.76 | −0.37 | 10.00 | −0.39 | 7.86 | 0.55 |

Cluster 8 | 4.00 | 0.68 | 9.84 | 0.51 | 9.45 | 0.80 | 11.00 | −0.63 | 8.06 | −0.41 | |

Cluster 10 | 4.50 | 0.21 | 16.83 | −1.14 | 15.49 | −1.28 | 18.50 | −0.63 | 11.65 | 1.31 | |

Cluster 11 | 4.00 | 0.17 | 7.86 | 0.67 | 6.25 | 1.51 | 9.00 | 0.18 | 8.00 | −0.53 |

**Table 6.**Sensitivity analysis of peak timing estimates based on harmonic regression models for health outcomes.

Infection | Peak Timing Estimates | Model A | Model A1 | Model A2 |
---|---|---|---|---|

Salmonellosis (A02.8) | RR (SE) | 187.0 (7.4) | 186.6 (8.6) | 186.5 (8.6) |

LCI; UCI | [179.7;194.4] | [178.0;195.3] | [177.9;195.0] | |

Enteritis (A04.9) | RR (SE) | 103.0 (9.5) | 105.4 (12.3) | 104.7 (11.9) |

LCI; UCI | [93.5;112.5] | [93.1;117.8] | [92.9;116.6] |

**Table 7.**Results of log-linear regression model for health outcomes: estimates of relative risk associated with weather parameters.

Infection | Risk Estimates | Temperature | Dew Point | Humidity | Pressure |
---|---|---|---|---|---|

Salmonellosis (A02.8) | RR | 1.278 | 1.309 | 0.915 | 0.772 |

LCI; UCI | [1.228;1.330] | [1.252;1.370] | [0.877;0.954] | [0.731;0.815] | |

Enteritis (A04.9) | RR | 0.994 | 0.986 | 0.965 | 0.986 |

LCI; UCI | [0.981;1.007] | [0.972;1.001] | [0.950;0.979] | [0.968;1.004] |

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## Share and Cite

**MDPI and ACS Style**

Stashevsky, P.S.; Yakovina, I.N.; Alarcon Falconi, T.M.; Naumova, E.N.
Agglomerative Clustering of Enteric Infections and Weather Parameters to Identify Seasonal Outbreaks in Cold Climates. *Int. J. Environ. Res. Public Health* **2019**, *16*, 2083.
https://doi.org/10.3390/ijerph16122083

**AMA Style**

Stashevsky PS, Yakovina IN, Alarcon Falconi TM, Naumova EN.
Agglomerative Clustering of Enteric Infections and Weather Parameters to Identify Seasonal Outbreaks in Cold Climates. *International Journal of Environmental Research and Public Health*. 2019; 16(12):2083.
https://doi.org/10.3390/ijerph16122083

**Chicago/Turabian Style**

Stashevsky, Pavel S., Irina N. Yakovina, Tania M. Alarcon Falconi, and Elena N. Naumova.
2019. "Agglomerative Clustering of Enteric Infections and Weather Parameters to Identify Seasonal Outbreaks in Cold Climates" *International Journal of Environmental Research and Public Health* 16, no. 12: 2083.
https://doi.org/10.3390/ijerph16122083