# Rotavirus Seasonality: An Application of Singular Spectrum Analysis and Polyharmonic Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Exploratory Analysis

#### 2.2. SSA Method

**Y**of the series ${Y}_{t}$, defined as:

**Y**:

#### 2.3. Poisson Polyharmonic Regression (PPHR) Model

#### 2.4. Analysis of the Relationship between Daily RI Rates and Ambient Temperature

## 3. Results

_{RR2}and UCI

_{RR2}). However, the percentage of the variance of the time series explained by the temperature components was low: 0.01%–0.04% (immediate effect) and 0.26%–0.49% (lagged effect), depending on the city (Table 4).

## 4. Discussion

`.`The existing theories for RI seasonality stipulate that many socio-demographic and environmental factors contribute to the seasonal change in RI incidence, including the effects of vaccination [36], age-specific symptomatology [14], population composition and birth rates [18], genotypic profile of locally circulated strains [37], extreme environmental temperature on viral composition and stability [38], and the effects of overall climatic changes in continental climates [33]. Yet, the absolute and relative contributions of these factors are unknown and require further investigations. It is easy to speculate that the RI seasonality is complex and depends on many factors. However, it is not easy to find an explanation why it is complex and how those factors affect seasonality. At the moment, no studies yet offer a solid theory on disease seasonality. With this study, we had started with temperature as the most pronounced environmental indicator in cold climates, yet the suggested method has the strong potential to detect complex variations in a temporal behavior of a seasonal disease and consider more than one factor.

## 5. Conclusions

## 6. Registration

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

^{th}, 2011), including researchers of the Novosibirsk State Technical University (Gubarev V.V. - lead): Khitsenko V.E., Chistyakov N.A., Yun S.G., Yakoviva I.N., Kazanskaya O.V.; scientists of the State Research Center of Virology and Biotechnology “VECTOR” (SRC VB “VECTOR”, Loktev V.B. (lead): Grazhdantseva A.A., Kochneva G.V., Protopopova E.V., Razumov I.A., Sivolobova G.F., Shvalov A.N., Zhirakovskaya E.V.; specialists of the Russian Federal Service for Surveillance on Consumer Rights Protection and Human Wellbeing (Rospotrebnadzor): Aksenova V.I., Belova T.V., Belozerceva N.B., Brusnicyna L.A., Ivanova L.V., Kozlovskii L.I., Mironova O.V., Ozerskaya L.V., Makshanceva S.N., Novoshincev V.N., Rad’kova N.N., Tarasov B.N.; Vaneeva G.K., Yakovleva T.S. and technical advisors: Egorov A.I. and Wright J.M. The authors are also thankful to Aishwarya Venkat for producing the climatic map and Yuri N. Naumov, Ryan Simpson, and Tania M. Alarcon Falconi for technical and editorial assistance. Authors are thankful to the reviewers for providing thoughtful comments and suggestions.

## Conflicts of Interest

## References

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**Figure 1.**Köppen–Geiger climate classification and locations for three Russian cities: Chelyabinsk, Yekaterinburg, and Barnaul (adapted from Reference [33]).

**Figure 2.**Time series of daily ambient temperature (red line) and daily rates of rotaviral infection (RI) (blue line) (

**a**–

**c**) along with the RI rate empirical distribution (blue color bars) and fitted Poisson distribution (black line) (

**d**–

**f**): (

**a**,

**d**) Chelyabinsk; (

**b**,

**e**) Yekaterinburg; and (

**c**,

**f**) Barnaul, Russia, 2005–2011.

**Figure 3.**The periodogram spectral analysis for time series of daily rates of rotaviral infection (RI) (

**a**–

**c**) and daily ambient temperature (

**d**–

**f**) in three Russian cities: (

**a**,

**d**) Chelyabinsk; (

**b**,

**e**) Yekaterinburg; and (

**c**,

**f**) Barnaul, 2005–2011.

**Figure 4.**Plots of the eigenvectors (EV) pairs: 1-EV and lag, 2–EV and 3-EV, and 4-EV and 5-EV for Chelyabinsk (

**a**–

**c**); Yekaterinburg (

**d**–

**f**); and Barnaul (

**g**–

**j**).

**Figure 5.**The predicted RI rates based on Model A (red line), Model B’ (green line), and Model C’ (blue line) in: (

**a**) Chelyabinsk, (

**b**) Yekaterinburg, and (

**c**) Barnaul, Russia, 2005–2011.

**Figure 6.**The average seasonal curves for daily time series rotaviral infection (RI) rate based on Model C’ for three cities, Chelyabinsk (blue line), Yekaterinburg (red line), and Barnaul (green line), and for the average daily temperature which was identical for all three cities (black line).

**Table 1.**General study site characteristics: population, location, study period, and total number of rotavirus cases for three Russian cities: Chelyabinsk, Yekaterinburg and Barnaul, 2005–2011.

Parameter | Chelyabinsk | Yekaterinburg | Barnaul |
---|---|---|---|

General Characteristics: | |||

Population (mean) | 1,103,862 | 1,326,133 | 607,785 |

2005 | 1,095,100 | 1,304,300 | 631,200 |

2006 | 1,093,000 | 1,308,400 | 604,200 |

2007 | 1,091,500 | 1,315,100 | 600,100 |

2008 | 1,092,500 | 1,323,000 | 597,200 |

2009 | 1,093,699 | 1,332,264 | 597,296 |

2010 | 1,130,132 | 1,349,772 | 612,401 |

2011 | 1,131,108 | 1,350,100 | 612,100 |

Latitude Longitude | 55.1644° N 61.4368° E | 56.8389° N 60.6057° E | 53.3548° N 83.7698° E |

Study period start Study period end | 1 Jan 2005 30 Jun 2011 | 1 Jan 2005 31 Jul 2011 | 1 Jan 2006 31 Dec 2011 |

Number of days | 2372 | 2403 | 2191 |

Number of cases | 12,423 | 13,342 | 3307 |

**Table 2.**Statistical characteristics of daily rates of rotaviral infection (RI) (outcome) and daily average temperature (exposure) for three Russian cities: Chelyabinsk, Yekaterinburg, and Barnaul, 2005–2011.

Parameter | Chelyabinsk | Yekaterinburg | Barnaul |
---|---|---|---|

Outcome: Daily RI rate per 1 M population | |||

Number of cases | 12,423 | 13,342 | 3307 |

Mean $\pm $ SE | 4.72 ± 0.08 | 4.16 ± 0.08 | 2.48 ± 0.07 |

1st; 3rd quartile | 1.83; 6.40 | 0.77; 6.06 | 0; 3.34 |

Median; Maximum | 3.66; 27.39 | 3.02; 31.71 | 1.63; 27.57 |

Skewness; Kurtosis | 1.47; 3.07 | 1.43; 2.65 | 1.88; 4.46 |

Exposure: Daily ambient temperature in °C (Mean $\pm $ SE) | |||

Winter: Dec–Feb Spring: Mar–May Summer: Jun–Aug Fall: Sep–Nov | −13.27 ± 0.31 4.85 ± 0.35 17.91 ± 0.17 4.47 ± 0.35 | −13.78 ± 0.32 3.93 ± 0.33 17.25 ± 0.18 3.55 ± 0.34 | −15.84 ± 0.35 4.03 ± 0.41 18.02 ± 0.17 3.91 ± 0.37 |

Cities Parameters | Models | |||||
---|---|---|---|---|---|---|

A | B | B’ | C | C’ | D | |

Chelyabinsk | ||||||

Minimum | −7.715 | −7.796 | −7.107 | −7.610 | −7.178 | −22.91 |

Mean | 0.027 | 0.011 | 0.042 | 0.008 | 0.039 | 0.000 |

Median | −0.266 | −0.317 | −0.280 | −0.310 | −0.274 | 0.218 |

Maximum | 15.42 | 15.97 | 16.58 | 16.28 | 16.82 | 16.24 |

Std. deviation | 2.669 | 2.657 | 2.688 | 2.656 | 2.687 | 5.499 |

MAE * | 1.986 | 2.013 | 2.029 | 2.012 | 2.032 | 4.317 |

D ** | 52.70 | 48.97 | 47.75 | 49.43 | 48.25 | 82.93 |

Yekaterinburg | ||||||

Minimum | −9.246 | −10.57 | −10.90 | −10.71 | −11.57 | −21.43 |

Mean | 0.027 | −0.026 | 0.016 | −0.027 | 0.010 | 0.000 |

Median | −0.180 | −0.365 | −0.258 | −0.374 | −0.276 | 0.301 |

Maximum | 20.10 | 19.19 | 19.31 | 19.48 | 19.74 | 16.32 |

Std. deviation | 2.475 | 2.531 | 2.575 | 2.533 | 2.582 | 5.561 |

MAE | 1.737 | 1.778 | 1.789 | 1.779 | 1.795 | 4.369 |

D | 62.77 | 58.85 | 57.45 | 59.39 | 57.80 | 82.36 |

Barnaul | ||||||

Minimum | −6.955 | −8.232 | −8.640 | −8.621 | −10.02 | −23.15 |

Mean | 0.003 | −0.054 | −0.086 | −0.051 | −0.091 | 0.000 |

Median | −0.264 | −0.455 | −0.680 | −0.474 | −0.692 | 0.124 |

Maximum | 24.27 | 24.88 | 24.13 | 24.16 | 24.16 | 16.405 |

Std. deviation | 2.678 | 2.626 | 2.728 | 2.625 | 2.741 | 5.954 |

MAE | 1.810 | 1.774 | 1.914 | 1.761 | 1.929 | 4.707 |

D | 43.06 | 40.31 | 35.54 | 40.84 | 35.84 | 82.94 |

**Table 4.**Variance explained by the Model B and C’ components and the immediate and lagged effects of ambient temperature estimated from Model C’ in three cities.

Characteristics | Chelyabinsk | Yekaterinburg | Barnaul |
---|---|---|---|

Variance explained by the Model B by components: | |||

Trend | 11.48 | 11.67 | 21.46 |

Annual seasonal component | 36.73 | 46.49 | 18.51 |

Semi-annual seasonal component | 0.759 | 0.690 | 0.343 |

Total variance | 48.97 | 58.85 | 40.31 |

Variance explained by the Model C’ by components: | |||

Trend | 10.35 | 10.47 | 15.03 |

Annual seasonal component | 37.39 | 46.98 | 20.51 |

Temperature (immediate effect) | 0.011 | 0.012 | 0.042 |

Temperature (lagged effect) | 0.499 | 0.341 | 0.262 |

Total variance | 48.25 | 57.80 | 35.84 |

Effect of ambient temperature: | |||

RR1 | 1.00063 | 1.00166 | 0.99517 |

LCI_{RR1}UCI _{RR1} | 0.99726 1.00401 | 0.99825 1.00509 | 0.98964 1.00072 |

RR2 | 0.99585 | 0.99608 | 0.98996 |

LCI_{RR2}UCI _{RR2} | 0.99276 0.99895 | 0.99309 0.99909 | 0.98510 0.99484 |

**Table 5.**The estimates of peak timing in rotaviral infection for a non-leap year for models A, B’, and C’.

Cities Models | Chelyabinsk | Yekaterinburg | Barnaul | ||||||
---|---|---|---|---|---|---|---|---|---|

A | B’ | C’ | A | B’ | C’ | A | B’ | C’ | |

Peak * | 55.17 (3.21) | 55.14 (0.14) | 53.16 (3.4) | 64.17 (5.12) | 62.14 (0.14) | 60.43 (1.25) | 71.11 (7.48) | 74.17 (0.15) | 72.83 (3.52) |

Dates | 24 Feb | 24 Feb | 22 Feb | 5 Mar | 3 Mar | 1 Mar | 12 Mar | 15 Mar | 14 Mar |

Range (days) | 42–62 | 55–56 | 40–65 | 53–80 | 62–63 | 56–70 | 56–99 | 74–75 | 63–84 |

Dates | 11 Feb 3 Mar | 24 Feb 25 Feb | 9 Feb 6 Mar | 22 Feb 21 Mar | 3 Mar 4 Mar | 25 Feb 11 Mar | 25 Feb 9 Apr | 15 Mar 16 Mar | 4 Mar 25 Mar |

Lag in days ** | 40 | 40 | 38 | 49 | 47 | 45 | 56 | 59 | 58 |

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**MDPI and ACS Style**

Alsova, O.K.; Loktev, V.B.; Naumova, E.N. Rotavirus Seasonality: An Application of Singular Spectrum Analysis and Polyharmonic Modeling. *Int. J. Environ. Res. Public Health* **2019**, *16*, 4309.
https://doi.org/10.3390/ijerph16224309

**AMA Style**

Alsova OK, Loktev VB, Naumova EN. Rotavirus Seasonality: An Application of Singular Spectrum Analysis and Polyharmonic Modeling. *International Journal of Environmental Research and Public Health*. 2019; 16(22):4309.
https://doi.org/10.3390/ijerph16224309

**Chicago/Turabian Style**

Alsova, Olga K., Valery B. Loktev, and Elena N. Naumova. 2019. "Rotavirus Seasonality: An Application of Singular Spectrum Analysis and Polyharmonic Modeling" *International Journal of Environmental Research and Public Health* 16, no. 22: 4309.
https://doi.org/10.3390/ijerph16224309