# Time-Series Analysis for the Number of Foot and Mouth Disease Outbreak Episodes in Cattle Farms in Thailand Using Data from 2010–2020

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Outbreak Episode Definition, Data, and Data Analysis Steps

#### 2.2. Decomposition of Time-Series Data

#### 2.3. Model Development

#### 2.3.1. SARIMA Model

#### 2.3.2. NNAR Model

_{t}and $\left({y}_{t-i,...,}{y}_{t-p}\right)$ are the output and the input, ${\omega}_{i,j}\left(i=0,1,2,...,P,j=1,2,...,Q\right)$ and ${\omega}_{j}\left(j=0,1,2,...,Q\right)$ are model parameters, which are known as connection weights; the number of input nodes is represented by $P$ while the number of hidden nodes is indicated by $Q$.

#### 2.3.3. ETS Model

#### 2.3.4. TBATS Model

#### 2.3.5. Hybrid Model

#### 2.4. Forecast and Model Performances

## 3. Results

#### 3.1. Trends and Seasonality

#### 3.2. Fitted Time-Series Models, Model Performances, and Forecasts

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Time-series modeling procedure. A full dataset of the number of FMD outbreak episodes was split into training and validation datasets. Forecast models were developed using seasonal autoregressive integrated moving average (SARIMA), error trend seasonality (ETS), neural network autoregression (NNAR), and Trigonometric Exponential smoothing state–space model with Box–Cox transformation, ARMA errors, Trend and Seasonal components (TBATS), and hybrid methods. With the validation data, error measures, including root mean squared error (RMSE), mean absolute error (MAE), and mean absolute scaled error (MASE), were determined in order to compare the performances of prediction models.

**Figure 2.**Decomposition of the number of time-series FMD outbreak episodes from January 2010 to December 2020 in actual (data), trend, decomposed seasonal trait (seasonal), and random fluctuation (remainder) of FMD outbreak episodes were illustrated.

**Figure 3.**Actual, fitted, and forecast value from SARIMA, NNAR, ETS, TBATS. SARIMA-NNAR, SARIMA-ETS, SARIMA-TBATS, NNAR-ETS, NNAR-TBATS, and ETS-TBATS models. The x-axis and y-axis are the year and number of FMD outbreak episodes, respectively.

**Figure 4.**Forecasts of the number of FMD episodes (red line) from SARIMA, NNAR, ETS, TBATS, SARIMA-NNAR, SARIMA-ETS, SARIMA-TBATS, NNAR-ETS, NNAR-TBATS, and ETS-TBATS models. The x-axis and y-axis are the year and number of FMD outbreak episodes, respectively. The yellow band indicates a 95% confidence interval of forecast values.

Model ^{1} | Training Data (Data: 2010–2019) | Validation Data (Data: 2020) | ||||
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RMSE ^{2} | MAE | MASE | RMSE | MAE | MASE | |

SARIMA | 6.21 | 3.81 | 0.64 | 11.47 | 8.75 | 1.47 |

NNAR | 5.21 | 3.39 | 0.56 | 12.61 | 10.46 | 1.76 |

ETS | 6.59 | 4.44 | 0.75 | 36.11 | 35.26 | 5.94 |

TBATS | 6.38 | 4.26 | 0.89 | 16.24 | 13.38 | 2.25 |

SARIMA-NNAR | 5.59 | 3.63 | 0.61 | 11.43 | 8.79 | 1.48 |

SARIMA-ETS | 5.94 | 3.82 | 0.64 | 20.89 | 19.2 | 3.23 |

SARIMA-TBATS | 5.91 | 3.75 | 0.63 | 12.56 | 10.4 | 1.75 |

NNAR-ETS | 5.38 | 3.52 | 0.60 | 17.49 | 15.17 | 2.60 |

NNAR-TBATS | 5.40 | 3.53 | 0.60 | 11.41 | 10.02 | 1.69 |

ETS-TBATS | 6.42 | 4.29 | 0.72 | 24.44 | 22.14 | 3.73 |

^{1}SARIMA = seasonal autoregressive integrated moving average, ETS = error trend seasonality, NNAR = neural network autoregression, TBATS = Trigonometric Exponential smoothing state-space model with Box–Cox transformation, autoregressive integrated moving errors, Trend and Seasonal components.

^{2}RMSE = root mean squared error, MAE = mean absolute error and MASE = mean absolute scaled error.

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## Share and Cite

**MDPI and ACS Style**

Punyapornwithaya, V.; Mishra, P.; Sansamur, C.; Pfeiffer, D.; Arjkumpa, O.; Prakotcheo, R.; Damrongwatanapokin, T.; Jampachaisri, K. Time-Series Analysis for the Number of Foot and Mouth Disease Outbreak Episodes in Cattle Farms in Thailand Using Data from 2010–2020. *Viruses* **2022**, *14*, 1367.
https://doi.org/10.3390/v14071367

**AMA Style**

Punyapornwithaya V, Mishra P, Sansamur C, Pfeiffer D, Arjkumpa O, Prakotcheo R, Damrongwatanapokin T, Jampachaisri K. Time-Series Analysis for the Number of Foot and Mouth Disease Outbreak Episodes in Cattle Farms in Thailand Using Data from 2010–2020. *Viruses*. 2022; 14(7):1367.
https://doi.org/10.3390/v14071367

**Chicago/Turabian Style**

Punyapornwithaya, Veerasak, Pradeep Mishra, Chalutwan Sansamur, Dirk Pfeiffer, Orapun Arjkumpa, Rotchana Prakotcheo, Thanis Damrongwatanapokin, and Katechan Jampachaisri. 2022. "Time-Series Analysis for the Number of Foot and Mouth Disease Outbreak Episodes in Cattle Farms in Thailand Using Data from 2010–2020" *Viruses* 14, no. 7: 1367.
https://doi.org/10.3390/v14071367