Methods and Applications: An Autoregressive Integrated Moving Average Model for Predicting Varicella Outbreaks — China, 2019

Per the “National Public Health Emergency Related Information Reporting Management Standards” distributed by the Ministry of Health’s General Office on December 27, 2005, any instance of more than ten varicella cases within the same school, kindergarten, and other related units in a single week is classified as a varicella outbreak. Such outbreaks are mandated to be reported via the public health emergency information reporting system. Our research involved the extraction of varicella outbreak surveillance data from January 2005 to November 2019. This data was divided into segments for model development and model validation. We used the monthly varicella outbreak cases from 2005 to 2018 to construct the model, while the 2019 monthly data was employed to validate the model and generate predictions.

ARIMA models take the form of ARIMA (p, d, q)×(P, D, Q)s. Parameters d (the degree of differencing) and D (moving average) are numbers of differences required to stabilize the time series. Parameters p (the order of autoregression) and q (the order of moving average) are simple numeric parameters. Parameters P (seasonal autoregression) and Q (seasonal integration) are seasonal parameters, and s is the length of the seasonal period.

The construction and prediction of the ARIMA model consist of three steps. First, Time series stabilization: we assessed stationarity and seasonality by graphing a time series plot of the monthly varicella outbreak cases. The trend and seasonality of the initial sequence were eliminated by taking the ordinary and seasonal differences. The time series’ stationarity was then determined through the analysis of the stabilized sequence graph as well as the autocorrelation function (ACF) and partial autocorrelation function (PACF). Second, model identification and diagnosis: the values of d and D were determined based on the trend differences and seasonal variations. The values for p and q, and P and Q were permitted to vary between 0 and 2, and were assessed individually in model construction. Each proposed model had to pass the Ljung–Box and parameter tests. The most suitable model was subsequently selected based on the highest coefficients of determination (R²) and the lowest normalized Bayesian Information Criterion (BIC). Lastly, prediction: the fitted model was used to project the number of monthly varicella outbreaks for 2019 (4).

The analysis of the data was performed utilizing the SPSS software (version 26.0, IBM, Armonk, NY, USA). The Mann-Kendall trend test was utilized to evaluate the outbreak trends. A significance level was established at P<0.05.

From 2005 to 2018, China reported 246,772 outbreak cases in 8,545 varicella outbreaks. The time series mapping of these cases revealed a statistically significant decline from 2007 to 2011 (Z=−2.25, P<0.05). However, from 2012 to 2018, there was a notable increase (Z=2.63, P<0.05). When decomposed, the time series exhibited three components: random errors, periodic factors, and long-term trend factors. The data demonstrated seasonal characteristics, with major and minor epidemic peaks recurring annually (Figure 1).

Figure 1. 

The time series graph of monthly varicella outbreak cases in China, 2005–2018. (A) Original time series; (B) seasonal effect; (C) random fluctuation effect; (D) long-term trend effect.

The ARIMA model was developed using monthly instances of varicella outbreaks spanning from January 2005 to December 2018. Upon inspection of Figure 1, it became apparent that the series displayed non-stationary characteristics, thus necessitating the stabilization of the series through the incorporation of one-order ordinal and seasonal differences. The stabilized sequence, as depicted in Supplementary Figure S1, did not exhibit a pronounced upward or downward trend. Supplementary Figure S2 illustrates the autocorrelation coefficient of the stationary series experiencing a swift decrease following a brief delay period. This observation suggests that the modified time series tended toward stationarity subsequent to differencing adjustment. Consequently, this procedure assigned the parameters d and D as 1.

The initial model was designated as ARIMA (p, 1, q)×(P, 1, Q)₁₂. The individual values for p, q, P, and Q were adjusted independently, ranging from 0 to 2. Following this iterative adjustment, four models successfully passed the parameter tests, all with P<0.05, and the Ljung-Box tests, all with P>0.05: ARIMA (2, 1, 0)×(0, 1, 1)₁₂, ARIMA (1, 1, 0)×(0, 1, 1)₁₂, ARIMA (1, 1, 1)×(0, 1, 1)₁₂, and ARIMA (1, 1, 1)×(2, 1, 0)₁₂. Due to its superior R² and the minimal standardized BIC, ARIMA (1, 1, 1)×(0, 1, 1)₁₂ was selected as the optimal model (Table 1). The autoregressive coefficients of the model residuals all fell within the control line, as depicted in Supplementary Figure 3, suggesting that the residual error was random and affirming the validity of the chosen model.

Utilizing the optimal ARIMA (1, 1, 1)×(0, 1, 1) ₁₂ model, we predicted varicella outbreak cases from January through November 2019. The actual values aligned closely with the fitted values preceding October 2019 (Figure 2). Even though the subsequent fitted values did not align as closely, they remained within the predicted 95% confidence intervals. The average relative error between the predicted and actual values was 15.2% (Table 2), inferring that the model was deemed suitable for prediction purposes.

Figure 2. 

Time-series plots of predicted monthly varicella outbreak cases using the ARIMA model, January 2005–December 2019.

Note: The dotted lines represent the 95% CIs, with UCL denoting the upper limit and LCL indicating the lower limit of the 95% CI.

Abbreviation: ARIMA=autoregressive integrated moving average model; CI=confidence interval; UCL=upper confidence limit; LCL=lower confidence limit.

This study may be the premiere use of an ARIMA model to delineate the epidemic trajectory of varicella outbreaks in China, as it offers a predictive overview of imminent varicella trends. This provides valuable insight for preemptive measures and public health guidance (5). Our research reveals an uninterrupted increase in reported varicella outbreak cases since 2012, with a significant surge from 2017 that peaked in 2018. Projections for 2019 continue this rising trend, suggesting that varicella outbreaks have not been fully contained. The low Varicella vaccine (VarV) coverage in China could be a potential catalyst for these increases (6). Previous research supports the efficacy of both single and double dose varicella vaccination schedules in mitigating varicella outbreaks (7–9). A separate study conducted in China (10) utilized a modified Delphi technique to gather expert opinions on the potential inclusion of non-program vaccines into China’s Expanded Program on Immunization (EPI). VarV emerged as the top non-program vaccine recommended for incorporation into the EPI. Thus, these findings underscore the importance and urgency of integrating VarV into the national immunization framework.

After conducting numerous adjustments to the parameters and running goodness-of-fit tests, it was conclusively determined that the ARIMA (1, 1, 1)×(0, 1, 1)₁₂ model was the most compatible with the original time-series data of monthly varicella outbreak cases gathered from 2005 to 2018. This optimal model was subsequently used to predict monthly varicella outbreak cases in 2019. The results revealed that the estimated cases of outbreaks were congruent with the actual reported cases, particularly from January to September. This correspondence indicated the model’s ability to accurately predict varicella outbreak cases. From October 2019 onwards, the fitted values did not align as closely, albeit still falling within the 95% confidence intervals. This points to potential influences of large seasonal fluctuations or changes in policy on the model’s accuracy, a factor which warrants further analysis. Consequently, it is recommended that the model’s data be regularly updated with the most current information to ensure optimal accuracy.

Time series models are instrumental in the prediction of varying trends in infectious diseases such as hand foot and mouth disease (HFMD) (11), coronavirus disease 2019 (COVID-19) (12), and influenza (3). Our research reinforces the scientific consensus deeming the ARIMA models as proficient tools for synchronous surveillance and forecasting of evolving trends in infectious diseases. Notably, a study conducted in Bulgaria (13) illustrated the appropriateness of an ARIMA model in describing varicella incidence trends, and its suitability in projecting near-future disease dynamics, although it didn’t account for varicella seasonality. In relation to China, there have been limited studies conducted on varicella incidence prediction, with existing studies only forecasting sporadic varicella incidence in specific regions. For the first time, our study utilizes varicella outbreak data to forecast varicella outbreak occurrences in China on a monthly basis, effectively eliminating the influence of seasonality.

Our study is subject to some limitations. Initially, the use of passive surveillance data can potentially result in an underestimation of the disease burden, which could consequently impact the precision and accuracy of our analyses. Furthermore, the accuracy of our ARIMA model might be subjected to the dynamic changes in key influencing factors such as policy alterations and climate changes. Therefore, the establishment of a dynamic adjustment model is essential to enhance the accuracy of long-term predictions.

In conclusion, the findings from our research indicate the practicality of employing ARIMA models for predicting varicella outbreaks in China. Consequently, these models pose a valuable tool for enhancing varicella prevention and control measures, offering forecasting capabilities for future varicella outbreaks and trend identification within the nation.

No conflicts of interest.