Time series modeling of pertussis incidence in China from 2004 to 2018 with a novel wavelet based SARIMA-NAR hybrid model

Abstract

Background: It is a daunting task to discontinue pertussis completely in China owing to its growing increase in the incidence. While basic to any formulation of prevention and control measures is early response for future epidemic trends. Discrete wavelet transform(DWT) has been emerged as a powerful tool in decomposing time series into different constituents, which facilitates better improvement in prediction accuracy. Thus we aim to integrate modeling approaches as a decision-making supportive tool for formulating health resources.

Methods: We constructed a novel hybrid method based on the pertussis morbidity cases from January 2004 to May 2018 in China, where the approximations and details decomposed by DWT were forecasted by a seasonal autoregressive integrated moving average (SARIMA) and nonlinear autoregressive network (NAR), respectively. Then, the obtained values were aggregated as the final results predicted by the combined model. Finally, the performance was compared with the SARIMA, NAR and traditional SARIMA-NAR techniques.

Results: The hybrid technique at level 2 of db2 wavelet including a SARIMA(0,1,3)(1,0,0)12modelfor the approximation-forecasting and NAR model with 12 hidden units and 4 delays for the detail d1-forecasting, along with another NAR model with 11 hidden units and 5 delays for the detail d2-forecasting notably outperformed other wavelets, SARIMA, NAR and traditional SARIMA-NAR techniques in terms of the mean square error, root mean square error, mean absolute error and mean absolute percentage error. Descriptive statistics exhibited that a substantial rise was observed in the notifications from 2013 to 2018, and there was an apparent seasonality with summer peak. Moreover, the trend was projected to continue upwards in the near future.

Conclusions: This hybrid approach has an outstanding ability to improve the prediction accuracy relative to the others, which can be of great help in the prevention of pertussis. Besides, under current trend of pertussis morbidity, it is required to urgently address strategically within the proper policy adopted.

Fig 1. The notified monthly cases of pertussis and decomposed long-term trend and cyclicity with Hodrick-Prescott filter methodology from January 2004 to May 2018 in China.

Blue line represents the reported cases; red line stands for the decomposed long-term trend of the notified monthly cases of pertussis, it can be seen that a remarkably upward trend was observed from 2013 to 2017; green line is the decomposed cyclical process with a length of 12 months.

Fig 2. The resulting autocorrelation function (ACF) and partial autocorrelation function (PACF) graphs of residuals from SARIMA(2,1,0)(0,1,1)₁₂ model for the observatory pertussis morbidity time series.

Almost all spikes fell within the estimated 95% uncertainty bounds at varying lags apart from the correlations at 17 and 23 lags, exhibiting that there appeared to be no correlated residuals. With respect to the spikes at 17 and 23 lags, it is also reasonable because two higher-order correlation functions may occasionally exceed the estimated confidence intervals.

Fig 3. Autocorrelation function (ACF) graph of residuals for the observatory pertussis morbidity time series.

(a) ACF graph of residuals from the basic NAR model; (b) ACF graph of residuals from the traditional SARIMA-NAR hybrid method.Theobservatory spikes aside from the one at zero lag failed to exceed the estimated 95% uncertainty intervals, so these two networks appear to be suitable.

Fig 4. Response results of outputs and targets for the observatory pertussis morbidity time series.

(a) Response graph from the basic NAR model; (b) Response graph from the traditional SARIMA-NAR hybrid method. The plots exhibited which time points were chosen as the training, validation and testing datasets, coupled with their corresponding errors between inputs and targets. In view of the small errors suggesting the prediction is accurate.

Fig 5. The decompositions at level 2 of db2 wavelet for the observatory pertussis morbidity cases time series.

This plot displayed that the approximate element a2 stored the entire form of the pertussis morbidity cases time series, while the detailed components d1 and d2 showed the noise information of the pertussis reported series.

Fig 6

Diagnostic tests of the corresponding individual models for the approximation and details yielded by db2 wavelet: (a) Autocorrelation function (ACF) graph of residuals from the SARIMA(0,1,3)(1,0,0)₁₂ model for the approximation; (b) Partial autocorrelation function (PACF) graph of residuals from the SARIMA(0,1,3)(1,0,0)₁₂ model for the approximation; (c) Autocorrelation function (ACF) graph of residuals from the NAR model for the detailed component d1; (d) The response results of outputs and targets from the NAR model for the detailed component d1; (e) Autocorrelation function (ACF) graph of residuals from the NAR model for the detailed component d2; (f) The response results of outputs and targets from the NAR model for the detailed component d2.

Fig 7. Autocorrelation function (ACF) and partial autocorrelation function (PACF) graphs of residuals from the hybrid model for the observatory pertussis morbidity time series.

All spikes aside from the one at 12lag failed to exceed the estimated 95% uncertainty intervals, so the method appears to be suitable.

Fig 8. Comparison of incidence cases simulated and estimated between the selected four models and observations.

The shaded area represents the validation data points from December 2017 to May 2018, in which the curve forecasted by the hybrid approach better tracked the actual; the black solid line stands for the trends from June 2018 to December 2019 projected by the hybrid method, there is a notably rising trend(purple dotted line).