ARIMA and ARIMA-ERNN models for prediction of pertussis incidence in mainland China from 2004 to 2021

Pertussis (whooping cough) is an acute and highly contagious pulmonary disease caused by a small aerobic Gram-negative bacterium, Bordetella pertussis [1]. Pertussis can occur in adults and children, but is often more serious in children, particularly very young infants. Worldwide, pertussis is one of the top ten causes of death during childhood [2]. A 2012 study of pertussis estimated that there were about 30 to 50 million cases and 300,000 deaths per year globally [3], and a 2014 study estimated that there were 24.1 million cases and 160,700 deaths per year globally in children younger than 5 years [4]. In 2018, the WHO estimated that there were approximately 150,000 cases of pertussis worldwide [5]. However, pertussis is often overlooked or misdiagnosed because in many patients it presents with only mild clinical symptoms [6], leading to a possible underestimation of its morbidity [7]. Recent studies of the epidemiology of pertussis reported an epidemic cycle, with increasing numbers of patients every 3 years (on average) [8] in countries such as Canada, Australia, and China [9, 10]. Several other recent studies reported that the incidence of pertussis in China has risen sharply during recent years [11, 12]. In China, for example, the median total economic burden for each case of pertussis in 2017 and 2018 was 8603 Yuan in Yantai (Shangdon) [13], and the average direct economic burden of each inpatient with pertussis in 2019 was 13,291 Yuan in Chongqing[14]. Thus, the resurgence of pertussis is a major financial and public health problem in China.

It is necessary to forecast changes in the morbidity of pertussis so that effective strategies can be implemented for prevention and control, and so that associated health hazards and economic losses can be reduced. There are currently two general types of time series forecasting models that are widely used in epidemiological forecasting. Conventional time series analysis models construct a model using historical data and mainly rely on the linear features of the data; these include the Grey model, Markov model, and autoregressive integrated moving average (ARIMA). A time series may also be analyzed using machine learning theory, in which a model is constructed using an artificial neural network (ANN) to capture the nonlinear features of the data. ARIMA models are the best-known model for time series forecasting, and have been used by many researchers to predict infectious diseases that have characteristic seasonal outbreaks [15]. However, an ARIMA model does not consider nonlinearities in a time series [16].

Given the shortcomings of ARIMA models, there is increasing interest in using ANN models for epidemiological time series forecasting [17] because these models account for nonlinearities in the data. Most of the ANN models used in epidemiological forecasting are based on feed-forward ANNs (static neural networks), such as the back-propagation neural network and the generalized regression neural network. Due to the aggregation and variation of infectious diseases, feed-forward ANNs may not be suitable for analyzing epidemiological data [18]. Unlike feed-forward neural networks, the Elman recurrent neural network (ERNN) can model dynamic information because it uses of additional memory neurons and local feedback [3]. The ability of the ERNN to model dynamic information and its strong sensitivity to time series data thus make it suitable for modeling infectious diseases. Although ANNs can successfully model nonlinear data, they often fail to capture the linear features of the data. Real world time series often contain linear and nonlinear components [19] hence, a model should capture both of these patterns [20]. Therefore, the combined use of an ARIMA model and an ERNN model may provide superior performance [21].

A wide range of epidemiological research has been conducted on pertussis, with most studies focusing on factors that influenced its incidence [22‐26]. Very few reports have focused on predicting the incidence of pertussis. Two recent studies used ARIMA to predict the incidence of pertussis. Raycheva R et al. [27] developed an ARIMA (3, 0, 0) model that adequately reflected trends in pertussis incidence and predicted recent disease dynamics with acceptably low errors. Zeng et al. [12] used ARIMA to analyze pertussis data from January 2005 to June 2016 in China; they found that an ARIMA(0,1,0)(1,1,1)12 model showed the best performance. Another study used a seasonal ARIMA model combined with a nonlinear autoregressive network (SARIMA-NAR) model to forecast the incidence of pertussis in China, and found that using this combination of models greatly improved the accuracy of predictions [11]. In this research, we compared the abilities of an ARIMA-ERNN model and an ARIMA model to predict incidence of pertussis in China. We evaluated the performance of these models by calculating the mean squared error (MSE), mean absolute error (MAE), and mean absolute percentage error (MAPE).

The introduction of the pertussis vaccine greatly reduced the threat of this disease. However, a resurgence of pertussis has occurred in many countries, including China, and pertussis remains a challenging public health problem in China and elsewhere. Therefore, the ability to accurately predict the incidence of pertussis would assist in the implementation of appropriate public health interventions. This study compared an ARIMA model with an ARIMA-ERNN model in predicting the incidence of pertussis in mainland China. We found that an ARIMA (1,1,1) (0,1,1) 12 model provided highly accurate predictions of the incidence of pertussis in mainland China from January 2018 to June 2019. This is not consistent with the best ARIMA used in the previous two studies [12, 27], presumably due to the use of data from different years.

In other fields, such as economics and transportation, the ARIMA-ERNN model has been found to provide better predictive accuracy than other models [30, 31]. However, epidemiologists have only rarely used the ARIMA-ERNN model for the prediction of infectious diseases [32]. To the best of our knowledge, the present study constitutes the first use of a combined ARIMA-ERNN model to predict the incidence of pertussis. Compared with the ARIMA model, the statistical fit of our ARIMA-ERNN model had an 81.43% lower MAE, 95.97% lower MSE, and 80.86% lower MAPE, and the model predictions had a 37.75% lower MAE, 56.88% lower MSE, and 43.75% lower MAPE. Thus, the statistical fit and predictions of the combined model were better than those of the single ARIMA model, consistent with previous researches [11, 29]. We attribute these findings to the superior ability of the ARIMA-ERNN model to capture the linear and nonlinear characteristics of the sequence, and to reduce the loss of information. At the same time, the ERNN contains a local topological recursive structure, which makes it more tolerant [20] and provides certain advantages in dynamic modeling compared with a static neural network [3, 33]. We believe that these characteristics of the ERNN give the ARIMA-ERNN model a better ability to characterize the dynamic information in the time series data.

Compared with the results of two other studies [11, 12], our ARIMA-ERNN model also provided better accuracy. The MAPE is the most commonly used measure of model accuracy due to its scale-independency and easy interpretability [34]. Analysis of the statistical fit indicated that the MAPE value of our ARIMA-ERNN model was 76.96% lower than reported for an ETS model and 52.59% lower than reported for a novel wavelet-based SARIMA-NAR hybrid model. This increased prediction accuracy may be due to our use of more monthly data. Specifically, we used 18 months as the forecast set, whereas previous studies [11, 12] used only 6 months as the forecast set. We also calculated the MAPE of the forecast set from January to June 2018 to ensure the accuracy of comparison. Our MAPE was 6.53%, slightly lower than reported in the previous study (6.70%), confirming that our model was more accurate. Thus, our research indicated that the ARIMA-ERNN model was highly effective in predicting the incidence of pertussis, suggesting it may also have potential for predicting the incidence of similar infectious diseases.

The present research indicated that the incidence of pertussis in China did indeed increase, especially during 2018. This is consistent with previous research findings in China [11, 12, 35]. From 2004 to 2013, the incidence of pertussis in China had an overall downward trend. However, after 2014, there was a huge increase up to a rate of 2.15 per 100,000 in 2019, providing an important reminder that pertussis remains a threat in China. Similar to countries such as Canada, the United States, and Australia [36], the recurrence of pertussis has become an increasing problem in China. Previous studies indicated that the appearance of erythromycin-resistant B. pertussis and the evolution of B. pertussis might be the responsible for the increasing incidence in China [35]. In 2013, China completed the switch from the whole-cell pertussis vaccine (DTwP) to the diphtheria tetanus pertussis (DTaP) vaccine. Since 2013, three anti-PT IgG antibody detection kits have been approved in China, and nucleic acid PCR detection reagents were approved in 2019 [37]. We speculate that the change of vaccine type and the increased use of diagnostic testing may have contributed to the increased identification of cases, as in some developed countries [38]. In addition, unlike some developed countries, China does not implement a “cocooning strategy” [39, 40] for immunization and it does not have separate pertussis vaccines for adolescents and adults; these, two factors may also have contributed to the increase in incidence of pertussis. In general, we believe that the resurgence of pertussis cannot be attributed to any single factor, and that further studies are needed to determine the potential reasons for the increasing incidence in China.

We found a significant seasonality in the incidence of pertussis, with the greatest incidence during March to September. This result is consistent with several other studies [11, 12], but the nature of the seasonality of pertussis differs in different regions. For example, Leong et al. reported a peak incidence in Australia during spring and summer (November to January) [41], Guimarães et al. reported a peak incidence in Brazil during spring and autumn [42] and Hitz et al. reported a peak incidence in Germany during summer. Unfortunately, the reasons for the seasonality of pertussis remain mostly unknown. Some seasons may provide a more optimal environment for the pathogen, and the human immune response may also vary with the seasons. Thus, further studies are needed to examine the distribution and survival of B. pertussis and the mechanisms of underlying pathogenic factors [43].

This study had some limitations. All of our primary data were from a national database. Although China classifies pertussis as a Class B statutory infectious disease, the actual incidence of the disease is probably underestimated. Our research predicted the incidence rate for China overall, although there are likely to be large differences in incidence within China due to its large area and many regional differences. Moreover, we were unable to include some factors in our models that may affect the incidence of pertussis because the available data were not comprehensive. Future studies should seek to overcome these limitations.

The present study compared predictions of pertussis incidence in mainland China obtained using an ARIMA model and an ARIMA-ERNN model. The results indicated that an ARIMA-ERNN model should be considered for monitoring the incidence of pertussis in China.