Tuberculosis (TB) remains a major deadly threat in mainland China. Early warning and advanced response systems play a central role in addressing such a wide-ranging threat. The purpose of this study is to establish a new hybrid model combining a seasonal autoregressive integrated moving average (SARIMA) model and a non-linear autoregressive neural network with exogenous input (NARNNX) model to understand the future epidemiological patterns of TB morbidity.

We develop a SARIMA-NARNNX hybrid model for forecasting future levels of TB incidence based on data containing 255 observations from January 1997 to March 2018 in mainland China, and the ultimate simulating and forecasting performances were compared with the basic SARIMA, non-linear autoregressive neural network (NARNN) and error-trend-seasonal (ETS) approaches, as well as the SARIMA-generalised regression neural network (GRNN) and SARIMA-NARNN hybrid techniques.

In terms of the root mean square error, mean absolute error, mean error rate and mean absolute percentage error, the identified best-fitting SARIMA-NARNNX combined model with 17 hidden neurons and 4 feedback delays had smaller values in both in-sample simulating scheme and the out-of-sample forecasting scheme than the preferred single SARIMA(2,1,3)(0,1,1)_{12} model, a NARNN with 19 hidden neurons and 6 feedback delays and ETS(M,A,A), and the best-performing SARIMA-GRNN and SARIMA-NARNN models with 32 hidden neurons and 6 feedback delays. Every year, there was an obvious high-risk season for the notified TB cases in March and April. Importantly, the epidemic levels of TB from 2006 to 2017 trended slightly downward. According to the projection results from 2018 to 2025, TB incidence will continue to drop by 3.002% annually but will remain high.

The new SARIMA-NARNNX combined model visibly outperforms the other methods. This hybrid model should be used for forecasting the long-term epidemic patterns of TB, and it may serve as a beneficial and effective tool for controlling this disease.

This work showed the long-term temporal patterns and characteristics in tuberculosis (TB) incidence series through a 29-year analysis.

The seasonal autoregressive integrated moving average-non-linear autoregressive neural network with exogenous input (SARIMA-NARNNX) hybrid model can be employed to implement long-term forecasting of TB in mainland China.

The time variable is a significantly useful parameter that fails to be ignored during the process of constructing prediction models, particularly when a clear seasonality is included in the time series.

The SARIMA-NARNNX hybrid model has the potential for far-reaching implications for further prevention and control of TB incidence.

The SARIMA-NARNNX hybrid model only relies on the retrospective responses and time factors without considering additional explanatory variables.

Tuberculosis (TB) is a worldwide chronic infectious disease caused by the aetiological agent

Currently, numerous useful statistical techniques have been extensively employed in the forecasting domain, including linear methods such as the seasonal autoregressive integrated moving average (SARIMA) method,

In this observational study, the longitudinal monthly morbidity data from January 1997 to March 2018 were extracted from the notifiable infectious disease reporting system supplied by the Chinese Center for Disease Control and Prevention (

The NARNN model only uses the known past input values to estimate the present output results (online

Here, f represents a function that relies on the structure and connection weights of the NARNNX model,

In this hybrid model, the modelling steps were as follows. Initially, as mentioned above, the time variable and the estimated values of the SARIMA model were regarded as input variables, while the corresponding reported cases of TB were used as the output variable. Subsequently, the random divider and function was applied to classify the in-sample data into three subsets: a training dataset (80% of the data), a validation dataset (10%) and a testing dataset (10%). Then, the number of hidden neurons and delays d was adjusted by trial and error using the Levenberg-Marquardt algorithm in an open feedback loop mode. The response plot of the outputs and targets and the residual autocorrelation function (ACF) plot, together with the mean square error (MSE) and correlation coefficient (R), were used to find the best-performing SARIMA-NARNNX model. Finally, the training open-loop architecture was transformed into closed-loop mode to make multistep-ahead forecasts (The code used in our experiments is included in the online

Moreover, the single SARIMA and the traditional SARIMA-GRNN and SARIMA-NARNN models were constructed as described in the online

In this work, the SARIMA model was built with the R statistical package (V.3.4.3, R Development Core Team, Vienna, Austria), and the selected three hybrid models, including the SARIMA-GRNN, SARIMA-NARNN and SARIMA-NARNNX, were developed with MATLAB (V.R2014a, MathWorks, Natick, Massachusetts, USA). A two-sided p <0.05 was considered statistically significant.

Of the mathematical methodologies mentioned above, the mimic and predictive performances were judged by two types of measures: scale-dependent indices (ie, the root mean square error (RMSE) and the mean absolute error (MAE)) and indices that depend on percentage errors (ie, the mean error rate (MER) and the mean absolute percentage error (MAPE)). For these measures, the smallest values correspond to the optimal method.

where X_{i} denotes the observed values,

Patients and the public were not involved in our present study, as the TB incidence series from the notifiable infectious disease reporting system was aggregated as secondary data and does not contain personal identifying information. Thus, these data are publicly available.

A total 17 926 271 cases were reported during the period between January 1997 and March 2018, with a monthly average morbidity of 88 304 cases, resulting in a yearly average incidence rate of 63.724 cases per 100 000 people. The incidence rate has remarkably risen from 30.836 cases per 100 000 persons in 1997 to 60.082 cases per 100 000 persons in 2017, with an increase of 94.847%. The highest incidence peak was at a maximum in 2005 with 96.310 cases per 100 000 population, which was a marginal increase of 212.332% compared with 2008 (online

Decomposition of monthly TB time series in mainland China from 1997 to 2018 into trend and cyclical components using the Hodrick-Prescott filter. TB, tuberculosis.

Before modelling, an augmented Dickey-Fuller (ADF) test was performed in the reported TB incidence series that indicated that the data are irregular and non-stationary (ADF=−1.705, p=0.427). Consequently, according to the results of the ADF test and TB incidence periodicity, the first-order seasonal and non-seasonal differences were taken to remove the instabilities in the variance and mean (ADF=−4.175, p<0.001), which indicated that the differenced series was stationary. Subsequently, by analysing the spikes of the ACF and partial ACF (PACF) plots from the transformed TB morbidity series (online _{12} model was selected based on the residual correlations in both the ACF and PACF plots, as well as the akaike information criterion (AIC), bias-corrected AIC (AICc) and schwarz bayesian criterion (SBC) values. The AIC, AICc and SBC values of 4914.44, 4914.94 and 4938.60, respectively, were the smallest among those candidate models. The ACF and PACF residual plots demonstrated that the error correlations at lags almost fell into the estimated threshold limits, and the Ljung-Box Q-test also revealed that the residuals were a white noise series (_{12} model can be written as (1-B)(1-B)^{12} X_{t}=(1+2.161B-1.938B^{2} +0.633B^{3} (1+0.7B^{12} ɛ_{t}/(1–1.52B+0.909B^{2}). Ultimately, the preferred model can be applied to predict the incident cases from January 2017 to March 2018 (

Diagnostic checking for the residuals generated by the SARIMA(2,1,3)×(0,1,1)_{12} method. (A) Standardised residual plot; (B) autocorrelation function (ACF) of the errors at various lags; (C) Partial ACF (PACF) of the errors at various lags. SARIMA, seasonal autoregressive integrated moving average.

Ljung-Box Q tests of the errors series for the chosen best-undertaking methods at different lags

Lags | SARIMA | SARIMA-GRNN | SARIMA-NARNN | SARIMA-NARNNX | ||||

Box-Ljung Q | P value | Box-Ljung Q | P value | Box-Ljung Q | P value | Box-Ljung Q | P value | |

1 | 0.079 | 0.779 | 2.287 | 0.130 | 1.769 | 0.183 | 0.057 | 0.811 |

3 | 0.084 | 0.994 | 3.308 | 0.347 | 6.010 | 0.111 | 0.336 | 0.953 |

6 | 0.970 | 0.987 | 3.969 | 0.681 | 7.375 | 0.288 | 0.436 | 0.999 |

9 | 4.065 | 0.907 | 4.850 | 0.847 | 8.940 | 0.443 | 1.348 | 0.998 |

12 | 7.396 | 0.830 | 6.097 | 0.911 | 10.706 | 0.554 | 3.160 | 0.994 |

15 | 8.480 | 0.903 | 7.667 | 0.936 | 14.006 | 0.525 | 5.059 | 0.992 |

18 | 9.763 | 0.939 | 11.351 | 0.879 | 14.478 | 0.697 | 9.189 | 0.955 |

21 | 10.959 | 0.964 | 11.446 | 0.953 | 14.997 | 0.823 | 10.320 | 0.975 |

24 | 15.399 | 0.909 | 17.626 | 0.821 | 17.174 | 0.841 | 18.569 | 0.775 |

27 | 20.056 | 0.828 | 21.015 | 0.786 | 19.064 | 0.868 | 30.082 | 0.311 |

30 | 20.803 | 0.894 | 21.071 | 0.886 | 20.029 | 0.916 | 33.156 | 0.316 |

33 | 23.485 | 0.889 | 22.172 | 0.924 | 21.507 | 0.938 | 38.354 | 0.240 |

36 | 25.866 | 0.894 | 25.066 | 0.914 | 22.383 | 0.963 | 40.113 | 0.293 |

GRNN, generalised regression neural network; NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; SARIMA, seasonal autoregressive integrated moving average.

ARCH effects of the observations and errors series for the chosen best-undertaking method at various lags

Lags | Original values | SARIMA | SARIMA-GRNN | SARIMA-NARNN | SARIMA-NARNNX | |||||

LM-test | P value | LM-test | P value | LM-test | P value | LM-test | P value | LM-test | P value | |

1 | 192.37 | <0.001 | 44.973 | <0.001 | 58.393 | <0.001 | 5.522 | 0.019 | 0.358 | 0.550 |

3 | 192.95 | <0.001 | 47.413 | <0.001 | 67.078 | <0.001 | 6.659 | 0.084 | 4.639 | 0.200 |

6 | 190.52 | <0.001 | 46.980 | <0.001 | 66.140 | <0.001 | 8.581 | 0.199 | 4.791 | 0.571 |

9 | 192.910 | <0.001 | 46.254 | <0.001 | 65.552 | <0.001 | 9.030 | 0.435 | 5.654 | 0.774 |

12 | 199.100 | <0.001 | 71.910 | <0.001 | 71.985 | <0.001 | 11.386 | 0.496 | 7.08 | 0.852 |

15 | 203.180 | <0.001 | 72.409 | <0.001 | 71.654 | <0.001 | 11.841 | 0.691 | 8.205 | 0.915 |

18 | 200.2 | <0.001 | 71.245 | <0.001 | 71.075 | <0.001 | 12.991 | 0.792 | 9.272 | 0.953 |

21 | 197.06 | <0.001 | 70.719 | <0.001 | 70.505 | <0.001 | 14.318 | 0.856 | 9.193 | 0.988 |

24 | 194.57 | <0.001 | 69.891 | <0.001 | 72.594 | <0.001 | 15.646 | 0.900 | 9.947 | 0.995 |

27 | 191.580 | <0.001 | 70.122 | <0.001 | 74.501 | <0.001 | 21.634 | 0.756 | 10.339 | 0.998 |

30 | 188.550 | <0.001 | 69.301 | <0.001 | 73.457 | <0.001 | 24.734 | 0.738 | 10.681 | 1.000 |

33 | 185.330 | <0.001 | 68.289 | <0.001 | 72.563 | <0.001 | 26.942 | 0.762 | 11.038 | 1.000 |

36 | 182.53 | <0.001 | 67.643 | 0.001 | 72.764 | <0.001 | 32.373 | 0.642 | 13.956 | 1.000 |

ARCH, autoregressive conditional heteroscedastic; GRNN, generalised regression neural network; NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; SARIMA, seasonal autoregressive integrated moving average.

The projected cases of TB incidence using the best-performing approaches chosen from January 2017 to March 2018 in mainland China

Time | Original values | SARIMA | SARIMA-GRNN | SARIMA-NARNN | SARIMA-NARNNX | ||||

Forecasts | MAE | Forecasts | MAE | Forecasts | MAE | Forecasts | MAE | ||

January-2017 | 80 911 | 84 673 | 0.046 | 86 123 | 0.064 | 81 502 | 0.007 | 80 411 | 0.006 |

February-2017 | 92 037 | 78 429 | 0.148 | 84 772 | 0.079 | 82 383 | 0.105 | 91 943 | 0.001 |

March-2017 | 105 633 | 1 14 580 | 0.085 | 110 652 | 0.048 | 103 932 | 0.016 | 112 460 | 0.065 |

April-2017 | 97 296 | 1 06 435 | 0.094 | 108 340 | 0.114 | 93 322 | 0.041 | 104 869 | 0.078 |

May-2017 | 101 628 | 97 474 | 0.041 | 99 846 | 0.018 | 105 292 | 0.036 | 102 436 | 0.008 |

June-2017 | 99 001 | 92 719 | 0.063 | 94 623 | 0.044 | 89 143 | 0.100 | 93 127 | 0.059 |

July-2017 | 96 471 | 94 806 | 0.017 | 96 130 | 0.004 | 94 522 | 0.020 | 96 794 | 0.003 |

August-2017 | 100 076 | 92 419 | 0.077 | 93 497 | 0.066 | 91 733 | 0.083 | 93 969 | 0.061 |

September-2017 | 92 494 | 89 344 | 0.034 | 92 583 | 0.001 | 81 409 | 0.120 | 92 088 | 0.004 |

October-2017 | 81 554 | 82 947 | 0.017 | 88 642 | 0.087 | 80 656 | 0.011 | 85 865 | 0.053 |

November-2017 | 89 976 | 86 118 | 0.043 | 84 990 | 0.055 | 87 067 | 0.032 | 86 295 | 0.041 |

December-2017 | 87 630 | 87 387 | 0.003 | 93 857 | 0.071 | 84 549 | 0.035 | 88 608 | 0.011 |

January-2018 | 96 125 | 81 167 | 0.156 | 89 054 | 0.074 | 85 574 | 0.110 | 84 287 | 0.123 |

February-2018 | 77 224 | 80 301 | 0.040 | 86 698 | 0.123 | 80 071 | 0.037 | 80 715 | 0.045 |

March-2018 | 110 124 | 1 06 125 | 0.036 | 105 440 | 0.043 | 102 342 | 0.071 | 110 760 | 0.006 |

GRNN, generalised regression neural network; MAE, mean absolute error; NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; SARIMA, seasonal autoregressive integrated moving average; TB, tuberculosis.

The first-order differences taken for the TB morbidity series when building the SARIMA model caused 13-month missing data. Therefore, the modelling values of the SARIMA model from February 1998 to December 2016 were used for the inputs, while the original values of the TB incidence in the same months were used as the expected outputs to obtain the modelling results of the SARIMA-GRNN hybrid technique. To find the preferred GRNN model in which the smoothing factor can generate the smallest value of RMSE on the randomly selected testing set, after running the random integer function randint(1,2,(1 227)) in MATLAB, two sample points of 29 and 208 corresponding to the values of June 2000 and May 2015, respectively, were chosen for the SARIMA-GRNN hybrid technique modelling. Then, we used the dataset that removed the two above-mentioned sample points to develop SARIMA-GRNN combined models with smoothing factors between 0 and 1 (incremented by 0.001). The results are depicted in

The RMSE values corresponding to different smoothing factors for the SARIMA-GRNN combined technique. It can be seen that when smoothing factor is 0.006, the lowest RMSE value is 0.0024. GRNN, generalised regression neural network; SARIMA, seasonal autoregressive integrated moving average.

To find the best-fitting NARNN model for the SARIMA error series, the hidden units and feedback delays ranged from 10 to 40 and 2 to 7, respectively, were iterated one by one. Ultimately, comprehensively taking all the performance indices into consideration, we identified the optimum model with 32 hidden neurons and 6 feedback delays. As presented in online

The resultant error autocorrelation function (ACF) plots for the three optimal hybrid models selected. (A) ACF plot of errors for the best-performing SARIMA-GRNN hybrid technique across varying lags; (B) ACF plot of errors for the best-performing SARIMA-NARNN hybrid technique across varying lags; (C) ACF plot of errors for the best-performing SARIMA-NARNNX hybrid technique across varying lags. GRNN, generalised regression neural network; NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; SARIMA, seasonal autoregressive integrated moving average; TB, tuberculosis.

The corresponding time series response plots of outputs and targets for the best-undertaking SARIMA-NARNN and SARIMA-NARNNX hybrid models at various time points. (A) Response plot of the outputs and targets for the best-undertaking SARIMA-NARNN hybrid model; (B) Response plot of the outputs and targets for the best-undertaking SARIMA-NARNNX hybrid model. NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; SARIMA, seasonal autoregressive integrated moving average; TB, tuberculosis.

To deeply mine the linear and non-linear patterns included in the TB incidence series, the preferred SARIMA-NARNNX hybrid model was built by trial and error. Finally, the SARIMA-NARNNX model with 17 hidden neurons and 4 feedback delays was determined to be the best-fitting model based on the minimum MSE for the training subset (18907817.559), the validation subset (19921017.940) and the testing subset (36872592.071), as well as with the maximum R values of the training, validation testing datasets and the entire dataset (0.994, 0.982, 0.986 and 0.992, respectively) (online

As shown in

Comparison of in-sample fitting and out-of-sample predicting performances among the best-performing approaches chosen

Models | Simulating power | Predictive power | ||||||

MAE | MER | MAPE | RMSE | MAE | MER | MAPE | RMSE | |

SARIMA | 5636.303 | 0.062 | 0.067 | 8781.186 | 5726.262 | 0.061 | 0.060 | 7104.34 |

SARIMA-GRNN | 4437.958 | 0.049 | 0.054 | 6939.078 | 5415.985 | 0.058 | 0.059 | 6155.964 |

SARIMA-NARNN | 3283.274 | 0.035 | 0.043 | 5265.82 | 5259.556 | 0.056 | 0.055 | 6418.445 |

SARIMA-NARNNX | 2878.484 | 0.031 | 0.038 | 4468.578 | 3563.179 | 0.038 | 0.038 | 4917.829 |

Percentage reductions (%) | ||||||||

D versus A | 48.930 | 50.000 | 43.284 | 49.112 | 37.775 | 37.705 | 36.667 | 30.777 |

D versus B | 27.668 | 29.032 | 23.881 | 28.134 | 32.356 | 32.787 | 35.000 | 17.428 |

D versus C | 7.182 | 6.452 | 7.463 | 9.079 | 29.625 | 29.508 | 28.333 | 21.123 |

GRNN, generalised regression neural network; MAPE, mean absolute percentage error; MER, mean error rate; MAE, mean absolute error; NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; RMSE, root mean square error; SARIMA, seasonal autoregressive integrated moving average.

The comparison graph of the fitting and forecasting results among various models. GRNN, generalised regression neural network; NARNN, non-linear autoregressive neural network; NARNNX, non-linear autoregressive neural network with exogenous input; SARIMA, seasonal autoregressive integrated moving average.

The comparison graph between the estimated epidemic trends of TB incidence from 2018 to 2025 and the milestones goals suggested by WHO. TB, tuberculosis.

A refers to the SARIMA model; B stands for the SARIMA-GRNN hybrid model; C signifies the SARIMA-NARNN hybrid model; D represents the SARIMA-NARNNX the hybrid model.

As one of the oldest infectious diseases, many countries have been fighting TB for years, but TB is still by far one of the foremost public health problems in China and worldwide.

It is well accepted that the accurate identification of seasonality plays a major role in timely responses and reasonably allocated resources for TB epidemics._{2.5}, PM_{10}, NO_{2} and SO_{2}, the key indicators of air pollution, increasingly hits new records in the winter in almost all the large cities. Importantly, few studies have revealed a positive correlation between air pollution and the seasonal risk of TB, and the potential hazards of air pollution on health exhibit an obvious lagged effect.

By characterising the TB notified cases in mainland China, a slight downturn since 2006 has been noted. Moreover, China has been on track to achieve the goal of reducing TB morbidity and mortality rates by 50% in 2015 relative to 1990.

In this work, we concentrate on the epidemic trend analysis of TB incidence and have succeeded in developing and assessing a hybrid technique with a potential for forecasting the long-term TB incidence data in mainland China. Moreover, the important findings drawn from this work are based on a sufficiently large TB incidence dataset spanning 29 years and a comprehensive comparison of models that are, currently, either the most extensively adopted or the most efficient for predicting infectious diseases incidence data. Nonetheless, several limitations still need to be considered. First, there is a lack of standardised methods that can be employed to identify the best-performing configuration and key parameters of ANNs; in applications, repeated attempts are required. Second, the underestimation of the total number of monthly incident cases is inevitable in passive contagious disease reporting systems as a result of a mixture of under-reporting of detected cases and underdiagnosis (eg, individuals may fail to access healthcare or they may fail to be diagnosed when they do). Third, weekly data may allow a greater examination of the temporal differences between years. Nevertheless, we do not perform further analysis due to the lack of available data. Fourth, the model was established without taking other drivers related to TB occurrence and development into consideration in addition to the case numbers and months. Fifth, this model should be regularly updated with new notified data to ensure its prediction accuracy. Lastly, this work is only focused on the TB incidence data forecasting in mainland China. Further studies involving predictions for various regions and different types of infectious diseases exhibiting marked seasonal and cyclic variations are required to verify the potential application of the SARIMA-NARNNX hybrid technique.

In summary, our proposed SARIMA-NARNNX method offers more accurate predictions for TB case notifications than that of the basic SARIMA, NARNN and ETS(M,A,A) methods, as well as the traditional SARIMA-GRNN and SARIMA-NARNN hybrid approaches. This model may be conducive and instrumental for government officials to rationally allocate health resources and appropriately formulate long-term preventive and control plans for TB. Additionally, the projected incidents display a potential slight downturn but still retain a fairly high morbidity level, so urgent action is needed to formulate additional comprehensive prevention, control and intervention strategies.

We thank all members involving the collection of TB data.

YW and CX contributed equally.

YW, CX and JY conceived and proposed this work. SZ and ZW collected and analysed the data. LY and YZ improved the paper. All authors agreeed to submit this article.

This work was supported by the Graduate Student Innovation Fund of Hebei Province (no. CXZZBS2017130).

None declared.

The ethical approval is not warranted for our present work as the monthly monitoring data of TB morbidity are publicly available in China.

Not commissioned; externally peer reviewed.

All data were enclosed to the online supplementary materials.

Not required.

_{ 2.5 }concentration and the seasonality of tuberculosis for Beijing and Hong Kong