## Related Methodology
### Structure of the Proposed Integrated Forecasting Framework  
Mathematical models (in Table 1) consist of compartments which represent sets of individuals grouped by disease status. The links between compartments represent transitions from one state of disease to another state and different compartments can be included or excluded according to the assumptions of the mathematical models. However, the combined model based on polynomial regression proposed in this paper aims at the incidence of TB; no other assumptions are needed in the modeling process. In the process of forecasting, it avoids the deviation of forecasting results caused by the invalid assumptions of the mathematical model. This is different from the mathematical models whose goal of a combined model is to model a non-linear relationship between the independent and dependent variables (technically, between the independent variable (year) and the conditional mean of the dependent variable (tuberculosis rates), and the combined forecasting model is the same as other common forecasting models, which mainly reflects the statistical regularity of diseases from data.  
Hierarchical tuberculosis prevalence rate data were collected from the World Health Organization (WHO) and the data were collected into four economic groups: high income, upper-middle income, lower-middle income, and low income groups. Given these data, we first employ the KW test to check whether tuberculosis prevalence rates are significantly different among the four income groups.  
After the hierarchical tuberculosis prevalence rate data analysis, the TB time series data is input into the five different regression models. The overall flowchart of the proposed integrated model is depicted in Figure 1.  
The parameters of the different regression models are determined by employing an RBF neural network; the RBF neural network is used to fit unknown function. Given a nonlinear function, such as \(y=ae^{bx}\), the parameters of the function \(a\) and \(b\) are not known. To determine them, first randomly generate two trial parameters of \(a\) and \(b\). With these two parameters, \(y\) is calculated by \(y=ae^{bx}\), and is used as the output data of the RBF neural network. Thus, the RBF neural network establishes approximate and exact regression analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Pairwise Income Group** & **Lower Bound of a 95\%** & **Estimated Difference in** & **Upper Bound of a 95\%** \\  & **Confidence Interval** & **Mean Ranks** & **Confidence Interval** \\ \hline High vs upper-middle & \(-34.4219\) & \(-17.0000\) & \(0.4219\) \\ High vs lower-middle & \(-53.5689\) & \(-36.1471\) & \(-18.7252\) \\ High vs low & \(-66.2748\) & \(-48.8529\) & \(-31.4311\) \\ Upper-middle vs & \(-36.5689\) & \(-19.1471\) & \(-1.7252\) \\ lower-middle & \(-49.2748\) & \(-31.8529\) & \(-14.4311\) \\ Lower-middle vs low & \(-30.1277\) & \(-12.7059\) & \(4.7160\) \\ \hline \end{tabular}
\end{table}
Table 4: Results of multiple comparison test.  
With the different regressions determined, a combined forecasting model is employed. The combined forecasting model is based on multiple different forecasting models for the same problem. It can be a combination of several quantitative methods or a combination of several qualitative methods. In this paper, a quantitative method is used to combine six regression models. The main purpose of combination is to make full use of the information provided by various forecasting models. To improve the forecasting accuracy as much as possible, this paper uses the cuckoo search algorithm to optimize and determine the combination weights in the combined model.  
It is worth noting that there are four steps of future forecasting, \(h=4\), for the different income groups studied in this paper, with the forecasting values from 2013 to 2016.