## 2 Material and Methods
### Mathematical Modeling of the Expanded S-Curve  
The expanded S-curve model illustrating the relationship between DWPC and GDP per capita offers a tool to identify critical transitions from one stable state to another during economic development. A mathematical technique is employed to describe the expanded S-curve model. According to the expanded S-curve in previous studies [20; 21; 29], the relationship between DWPC (\(W\)) and GDP per capita (\(G\)) can be expressed as follows:  
\[W-W_{i}=A\frac{\exp[\alpha_{1}(G-G_{i})]-\exp(-\alpha_{3}(G-G_{i}))]}{2\cos h[ \alpha_{2}(G-G_{i})]} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Fitting Curve**} & \multicolumn{2}{c}{**(R-Squared)**} & \multirow{2}{*}{**Applicable Countries**} \\  & & \(\mathbf{R^{2}}\) & \\ \hline Model 1 & y = 0.0208*\(2.1216\) & 0.8908 & USA, CA \\ Model 2 & y = 0.49617*\(1.22812\) & 0.8689 & KR, IND, IN, UK \\ Model 3 & y = 48.314 In(\(\chi\)) \(-\) 128.18 & 0.8743 & JP, FR, CN \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulated models between urbanization and DWPC.  
Figure 3: Relationship between GDP per capita and DWPC for 10 typical countries.  
Figure 2: Relationship between urbanization and DWPC.  
where \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are the exponential constraints, and \(A\) is the amplitude of the equation. \(W_{i}\) and \(G_{i}\) are the corresponding turning points on the expanded S-curve for DWPC and GDP per capita, respectively. Equation (1) is expressed as a hyperbolic tangent function.  
Then, the linearity changes before the takeoff point, around the turning point, and after the zero-growth point are derived from Equation (1) as Equations (2)-(4).  
\[W-W_{i}=A+A(\alpha_{2}-\alpha_{3})(G-G_{i})=A+\rho_{l}(G-G_{i}) \tag{2}\]  
\[W-W_{i}=0.5A(\alpha_{1}+\alpha_{3})(G-G_{i})=\rho_{i}(G-G_{i}) \tag{3}\]  
\[W-W_{i}=A+A(\alpha_{1}-\alpha_{2})(G-G_{i})=A+\rho_{v}(G-G_{i}) \tag{4}\]  
where \(\rho_{l}\), \(\rho_{i}\), and \(\rho_{v}\) are the slopes of the curve before the takeoff point, around the turning point, and after the zero-growth point, respectively. They can be calculated from the systems of Equations (5)-(7):  
\[\alpha_{1}=\frac{\rho_{l}+2\rho_{i}+\rho_{v}}{2A} \tag{5}\]  
\[\alpha_{2}=\frac{\rho_{l}+2\rho_{i}-\rho_{v}}{2A} \tag{6}\]  
\[\alpha_{3}=\frac{-\rho_{l+}2\rho_{i}-\rho_{v}}{2A} \tag{7}\]  
Equation (1) has a first-order partial derived from 0 at the zero-point of the S-curve:  
\[\tanh[\alpha_{1}(G_{v}-G_{i})]\tanh\left[\alpha_{2}(G_{v}-G_{i})\right]=\alpha _{1}\alpha_{2}^{-1},\frac{dW}{dG}=0 \tag{8}\]  
By substituting Equations (5) and (6) into Equation (8), Equation (9) can be obtained  
\[\tanh\left(\varphi_{1}A^{-1}\right)\tanh\left(\varphi_{2}A^{-1}\right)=\varphi _{3} \tag{9}\]  
where  
\[\varphi_{1}=0.5(\rho_{l}+\rho_{i}+\rho_{v})(G_{v}-G_{i}) \tag{10}\]  
\[\varphi_{2}=0.5(\rho_{l}+2\rho_{i}-\rho_{v})(G_{v}-G_{i}) \tag{11}\]  
\[\varphi_{3}=\frac{\rho_{l}+\rho_{i}+\rho_{v}}{\rho_{l}+2\rho_{i}-\rho_{v}} \tag{12}\]  
In summary, the \(W_{i}\), \(G_{i}\), \(\rho_{l}\), \(\rho_{i}\), and \(\rho_{v}\) were from research data, and the \(A\), \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) were from the equations.