## 3 Results  
### Expanded S-Curve in Typical Developed Countries  
According to the universal equation, the expanded S-curve equations of the DWPC were established for four typical countries as Equations (13)-(16), which were the US, Japan, UK, and France, respectively.  
(1) US  
\[W=170+181\times\frac{\exp[0.000311\times(G-11,500)]-\exp[-0.0000872(G-11,500)]} {2\cosh[0.000088(G-11,500)]} \tag{13}\]  
(2) Japan  
\[W=95+90\times\frac{\exp[0.0000338\times(G-11,000)]-\exp[-0.0000668(G-11,000)]} {2\cosh[0.0000673(G-11,000)]} \tag{14}\]  
(3) UK  
\[W=120+30\times\frac{\exp[0.000395\times(G-13,000)]-\exp[-0.000311(G-13,000)]} {2\cosh[0.000395(G-13,000)]} \tag{15}\](4)  
The changing trajectories of DWPC with GDP per capita in these countries generally experienced three stages.  
For the US in Figure 4a, the first stage was before the GDP per capita of USD 6500 in 1930 during Great Depression, and its economy sequence entered a special period with a winding curve until World War II. The DWPC maintained a flat trend during the first stage. In the second stage, the DWPC in the US kept growing to a high level of 230 m\({}^{3}\) in 1990, with a GDP per capita of USD 20,000. After the GDP per capita of USD 35,000 in 2000, the DWPC started to decrease due to technology improvement, with the wide use of dishwashers and water-saving toilets. The efficiency improvements dramatically reduced the water usage.  
Japan, in Figure 4b, showed a similar evolution pattern. The DWPC decreased distinctly during World War II in 1940s with a GDP per capita of around USD 2500, and it dropped to 32 m\({}^{3}\) with an annual decreasing rate of 7%. Then, with the post-war construction in 1950 with GDP per capita around USD 3500, the economic development model enabled Japan to enter a rapid development process of urbanization. From 1990 to 2000, after 40 years of linear growth, the DWPC in Japan peaked at approximately 120 m\({}^{3}\) with a GDP per capita of USD 20,000. After 2000, more efficient appliances and fixtures contributed to significant reductions in DWPC in Japan, with the same reduction trends in US.  
France's and Britain's economic development was similar to that of US and Japan in the first stage, as shown in Figure 4c, d. They were stagnant for a long time after World War I and World War II before the 1950s. Meanwhile, the DWPC showed stationary trends until 1950. Then the DWPC in these two countries developed differently. For France, the DWPC showed no variation around 65 m\({}^{3}\) until 1970, with a GDP per capita of USD 11,000. Then it started to experience a slight increase due to the soaring urbanization rate. In the third stage, the DWPC in France diminished from 106 m\({}^{3}\) after 2000 with a GDP per capita  
Figure 4: Expanded S-curve simulation of DWPC and GDP for (**a**) the United States (**b**) Japan, (**c**) France, and (**d**) the United Kingdom.  
of USD 20,000 due to the application of water-saving machines on a large scale. The DWPC variations in UK were directly related to the evolution of water bureau management, which was further propelled by urbanization development. In the late 1960s with a GDP per capita above USD 10,000, the DWPC increased from 113 m\({}^{3}\) with the increasing urbanization rate and population, and the management framework of water in the UK was optimized to improve the water efficiency. So, from 1970 to 1990, with a GDP per capita of between USD 11,000 and USD 18,000, an increase in the DWPC of between 115 m\({}^{3}\) to 138 m\({}^{3}\) was evident in the UK. In 1980s, with the stagflation in economics, the UK government published the Government White Paper on Privatization of Water Industry in 1986 and the top 10 water industries in the UK completed the privatization in 1989, which led to an increase in water prices and a decrease in DWPC in the 1990s with a GDP per capita of above USD 18,000 [30].