## 4 Numerical Examples
### Analysis and Discussion of Model Performance Comparison Using LSE Estimation Parameter Values
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & _MSE_ & \(R^{2}\) & _RMSE_ & _KD_ & _TS_ & _Bias_ \\ \hline G-O model & \(a\) = 7294.4; \(b\) = 0.000327 & 84.69 & 0.8715 & 9.2 & 0.2494 & 19.49 & 7.66 \\ \hline Delayed S-shaped model & \(a\) = 281.75; \(b\) = 0.034538 & 74.06 & 0.8876 & 8.61 & 0.1823 & 18.22 & 7.62 \\ \hline Inflection S-shaped model & \(a\) = 968.32; \(b\) = 0.042453; _β_ = 28.28 & 52.09 & 0.921 & 7.22 & 0.2192 & 15.28 & 6.16 \\ \hline Yamada Imperfect-2 model & \(a\) = 920; \(b\) = 0.001384; _α_ = 0.071173 & 55.15 & 0.9163 & 7.43 & 0.2011 & 15.73 & 6.29 \\ \hline P-N-Z model & \(a\) = 126.29; \(b\) = 0.0316; _β_ = 1.6331; _α_ = 0.035292 & 56.16 & 0.9148 & 7.49 & 0.1854 & 15.87 & 6.05 \\ \hline Weibull distribution model & \(a\) = 828.68; \(b\) = 0.002392; \(c\) = 1.0687 & 80.84 & 0.8773 & 8.99 & 0.2405 & 19.04 & 7.39 \\ \hline Wang model & \(a\) = 133.01; \(b\) = 0.000023; _β_ = 11.485; \(d\) = 2.8811 & 84.98 & 0.8711 & 9.22 & 0.2484, & 19.52 & 7.7 \\ \hline Li model & \(a\) = 132.92; \(b\) = 0.027505; \(b\) = 0.10282; \(N\) = 12.143 & 178.49 & 0.7292 & 13.36 & 0.3972 & 28.29 & 11.25 \\ \hline PM & \(a\) = 100.01; \(b\) = 0.047628; _β_ = 0.00013; \(d\) = 2.6091; _β_ = 1.0922 & **35.85** & **0.9456** & **5.99** & **0.1438** & **12.68** & **4.82** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model performance comparison using 100% of the data (DS1-1). Tables 6-8 show that using 100% of the data (DS2-1, DS2-2, and DS2-3), the \(MSE\), \(R^{2}\), \(RMSE\), \(KD\), \(TS\), and \(Bias\) values of the PM are 1040, 0.9619, 32.25, 0.2497, 14.95, and 22.76; 356.92, 0.8562, 18.89, 0.3842, 29.18, and 12.93; and 1142.3, 0.7558, 33.8, 0.392, 40.18, and 27.67, respectively. According to Table 6, the \(MSE\), \(RMSE\), \(TS\), and \(Bias\) values of the second-placed GGO model are roughly 0.45, 0.53, 0.54, and 0.58 times higher than those of the PM, respectively. In Table 7, we can see that the \(MSE\), \(RMSE\), \(TS\), and \(Bias\) values of the DSS model which ranked second are about 1.33, 1.16, 1.15, and 1.33 times as large as those of the PM, respectively. The performance of fitting is the highest for the PM. Using 100% of the data (DS2-1, DS2-2, and DS2-3), the second best is the GGO model, and then, the DSS model and the P-N-Z model, respectively, follow. The worst is the Li model, which is three times worse in terms of its ranking. Figure 2a,c,e show the fitting comparison of the models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & \(MSE\) & \(R^{2}\) & \(RMSE\) & \(KD\) & \(TS\) & \(Bias\) \\ \hline G-O model & \(a=57\),619; \(b=0.000146\) & 9378.4 & 0.6567 & 96.84 & 0.4692 & 44.9 & 87.82 \\ \hline Delayed S-shaped model & \(a=208\),930; \(b=0.001712\) & 3622.7 & 0.8674 & 60.19 & 0.3548 & 27.91 & 48.38 \\ \hline Infection S-shaped model & \(a=398.82\); \(b=0.069061\); \(\beta=3.5\) & 9261.9 & 0.661 & 96.24 & 0.4943 & 44.62 & 86.63 \\ \hline Yamada Imperfect-2 model & \(a=50\),948; \(b=0.000013\); \(a=0.73841\) & 4249.1 & 0.8445 & 65.19 & 0.35 & 30.22 & 52.4 \\ \hline P-N-Z model & \(a=114.8\); \(b=0.15004\); \(\beta=29.315\); \(a=0.10016\) & 2659.7 & 0.9026 & 51.57 & 0.3059 & 23.91 & 39.05 \\ \hline Weibull distribution model & \(a=436.11\); \(b=1\times 10^{-5}\); \(c=3.389\) & 2450.7 & 0.9103 & 49.5 & 0.2516 & 22.95 & 35.99 \\ \hline Wang model & \(a=420\); \(b=0.016864\); \(\beta=1.6113\); \(d=1.2419\) & 13,986 & 0.488 & 118.26 & 0.6285 & 54.83 & 108.21 \\ \hline Li model & \(a=419.98\); \(b=0.073397\); \(\phi=1.2663\); \(N=1.2736\) & 16,313 & 0.4028 & 127.72 & 0.6895 & 59.22 & 111.16 \\ \hline PM & \(a=401.99\); \(b=0.24812\); \(\theta=0.008026\); \(d=0.78784\); \(\sigma=0.000067\); \(f=850.84\) & **1040** & **0.9619** & **32.25** & **0.2497** & **14.95** & **22.76** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model performance comparison using 100% of the data (DS2-1).