## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(\mathbf{x_{1}}(\mathbf{\beta})\) & \(\mathbf{x_{2}}(\mathbf{\gamma})\) & \(\mathbf{f}(\mathbf{x})\) **(SSE)** \\ \hline SHS & 0.3426 & 0.1044 & 970.02 \\ IHS & 0.3427 & 0.1044 & 1078.34 \\ NGHS & 0.3436 & 0.1042 & 1233.88 \\ SGHS & 0.3425 & 0.1043 & 703.22 \\ ITHS & 0.3441 & 0.1042 & 1827.38 \\
**NSHS** & **0.3420** & **0.1043** & **662.22** \\ GDHS & 0.3432 & 0.1043 & 1202.47 \\ PAHS & 0.3425 & 0.1043 & 701.58 \\ ESHS & 0.3436 & 0.1041 & 1247.19 \\ IBGHS & 0.3425 & 0.1044 & 971.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: IR: Optimized parameters and corresponding fitness. #### 4.1.5 SG Data Set (Singapore)

Table 7 provides the optimized epidemiological parameters and the corresponding fitness values (average from 10 independent runs) for the SG data set. Figure 6 displays the visualization of the 220 days simulation, together with the 20 days projected simulation using the optimized parameters. The simulations are distinguishable, thus the visualization is informative, except for a slight overlap between ITHS and IBGHS. The simulations band begin to diverge from the observed cumulative cases approximately around the hundredth day of the outbreak, except for the simulations of ESHS, PAHS, and IHS, which diverged gradually. While other simulations deviated from the rest, ESHS, PAHS, and IHS were close to each other until day 200 of the outbreak. Only after that, the simulations diverged from each other in a small amount up to the projection period. All of the simulations were far from the observed values, resembling a similarity with the case of simulations in the SK data set. The projected simulation for subsequent 20 days indicated an even larger deviation of the simulations from the observed values. It is observed from the fitness values (SSE) presented in Table 7 that IHS appears to best performing estimator for SG data set, while ITHS is the least performing estimator, similar to IR data set. For the SG data set, it is observed that the simulations are quite far off the actual cumulative cases, with ITHS being the farthest and IHS being the nearest. Nevertheless, IHS's simulation is still far off the observed cumulative cases. The simulations were not able to resemble the observed trend well and it is undoubtedly not as good as for USA, FR, and IR data sets. The quality of simulations are similar to the SK data set. The simulations of ITHS and IBGHS are really far off the observed values. We note that the simulations of ITHS and IBGHS are similar, NSHS and SGHS are close to each other, GDHS and NGHS are approximately close, ESHS and PAHS are close to each other, while SHS and IHS were not similar or close to the rest. Figure 5: IR: Simulation for 220 days (solid line) and projected simulation for 20 days (dashed line). ### Performance Comparison

Following the SIR simulation experiments that were performed on the five data sets, it is noted that the performance of each algorithm (based on the fitness values (SSE)) varies across the data sets. Based on the parameter estimates for each data set presented in Tables 3-7, the estimates produced by each algorithm are fairly similar, which indicates that the optimization performed is consistent due to the underlying nature of HS, regardless of the type of HS variant. Table 8 lists the best performing algorithm for each data set. Apparently, there is no one clear winner algorithm for this particular application of HS. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(\mathbf{x_{1}(\beta)}\) & \(\mathbf{x_{2}(\gamma)}\) & \(\mathbf{f(x)}\) (SSE) \\ \hline SHS & 0.3604 & 0.1331 & 2015.77 \\
**IHS** & **0.3582** & **0.1334** & **1002.64** \\ NSHS & 0.3600 & 0.1332 & 1799.32 \\ SGHS & 0.3607 & 0.1331 & 2238.65 \\ ITHS & 0.3612 & 0.1328 & 2564.23 \\ NSHS & 0.3608 & 0.1331 & 2241.29 \\ GDHS & 0.3600 & 0.1333 & 1805.79 \\ PAHS & 0.3585 & 0.1333 & 1542.52 \\ ESHS & 0.3588 & 0.1332 & 1622.38 \\ IBGHS & 0.3611 & 0.1328 & 2557.87 \\ \hline \hline \end{tabular}
\end{table}