## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
However, in the projection period, IHS's simulation seems to be closer to ESHS and PAHS. The other groups of algorithms that produced similar simulations are ITHS, NSHS, SGHS and SHS, GDHS, IBGHS. The simulations started diverging from each after day 60 of the outbreak, and gradually digressed from the observed cumulative values all the way up to the projection period. The projected simulation for subsequent 20 days depicted a further deviance of the simulations from the observed values. It is observed from the fitness values (SSE) in Table 4, that IBGHS appears to best performing estimator for FR data set, while NGHS is the least performing estimator, just as the case of USA data set. Figure 2: USA: Simulation for 220 days (solid line) and projected simulation for 20 days (dashed line). For the FR data set, observe that the simulations are slightly far off the observed cumulative cases. The simulations were not able to accurately mimic the observed trend. NGHS's simulation was apart all the way, whereas we can infer that the simulations of IHS, ESHS, and PAHS are similar; ITHS, NSHS, and SGHS are alike; and finally, SGHS, GDHS, SHS, and IBGHS are close to each other. NGHS's simulation is again the farthest and different from the rest, which is probably due to the use of genetic mutation, which sets it apart from the rest. Genetic mutation is also used in IBGHS, but the standard _PAR_ setting in IBGHS could have contributed to the high similarity in behavior as SHS than NGHS. Hence, for this particular data set, the _PAR_ parameter was more influential then genetic mutation, as compared to the USA data set. #### 4.1.3 SK Data Set (South Korea)

Table 5 presents the optimized epidemiological parameters and the corresponding fitness values (average from 10 independent runs) for the SK data set. Figure 4 displays \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & \(x_{1}(\beta)\) & \(x_{2}(\gamma)\) & \(f(\text{x})\) (SSE) \\ \hline SHS & 0.2208 & 0.1227 & 1002.11 \\ IHS & 0.2221 & 0.1223 & 1211.77 \\ NGHS & 0.2224 & 0.1220 & 1427.31 \\ SGHS & 0.2213 & 0.1225 & 1101.25 \\ ITHS & 0.2214 & 0.1226 & 1102.71 \\ NSHS & 0.2213 & 0.1226 & 1102.42 \\ GDHS & 0.2208 & 0.1228 & 992.04 \\ PAHS & 0.2218 & 0.1226 & 1125.00 \\ ESHS & 0.2218 & 0.1226 & 1125.36 \\
**IBGHS** & **0.2207** & **0.1228** & **987.31** \\ \hline \end{tabular}
\end{table} 
Table 4: FR: Optimized parameters and corresponding fitness. Figure 3: FR: Simulation for 220 days (solid line) and projected simulation for 20 days (dashed line). the visualization of the 220 days simulation, together with the 20 days projected simulation using the optimized parameters. The simulations can be well visualized as the lines representing each algorithm is distinct, except for SHS, which overlaps with ESHS. The simulations started departing from the observed cumulative cases as early as the fiftieth day of the outbreak. The band of simulations also started deviating around the same time and was gradually separated up to the projection period. All of the simulations were distinguishable, except in the case of IHS and NSHS, which were close to each other and SHS and ESHS that were similar. The projected simulation for subsequent 20 days indicated a larger deviation of the simulations from the observed values. We observe from the fitness values (SSE) in Table 5 that \(\Gamma\)THS appears to be the best performing estimator for SK data set, while NGHS is the least performing estimator, just as the case of USA and FR data sets.