## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(\mathbf{x_{1}(\beta)}\) & \(\mathbf{x_{2}(\gamma)}\) & \(\mathbf{f(x)}\) **(SSE)** \\ \hline SHS & 0.3125 & 0.1012 & 2382.52 \\ IHS & 0.3137 & 0.1011 & 2510.07 \\ NGHS & 0.3142 & 0.1014 & 2531.25 \\ SGHS & 0.3120 & 0.1013 & 2203.09 \\
**ITHS** & **0.3140** & **0.1012** & **2172.45** \\ NSHS & 0.3137 & 0.1011 & 2508.21 \\ GDHS & 0.3130 & 0.1012 & 2500.66 \\ PAHS & 0.3140 & 0.1014 & 2528.17 \\ ESHS & 0.3125 & 0.1011 & 2381.76 \\ IBGHS & 0.3135 & 0.1012 & 2504.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: SK: Optimized parameters and corresponding fitness. Figure 4: SK: Simulation for 220 days (solid line) and projected simulation for 20 days (dashed line). For the SK data set, observe that the simulations are quite far off the observed cumulative cases, with NGHS being the farthest and ITHS being the nearest. Yet, ITHS's simulation is still far off the observed cumulative cases comparatively. The simulations were not able to resemble the observed trend well. The simulation for SK data set is clearly not as good as USA or FR data sets. the simulations of all the algorithms were distinct except for the overlapping SHS and ESHS. NGHS's simulation is again very different from the rest, probably due to use of genetic mutation, which sets it apart from the rest. As far as this data set is concerned, it is also interesting to observe ESHS that requires zero parameter setting and uses Gaussian mutation in the place of random generation of harmony, behaves somewhat similar to SHS. #### 4.1.4 IR Data Set (Ireland)

Table 6 provides the optimized epidemiological parameters and the corresponding fitness values (average from 10 independent runs) for the IR data set. Figure 5 depicts the visualization of the 220 days simulation, together with the 20 days projected simulation using the optimized parameters. The visual inspection of the simulations is informative, although the lines representing most of the algorithms are quite close to each other, with only SGHS and PAHS overlapping. The simulations drifted away from the observed cumulative cases as early as before the fiftieth day of the outbreak itself. However, the simulations of SGHS, PAHS, and NSHS managed to stay intact with the observed values, even during the projection period. On the other hand, the simulations of IHS, IBGHS, and SHS were still close to the observed values and they became closer in the projection period. ITHS remained steadily far from the observed values, while ESHS, NGHS, and GDHS maintained a constant separation from the observed values. The projected simulation for subsequent 20 days indicated a smaller deviation of the simulations from the observed values, except for ITHS. It is observed from the fitness values (SSE) in Table 6 that NSHS appears to be the best performing estimator for IR data set, while ITHS emerged as the least performing estimator. For the IR data set, it is observed that the simulations are reasonably close to the observed cumulative cases, except for ITHS. This shows that most of the simulations managed to mimic the observed trend well. Excluding ITHS, we can group the algorithms with similar simulations as (i) ESHS, NGHS, and GDHS; (ii) IHS, IBGHS, and SHS; and, (iii) SGHS, PAHS, and NSHS. Note that the difference in the simulation of ITHS was better in the previous three data sets, but it appeared to be the worst in this data set. The combination of algorithms that produced similar simulations is also different from combinations in previous data set, notably the FR data set.