## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
The comparison is performed while using the Friedman test to see whether there is an overall significant difference in the estimations produced by each of the algorithms. Furthermore, if the performance of SHS is found to be comparable with the rest of the HS variants, then the post-hoc procedure of Wilcoxon signed-rank tests is conductedin order tto determine whether there is any statistically significant difference between the estimation performance of SHS and the rest of the HS variants individually. ## 4 SIR Simulation Experiments and Discussion

In this section, SIR simulation experiments are performed to illustrate the ability of HS algorithms as an efficacious estimator of SIR parameters. All of the variants of HS algorithm were coded in MATLAB R2017b on a laptop computer with 2.50 GHz Intel i7-4710 HQ CPU with 32 GB of RAM. The discussion is discretely presented for each data set used in this study. ### Simulations and Projected Simulations

#### 4.1.1 USA Data Set (United States of America)

Table 3 presents the optimized epidemiological parameters and corresponding fitness values (average from ten independent runs) for the USA data set. Figure 2 displays the visualization of the 220 days simulation together with the 20 days projected simulation using the optimized parameters. Visual inspection for USA data set is not informative enough as some of the lines representing the algorithms overlapped, which suggested that the estimates are very close to each other, except for NGHS, IBGHS, SHS, ITHS, ESHS, PAHS, and IHS that are visible. Simulations of each algorithms for approximately the first 100 days were indeed close to each other as well as to the observed cumulative cases. Beginning from the hundredth day of the outbreak, the simulations of each algorithm showed differences while they started deviating from the observed values, except for PAHS and IHS, which are still intact with the observed values. The projected simulation for subsequent 20 days were consistent in terms of the pattern in the calibration period. It is observed from the fitness values (SSE) in Table 3 that IHS appears to be the best performing estimator for USA data set, while NGHS is the least performing estimator. \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & \(\mathbf{x_{1}}(\beta)\) & \(\mathbf{x_{2}}(\gamma)\) & \(\mathbf{f(x)}\) **(SSE)** \\ \hline SHS & 0.4208 & 0.0931 & 1098.33 \\
**IHS** & **0.4198** & **0.0938** & **1072.83** \\ NGHS & 0.4212 & 0.0924 & 1174.67 \\ SGHS & 0.4202 & 0.0934 & 1076.62 \\ ITHS & 0.4203 & 0.0933 & 1082.46 \\ NSHS & 0.4202 & 0.0934 & 1077.16 \\ GDHS & 0.4202 & 0.0934 & 1077.14 \\ PAHS & 0.4198 & 0.0938 & 1073.06 \\ ESHS & 0.4202 & 0.0934 & 1077.22 \\ IBGHS & 0.4210 & 0.0921 & 1132.08 \\ \hline \end{tabular}
\end{table} 
Table 3: USA: Optimized parameters and corresponding fitness. For the USA data set, it is observed that the simulation from each HS algorithms are not very far off the observed cumulative cases, even in the projection period. Most of the simulations are similar as far as the parameters' values are concerned and they were able to approximately resemble the observed trend. Simulations of NGHS and IBGHS are the farthest deviation from the actual values. The similar behavior between these two algorithms may be due to the use of genetic mutation in the improvisation step that sets them apart from the rest of the algorithms. #### 4.1.2 FR Data Set (France)

Table 4 provides the optimized epidemiological parameters and the corresponding fitness values (the average from 10 independent runs) for the FR data set. The visualization of the 220 days simulation together with the 20 days projected simulation using the optimized parameters are depicted in Figure 3. The simulations can be visually inspected as the lines representing the algorithms do not overlap each other except for SHS, which overlaps with GDHS. During the calibration period the simulation of IHS was similar but not so close to ESHS and PAHS.