## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)

Table 7: SG: Optimized parameters and corresponding fitness. Figure 6: SG: Simulation for 220 days (solid line) and projected simulation for 20 days (dashed line). We obtain the RMSE values from the 20 days projected SIR simulation while using Equation (24) in order to statistically compare the performance of each algorithm within each of the data sets. The Friedman test is conducted to determine whether there is any statistically significant differences among the estimation performances (in terms of RMSE) of each algorithm. Table 9 displays the RMSE values. The low RMSE values within each data set indicates that the predictive capability of SIR model while using the HS optimized parameters are fairly good. Eventually, it attests that the parameter estimation accuracy of HS variants is satisfactory. The Friedman test elicited no statistically significant difference in the estimation performance of the HS algorithms at significance level of 0.05 (\(\chi^{2}(9)=11.749,p=0.228\)). It is noteworthy to observe that the estimation performance of SHS is comparable with the rest of the HS variants and fairly consistent across the data sets. The simulations of SHS are also reasonably close to the observed cumulative cases in FR, SK, IR, SG, and USA data sets (in the closest order). The post-hoc analysis for SHS is performed using the Wilcoxon signed-rank tests in order to identify whether there is any statistically significant difference between the estimation performance of SHS and the rest of the HS variants individually. Table 10 displays the test results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test** & \(\mathbf{Z}\) & \(\mathbf{p}\)**-Value** \\ \hline SHS-IHS & \(-0.405\) & 0.686 \\ SHS-NGHS & \(-1.753\) & 0.080 \\ SHS-SGHS & \(-1.214\) & 0.225 \\ SHS-ITHS & \(-0.674\) & 0.500 \\ SHS-NSHS & \(-0.405\) & 0.686 \\ SHS-GDHS & \(-0.135\) & 0.893 \\ SHS-PAHS & \(-1.214\) & 0.225 \\ SHS-ESHS & \(-0.135\) & 0.893 \\ SHS-IBGHS & \(-1.483\) & 0.138 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Post-hoc analysis for standard HS (SHS). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **USA** & FR & **SK** & IR & **SG** \\ \hline SHS & 25.39 & 12.86 & 29.73 & 16.66 & 28.44 \\ IHS & 12.95 & 18.73 & 34.12 & 18.55 & 17.23 \\ NGHS & 38.12 & 27.32 & 39.64 & 25.44 & 24.22 \\ SGHS & 13.85 & 13.63 & 26.24 & 13.45 & 30.66 \\ ITHS & 15.34 & 14.75 & 25.44 & 37.89 & 39.14 \\ NSHS & 14.76 & 14.28 & 34.08 & 10.98 & 31.28 \\ GDHS & 14.22 & 12.19 & 33.64 & 22.62 & 25.22 \\ PAHS & 13.02 & 16.38 & 30.08 & 13.03 & 19.86 \\ ESHS & 14.92 & 16.59 & 38.7 & 26.12 & 21.46 \\ IBGHS & 33.69 & 11.35 & 33.78 & 16.94 & 38.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: RMSE. The Wilcoxon signed-rank tests that were conducted with a Bonferroni correction applied at the resulting significance level of \(\frac{0.05}{9}=0.006\) elicited no statistically significant difference in the estimation performance of SHS when compared to the rest of the HS variants individually. Indeed, the insignificant difference supports that the performance of SHS is comparable. Therefore, although SHS may not be the best performing algorithm for any of the data sets, the consistency and statistical tests' results elucidate that SHS is competent enough to be a potential efficacious estimator for the epidemiological parameters of SIR model. A slight manual fine-tuning of the control parameters may do the job of increasing the estimation accuracy of SHS. In essence, the primary advantages of applying HS to estimate the epidemiological parameters of compartmental models are as follows:

1. **No initial values for the epidemiological parameters are required. ** One does not need to adapt the values of epidemiological parameters from previous studies, so as to alleviate any bias in the estimation process. 2. **No specified upper and lower bounds for the epidemiological parameters are required to suit the data sets.