## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
Ten initial HMs (\(HM_{1},HM_{2},\ldots,HM_{10}\)) and ten corresponding initial fitness vectors (\(f_{1},f_{2},\ldots,f_{10}\)) are generated to be used in each run according to the initialization of HM (Step 2) in Section 2. 3. Optimization is performed for ten independent runs for each data set while using the identical \(HM\) and \(f\) using each of the algorithms (Algorithm 1 to Algorithm 10), as described in Section 2. For instance, the first run for the USA data set will be performed while using the same \(HM_{1}\) and \(f_{1}\) for all ten HS algorithms. The specific control parameters that are shared among some algorithms are also set to be identical as displayed in Table 2.
4. Repeat **Step 3** until the runs are completed for all five data sets. The combination of parameters' values that yields the best fitness (lowest SSE) are designated as the optimized parameters. 5. The average values of \(\{\mathbf{x}_{1},\mathbf{x}_{2}\}=\{\mathbf{\hat{x}}_{1},\mathbf{\hat{x}}_{2}\}\) from each run for each HS algorithm are obtained, where, \[\mathbf{\hat{x}}_{i}=\frac{\sum_{k=1}^{k=10}x_{ik}}{10},\quad(i=1,2),(k=1,2, \ldots,10).\] (23) Subsequently, \(\{\mathbf{\hat{x}}_{1},\mathbf{\hat{x}}_{2}\}\) are designated as the overall optimized SIR parameters with respect to each algorithm, thus \(\mathbf{x}_{optimized}=\{\mathbf{\hat{x}}_{1},\mathbf{\hat{x}}_{2}\}\). Note that the identical common and specific control parameters, as well as the identical HM and \(f\), are used in this study to ensure a fair comparison among the HS algorithms. ### Evaluation of HS Estimation and Performance Comparison

The accuracy of the SIR parameter estimates is deduced from the fitness values (SSE), where the algorithm with lowest fitness is designated as the best performing estimator and vice-versa. The optimized epidemiological parameters \(\mathbf{x}_{optimized}\) are supplied to the SIR model once again to produce a projection of simulation for a period of 20 subsequent days from the end date of calibration period in order to evaluate the predictive capability of SIR model while using the parameters that were estimated from HS algorithms. Let \(C_{T}\) be the observed cumulative infectious cases on the projection period and \(\widehat{C}_{T}\) be the projected SIR model simulated cumulative infectious cases using \(\mathbf{x}_{optimized}\), then the accuracy of estimation is evaluated by computing the Root Mean Squared Error (RMSE) between \(C_{T}\) and \(\widehat{C}_{T}\) in the period of projection using:

\[RMSE=\sqrt{\frac{\sum_{T=220}^{T=229}(C_{T}-\widehat{C}_{T})^{2}}{20}}. \tag{24}\]

The accuracy of the SIR model's projected simulation is decided based on the RMSE value, where a lower RMSE value indicates a better prediction and vice-versa. Eventually, \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Algorithm** & **HMCR** & **PAR** & **HMCRnn** & **HMCRnnx** & **PARmin** & **PARmax** & **BWmin** & **BWmax** & **Pm** & **Pc** & **LP** & **HMCRn** & **PARm** \\ \hline SHS & 0.95 & 0.3 & 0.01 & - & - & - & - & - & - & - & - & - & - & - \\ IHS & 0.95 & - & - & - & - & 0.99 & 0.01 & 0.001 & 1/(20x(UB-LB)) & - & - & - & - & - \\ NGRS & - & - & - & - & - & - & - & - & - & - & - & - & - \\ SGRS & - & - & - & - & - & - & - & 0.001 & 1/(20x(UB-LB)) & - & - & 100 & 0.98 & 0.9 \\ ITHS & 0.95 & - & - & - & - & 0.99 & 0.01 & - & - & - & - & - & - & - \\ NSFS & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\ GDHS & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\ PAHs & - & - & - & 0.99 & 0.7 & 0.99 & 0.01 & 0.001 & 1/(20x(UB-LB)) & - & - & - & - & - \\ ESHS & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\ IBGHS & - & 0.3 & 0.01 & - & - & - & - & - & - & 0.005 & 0.9 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specific control parameters of Harmony Search (HS) algorithms. the predictive capability indicates the parameter estimation accuracy of the HS variants. The corresponding RMSE for each algorithm within the same data set will be statistically compared.