## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
Gaussian mutation is claimed to be more efficient in exploring the global optimum solution as compared to uniform randomization. Thus, the random selection in ESHS is performed while using:

\[x_{i}^{\prime}=N(\mu_{i},\sigma),\quad\mu_{i}=x_{i}^{best},\quad\sigma=1-( \frac{t-1}{MaxImp})^{2}, \tag{21}\]with the probability of \(1-HMCR\). ESHS is detailed in Algorithm 9. ```
1:Set \(HMS\), \(HMCR\) using (18), \(PAR\) using (19), \(BW\) using (20), and \(MaxImp\)
2:\(x_{i}^{i}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i}),\forall i=1,2,...,n\) and \(\forall j=1,2,...,HMS\)
3:Compute \(f(\mathbf{x}^{i}),\forall j=1,2,...,HMS\)
4:while\((t\leq MaxImp)\)do
5:for each \(i\in[1,n]\)do
6:if\(U(0,1)<HMCR_{t}\)then
7:\(x_{i}^{\prime}=x_{i}^{\prime}\) where \(j\sim U(1,HMS)\)
8:if\(U(0,1)<PAR_{t}\)then
9:\(x_{i}^{\prime}=x_{i}^{\prime}+U(-1,1)\times BW_{t}\)
10:endif
11:else
12: perform Gaussian mutation using (21)
13:endif
14:if\((x_{i}^{\prime}>UB_{i}\) or \(x_{i}^{\prime}<LB_{i})\)then
15:\(x_{i}^{\prime}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i})\)
16:endif
17:endfor
18:if\(f(\mathbf{x}^{\prime})<f(\mathbf{x}^{\prime lowest})\)then
19: replace \(\mathbf{x}^{\prime lowest}\) in HM with \(\mathbf{x}^{\prime}\)
20:endif
21:endwhile
```
**Algorithm 9**Enhanced Self-Adaptive Global Best Harmony Search

### Improved Binary Global Harmony Search (IBGHS)

The Improved Binary Global Harmony Search (IBGHS) [61] is a binary variant of the NGBHS [54] that aims to improve the two limitations of NGBS namely the local optima trap and slow convergence. The improvisation step is modified with the introduction of a control parameter \(Pc\) in the place of \(HMCR\) and a linear combination of the best and worst harmonies in order to improve the global search ability and convergence speed of NGBS. Algorithm 10 provides the computational procedure of IBGHS. ```
1:Set \(HMS\), \(Pc\), \(Pm\), \(PAR\), \(BW\), and \(MaxImp\)
2:\(x_{i}^{i}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i}),\forall i=1,2,...,n\) and \(\forall j=1,2,...,HMS\)
3:Compute \(f(\mathbf{x}^{i}),\forall j=1,2,...,HMS\)
4:while\((t\leq MaxImp)\)do
5:for each \(i\in[1,n]\)do
6:if\((U(0,1)\leq Pc)\)then
7:\(x_{R}=2\times x_{i}^{\mathit{best}}-x_{i}^{\mathit{unrest}}\)
8:\(x_{i}^{\prime}=min(x_{i}^{\prime},LB_{i}),\{UB_{i}\}\)
9:\(x_{i}^{\prime}=x_{i}^{\mathit{unrest}}+U(0,1)\times(x_{R}-x_{i}^{\mathit{ unrest}})\) %position updating
10:if\((U(0,1)\leq Pc)\) %genetic mutation
11:\(x_{i}^{\prime}=LB_{i}+U(0,1)\times(UB_{i}-LB_{i})\)
12:else
13:\(x_{i}^{\prime}=0.7x_{i}^{\mathit{best}}+0.3x_{i}^{\mathit{unrest}}\)
14:if\((U(0,1)\leq PAR)\)then
15:\(x_{i}^{\prime}=x_{i}^{\prime}+U(0,1)\times BW\)
16:\(x_{i}^{\prime}=min(max(x_{i}^{\prime},LB_{i}),\{UB_{i}\})\)
17:endif
18:endif
19:endfor
20:if\(f(\mathbf{x}^{\prime})<f(\mathbf{x}^{\mathit{unrest}})\)then
21: replace \(\mathbf{x}^{\mathit{unrest}}\) in HM with \(\mathbf{x}^{\prime}\)
22:endif
23:endwhile
```

**Algorithm 10**Improved Binary Global Harmony Search

Despite being successfully applied in various fields, to the best of our knowledge, HS algorithms have yet to be applied in epidemiology, particularly in epidemiological modeling. Thus, in this study, ten variants of HS algorithm is proposed to be applied in order to estimate the epidemiological parameters of interest in the prototypical compartmental epidemiological SIR model and compare the estimation performance of each algorithm. ## 3 Estimating the Epidemiological Parameters of SIR Model

HS algorithms are applied to the SIR parameter estimation problem while using the cumulative infectious cases (total cases) of the COVID-19 pandemic as the use-case. The optimized (final) values of parameters \(\beta\) and \(\gamma\) are estimated by calibrating the available COVID-19 data and the SIR model with the generated harmonies (candidate solutions) from HM.