## 2 Harmony Search Algorithm and Selected Variants
### Improved Harmony Search (IHS)
### COVID-19 Data Sets

The authors obtain the time series of cumulative infectious cases per day for five countries, namely the United States of America (USA), France (FR), South Korea (SK), Ireland (IR), and Singapore (SG), by web scraping the figures in [[https://www.worldometers.info/coronavirus](https://www.worldometers.info/coronavirus)]([https://www.worldometers.info/coronavirus](https://www.worldometers.info/coronavirus)) (which gathers data from various reliable sources including European Centre for Disease Prevention & Control and Johns Hopkins University & Medicine Coronavirus Resource Center). The data collected are for a period of 240 days, beginning from the first day of the outbreak in each country. Next, the first 220 days data is used for calibration to obtain the estimates of the parameters. The remaining 20 days of data are used for validation against the projection of simulation produced while using the HS optimized parameter values. ### SIR Model Setup

The SIR compartments are initialized (at time \(t=0\)) with initial conditions \(I_{0}\) and \(R_{0}\), according to the actual number of infectious and removed cases, respectively, on the first day of the outbreak in each country. Meanwhile, \(S_{0}\) is set as the remaining individuals in the population \(N\) who are yet to be infected and removed, i.e., \(S_{0}=N-I_{0}-R_{0}\). Table 1 provides the modeling initialization. ### SIR Parameters Estimation as Optimization Problem

The estimation of SIR parameters is formulated as an optimization problem with decision variables \(\mathbf{x}=\{x_{1},x_{2}\}\), where \(\beta=x_{1}\) and \(\gamma=x_{2}\). The upper and lower bounds of the decision variables are set according to the complete possible ranges of \(\beta\) and \(\gamma\), which are essentially the epidemiological dynamics' rates and, hence, they share the same bounds, i.e., \(\beta,\gamma\in(0,1)\). The objective function is formulated, as follows:

1. Let \(C_{T}=\sum_{t=0}^{t=T}I_{t}\) be the observed cumulative infectious cases on day \(t=T,(t,T\in[0,219])\), where \(I_{t}\) is the observed infectious cases on day \(t\). 2. Let \(\widehat{C}_{T}=\sum_{t=0}^{t=T}\widehat{I}_{t}\) be the simulated cumulative infectious cases (rounded to the corresponding integers) on day \(t=T_{s}(t,T\in[0,219])\), where \(\widehat{I}_{t}\) is the SIR model simulated infectious cases on (rounded to the corresponding integers) day \(t\) while using \(\mathbf{x}\in\text{HM}\). 3. Subsequently, an objective function \(f(\cdot)\) that minimizes the SSE between \(C_{T}\) and \(\widehat{C}_{T}\) can be formulated as: \[\text{minimize}\ f(\mathbf{x})=\sum_{T=0}^{T=219}(C_{T}-\widehat{C}_{T})^{2} \quad\text{ s.t}\quad\mathbf{x}\in(0,1).\] (22) \begin{table}
\begin{tabular}{c c c c c c} \hline
**Data Set** & **Calibration Period** & **Projection Period** & **I0** & **R0** & **N * \\
**(Country)** & **(220 Days)** & **(20 Days)** & **(20 Days)** & **1** & 0 & 331,002,651 \\ \hline USA & 21 January 2020–27 August 2020 & 28 August 2020–16 September 2020 & 1 & 0 & 331,002,651 \\ FR & 25 January 2020–31 August 2020 & 1 September 2020–20 September 2020 & 3 & 0 & 65,273,511 \\ SK & 20 January 2020–26 August 2020 & 27 August 2020–15 September 2020 & 1 & 0 & 51,269,185 \\ IR & 29 February 2020–5 October 2020 & 6 October 2020–25 October 2020 & 1 & 0 & 4,937,786 \\ SG & 24 January 2020–30 August 2020 & 31 August 2020–19 September 2020 & 3 & 0 & 5,850,342 \\ \hline \end{tabular}
* estimated at mid year according to UN data.

\end{table} 
Table 1: Susceptible-Infectious-Removed (SIR) modeling initialization. A set of ten independent runs is performed for each HS algorithm while using each of the data sets. The optimization steps are detailed, as follows:

1. The common control parameters shared among all ten algorithms are set to be identical, \(HMS=30\) and \(MaxImp=50,000\), following the recommendation in [47]. 2.