## 4 Experiment, Results & Discussion
### PSA for Vaccine Distribution Optimization

Table 6: Contact frequency matrix. \[ \begin{split} fit_{i}=min\left(\sum\limits_{i=1}^{5}I_{g}(peak) \right)\\ \text{subject to : }\sum\limits_{g=1}^{5}V_{g}=V_{max},\quad V_{g}=P_{i}^{g} \times V_{max}\\ \sum\limits_{g=1}^{5}P_{i}^{g}=1\\ 1\leq peak\leq end_{ob}\end{split}  \tag{9}\]  
where \(fit_{i}\) is fitness of agent \(i\), \(I_{g}(peak)\) is the peak of infection number from the start of the outbreak to the end of the outbreak, \(end_{ob}\). \(V_{max}\) is the total number of vaccines available. The infection number is calculated using Equation (5). Since day of the outbreak, \(t\), is a discrete value (\(t\in(1,\,end_{ob})\)), the differential equation is approached using the discrete calculus method, where the \(I\) is calculated using summation. The performance of the PSA in solving the vaccine distribution problem is compared with the three traditional strategies implemented in [31], which are based on transmissibility (S1), vulnerability (S2), and infection risk (S3). Three conditions are used here, where the number of vaccines is set to small, medium, and large quantity which are 5%, 10%, and 20% of the total population. The vaccine is assumed to be given during the infection dispersing stage, which is at day 50. The duration of the outbreak is \(end_{ob}=300\) days and the initial exposed individual are 30 from \(A_{2}\). The experimental parameter settings are tabulated in Table 8.  
The results of vaccine distribution using PSA are illustrated in Figures 5-7. The results show that for small and medium vaccination coverage, PSA can find much better vaccination distribution percentages that the other three traditional strategies. The peaks of the infection outbreak are lowest using the percentage determined by PSA compared to the other strategy. Meanwhile, when the number of vaccines is large, the performance of PSA, S2, and S3 are close to one another. Overall, the findings show that PSA is a good optimization algorithm that can solve a complex nonlinear differential problem like the vaccination distribution problem. In addition, it is observed that increasing the number of vaccines lowers and flattens the infection curve faster. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Age Group** & **Population \(P_{i}\)** & **Infection Rate \(\beta_{i}\)** & **Incubation Rate \(\gamma\)** & **Recovery Rate \(\tau\)** \\ \hline \(A_{1}\) & 0.94 m & 0.434 & 0.25 & 0.334 \\ \(A_{2}\) & 0.94 m & 0.158 & 0.25 & 0.334 \\ \(A_{3}\) & 2.30 m & 0.118 & 0.25 & 0.334 \\ \(A_{4}\) & 1.86 m & 0.046 & 0.25 & 0.334 \\ \(A_{5}\) & 0.85 m & 0.046 & 0.25 & 0.334 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters of 2009 Hong Kong’s H1N1 epidemic. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline No. of vaccine & (0.3 m = 5\%, 0.6 m = 10\%, 1.2 m = 20\%) \\ Administered day & 50 \\ Outbreak duration & 300 \\ Initial exposed individuals \{\(E_{1}E_{2},\,E_{3},\,E_{4},\,E_{5}\)\} & \{0, 30, 0, 0, 0\} \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Experimental parameter settings. Figure 5: Infection dynamic for 5% vaccination coverage. Figure 6: Infection dynamic for 10% vaccination coverage. Figure 7: Infection dynamic for 20% vaccination coverage.