## 4 Experiment, Results & Discussion
### PSA for Vaccine Distribution Optimization  
During a pandemic, an efficient vaccine distribution strategy is important to ensure that objectives such as achieving heard immunity and lowering the transmissibility rate is achieved. As observed during the current COVID-19 pandemic as well as the 2009 H1N1 outbreak, vaccination is an important measure to combat the outbreak of infectious diseases. However, vaccine production is frequently costly and time-consuming, causing limited supply [29]. Therefore, the distribution of vaccines to susceptible persons must be optimized in order to achieve the optimum effects [30]. Figure 4: Convergence Curves. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(i\) & **Algorithm** & \(p\) & **Holm** \\ \hline
2 & SCA & 0.000194 & 0.025 \\
1 & PSO & 0.3268 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Holm Post Hoc for Number of Agents. In this work, we proposed the application of PSA for vaccine distribution optimization. The 2009 Hong Kong's H1N1 epidemic [31] is used here as the case study. In [31], an extended SEIR model with vaccination (SEIR-V) is proposed. The SEIR model where, S--susceptible, E--exposed, I--infected, and R--recovered, is a compartmental mathematical model frequently adopted to model dynamics of infectious diseases. The SEIR model can be extended to include other aspects such as vaccination effect (V), death (D), and hospitalization (H). The model adopted here characterizes the infection dynamics according to five age groups and the contact rate of the individuals. The SEIR-V model is represented by nonlinear differential Equations (3)-(7). \[\frac{dS_{g}}{dt}=\big{(}-\lambda_{g}\big{)}\cdot\big{(}S_{g}-\Delta v_{g} \big{)}+\big{(}-\Delta v_{g}\big{)} \tag{3}\]  
\[\frac{dE_{g}}{dt}=(-\gamma)\cdot E_{g}+\lambda_{g}\cdot\big{(}S_{g}-\Delta v_{ g}\big{)} \tag{4}\]  
\[\frac{dI_{g}}{dt}=(-\tau)\cdot I_{g}+\gamma\cdot E_{g} \tag{5}\]  
\[\frac{dR_{g}}{dt}=\tau\cdot I_{g} \tag{6}\]  
\[\frac{dV_{g}}{dt}=\Delta v_{g} \tag{7}\]  
The subscript \(g\) in the equations represents age group, where \(g=\{1,2,3,4,5\}\). The population is divided to five age groups as follows; \(A_{1}(5-14)\), \(A_{2}(15-24)\), \(A_{3}(25-44)\), \(A_{4}(45-64)\), and \(A_{5}(65+)\). Children between the age of 0-4 are not included as they do not have independent social contact. The contact frequency matrix, \(C=\big{\{}c_{g1,g2}\big{|}g1,g2\in(1,5)\big{\}}\), between individuals from different and same age groups is presented in Table 6. The numbers are decided based on the work conducted in [32]. In Equation (3), (4), and (7), \(\Delta v_{g}\) is the amount of vaccine released in the current step. Meanwhile, \(\lambda_{g}\) represents infection risk of the susceptible individuals in group \(A_{g}\) which is formulated in Equation (8). \[\lambda\text{g}=\frac{1}{5}\cdot\left(\sum_{j=1}^{5}\bigg{(}c_{g},g^{2}\cdot \frac{I\text{g}2}{P\text{g}2}\ \bigg{)}\right)\cdot\frac{S\text{g}}{P\text{g}}.\beta\text{g} \tag{8}\]  
The size of population according to age group is represented by \(P\text{g}\) while \(\beta\text{g}\) is the age-based infection rate. \(\gamma\) and \(\tau\) represent the incubation and recovery rate. These parameters are set according to the setting of [31], which is shown in Table 7.  
The objective of the PSA is to minimize the total infections by minimizing the peak of the infection curve. The position of PSA's agent, \(P_{i}^{g}\), represents the distribution percentage of the vaccine according to age group, \(g\). The dimension of the agent follows the number of age group. The objective function is as shown in Equation (9). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) & \(A_{5}\) \\ \hline \(A_{1}\) & 8.27 & 1.395 & 4.165 & 1.51 & 0.715 \\ \(A_{2}\) & 1.395 & 5.65 & 2.385 & 1.83 & 0.895 \\ \(A_{3}\) & 4.167 & 2.385 & 6.55 & 3.425 & 1.383 \\ \(A_{4}\) & 1.51 & 1.83 & 3.425 & 4.2 & 2.055 \\ \(A_{5}\) & 0.715 & 0.895 & 1.383 & 2.055 & 2.66 \\ \hline \hline \end{tabular}
\end{table}