## 4 Discussion  
Combining CSIRQ and CSIRV leads to the CSIRVQ model (Figure 5). Many runs were conducted to check stability from phase portraits. It was found that all equilibriums were unstable as shown in Figure 12.  
A comparison of the results obtained from the basic CSIR, CSIRQ and CSIRV models presented in Figures 7, 11 and 13, respectively, indicate that either quarantine or vaccination effectively reduces viral load in relation to the basic CSIR scheme (29.48), with quarantine being more powerful than vaccination under the specific conditions studied (4.32 vs. 7.56). Undoubtedly, combining both quarantine and vaccination policies drastically reduces viral load to 0.23 (Figure 14) under the same conditions. Table 3 also presents two alternative scenarios for the CSIRQV model. In the CSIRQV_1 scenario, the rates \(\delta\) (S to Q) and \(\mu\) (Q to S) are increased by an order of magnitude to describe a more se \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Rates** & **Cumulative Viral Load** \\ \hline \multirow{3}{*}{CSIRQ} & \(\alpha=0.18\), \(\beta=0.06\), \(\gamma=0.006\) & \multirow{3}{*}{7.56} \\  & \(\varepsilon=0.008333\), \(\theta=0.0006\), & \\  & \(\kappa=0.00008333\) & \\ \hline \multirow{3}{*}{CSIRV\_1} & \(\alpha=0.18\), \(\beta=0.06\), \(\gamma=0.006\) & \multirow{3}{*}{1.64} \\  & \(\epsilon=0.016666\), \(\theta=0.0006\), & \\ \cline{1-1}  & \(\kappa=0.0001666\) & \\ \hline \multirow{3}{*}{CSIRV\_2} & \(\alpha=0.18\), \(\beta=0.06\), \(\gamma=0.006\) & \multirow{3}{*}{14.01} \\  & \(\varepsilon=0.004167\), \(\theta=0.0006\), & \\ \cline{1-1}  & \(\kappa=0.00004167\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cumulative Viral load estimations for different scenarios of the CSIRV model (t = 365 days). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Rates** & **Cumulative Viral Load** \\ \hline \multirow{3}{*}{CSIRV} & \(\alpha=0.18\), \(\beta=0.06\), \(\gamma=0.006\) & \multirow{3}{*}{4.32} \\  & \(\delta=0.001\), \(\mu=0.0001\), \(\zeta=0.06\), & \\  & \(\iota=0.006\) & \\ \hline \multirow{3}{*}{CSIRQ\_1} & \(\alpha=0.18\), \(\beta=0.06\), \(\gamma=0.006\), & \multirow{3}{*}{3.02} \\  & \(\delta=0.002\), \(\mu=0.0002\), \(\zeta=0.06\), & \\ \cline{1-1}  & \(\iota=0.006\) & \\ \hline \multirow{3}{*}{CSIRQ\_2} & \(\alpha=0.18\), \(\beta=0.06\), \(\gamma=0.006\), & \multirow{3}{*}{4.99} \\  & \(\delta=0.0005\), \(\mu=0.00005\), \(\zeta=0.06\), & \\ \cline{1-1}  & \(\iota=0.006\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cumulative Viral load estimations for different scenarios of the CSIRQ model (t = 365 days). than the basic one. The resultant viral load is 0.09, demonstrating the pivotal role that social distancing can play in diminishing the infected subpopulation. In the CSIRQV_2 scenario, there is no interaction between the S and Q compartments (\(\delta\) = 0 and \(\mu\) = 0). In this case, the viral load obtains a value of 0.27, close to the corresponding value of the basic scenario. It is assumed, therefore, that social distancing measures on the susceptible individuals have a minor impact on the overall viral load in a comprehensive scenario also incorporating vaccination. This research attempts to provide a generalized approach that may be of interest in studying epidemics and in providing insights with regard to the effect that various policies, such as vaccination and quarantine, may have on the cumulative viral load. The proposed approach allows for the consideration of combined policies and different levels of vaccination compliance and quarantine stringency. It also allows for studying the crucial role of time parameters, such as the timepoint of adoption of constraining measures, or other factors, such as vaccination coverage and the emergence of vaccine resistance, that may play in epidemic control. By incorporating realistic statistical data and/or data from molecular biology into the model, the proposed approach may provide a useful decision-making tool for mitigating or suppressing a viral epidemic or pandemic.