## 4 Control Authority and Regulation  
This section discusses how to apply control strategies to the mathematical model, in Equation (8), and predict the best actions and their influence to slow down the spread rate of the coronavirus. There are two control schemes that are proposed to interact with the population model and provide system stability by reducing the number of infected cases per day, as shown in Figure 12. First, the high level control represents the effectiveness of any potential vaccine to avoid infection of COVID-19 or potential medication to treat infected cases. Both control inputs are tuned to decrease the size of the infected population to the desired reference, zero infection. Second, the low level control represents the actions that individuals should follow to reach to the same desired reference. Since there is no vaccine nor medication that has been proven to protect or treat individuals from this disease, let us discuss how other model parameters provide control on the spread rate. In the dynamic model, the values of \(\gamma\), \(\eta\), and \(\delta\) are uncontrollable and are normally estimated using the statistical data. However, the values of \(\alpha^{H}\), \(\alpha^{I}\), and \(\beta\) are controllable. Working to minimize the values of \(\alpha^{H}\), \(\alpha^{I}\), and \(\beta\) improves the situation and results in less number of infected population. Figure 11: The Extended Kalman Filter (EKF) estimated parameters used for the predictions. Figure 12: Low level and high level control structure considered in this paper. Optimum case is \(a^{H}=\alpha^{I}=\beta=0\). Let us simulate the dynamics by tuning these variables and investigate the effect in the healthy and infected populations. Based on the statistical data available, we ran experimental simulation for five test cases described below and shown in Table 1 and Figure 13. We assume that country (A) has 100 people total population with 50% infected with COVID-19. Case 1 investigates the impact of the infection rate \(\beta\) on the population (e.g., the country imposes a travel ban and no immigration allowed). As shown in Figure 13a, country (A) could reach to zero infected cases in less than two weeks if it controlled the infection rate (\(\beta=0\)); e.g., social distancing. However in case 2, the country might need about three months to reach a steady state zero infection if it allowed infection rate and immigration to continue as is, as shown in Figure 13b. Case (3) shows the extreme condition when higher infection rate is allowed. Thus, the majority of the population got infected in the first week before the entire population dropped to small numbers due to the death rate, as shown in Figure 13c. Note that we assumed that any recovered case could still get the infection if it was exposed to infected individuals. Case 4 investigates the infected population changes to different values of the infection rate. It is clear in Figure 13d that the infected population increased with the increase of the infection rate. The elevation reached the peak at earlier week, when the entire community got infected. Then, the population started dropping down due to the increase in death rate. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case** & \(\mathbf{\alpha^{H}}\) & \(\mathbf{\alpha^{I}}\) & \(\mathbf{\beta}\) & \(\mathbf{\gamma}\) & \(\mathbf{\eta^{H}}\) & \(\mathbf{\eta^{I}}\) & \(\mathbf{\xi^{H}}\) & \(\mathbf{\xi^{I}}\) \\ \hline
1 & 0 & 0 & 0 & 0.25 & 0.01 & 0 & 0.001 & 0.05 \\
2 & 0.001 & 0.001 & 0.005 & 0.25 & 0.01 & 0 & 0.010 & 0.05 \\
3 & 0.001 & 0.001 & 0.050 & 0.25 & 0.01 & 0 & 0.010 & 0.05 \\
4 & 0 & 0 & vary & 0.25 & 0.05 & 0.025 & 0.010 & 0.05 \\
5 & 0 & 0.100 & control & 0.25 & 0.01 & 0.010 & 0.010 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test cases used to evaluate the model. Figure 13: Experimental results based on the assumed 100 subjects evaluated for this study.