## 6 Worked Numerical Examples

Table 3: Available hospital resources and target values. **Example 2**. : _Dynamics of the controlled model when Programme 2 is employed and the transmission rate is constant (i.e., it is not a manipulated control variable)._  
Now, Programme 2 given in Section 4 is used with no iterations (i.e., \(r=1\)) and \(\lambda_{k}\equiv 1\) to design the external counteracting measurements in order to avoid the spread of COVID-19 while particularly guaranteeing that the number of individuals in ICU does not overflow the maximum available resources. Figure 2 depicts the evolution of all subpopulations when this algorithm is employed with the transmission rate being kept constant, that is, it is not a control variable to be designed. Figures 3 and 4 display the control gains and the control effort, respectively. The peak of hospitalized and ICU individuals reach the values of 25,830 and 1431, respectively. Figure 1: Evolution of all the populations of the model in the absence of external control actions. Figure 2: Evolution of the populations of the model when Programme 2 is used. **Example 3**. : _Dynamics of the controlled model when Programme 2 is employed and the transmission rate is indeed a manipulated control variable._  
Figure 4: Control efforts corresponding to each action. Figure 3: Control gains provided by Programme 2. It is observed a substantial reduction in the number of hospitalized cases in Figure 5 with respect to the levels of Figure 1 while Figure 2 shows that the population becomes immune faster than in the absence of control actions, as it could have been intuitively expected. **Example 4**. : _Dynamics of the controlled model when Programme 6 is employed and the transmission rate is constant (i.e., it is not a manipulated control variable)._  
Figures 8-11 display some obtained results referred to the use of Programme 6 without iterations. The death toll is also reduced as concluded by comparing Figures 2 and 8 regarding the evolution of the total population. Figure 9 displays the vaccination and treatment control actions needed. The great improvement in the disease incidence is achieved at the expense of a high effort in vaccination as it is concluded from Figures 4 and 10. As vaccination increases, the number of hospitalized cases declines. As claimed in Section 3, the use of vaccination improves the behavior of the coronavirus spread. The peak of hospitalized and ICU individuals reach the values of 11,932 and 651, respectively. Figure 7: Control efforts when Programme 2 is employed and \(\beta\) is a control variable. Figure 8: Evolution of all population when Programme 6 is employed with \(\beta\) being a control variable. Figure 9: Control gains when Programme 6 is employed. Figure 11: Evolution of all population when Programme 6 is employed with \(\beta\) being a control variable. Figure 10: Control actions when Programme 6 is employed. **Example 5**. : _Dynamics of the controlled model when Programme 6 is employed and the transmission rate is indeed a manipulated control variable._  
Finally, Figures 11-13 show the evolution of the system when Programme 6 is used, with \(\lambda_{4}\equiv 0.7\) and \(\lambda_{5}\equiv 1\) and without iteration, while beta is now a control variable. The value of \(N(0)\beta\) is constrained to the interval [0.5, 1] in order to capture the fact that the transmission rate cannot be reduced arbitrarily. The saturated lower bounds of the remaining control variables are fixed to zero. The proposed algorithm designs the control parameters so as to achieve the hospitalized and ICU constraints as it is observed in Figure 1:0 As it is deduced by comparing Figures 9 and 12, the vaccination effort reduces when the transmission factor beta is also employed as a control parameter. In this way, social distance, use of face masks, etc. can be used as effective control measurements, of course, at the expense of a very atypical lifestyle, as an alternative to vaccination in order to control the virus spreading.