## 3 Supervised Selection of the Control Gains
### Updating Algorithms
(**b**) Index representing the active vaccination control gain versus time from \(S_{KV}\), parameterizing the vaccination effort. (**c**) Index representing the active treatment control gain versus time from \(S_{KT}\), parameterizing the treatment effort. (**d**) Evolution of model’s populations under constant control gains of \(k_{V}=k_{T}=0.001\). (**e**) Evolution of model’s populations under constant gains of \(k_{V}=k_{T}=0.035\). as concluded from Figure 2a,e where it is seen that the infectious subpopulation vanishes at a later time when \(k_{V}=k_{T}=0.035\) while the infection peak is smaller. Note also that larger constant control efforts can give fewer values of the infection peak but effectively damping the infection takes a longer time. See Figure 2a versus Figure 2e. Thus, the proposed algorithm allows designing epidemic counteracting actions without the necessity of performing extensive trial-error work in order to design appropriate gains to obtain a desired vanishing rate for the infectious, fulfilling hence the Objective 1 stated in Section 3 regarding the updated selection of appropriate control gains. Figure 3 displays the evolution of the system (the time evolution of Susceptible, Infectious and Immune) for different values of the weighting factor \(\lambda=0,0.25,0.5,0.75,1\). It is observed that the evolution of the system is the same for \(\lambda=0.5,0.75,1\) being all the plots superimposed for Susceptible, Infectious and Immune. Moreover, the higher vanishing rates for both the Susceptible and the Infectious are given for these latter values. Thus, a value of \(\lambda\)\(\geq\)0.5 is recommended for the algorithm and \(\lambda=0.5\) is used along this section. Figure 3: Evolution of the model’s populations for different values of \(\lambda\) and Algorithm 1. The model’s parameters are \(\beta=2.5\) and \(\gamma=0.79\) days \({}^{-1}\) while \(T_{\min}=2\) days and \(S_{KV}=S_{KT}=\{0\quad 0.001\quad 0.0035\quad 0.0075\quad 0.01\}\). (**a**) Susceptible. (**b**) Infectious. (**c**) Immune. It is interesting to notice that the switching mechanism based on the supervisory loss function selects the highest values for the feedback control gains (implying that Figure 2b,c are the same for both gains). This happens because a higher value for the control actions provokes a higher vanishing rate for Susceptible and Infectious, making the loss function to converge to zero as well (the minimum value of the loss function). Thus, the algorithm quickly selects the highest values. A variant of Algorithm 1 may be considered in order to avoid the fast convergence of the feedback gains to the maximum available value, implying the potential fast consumption of the available resources Thus, if the sets \(S_{KV}\) and \(S_{KT}\) are ordered we may only allow the switching in steps of one. This means that if the algorithm selects a gain that is more than one step ahead from the current active one, the algorithm is only allowed to move one position in the direction of the selected gain. If this variant is added in Algorithm 1 the closed-loop trajectories depicted in Figure 4a are obtained. It can be seen in Figure 4a that the Susceptible and the Infectious vanish while all the population becomes immune asymptotically. Figure 4b,c show the indexes of the sets representing the gains at each time interval. It is observed in these figures that the convergence to the highest gain occurs at steps of size one. However, the transient behavior is worse for this case, as it can be deduced by comparing Figures 2a and 4a. This happens because of the lower values for the control actions during the transient response in this last case in comparison with the standard Algorithm 1, as shown in Figure 5. The total vaccination and treatment efforts for Algorithm 1 are given by \(\int_{0}^{40 days}V(\tau)d\tau=0.60\) and \(\int_{0}^{40 days}T(\tau)d\tau=0.40\), respectively, while the efforts when the above variant is included are given by \(\int_{0}^{40 days}V(\tau)d\tau=0.57\) and \(\int_{0}^{40 days}T(\tau)d\tau=0.41\).