## 3 Supervised Selection of the Control Gains
### Updating Algorithms
These values have been inspired from [42] while the parameter \(\beta\) has been increased in order to attain and endemic equilibrium point in the presented simulation examples. The model is normalized to unity and the initial conditions are \(S(0)=0.95\) and \(I(0)=0.05\), implying that the Susceptible are initially the 95% of the total population while the Infectious represent the 5%. In the absence of any external control action, the evolution of the system is displayed in Figure 1. As it can be observed in Figure 1, the compensatory immune subpopulation remains at zero at all time while the Susceptible and infectious converge to the endemic equilibrium point, characterized by an asymptotic nonzero infectious subpopulation. This situation corresponds to the case when the infection is persistent, since there exist a percentage of infectious individuals at the equilibrium, and external counteracting measurements have to be considered in order to eradicate the infection. It is observed in Figure 1 that the infection is persistent since there exists a percentage of infectious individuals at the equilibrium. To this end, vaccination and treatment are introduced in the model and orchestrated following the guidelines provided in Section 3. In this way, consider the sets \(S_{KV}=S_{KT}=\left\{ \begin{array}{cccc}0&0.001&0.0035&0.0075&0.01\end{array} \right\}\) of potential feedback gains of vaccination and treatment, respectively. The initial value is zero, representing the case when no control action is taken, while the maximum value of these sets might be selected according to the maximum available resources (regarding the number of available vaccines or treatment doses) or by trial-error work. In this case, the sets are selected as example just to show the efficiency of the algorithms. With this choice there are 25 different elements in \(S_{KV}\times S_{KT}\) leading to 25 different models, all of them parameterized by the same values for \(\beta=2.5\) and \(\gamma=0.79\). The initial conditions for all these models are \(S_{ij}(0)=0.95\) and \(I_{ij}(0)=0.05\) with \(i,j=1,2,...,5\), where \(i\) is the index denoting the position of the gain from \(S_{KV}\) while \(j\) is the index denoting the position of the gain from \(S_{KT}\). If the Algorithm 1 of Section 3 with \(T_{min}=2\) days and constant \(\lambda=0.5\) is used, the evolution of the system is given in Figure 2a while the indexes that define the employed gains from the sets \(S_{KV}\) and \(S_{KT}\) are showed in Figure 2b,c, respectively. Figure 2d shows the evolution through time of the system under constant control gains of \(k_{V}=k_{T}=0.001\). Figure 1: Evolution of the populations for \(\beta=2.5\) and \(\gamma=0.79\) days \({}^{-1}\) in the absence of external control actions, leading to a persistent infection. It can be observed in Figure 2a that the control actions driven by Algorithm 1 are able to become the whole population immune asymptotically, removing the infection and making the Susceptible and the Infectious to vanish. Moreover, the evolution of the system under constant control gains of \(k_{V}=k_{T}=0.001\) (the minimum non-zero values of the sets \(S_{KV}\) and \(S_{KT}\)) is displayed in Figure 2d while the evolution of the system when \(k_{V}=k_{T}=0.035\) is depicted in Figure 2e. It can be concluded that the selection of some constant values for the gains might lead to slower vanishing rates for the infectious, as it is observed from Figure 2a,d, while it might also lead to larger infective periods,

Figure 2: Performance of the system for \(\beta=2.5\) and \(\gamma=0.79\) days \({}^{-1}\) when vaccination and treatment are included in the model. The control parameters are \(S_{KV}=S_{KT}=\{0\quad 0.001\quad 0.0035\quad 0.0075\quad 0.01\}\), \(T_{min}=2\) days and \(\lambda=0.5\). The vaccination and treatment efforts are selected online via Algorithm 1. (**a**) Evolution of model’s populations. The infection is asymptotically removed while the whole population becomes immune asymptotically.