## 3 Supervised Selection of the Control Gains
### Updating Algorithms
From these values, calculated as the referred to time-integrals calculated over the control efforts which are obtained numerically from the simulation, it can be concluded that the total effort is not substantially improved when variations of one step are considered. Moreover, if vaccination and treatment resources are available it is recommendable to employ them as soon as possible since the maximum number of Infectious (peak of the infection) will be reduced and the duration of the infection will also be improved. Thus, the original description of Algorithm 1 outperforms the proposed variant. Now, we consider the typical case when the transmission rate \(\beta\) exhibits seasonal behavior. To this end, we consider the well-known [PERSON]'s model given by \(\beta(t)=\beta_{0}(1+Acos(2\pi ft))\) in order to model the transmission rate with \(\beta_{0}=2.5\), \(A=0.6\) and \(f=0.2\). The evolution of the system in the absence of external actions, when this incidence rate is used, is depicted in Figure 6, where the Immune are identically zero. As before, the infection is persistent since there always exist a positive number of infectious individuals and counteracting measurements are needed. Figure 4: 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 \(\mathcal{S}_{KV}=\mathcal{S}_{KT}=\{0\quad 0.001\quad 0.0035\quad 0.0075\quad 0.01\}\), \(T_{\min}=2\) days and \(\lambda=0.5\). The vaccination and treatment efforts are selected online via the variation of Algorithm 1 that only allows changes in steps of unity. (**a**) Evolution of model’s populations. (**b**) Index representing the active vaccination control gain versus time from \(\mathcal{S}_{KV}\), parameterizing the vaccination effort. (**c**) Index representing the active treatment control gain versus time from \(\mathcal{S}_{KT}\), parameterizing the vaccination effort. We consider now the set \(S_{\beta}\) composed of 10 elements linearly spaced between \(\beta_{min}=\beta_{0}(1-A)\) and \(\beta_{max}=\beta_{0}(1+A)\). Under these circumstances, the Algorithm 2 may be used to select the most appropriate control gains. Thus, consider the 250 models contained in \(S_{\beta}\times S_{KV}\times S_{KT}\) running in parallel. The same parameters as in the previous example are used and no variation is included in the algorithm. Figure 7 displays the evolution of the populations while the indexes corresponding to the active control gains and are given in Figure 7. From Figure 7 it can be deduced that the switching rule is able to select the appropriate gains in order to eradicate the infection asymptotically. A simulation example for Algorithm 3 is considered now. To this end, take \(k_{V}^{0}=k_{T}^{0}=0\), \(\Delta k_{V}(0)=\Delta k_{T}(0)=0.003\), the number of models is 6 for each gain set, \(\rho_{V}=\rho_{T}=0.001\) days \({}^{-1}\) and the same parameters as in the previous example. Therefore, the case of seasonal incidence rate is also considered here. The evolution of the system is depicted in Figure 8. As before, the switching mechanism selects appropriate gains in order to remove the Infectious asymptotically. However, in this case the Susceptible do not converge to zero. Figure 8 displays the control gains applied to the actual system. It can be seen that the vaccination remains zero at all time and the entropy-based approach selects only the application of treatment. This is the reason why the Susceptible do not converge to zero. Moreover, Figure 8 shows the time evolution of the treatment gain pointing out the fact that in this case the algorithm adapts the gains' values through time. Figure 5: Comparison in the control efforts between Algorithm 1 and its proposed variation. (**a**) Vaccination. (**b**) Treatment. Figure 6: Evolution of the system when a seasonal incidence rate is employed. Figure 8: Performance of the system with seasonal incidence rate and vaccination and treatment actions tuned by Algorithm 3.