## 3 Supervised Selection of the Control Gains
### Updating Algorithms  
**Remark 1**. : _The case of \(\textsf{S}_{KY}\cap\textsf{S}_{KT}=\{0\}\) is that of absence of intervention controls with the Cartesian product \(\textsf{S}_{K_{V}}\times\textsf{S}_{K_{T}}=\{(0,\,0)\}\), trivially leading to \(\text{card}\big{(}\textsf{S}_{K_{V}}\times\textsf{S}_{K_{T}}\big{)}=1\), (both gains are identically zero for all time)._  
**Remark 2**. : _The case of \(\text{card}\big{(}\textsf{S}_{K_{V}}\times\textsf{S}_{K_{T}}\big{)}\geq 2\) allows a potential switching among different controllers if necessary._  
**Remark 3**. : _The supervisory loss functions \(\textsf{J}_{ij}(t)\) are specifically useful to deal with normalized models for their maximizing by choosing the appropriate controller gains so that the infection be correspondingly minimized._  
**Remark 4**. : _The use of normalized models is not crucial in the sense that the supervisory algorithm could be easily adapted to the un-normalized case and other loss function structures. However, such a loss function (10), with an incorporated negative sign, should be maximized to seeking for small infection force along time if the infectious subpopulation is normalized (that is, no larger than unity). Note that a particular interesting interpretation of loss functions \(\textsf{J}_{ij}(t)\) as \(t\to+\infty\) relies on the continuous Shannon's entropy of the infection if, for instance, the weighting functions satisfy \(\lambda(t)\equiv 1\) so that the objective is just to dump the infection as much as possible. For non-normalized models, another loss functions can be used which are not necessarily interpreted in an entropic context._  
**Remark 5**. : _Assume thats and \(\gamma\) are values which can belong to admissible discrete sets, that is \(\beta\in\textsf{S}_{\beta}=\big{\{}\beta_{1},\,\beta_{2},\ldots,\beta_{n_{\beta }}\big{\}}\) and \(\gamma\in\textsf{S}_{\gamma}=\big{\{}\gamma_{1},\,\gamma_{2},\ldots,\gamma_{n _{\gamma}}\big{\}}\), with at least one of them being of cardinality greater than one. An extension of Algorithm 1 can be easily dealt by redefining the set \(\mathbf{M}\) with \(n_{V}n_{T}\) normalized epidemic models \(m_{ij}=m_{ij}\big{(}p_{ij}\big{)}\) of the same structure as (1)-(3) parameterized by the parameter \(\text{vector}_{ij}=\big{(}\beta,\,\gamma\,,k_{V_{i}}\,,k_{T_{j}}\big{)}\in \textsf{S}_{\beta}\times\textsf{S}_{\gamma}\times\textsf{S}_{K_{V}}\times \textsf{S}_{K_{T}}\). If \(\beta(t)\) and \(\gamma(t)\) are time functions associated with seasonality or different conditions of the environments, they can first to be discretized to define similar discrete sets of values. In those cases, we can modify the dimensionality of the set of models \(\mathbf{M}\) of Algorithm 1 under the guidelines of Algorithm 2 below which includes the combinations of parameters of both the uncontrolled epidemic model and the set of potential controllers in the parallel structure:_  
```
1 Preparatory Step 1: Define two sets of constant vaccination and treatment control gains \(\textsf{S}_{KY}=\big{\{}k_{V_{1}},\,k_{V_{2}},\ldots,\,k_{V_{n_{V}}}\big{\}}\) and \(\textsf{S}_{KT}=\big{\{}k_{T_{1}},\,k_{T_{2}},\ldots,\,k_{T_{n_{T}}}\big{\}}\), respectively, with the constraints \(k_{V_{i}}\in\big{[}0\,,\,k_{V}\big{]}\); \(\forall i\in\overline{n}_{V}=\{1,2,\ldots,n_{V}\}\) and \(k_{T_{j}}\in\big{[}0\,,\,\overline{k}_{T}\big{]}\), \(\forall i\in\overline{n}_{T}=\{1,2,\ldots,\ldots,n_{T}\}\) such that \(\text{card}\big{(}\textsf{S}_{K_{V}}\times\textsf{S}_{K_{T}}\big{)}\geq 2\), so that there are at least two different choices of pairs of controlings guaranteed by the constraints: \(n_{V}+n_{T}\geq 3\) if \(\textsf{S}_{KY}\
eq\{0\}\) and \(\textsf{S}_{KT}\
eq\{0\}\), \(n_{V}\geq 2\) if \(n_{T}=0\) (that is, if \(\textsf{S}_{KY}\
eq\{0\}\) and \(\textsf{S}_{KT}=\{0\}\)), \(n_{T}\geq 2\) if \(n_{V}=0\) (that is, if \(\textsf{S}_{KY}=[0]\) and \(\textsf{S}_{KT}\
eq\{0\}\)).