###### Abstract
That research also mentions the feature that the time-varying coefficient defining the differential equation might potentially contain some complementary interacting environment information to make such an equation well-posed in order to feasibly describe in practice a concrete disease time evolution. This paper is devoted to the design of linear time-varying feedback vaccination controls on the susceptible and linear time-varying feedback treatment controls on the infectious. Each of those controls has its own control gain, namely, vaccination control gain and treatment control gain. A modified SI (susceptible-infectious) model is proposed with a compensatory recovered subpopulation (referred to as an SI (RC)-susceptible-infectious with compensatory recovered- epidemic model) which processes the dynamics of the vaccinated susceptible and treated infectious under zero initial conditions. The SI (RC) model is basically an SIS (susceptible-infectious-susceptible) model in the absence of controls which converts into an SIR (susceptible-infectious-recovered) one by the action of such controls. This implies that the only way for the population to become immune is through the action of vaccination and/or treatment controls. The design is basically performed as follows:  
1. A set of potential control gains is predefined (or updated through time from some starting generator initial values). Such control gains together with the model parameterization define a parallel structure where each element is processing its own data. One of the members of the structure is designed after the supervision strategy as the active controller which is really applied on the real subpopulations;
2. A supervisory loss function weighting the Infectious and Susceptible, with the option of removing one of the two, is processed by each member of the above-mentioned parallel structure. The one which is proposed in this paper, with no loss in generality against other potential alternative ones, can be interpreted as a continuous Shannon entropy [30; 31; 32] for the normalized model (that is, that with unity total population while the others representing fractions of it). Such an interpretation arises when the integral of the supervisory loss function runs from zero to infinity and only the infectious are weighted in such a function. In the general case, the supervisory loss function is simply interpreted as a nonlinear integral function which weights the infectious numbers affected by themselves as a power running on a finite sliding time horizon;
3. A higher-level decision maker decides about switching from the current active controller to another active controller candidate as soon as another member of the parallel structure exhibits a better time interval value of its associated loss function compared to that of the former active controller. Such an evaluation test on the supervisory loss functions is performed by respecting a minimum residence time of operation in the current active controller. A switching action from an active controller to another one at some time instant basically consists in changing the vaccination or treatment control gain at such a time instant. Several alternative algorithms are proposed for the selection of the active controller gains through time. The paper is organized as follows. Section 2 describes the proposed SI (RC) epidemic model under vaccination and treatment controls, in general, and its more relevant properties of solution boundedness for finite initial conditions, solution non-negativity for all time under any non-negative initial conditions and its existing equilibrium points. Section 3 develops the design of the controller gains under the processing of the evolution of the losses functions of all the elements of the parallel structure. The supervision is designed in the cases when the disease-free equilibrium point is reachable. In particular four design algorithms are designed for control supervision.