## 2 The SI (RC) Epidemic Model  
### The Controlled Epidemic Model  
Epidemic models are often based on differential, difference or hybrid differential-difference equations (see, for instance, [33] and References there in). In particular, we consider a modified SI epidemic model subject to linear feedback vaccination of the susceptible subpopulation and antiviral/antibiotic treatment of the infectious subpopulation and an incorporated recovered compensatory subpopulation. Such a recovered subpopulation is initialized to zero and evolvesaccording to the vaccinated and treated subpopulations which are transferred from the susceptible and infectious subpopulations. \[\dot{S}(t)=(\gamma-\beta S(t))I(t)-k_{V}S(t), \tag{1}\]  
\[\dot{I}(t)=(\beta S(t)-\gamma-k_{T})I(t), \tag{2}\]  
\[\dot{R}(t)=k_{V}S(t)+k_{T}I(t), \tag{3}\]  
\(\forall t\in\mathbf{R}_{0+}\) with \(S(0)=S_{0}\geq 0\), \(I(0)=I_{0}\geq 0\), such that \(S_{0}+I_{0}>0\), and \(R(0)=R_{0}=0\) with \(S(t)\) and \(I(t)\) are the susceptible and infectious subpopulations and \(R(t)\) is the recovered one from the control interventions which are vaccination \(V(t)=-k_{V}S(t)\); \(\forall t\in\mathbf{R}_{0+}\) (the set of nonnegative real numbers) on the susceptible subpopulation and treatment \(T(t)=-k_{T}I(t)\); \(\forall t\in\mathbf{R}_{0+}\) on the infectious subpopulation which have respective control gains \(k_{V}\geq 0\) and \(k_{T}\geq 0\). If any of both gains is zero then the corresponding control intervention is not applied. The parameters \(\beta>0\) and \(\gamma>0\) are, respectively, the transmission rate and the recovery rate. The above-mentioned recovered subpopulation is introduced simply for coherency purposes to complete the total population \(N(t)=S(t)+I(t)+R(t)=N(0)=N_{0}\); \(\forall t\in\mathbf{R}_{0+}\) under intervention controls and it is omitted otherwise. Thus, in the case that either \(k_{V}\) or \(k_{T}\) or both are nonzero then the recovered subpopulation is incorporated to give coherency to the total population versus the subpopulations and the model is referred to with the acronym SI (SI-RC) to reflect the feature that the recovered subpopulation is a compensatory one generated from a vaccination intervention control (on the Susceptible) and a treatment control intervention (on the Infectious) in the most general case. Note the following motivating features for the use of such controls:  
* The vaccination and treatment controls are generated via linear feedback laws instead through constant efforts. In this way, the intensities of the controls vary according to the levels of the respective subpopulations. * Note from inspection of (1)-(3) that the equilibrium values of the susceptible and the infectious are got by zeroing their time-derivatives are:
* \(I*=S*=0\) if and only if \(k_{V}>0\) with all the populations becoming asymptotically recovered. So, the disease-free equilibrium exists with zero equilibrium susceptible subpopulation only under vaccination control;
* The endemic equilibrium point satisfies \(I*=\frac{k_{V}}{k_{T}}\frac{\gamma+k_{T}}{\beta}\) which does not exist, since it is negative, if \(k_{V}>0\);
* If \(k_{V}=0\) and \(k_{T}>0\) then the endemic equilibrium \(I*>0\) does not exist since the susceptible equilibrium numbers should jointly satisfy the incompatible constraint \(S^{*}=\frac{\gamma}{\beta}=\frac{\gamma+k_{T}}{\beta}\) in order that both time derivatives could be zeroed. The simulated examples of Section 4 corroborate the above observations and, furthermore, show the advantages of using supervised time-varying control gains to improve the transient behavior of the subpopulations through time in terms of faster convergence and fewer maximum levels of infection. Two particular cases of the model with recovered compensatory subpopulation are: a) the SI(S-RC) with \(k_{T}=0\) and \(k_{V}>0\) so the only control is the vaccination; and b) the SI (I-RC) with \(k_{V}=0\) and \(k_{T}>0\) so the only control is the treatment.