## 2 The SI (RC) Epidemic Model
### About the Positivity and Equilibrium Points
Such an equilibrium point is globally asymptotically stable if_ \(\beta\in\left[0,\,\beta_{c}\right)\) _for some critical_ \(\beta_{c}>0\)_._
2. _The model (_1_)-(_3_) has an endemic equilibrium point_ \[x^{\star}_{une}=\left(S^{\star}_{une},I^{\star}_{une},R^{\star}_{une}\right)^{T }=\left(\frac{\gamma+k_{T}}{\beta},\,-\frac{k_{V}(\gamma+k_{T})}{k_{T}\beta},\,N_{0}-\frac{\gamma+k_{T}}{\beta}\left(1-\frac{k_{V}}{k_{T}}\right)\right)^{T}\] (9) _which is unattainable for any given control gain_ \(k_{V}>0\) _since it has then a negative infectious component. Such an equilibrium becomes a disease- free still unattainable point if_ \(k_{V}=0\) _and_ \(\beta\) _is large enough to satisfy_ \(\beta>\beta_{c1}=\gamma+k_{T}\)_._  
Proof. : The disease- free equilibrium point \(x^{\star}_{dfe}=\left(S^{\star}_{dfe},I^{\star}_{dfe},R^{\star}_{dfe}\right)^{T }=\left(0,0,N_{0}\right)^{T}\) exists as verified to be a solution to (1)-(3) by zeroing the time- derivatives. It is globally asymptotically stable from Theorem 1 (iii) if \(\beta\in\left[0,\,\beta_{c}\right)\) for some critical \(\beta_{c}>0\). Property (i) has been proved. On the other hand, note also by a similar procedure of zeroing the time-derivatives of (1)-(3) that \(x^{\star}_{une}\) in (9) is another equilibrium point of (1)-(3) which is unattainable for any given control gain \(k_{V}>0\) since it has then a negative infectious component. Additionally, such an equilibrium point becomes another disease-free one equilibrium point if \(k_{V}=0\) and \(\beta\) is large enough to satisfy \(\beta>\frac{\gamma+k_{T}}{N_{0}}=\frac{\gamma+k_{T}}{N^{\star}}\) since then \(0<S^{\star}_{une}=\frac{\gamma+k_{T}}{\beta}<1\) and \(R^{\star}_{une}=N^{\star}-\frac{\gamma+k_{T}}{\beta}\geq 0\). Such a point is unattainable if \(\beta\geq\beta_{c1}=\gamma+k_{T}\) as it is seen since the stability properties are independent of the initial conditions and under normalized total population \(N_{0}=N^{\star}=1\). Let be \(k_{V}=0\), \(I^{\star}_{une}=0\) and \(\beta\geq\beta_{c1}\), \(S^{\star}_{une}=\frac{\gamma+k_{T}}{\beta}\) then \(R^{\star}_{une}=1-\frac{\gamma+k_{T}}{\beta}=0\). So, from (6), one has furthermore that \(k_{T}=0\), \(\beta=\gamma\) and \(S^{\star}_{une}=\frac{\gamma}{\beta}\) but then \(I(t)\) does not converge to zero if \(I_{0}>0\) as time tends to infinity from (5.a) if \(S_{0}\geq S^{\star}_{une}\) which contradicts that \(I^{\star}_{une}=0\). One concludes that if \(\beta=\beta_{c1}\) then \(x^{\star}_{une}\) is unattainable if \(S_{0}\geq S^{\star}_{une}\). Now, if \(\beta<\beta_{c1}\) and \(S_{0}<S^{\star}_{une}\) then we still conclude that \(I(t)\) does not tend asymptotically to \(I^{\star}_{une}=0\) from (5.a). The following result is an immediate generalization of Theorems 1-2 if the control gains are time-varying instead of constant since the integrals which appear in the modified solutions (4)-(6) of the modified differential system (1)-(3) are finite under the given, and rather weak, assumptions. **Theorem 3**. : _Assume that the constant control gains \(k_{V}\) and \(k_{T}\) are replaced with bounded piecewise continuous control gains \(k_{V},k_{T}:\mathbf{R}_{0+}\rightarrow\mathbf{R}_{0+}\) in (1)-(3). Then, the following properties hold:_  
1. _A directly amended Theorem_ 1_(_i_) still holds by replacing_ \(k_{V}\to k_{V}(. )\) _and_ \(k_{T}\to k_{T}(. )\) _in (_4_)-(_6_)._
2. _Assume, in addition, that_ \(k_{V}(t)\to k^{\star}_{V}\)_,_ \(k_{T}(t)\to k^{\star}_{T}\) _as_ \(t\rightarrow+\infty\)_. Then, Theorem_ 2 _with the replacements_ \(k_{V}\to k^{\star}_{V}\) _and_ \(k_{T}\to k^{\star}_{T}\)_._