## 2 The SI (RC) Epidemic Model
### About the Positivity and Equilibrium Points  
The following result gives directly the analytic solution of the state-trajectory solution of (1) and it also discusses its positivity and boundedness properties:  
**Theorem 1**. : _The following properties hold:_  
_The state-trajectory solution of (1)-(3) is:_  
\[S(t)=e^{-\int_{0}^{t}(\beta I(\tau)+k_{V})dt}S_{0}+\gamma\int_{0}^{t}e^{-\int_{ \tau}^{t}(\beta I(\sigma)+k_{V})d\sigma}I(\tau)d\tau, \tag{4}\]  
\[I(t)=e^{\int_{0}^{t}(\beta S(\tau)-\gamma-\gamma_{T})dt}I_{0},\]  
\[=e^{-(\gamma+k_{T})t}e^{\beta\int_{0}^{t}(e^{-\int_{0}^{t}(\beta I(\sigma)+k_{V })dt}S_{0}+\gamma\int_{0}^{t}e^{-\int_{0}^{t}(\beta I(\sigma)+k_{V})dt}I(\sigma )d\sigma)}I_{0}, \tag{5}\]  
\[R(t)=\int_{0}^{t}(k_{V}S(\tau)+k_{T}I(\tau))dt,\]  
\[=k_{V}\int_{0}^{t}(e^{-\int_{0}^{t}(\beta I(\sigma)+k_{V})dt}S_{0}+\gamma\int _{0}^{t}e^{-\int_{0}^{t}(\beta I(\sigma)+k_{V})dt}I_{(\sigma)d\sigma})dt+k_{T} I_{0}\int_{0}^{t}e^{-(\gamma+k_{T})t}\beta\int_{0}^{t}e^{-\int_{0}^{t}(\beta I( \sigma)+k_{V})dt}S_{0}+\gamma\int_{0}^{t}e^{-\int_{0}^{t}(\beta I(\sigma)+k_{V })dt}I_{(\sigma)d\sigma}dt}, \tag{6}\]  
_for all \(t\in\mathbf{R}_{0+}\)._  
_\(0\leq S(t)\leq\overline{S}<+\infty\), \(0\leq I(t)\leq\overline{I}<+\infty\), \(0\leq R(t)=N_{0}-S(t)-I(t)\leq\overline{R}<+\infty\) for all \(t\in\mathbf{R}_{0+}\) and any given initial conditions \(N_{0}\geq S_{0}\geq 0\), \(I(0)=I_{0}=N_{0}-S_{0}\geq 0\) and \(R(0)=R_{0}=0\). As a result, the epidemic model is globally stable for any given finite and non-negative initial conditions._  
_There exists a critical transmission rate \(\beta_{c}>0\) such that \(I(t)\to 0\) and \(S(t)+R(t)\to N_{0}\) as \(t\to+\infty\) for any \(\beta\in[0\), \(\beta_{c})\) and any given initial conditions \(N_{0}\geq S_{0}\geq 0\), \(I(0)=I_{0}=N_{0}-S_{0}\geq 0\) and \(R(0)=R_{0}=0\)._  
Proof. : The solutions (4), (5) and (6) are directly obtained from (1)-(3) while (5) and (6) follow from (5) and (6) by using (4) in them. Property (i) has been proved. On the other hand, a direct inspection of (4)-(6) concludes that the state trajectory solution is non-negative for all time under any finite non-negative initial conditions since \(0\leq S(t)\), \(0\leq I(t)\), \(0\leq R(t)\) for all \(t\in\mathbf{R}_{0+}\) and any given initial conditions \(N_{0}\geq S_{0}\geq 0\), \(I(0)=I_{0}=N_{0}-S_{0}\geq 0\) and \(R(0)=R_{0}=0\). Since then the total population is non-negative and bounded for all time, that is,  
\[0\leq N(t)=S(t)+I(t)+R(t)=N(0)=N_{0}<+\infty\]  
for all \(t\in\mathbf{R}_{0+}\), one concludes also the trajectory solution component boundedness, that is, \(S(t)\leq\overline{S}<+\infty\), \(I(t)\leq\overline{I}<+\infty\), \(R(t)=N_{0}-S(t)-I(t)\leq\overline{R}<+\infty\) and any given initial conditions. Property (ii) has been proved. On the other hand, note that:  
\[f(t)=e^{-\int_{0}^{t}(\beta(\sigma)+k_{V})dt}S_{0}+\gamma\int_{0}^{t}e^{-\int_ {0}^{t}(\beta(\sigma)+k_{V})dt}I(\tau)d\tau\in L_{\infty}\cap L_{1}, \tag{7}\]  
Then, from (5)  
\[I(t)\leq e^{-(\gamma+k_{T}-\beta\overline{f})t}I_{0}=\overline{I},\forall t \in\mathbf{R}_{0+} \tag{8}\]  
where \(\overline{f}=\limsup_{t\to+\infty}\left(\sup_{0\leq\tau\leq t}f(\tau)\right)\)so that \(I(t)\to 0\) and \(S(t)+R(t)\to N_{0}\) as \(t\to+\infty\) for any given initial conditions \(N_{0}\geq S_{0}\geq 0\), \(I(0)=I_{0}=N_{0}-S_{0}\geq 0\) and \(R(0)=R_{0}=0\) and all \(\beta\in[0\), \(\beta_{c})\). Property (iii) has been proved. The attainable and unattainable equilibrium points are characterized in the next result:  
**Theorem 2**. : _The following properties hold:_  
1. _The model (_1_)-(_3_) has a disease-free equilibrium point_ \(x^{\star}_{dfe}=\left(S^{\star}_{dfe},I^{\star}_{dfe},R^{\star}_{dfe}\right)^{T} =\left(0,0,N_{0}\right)^{T}\) _irrespective of any given disease transmission rate_ \(\beta\geq 0\) _and any given recovery rate_ \(\gamma\geq 0\) _given control gains_ \(k_{V}>0\) _and_ \(k_{T}>0\)_.