## 2 The SE(Is)(Ih)(Iicu)AR Epidemic Model
Each solution is expressed in closed form as follows:_  
\[S(t)=e^{-\int_{0}^{t}\Phi(\tau)d\tau}S_{0}+\int_{0}^{t}e^{-\int_{t}^{t}\Phi( \xi)d\xi}(b_{1}+\eta\ R(\tau))\ d\tau;\ \forall t\in\mathbf{R}_{0+} \tag{12}\]  
\[E(t)=e^{-(b_{2}+\gamma)t}E_{0}+\int_{0}^{t}e^{-(b_{2}+\gamma)(t-\tau)}\Psi( \tau)S(\tau)d\tau;\ \forall t\in\mathbf{R}_{0+} \tag{13}\]  
_from (1) and (2), where_  
\[\Phi(t)=\Psi(t)+b_{2}+k_{V}(t);\ \Psi(t)=\beta(t)(I_{s}(t)+\beta_{nr}I_{h}(t)+ \beta_{nr}A(t)) \tag{14}\]  
\(\forall t\in\mathbf{R}_{0+}\)_. Also, one gets (3)-(6) that_  
\[I_{s}(t)=e^{-(b_{2}+\tau_{0})t}I_{s0}+\gamma p_{s}\int_{0}^{t}e^{-(b_{2}+\tau_ {0})(t-\tau)}E(\tau)d\tau;\ \forall t\in\mathbf{R}_{0+} \tag{15}\]  
\[I_{h}(t)=e^{-(b_{2}+\alpha+\tau_{0})t-\int_{0}^{t}k_{T}(\tau)d\tau}I_{h0}+ \gamma p_{h}\int_{0}^{t}e^{-(b_{2}+\alpha+\tau_{0})(t-\tau)-\int_{t}^{t}k_{T \tau}(\xi)d\xi}E(\tau)d\tau;\ \forall t\in\mathbf{R}_{0+} \tag{16}\]  
\[I_{icu}(t)=e^{-(b_{2}+\alpha+a_{i\alpha\tau}+\tau_{0})t-\int_{0}^{t}k_{T\tau \tau}(\tau)d\tau}I_{h0}+\gamma p_{icu}\int_{0}^{t}e^{-(b_{2}+\alpha+\tau_{0})( t-\tau)-\int_{t}^{t}k_{T\tau}(\xi)d\xi}E(\tau)d\tau;\forall t\in\mathbf{R}_{0+} \tag{17}\]  
\[A(t)=e^{-(b_{2}+\tau_{0})t}A(0)+\gamma p_{a}\int_{0}^{t}e^{(b_{2}+\tau_{0})(t -\tau)}E(\tau)d\tau;\ \forall t\in\mathbf{R}_{0+} \tag{18}\]  
\[R(t)=e^{-(b_{2}+\eta)t}R_{0}+\int_{0}^{t}e^{(b_{2}+\eta)\tau}\Omega(\tau)d \tau;\ \forall t\in\mathbf{R}_{0+} \tag{19}\]  
_where_  
\[\Omega(t)=\tau_{0}(I_{s}(t)+I_{h}(t)+I_{icu}(t)+A(t))+k_{T}(t)\ \ I_{h}(t)+k_{Ticu}(t)\ \ I_{ icu}(t)+k_{V}S(t);\ \forall t\in\mathbf{R}_{0+}\]  
_Now, one has from (7) that \(S_{0}\geq 0\Rightarrow S(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\) and since \(E_{0}\geq 0\) and \(S(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\) then \(E(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\) from (8). Then, it follows from (10)-(12) that \(I_{s}(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\) since \(E(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\) and \(I_{s0}\geq 0\), \(I_{h0}\geq 0\) and \(A_{0}\geq 0\). Finally, it follows from (13) that \(R(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\) since \(R_{0}\geq 0\) and \(\Omega(t)\geq 0\); \(\forall t\in\mathbf{R}_{0+}\). The proof is complete._  
Output Controllability Concerns and Basic Control Design Algorithms for Targeting Intensive Care Unit Levels  
This section and the next one describe the output controllability properties for the outputs being defined either by any of the two considered hospitalized subpopulations, or by their sum or, even, as distinct output components of a measurable output vector of dimension two. It is firstly argued as introductory motivation that a complete state controllability is not possible since all the state components cannot be jointly driven to prescribed suitable values. Therefore, the targeted components to prescribed suitable values are defined as the output of the system which has a smaller dimension than the state. The particular definition of the output to be considered depends on the hospital constraints to be satisfied in terms of beds availability, specialized staff needs and/or availability of other technical means, like, for instance, number of respirators. The hospital constraints are defined in terms of upper bounds of hospitalized infectious to be satisfied. The controls to be monitored from the output controllability issues are the vaccination on the susceptible patients, the treatments on the hospitalized ones with no intensive care needs, the treatment on the hospitalized ones staying in the intensive care unit. Eventually, the transmission rate is also a control which can be monitored by public interventions like, for instance, use of masks, limitation of numbers of attendees to meetings, rules on transportation, isolations of detected infectious or potential susceptible, partial or total quarantines, etc.