## 2 The SE(Is)(Ih)(Iicu)AR Epidemic Model
\right)^{T}\]  
_where_  
\[S^{*}_{df}=\frac{b_{1}+\eta R^{*}_{df}}{b_{2}+\eta+k_{V0}}=\frac{b_{1}(b_{2}+ \eta)}{b_{2}(b_{2}+\eta+k_{V0})};\;R^{*}_{df}=\frac{k_{V0}S^{*}_{df}}{b_{2}+ \eta}=\frac{k_{V0}b_{1}}{b_{2}(b_{2}+\eta+k_{V0})} \tag{11}\]  
_which is directly obtained by zeroing (1)-(7) together with the subpopulations of the exposed and all the subpopulations of the infectious. It turns out that the total population at the disease-free equilibrium point becomes \(N^{*}_{df}=S^{*}_{df}+R^{*}_{df}=\frac{b_{1}}{b_{2}}\)._  
The control design problem is being stated as an output controllability problem for some appropriate \"output\" to be then defined being of less dimension than the state \(x(t)\in\boldsymbol{R}^{n}\). It is firstly discussed with a very simple reasoning that the epidemic model (1)-(7) is not (state)-controllable. **Theorem 1**. : _Assume that \(\rho_{V}\)\((\mathbf{t})=1\); \(\forall t\in\boldsymbol{R}_{0+}\). Then, the SE(Is)(Ih)AR epidemic model (1)-(7) is not controllable through a control function of the form \(u(t)=\left(\beta_{c}(t)\;,\!k_{V}(t)\;,\!k_{lh}(t)\;,\!k_{lku}(t)\right)^{T}\) and so by any reduced control with less number of components. In particular, it is not controllable under vaccination and treatment controls nor by any combination of them._  
Proof. : Considered the linearized version of (1)-(7) around the disease-free equilibrium point. Since all the infective subpopulations (that is, the exposed and all the infectious subpopulations) are zero at the disease- free equilibrium point, it suffices to consider the subsystem of susceptible-recovered component in (8)-(10) with \(\rho_{V}(t)\equiv 1\). The remaining indicators can be zeroed, with no loss in generality, since they are irrelevant in the control matrix \(B(x(t))\) since they multiply to infective variables having zero value at the disease-free equilibrium point. It suffices to apply the Popov-Belevitch-Hautus eigenvalue controllability test to the \(S-R\) linearized subsystem around the disease-free equilibrium point, that is, to evaluate:  
\[rank[A_{SR}-\lambda I_{2}\;\;B_{SR}]=rank\left[ \begin{array}{cccc}A_{11}- \lambda&A_{17}&B_{12}\\ A_{71}&A_{77}-\lambda&B_{72}\end{array} \right]=rank\left[ \begin{array}{cccc}-b _{2}-\lambda&A_{17}&-S^{*}_{df}\\ A_{71}&-b_{2}-\eta-\lambda&S^{*}_{df}\end{array} \right]\]  
at the eigenvalues \(\lambda\in\left[-b_{2}\;,\;-(b_{2}+\eta)\right]\) of \(A_{SR}=\left[ \begin{array}{cc}A_{11}&A_{17}\\ A_{71}&A_{77}\end{array} \right]\) where \(B_{SR}=\left[ \begin{array}{c}B_{12}\\ B_{72}\end{array} \right]\) and to test if for at least one of them such a matrix is rank-defective. Since \(A_{17}=\eta\) and \(A_{71}=0\), since \(\rho_{V}(t)\equiv 1\), then if \(\lambda=-b_{2}\) then \(rank[A_{SR}-\lambda I_{2}\;\;B_{SR}]=rank\left[ \begin{array}{cccc}0&\eta&- S^{*}_{df}\\ 0&-\eta&S^{*}_{df}\end{array} \right]=1<2\) for any \(S^{*}_{df}\) (intuitively \(S^{*}_{df}\) cannot be prefixed arbitrarily if \(R^{*}_{df}\) is prefixed and vice-versa). Note that this linearized system is time-invariant, and the above test is then a necessary and sufficient condition for the controllability of such a linearized system. So, the whole non-linear system (8)-(10) cannot be driven by any control to the disease-free equilibrium point in any finite time so that it is uncontrollable since its subsystem \(S-R\) is uncontrollable. The following result relies on the solutions of the proposed SE(Is)(Ih)(Iicu)AR epidemic model and it will be subsequently also used to prove their non-negativity under arbitrary non-negative initial conditions:  
**Theorem 2**. : _Each solution of the SE(Is)(Ih)(Iicu)AR model (1) to (7) is uniquely defined and it is non-negative for all time for its corresponding given non-negative initial condition and any given vaccination and antiviral controls\(V(t)=k_{V}(t)S(t)\), \(T_{h}(t)=k_{Th}(t)I_{h}(t)\), \(T_{icu}(t)=k_{Ticu}(t)I_{icu}(t)\) of gains \(k_{V}\), \(k_{T},k_{Ticu}:\mathbf{R}_{0+}\rightarrow\mathbf{R}_{0+}\).