## 2 The SE(Is)(Ih)(Iicu)AR Epidemic Model
The proposed SE(Is)(Ih)(Iicu)AR model is the following one:  
\[\dot{S}(t)=b_{1}-[\ b_{2}+\beta(t)(I_{s}(t)+\beta_{br}I_{h}(t)+\beta_{icur}I_{ icu}(t)+\beta_{ur}A(t))\ \ +k_{V}(t)]S(t)+\eta R(t) \tag{1}\]  
\[\dot{E}(t)=-(b_{2}+\gamma)\ \ E(t)+\beta(t)(I_{s}(t)+\beta_{ir}I_{h}(t)+\beta_{ icur}I_{icu}(t)+\beta_{ar}A(t))S(t) \tag{2}\]  
\[\dot{I}_{s}(t)=-(b_{2}+\tau_{0})\ \ I_{s}(t)+\gamma p_{s}E(t) \tag{3}\]  
\[\dot{I}_{h}(t)=-(b_{2}+\alpha+\tau_{0}+k_{Tin}(t))\ \ I_{h}(t)+\gamma p_{h}E(t) \tag{4}\]  
\[\dot{I}_{icu}(t)=-(b_{2}+\alpha+\alpha_{icu}+\tau_{0}+k_{Ticu}(t))\ \ I_{icu}(t)+\gamma p_{icu}E(t) \tag{5}\]  
\[\dot{A}(t)=-(b_{2}+\tau_{0})\ \ A(t)+\gamma p_{a}E(t) \tag{6}\]  
\[\dot{R}(t)=-(b_{2}+\eta)R(t)+\tau_{0}(I_{s}(t)+I_{h}(t)+I_{icu}(t)+A(t))+k_{Th }(t)\ \ I_{h}(t)+k_{Ticu}(t)\ \ I_{icu}(t)+k_{V}S(t) \tag{7}\]  
\[\forall t\in\mathbf{R}_{0+}=\{z\in\mathbf{R}:z\geq 0\}\ \text{with initial conditions}\]  
\[S(0)=S_{0},E(0)=E_{0},I_{s}(0)=I_{s0},I_{h}(0)=I_{h0},I_{cu0}(0)=I_{icu0},A(0)=A _{0},\ R(0)=R_{0}\]  
subject to \(\min(S_{0}\,\ E_{0}\,\ I_{s0},I_{h0}\,\ I_{icu0}\,\ A_{0}\,\ R_{0}\ )\geq 0\), where the parameters are described below:  
* \(b_{1}\) is the recruitment rate,
* \(b_{2}\) is the natural average death rate,
* \(\beta(t)\), \(\beta_{br}\beta(t)\), \(\beta_{icu}\beta(t)\), \(\beta_{ur}\beta(t)\) are the transmission rates to the susceptible from the respective slight (un-hospitalized) symptomatic infectious, serious (hospitalized) symptomatic infectious, intensive care unit hospitalized infectious and asymptomatic infectious subpopulations,
* \(\eta\) is a parameter such that \(1/\eta\) is the average duration of the immunity period reflecting a transition from the recovered to the susceptible,
* \(\gamma\) is the transition rate from the exposed to all (i.e., both symptomatic and asymptomatic) infectious,
* \(\alpha\) is the average extra mortality associated with the symptomatic seriously infectious subpopulation,
* \(\alpha_{icu}\) is the average extra mortality over the above one registered in the subpopulation of the intensive care unit,
* \(\tau_{0}\) is the natural immune response rate for the whole infectious subpopulation (i.e., \(A+I\)), respectively, \(p_{s}\), \(p_{h}\), \(p_{icu}\), \(p_{a}\) are the fractions of the exposed which become slight symptomatic infectious, serious symptomatic infectious and asymptomatic infectious, respectively, whose sum equalizes unity. * \(1/\mu\) is the average period of infectiousness after death,
* \(V(t)=k_{V}(t)S(t)\) and \(T_{h}(t)=k_{Th}(t)I_{h}(t)\), \(T_{icu}(t)=k_{Ticu}(t)I_{icu}(t)\) are, respectively, the vaccination and antiviral treatment linear feedback controls on the susceptible, (non-intensive care) hospitalized infectious and intensive care hospitalized infectious, respectively, of feedback gains \(k_{V}\), \(k_{Th}\), \(k_{Ticu}:\mathbf{R}_{0+}\to\mathbf{R}_{0+}\). The vaccination control is applied to the susceptible individuals while the treatment controls to the hospitalized with no intensive care and those in the intensive care unit are, in general different and have different degrees of intensity. Note that the set of coupled first-order differential Equations (1)-(7) can be compacted into a 7-th ordinary differential equation with forcing terms, which are the vaccination and treatment controls. In this paper, it is also considered that the transmission rate can be partially monitored as a control variable. The above set of seven differential equations, with the controls being separately described as forcing terms, is given later on in (8) under the general description of a dynamic system. However, it can be also pointed out that the analysis of the above mentioned higher-order equation is more complex that its equivalent decomposition into a set of first-order ones. **Remark 1**. : _Note that, in the above parameterization, all the parameters are positive while it is assumed that \(\beta(t)\) can be time-varying, in general.