## 2 The \(\mu\)-SEIR Model
### Boundaries
Thus, in order to the endemic equilibrium to exist, \(S_{end}=0\) and \(\beta_{I}I_{end}+\beta_{E}E_{end}=c>0\). From \(S_{end}=0\) and Equation (2), one has that \(0=S_{end}c-\mu E_{end}=-\mu E_{end}\). Then, from Equation (3), one gets \(I_{end}=(1-\gamma)I_{end}+\mu E_{end}=(1-\gamma)I_{end}\), which implies that \(I_{end}=0\) if \(\gamma\in(0,1)\) and \(I_{end}>0\) is arbitrary if \(\gamma=1\). As a result, \(E_{end}=I_{end}=0\) if \(\gamma\in(0,1)\) and the equilibrium point is the disease-free one. Assume then that \(\gamma=1\), \(E_{end}=0\) and \(I_{end}>0\) is arbitrary. Thus, one gets from Equation (4) that \(R_{end}=R_{end}+I_{end}\), which implies that \(I_{end}=0\), a direct contradiction to \(I_{end}>0\). As a result, it is concluded that the endemic equilibrium point does not exist. The subsequent remark discusses the fact that under proportional linear feedback vaccination with a rather weaker condition on the control gain, the disease-free susceptible level can be zeroed so that all the population results asymptotically recovered in the disease-free equilibrium. Note that the given model does not guarantee the convergence to zero of the susceptible subpopulation, that is, it does not guarantee that \(S_{dfe}=0\) which would imply as a result that \(R_{dfe}=N_{0}\); namely, it is not guaranteed that all the population would become asymptotically recovered. The reason is that the coefficient sequence in parenthesis in Equation (1) converges asymptotically to one when \(i\to\infty\) as the infection vanishes so that the solution sequence converges asymptotically to the disease-free equilibrium point. This is not very surprising since former studies indicate that a small residual permanent nonzero susceptible population can exist even if the infection asymptotically extinguishes, implying that not all the population becomes asymptotically recovered. See, for instance, [51; 52]. The reason is that the coefficient sequence of Equation (1) of general term \(a_{i}=1-(\beta_{1}I_{i}+\beta_{E}E_{i});i\in\mathbb{Z}_{0+}\), leading to \(S_{i+1}=a_{i}S_{i}\) verifies \(a_{i}\in(0,1)\) for \(i\in\mathbb{Z}_{0+}\) provided that \(\min(E_{0},I_{0})>0\) with \(\{a_{i}\}_{0}^{\infty}\to 1\) since \(\{E_{i}\}_{i=0}^{\infty}\to 0\) and \(\{I_{i}\}_{i=0}^{\infty}\to 0\). However, if a feedback vaccination law proportional to the susceptible of the form \(V_{i}=k_{i}S_{i}\) is incorporated to the dynamic Equations (1)-(4), so that \(k_{i}S_{i}\) are removed from the right-hand side of Equation (1) and added to the right-hand side of Equation (4), then the model becomes modified as follows:  
\[S_{i+1} = (1-(k_{i}+\beta_{1}I_{i}+\beta_{E}E_{i}))S_{i} \tag{18}\] \[E_{i+1} = (1-\mu)E_{i}+(\beta_{1}I_{i}+\beta_{E}E_{t})S_{i}\] (19) \[I_{i+1} = (1-\gamma)I_{i}+\mu E_{i}\] (20) \[R_{i+1} = R_{i}+\gamma I_{i}+k_{i}S_{i} \tag{21}\]  
Now, if the sequence of control gains \(\{k_{i}\}_{i=0}^{\infty}\) is positive and it satisfies furthermore the rather weak constraint \(\limsup_{i\to\infty}(k_{i}+\beta_{1}I_{i}+\beta_{E}E_{i})<1\) then \(S_{dfe}=0\) and \(R_{dfe}=N_{0}\). Note that in order that \(\limsup_{i\to\infty}(k_{i}+\beta_{1}I_{i}+\beta_{E}E_{i})<1\), it suffices that some prefixed finite  
\[N\in\mathbb{Z}_{0+},\;k_{i}<1-(\beta_{1}I_{i}+\beta_{E}E_{i});\;\forall i(\geq N )\in\mathbb{Z}_{0+}\]  
Note also that the last sufficiency-type condition does not require that the control gain converges to a limit but, if it is the case, such limit has to be less than one. The vaccination law discussed in the above Remark 2 can be even weakened by certain vaccination laws whose gains asymptotically vanish at smaller rates than exponential ones. The subsequent result discusses a more general vaccination law which includes that of Remark 2 and allows the equilibrium susceptible subpopulation to reach a zero value under an asymptotically vanishing vaccination gain at lower rate than exponential:  
Assume that the following hypotheses hold:  
1.