## 2 The \(\mu\)-SEIR Model
### Boundaries
Basically, if the constraint \(\beta_{E}/\beta_{I}\geq\mu/\gamma\) of Theorem 1 for \(\gamma\in(0,1)\), then the properties of non-negativity of the solution and its asymptotic convergence to the disease-free equilibrium point are jointly guaranteed under the constraints for the initial conditions of Theorem 1. Theorem 2: Assume that \(\mu\in(0,1],\gamma\in(0,1]\), \(\beta_{I}>0\) and \(\beta_{E}>0\) and that conditions of Theorem 1 hold. Assume also that the initial conditions are subject to \(0\leq S_{0}\leq\frac{\mu}{2\beta_{E}}\leq\frac{\gamma\mu}{\mu\beta_{I}+\gamma \beta_{E}},0\leq E_{0}\leq\frac{1}{2\beta_{E}},0\leq I_{0}\leq\frac{1}{2\beta _{I}}\), and \(0\leq R_{0}\). Then, the vector sequence solution is non-negative and bounded for all samples and converges asymptotically to the disease-free equilibrium at exponential rate. **Proof. ** Equations (2) and (3) can jointly be expressed as follows:  
\[\left[ \begin{array}{c}E_{i+1}\\ I_{i+1}\end{array} \right]=\left[ \begin{array}{cc}1-\beta_{E}S_{i}-\mu&\beta_{1} S_{i}\\ \mu&1-\gamma\end{array} \right]\left[ \begin{array}{c}E_{i}\\ I_{i}\end{array} \right] \tag{11}\]  
The characteristic equation of the discrete system from Equation (11) for the case when \(S_{i}\) is constant, that is, assuming that the system (11) is time-invariant is  
\[q_{i}(z)= q_{i}(z,S_{i})=(z-1+\mu+\beta_{E}S_{i})(z+\gamma-1)-\mu\beta_{I}S_{i} \tag{12}\] \[= z^{2}+(\mu+\gamma+\beta_{E}S_{i}-2)z+(\gamma-1)(\mu-1+\beta_{E}S _{i})-\mu\beta_{I}S_{i}\]  
The system Equation (11), provided invariant, is stable if its modes are in \(|z|<1\). The conditions can be discussed by using the bilinear transformation \(z=\frac{1+\delta}{1-\delta}\) which is a comfortable transformation (except for \(s=1,z=+\infty\)) which transforms the complex region \(|z|\leq 1\) into the continuous complex region \(Re(s)\leq 0\). This transformation allows one to apply comformable the [PERSON] stability criterion to the transformed characteristic equation which is simpler to apply, especially for this simple second-order system, than discrete Jury's stability criterion. Thus, one defines \(\hat{q}_{si}(s)=q_{i}\Big{(}z=\frac{1+\delta}{1-\delta}\Big{)}\) so that its zeros being the solution of \(\hat{q}_{si}(s)=0\) are equivalent, after reducing to the common denominator, to the zeros of  
\[q_{si}(s) = (1+s)^{2}+(\mu+\gamma+\beta_{E}S_{i}-2)(1-s^{2}) \tag{13}\] \[+[(\gamma-1)(\mu-1)-S_{i}(\mu\beta_{I}-\beta_{E}(\gamma-1))](1-s )^{2}\] \[= a(S_{i})s^{2}+b(S_{i})s+c(S_{i})\]  
where  
\[ \begin{array}{rcl}a_{i}=a(S_{i})&=&(2-\gamma)(2-\mu)-S_{i}((2-\gamma)\beta_ {E}+\mu\beta_{I})\\ b_{i}=b(S_{i})&=&2[\gamma+\mu-\gamma\mu+S_{i}(\mu\beta_{I}+(1-\gamma)\beta_{E })]\\ c_{i}=c(S_{i})&=&\gamma\mu+S_{i}(\gamma\beta_{E}-\mu\beta_{I})\end{array}  \tag{14}\]  
It is now seen that the above three coefficients are positive for any \(i\in\mathbf{Z}_{0+}\) provided that the constraints of Theorem 1 hold, that is, (8), \(\beta_{1}/\beta_{E}\leq\gamma/\mu\), with \(\mu\in(0,1]\), and \(\gamma\in(0,1]\). Note that it is guaranteed from Equations (14) that:  
1. \(c_{i}\geq 0;\forall i\in\mathbf{Z}_{0+}\) if \(\gamma\beta_{E}-\mu\beta_{I}\geq 0\). Since the non-negativity conditions from Theorem 1 requires that \(\beta_{1}/\beta_{E}\leq\gamma/\mu\) it follows that \(c_{i}>0\). 2. \(b_{i}>0\) always, as the conditions \(\gamma\in(0,1)\) and \(\mu\in(0,1)\) implies that \(b_{i}=2[\gamma+\mu(1-\gamma)+S_{i}\mu\beta_{I}+S_{i}(1-\gamma)\beta_{E}]\geq 0\), as it is a sum of positive terms. 3. \(a_{i}>0;\forall i\in\mathbf{Z}_{0+}\) iff \(\frac{(2-\gamma)(2-\mu)}{(2-\gamma)\beta_{E}+\mu\beta_{I}}>S_{i}\). Since the condition of non-negativity from Theorem 1 \(\frac{\beta_{I}}{\beta_{E}}\leq\frac{\gamma}{\mu}\), or equivalently, \(\beta_{E}/\beta_{I}\geq\mu/\gamma\), has to hold then \[\frac{(2-\gamma)(2-\mu)}{(2-\gamma)\beta_{E}+\mu\beta_{I}}\geq\frac{(2-\gamma )(2-\mu)}{(2-\gamma)\beta_{E}+\gamma\beta_{E}}=\frac{(2-\gamma)(2-\mu)}{2\beta _{E}}>\frac{\mu}{2\beta_{E}}\geq S_{i}\] so that \(a_{i}>0\).