## 2 The \(\mu\)-SEIR Model
### Boundaries
It has been proved that the three coefficients of equations in (14) are positive for any \(i\in\mathbb{Z}_{0+}\). This implies that all the polynomials of \(q_{si}(s)\) Equation (13) satisfy the [PERSON]'s test for any \(i\in\mathbf{Z}_{0+}\). Since all those polynomials are of second order then all of them are from Routh-Hurwitz criterion (i.e., with both zeros in the open left-hand-side complex plane \(Re\,s<0\)). Then, all the discrete polynomials \(q_{i}(z)\) of Equation (12) for any \(i\in\mathbf{Z}_{0+}\) are stable (i.e., with both zeros in the open unit circle centered at zero, that is \(|z|<1\)). As a result, all the time-invariant second-order discrete systems of Equation (11) (each for an assumed constant \(S_{i}\)), for any \(i\in\mathbf{Z}_{0+}\) are exponentially stable. Thus, each norm of matrix of dynamics:\[\mathbf{A}_{i}=\mathbf{A}(S_{i})=\left[ \begin{array}{cc}1+\beta_{E}S_{i}-\mu& \beta_{I}S_{i}\\ \mu&1-\gamma\end{array} \right] \tag{15}\]  
whose eigenvalues are the zeros of their corresponding polynomial \(q_{i}(z)\) for each \(i\in\mathbb{Z}_{0+}\), satisfies \(||\mathbf{A}_{i}||^{k}\leq K_{i}\rho_{i}^{k}\) for some real constant \(\infty>K_{i}\geq 1\), which is norm dependent, and \(\rho_{i}\in(0,1)\) which is the spectral radius of \(\mathbf{A}_{i}\): That is the maximum absolute value of its two eigenvalues. Now define the real numbers:  
\[K=\lim\sup_{n\to\infty}\left(\sup_{0\leq i\leq n}K_{i}\right)\in[1,\infty); \rho=\lim\sup_{n\to\infty}\left(\sup_{0\leq i\leq n}\rho_{i}\right)\in(0,1) \tag{16}\]  
As a result, one gets from Equation (11) that:  
\[\left\| \begin{array}{cc}E_{i}\\ I_{i}\end{array} \right\| \leq \left\|A_{i-1}\right\|\left\| \begin{array}{cc}E_{i-1}\\ I_{i-1}\end{array} \right\| \tag{17}\] \[\leq \left\|A_{i-1}\right\|\left\|A_{i-2}\right\|\left\| \begin{array} []{cc}E_{i-2}\\ I_{i-2}\end{array} \right\|\] \[\leq \max\left(\|A_{i-1}\|^{2},\|A_{i-2}\|^{2}\right)\left\| \begin{array} []{cc}E_{i-2}\\ I_{i-2}\end{array} \right\|\] \[\leq \sup_{0\leq j\leq i-1}\left(\|A_{j}\|^{i}\right)\left\|\left( \begin{array}{cc}E_{0}\\ I_{0}\end{array} \right)\right\|\] \[\leq K\rho^{i}\left\|\left( \begin{array}{cc}E_{0}\\ I_{0}\end{array} \right)\right\|=0\]  
Then, \(\lim\limits_{i\to\infty}E_{i}=\lim\limits_{i\to\infty}I_{i}=0\) asymptotically at exponential rate and the solution sequences \(\{E_{i}\}_{i=0}^{\infty}\), \(\{I_{i}\}_{i=0}^{\infty}\), \(\{S_{i}\}_{i=0}^{\infty}\), \(\{R_{i}\}_{i=0}^{\infty}\) of Equation (1)-(4) are non-negative and bounded from Theorem 1 for any given finite initial conditions which satisfy Equation (8). In addition, the map of \(\left[0,\frac{\mu}{2B_{E}}\right]\) to itself defined by Equation (1) is non-expansive so that it has a fixed point which is the first component \(S_{dfe}\) of the disease-free equilibrium point and \(\lim\limits_{i\to\infty}S_{i}=S_{dfe}\). Finally, \(\lim\limits_{i\to\infty}R_{i}=R_{dfe}=N_{0}-S_{dfe}\) since \(\lim\limits_{i\to\infty}E_{i}=\lim\limits_{i\to\infty}I_{i}=0\) and the proof is complete. Note that the given model is claimed for its usefulness for short-term predictions in the evolution phase when the disease is blowing up. It is neither considered that vaccination is available for use or that there is immunity loss allowing for the increase again of the susceptible numbers after a certain delay. Therefore, the susceptible subpopulation is given by a decreasing sequence. It is now proved that the proposed model does not have an endemic equilibrium point. **Theorem 3**. : _Assume that \(\mu,\gamma\in(0,1]\). Then the delay-free model Equations (1)-(4) has no endemic equilibrium point \(x_{end}=(S_{end},E_{end},I_{end},R_{end}=N_{0}-(S_{end}+E_{end}+I_{end}))^{T}\)._  
Proof. : Assume that there is an endemic equilibrium point with \(S_{end}>0\). Then, from Equation (1), one gets \(\beta_{I}I_{end}+\beta_{E}E_{end}=0\) so that \(E_{end}=I_{end}=0\) which concludes that the eventual equilibrium point is the disease-free one.