## 3 Extended Discrete SEIR Epidemic Model with Delays
Thus, the application of the Routh-Hurwitz criterion is useful to investigate if all the zeros of the transformed polynomial \(\eta(s)=\eta(z=\frac{1+s}{1-s})\) belong to the open left-hand-side plane \(Re(s)<0\) or not (see Theorem 2). The next result relies on the convergence of the solution to the disease-free equilibrium point. **Theorem 6**. : _Assume that \(\eta(z)\) is a strictly stable polynomial and that all the hypotheses of Theorem 5 hold. Then \(\lim\limits_{i\to\infty}(1-(\beta_{I}I_{i}+\beta_{E}E_{i})\leq 1,\)\(\{S_{i}\}_{i=0}^{\infty}\to S_{dfe}\in[0,1],\)\(\{R_{i}\}_{i=0}^{\infty}\to R_{dfe}=(1-S_{dfe})\in[0,1]\) and \(\{E_{i}\}_{i=0}^{\infty}\to 0,\)\(\{I_{i}\}_{i=0}^{\infty}\to 0.\) If \(\lim\limits_{i\to\infty}(1-\beta_{I}I_{i}-\beta_{E}E_{i})<1\) then \(S_{dfe}=0\) and \(\liminf\min_{i\to\infty}(E_{i},I_{i})\geq 0.\)_  
Proof. : First note from Theorem 5 that the solution sequence vector is non-negative. Now, note from Equation (26) that \(\{S_{i}\}_{i=0}^{\infty}\to S_{dfe}\in[0,1]\) with \(1-\beta_{I}I_{i}-\beta_{E}E_{i}\leq 1;\forall i\in\mathbb{Z}_{0+}\)and \(\lim\limits_{i\to\infty}(1-\beta I_{i}-\beta_{E}E_{i})\leq 1,\) since otherwise, the susceptible subpopulation solution sequence would violate the non-negativity property of Theorem 5. On the other hand, note that:  
* If such limit is equal to unity then \(S_{dfe}\in[0,1]\) and \(\{E_{i}\}_{i=0}^{\infty}\to 0,\)\(\{I_{i}\}_{i=0}^{\infty}\to 0\) so that \(\{R_{i}\}_{i=0}^{\infty}\to R_{dfe}=1-S_{dfe}\)
* If the limit is less than unity then \(S_{dfe}=0\) and \(\liminf\min_{i\to\infty}(E_{i},I_{i})\geq 0\). If both \(\{E_{i}\}_{i=0}^{\infty}\to 0,\)\(\{I_{i}\}_{i=0}^{\infty}\to 0,\) the proof follows directly. Otherwise, assume that \(\liminf\min_{i\to\infty}(E_{i},I_{i})\geq 0\). Then, since \(\{S_{i}\}_{i=0}^{\infty}\to S_{dfe}=0\) one has from Equation (27) that \[\lim\limits_{i\to\infty}\left(E_{i+1}-E_{i}+\sum_{j=i-d}^{i}\mu_{j}E_{j}\right)=0\] Thus, since \(\eta(z)\) is strictly stable, then \(\{E_{i}\}_{i=0}^{\infty}\to 0\). From Equation (28), it follows that \(\{I_{i}\}_{i=0}^{\infty}\to 0\) since \(\{E_{i}\}_{i=0}^{\infty}\to 0\) and \(\gamma<1\). Also \(\{R_{i}\}_{i=0}^{\infty}\to R_{dfe}=1\). Furthermore, \(\{S_{i}\}_{i=0}^{\infty}\to S_{dfe}\in[0,1];\{R_{i}\}_{i=0}^{\infty}\to R_{dfe}= 1-S_{dfe}\) and \(\{E_{i}\}_{i=0}^{\infty}\to 0\), \(\{I_{i}\}_{i=0}^{\infty}\to 0\) so that the solution sequence converges to the disease-free equilibrium point. The subsequent result specifies further Theorem 6 in the sense that it relies on the monotonic convergence to zero levels of the infection in the case that the transmission rates are sufficiently small. In such a way, the levels of the exposed and infectious are strictly decreasing along the transient. Assume that all the hypotheses of Theorem 6 hold. Then \(\lim\limits_{i\to\infty}||z_{i}||\leq\frac{K(\beta_{I}+\beta_{E})}{1-\rho(\mathbf{A} )}\) where \(\rho(\mathbf{A})\in(0,1)\) is the spectral radius of \(\mathbf{A}\) which is \(max(1-\gamma,\eta_{M})\), where \(\eta_{M}\) is the maximum absolute value of the zeros of \(\eta(z)\) and \(K\geq 1\) is some norm-dependent real constant. Furthermore, if \(\rho(\mathbf{A})\) is sufficiently small and \(\beta_{I}+\beta_{E}\) is sufficiently small related to \(\rho(\mathbf{A})\) then the sequence \(\left\{\max_{i-d-1\leq j\leq i}\max\left(E_{j},I_{j}\right)\right\}_{i=0}^{\infty}\) converges monotonically to zero.