## 3 Extended Discrete SEIR Epidemic Model with Delays
: First, note that the constraint \(\gamma\geq\sum_{j=0}^{d}\mu_{j}\geq\beta_{I}+\beta_{E}\) implies that \(1-\sum_{j=0}^{d}\mu_{j}+\beta_{I}+\beta_{E}\leq 1\). Note also that from Equations (26)-(29) and the non-negativity/normalization constraints of the initial conditions for \(j=1\), one has, since \(\sum_{j=0}^{d}\mu_{j}\leq\gamma\leq 1\), that :  
\[ \begin{array}{rl}0&\leq\left(1-(\beta_{I}+\beta_{E})\max_{0\leq i\leq d+j-1}( max(I_{i},E_{i})\right)\min_{0\leq i\leq d+j-1}S_{i}\\ \leq S_{d+j}&\leq\left(1-(\beta_{I}+\beta_{E})\min_{0\leq i\leq d+j-1}(min(I_ {i},E_{i}))\right)\max_{0\leq i\leq d+j-1}S_{i}&\leq 1\end{array}  \tag{31}\]  
\[ \begin{array}{rl}0&\leq&(1-\sum_{i=j-1}^{j+d-1}\mu_{i})\min_{i\in[j-1,d+j-1] }E_{i}\\ +&(\beta_{I}+\beta_{E})\min_{i\in[j-1,d+j-1]}(\min(E_{i},I_{i}))\min_{i\in[j- 1,d+j-1]}S_{i}\leq E_{d+j}\\ \leq&E_{d+j-1}-\left(\sum_{i=j-1}^{j+d-1}\mu_{i}\right)\min_{i\in[j-1,d+j-1] }E_{i}\\ +&(\beta_{E}+\beta_{I})\max_{i\in[j-1,d+j-1]}(\max(I_{i},E_{i}))\max_{i\in[j- 1,d+j-1]}S_{i}\leq 1\end{array}  \tag{32}\]  
\[ \begin{array}{rl}0&\leq I_{d+j}&\leq 1-\gamma+\sum_{i=j-1}^{j+d-1} \mu_{i}\leq 1\\ 0&\leq R_{d+j}&=1-(S_{d+j}+E_{d+j}+i_{d+j})=R_{d+j-1}+\gamma I_{d+j-1}\end{array}  \tag{33}\]  
and by summing up all the subpopulations, one gets that, for \(j=1,N_{d+j}=N_{d+1}=N_{d}=N_{0}=1\). However, the above constraints also hold via recursive calculations for all \(j\in\mathbb{Z}_{0+}\) starting from the above equations for \(j=0\). The proof of non-negativity is complete. Equations (27) and (28) can be jointly described as evolution sequence vectors defined by sets of \(d+1\) consecutive samples by a real vector  
\[\mathbf{z}_{i}=(E_{i},E_{i-1},\ldots,E_{i-d},I_{i},I_{i-1},\ldots,I_{i-d})^{T}\]  
through the discrete dynamic system as follows:  
\[\mathbf{z}_{i+1}=\mathbf{A}\mathbf{z}_{i}+\mathbf{B}\mathbf{u}_{i};\forall i (\geq\mathbf{d})\in\mathbb{Z}_{\mathbf{0}+} \tag{35}\]  
subject to a point vector initial condition \(\mathbf{z}_{d}\in\mathbb{R}^{2(d+1)}\) built from the given \(d\) sets of initial conditions from Equations (27) and (28), whose partitioned parametrization is defined as follows:  
\[\mathbf{A}=\left[ \begin{array}{cc}\mathbf{A}_{11}&\mathbf{0}_{(d+1)\times(d +1)}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{array} \right]\in\mathbb{R}^{2(d+1)\times 2 (d+1)};\;\;\;\mathbf{B}=\left[ \begin{array}{cc}\mathbf{B}_{1}\\ \mathbf{0}_{d+1,1}\end{array} \right]\in\mathbb{R}^{2(d+1)};\]  
\[\mathbf{u}_{i}=\left[ \begin{array}{cc}(\beta_{I}I_{i}+\beta_{E}E_{i})S_{i} \\ 0_{2d+1,1}\end{array} \right] \tag{36}\]  
with\[\mathbf{A_{11}}=\left[ \begin{array}{cccc}1-\mu_{i}&...&-\mu_{i-d+1}&-\mu_{i}-d \\ &\mathbf{I}_{d}&&\mathbf{0}_{d\times 1}\end{array} \right]\in\mathbb{R}^{(\mathbf{d+1}) \times(\mathbf{d+1})};\mathbf{B_{1}}=\left[ \begin{array}{cccc}1\\ \mathbf{0}_{d\times 1}\end{array} \right]\in\mathbb{R}^{\mathbf{d+1}}\]  
\(I_{d}\) is the \(n\) identity matrix and \(0_{n,m}\in\mathbb{R}^{n\times m}\) is a zero matrix of order \(n\times m\). The recursive solution of Equation (35) becomes:  
\[\mathbf{z}_{i}=\mathbf{A}^{d}\mathbf{z}_{i-d}+\sum_{j=i-d}^{i-1}\mathbf{A}^{i -j-1}\mathbf{B}\ \mathbf{u}_{j} \tag{37}\]  
Note that, because of the triangular structure of the matrix \(\mathbf{A}\), its \(2(d+1)\) eigenvalues are those of \(A_{22}\), i.e., \(1-\gamma\) plus \(d\) zero eigenvalues, and those of \(A_{11}\) which, since it is a companion matrix, are the roots of the subsequent discrete polynomial:  
\[\eta(z)=z^{d+1}+(\mu_{i}-1)z^{d}+\sum_{j=i-d}^{i-1}\mu_{j}z^{j-i+d} \tag{38}\]  
whose stability property (i.e., its zeros lie in the open unit circle of the complex plane \(|z|<1\)) can be investigated through the Jury criterion or via the bilinear transformation \(z=\frac{1+s}{1-s}\), which transforms the open unit circle of the complex plane into the open left-hand-side of the complex plane.