## 3 Extended Discrete SEIR Epidemic Model with Delays  
An extension of the SEIR model, which includes a finite set of commensurate discrete delays in contrast to the delay-free infection model from Equations (1)-(4), is now discussed. Such a model is the following one:\[S_{i+1} = (1-\beta_{1}I_{i}-\beta_{E}E_{i})S_{i} \tag{22}\] \[E_{i+1} = (\beta_{1}I_{i}+\beta_{E}E_{i})S_{i}+E_{i}-\sum_{j=j_{\mu}}^{i-d_{1 }}(\mu_{j}E_{j})\] (23) \[I_{i+1} = (1-\gamma)I_{i}+\sum_{j=j_{\mu}}^{i-d_{1}}(\mu_{j}E_{j})\] (24) \[R_{i+1} = \gamma I_{i}+R_{i} \tag{25}\]  
for any integer \(i\in\mathbb{Z}_{0+}\) and any given initial conditions \(S_{0}\geq 0,E_{0}\geq 0\), \(I_{0}\geq 0\)_and_\(R_{0}\geq 0\), \(j_{d}=max(i-d-d_{1},0)\) with \(d\in\mathbb{Z}_{+}\), \(d_{1}\in\mathbb{Z}_{0+}\). The above model has \(d+2\) delays, namely, the one-step-delay of the discretization dynamics from \(E_{i+1}\) to \(E_{i}\) plus the contributions of \(d+1(\geq 0)\) more extra delays from the sample \(i-d_{1}\) to the sample \(i-d-d_{1}\). The integer \(d_{1}\in\mathbb{Z}_{0+}\) reflects a potential extra delay if \(d_{1}>0\), to start to account for extra former delays distinct than the discretization one in the delayed dynamics. If \(d_{1}=0\) then the delayed dynamics starts to influence directly immediately after the one-step discretization step. It becomes technically easier for discussion the mixed normalization and the solution sequence non- negativity to initialize values over \(d+1\) consecutive samples, instead for a point initialization for \(i=0\) and then to run the model for successive samples. Therefore, the model from Equations (22)-(25) is modified as follows with \(d_{1}=0\), i.e., such that the delayed dynamics starts in the previous sample to the current one and it is affected by a total of \(d\) consecutive previous delayed samples with \(d\geq 1\):  
\[S_{i+1} = (1-(\beta_{1}I_{i}+\beta_{E}E_{i}))S_{i} \tag{26}\] \[E_{i+1} = (\beta_{1}I_{i}+\beta_{E}E_{i})S_{i}+E_{i}-\sum_{j=i-d}^{i}(\mu_{ j}E_{j})\] (27) \[I_{i+1} = (1-\gamma)I_{i}+\sum_{j=i-d}^{i}(\mu_{j}E_{j})\] (28) \[R_{i+1} = \gamma I_{i}+R_{i} \tag{29}\]  
for any integer \(i(\geq d)\in\mathbb{Z}_{+}\) and given \(d+1\) sets of initial conditions \(S_{i}\geq 0,E_{i}\geq 0\), \(I_{i}\geq 0\)_and_\(R_{i}\geq 0\); \(i=0,1,\ldots,d\). In the above scheme, the total number of delays is \(d+1\) including that of the natural discretization. To distinguish this model from Equations (1)-(4), it is claimed that there is at least one extra delay than the discretization one, i.e., the total minimum number of delays \(d+1\) is two since \(d\geq 1\). If \(d=0\) then the model reduces to Equations (1)-(4) where the only delay is the discretization one. The normalized model from Equations (26)-(29) has a non-negative sequence vector solution provided that all the punctual initial conditions for \(i=0,1,\ldots,d\) are normalized as addressed in the next result. Such a result incorporates also some boundedness initial conditions for the infectious and the exposed subpopulations along the relevant number of consecutive delays to fix such initial conditions. Those constraints remember tightly those given in Theorems 1 and 2 for the simpler model from Equations (1)-(4) which possesses just the single one-step discretization delay. **Theorem 5**. : _Assume that \(d_{1}=0\), \(\beta_{1}>0\), \(\beta_{E}>0\), \(\gamma\geq\sum_{j=0}^{d}\mu_{j}\geq\beta_{1}+\beta_{E}\), \(\gamma\in(0,1]\) and \(\mu_{i}\in(0,1]\), \(0\leq E_{i}\leq\frac{1}{2\beta_{E}}\) and \(0\leq I_{i}\leq\frac{1}{2\beta_{1}}\); \(\forall i\in[1,d]\). Assume also that the initial conditions are non-negative and normalized (\(N_{0}=1\)) so that they fulfill the constraints:_  
\[\left\{ \begin{array}{l}\min(S_{i},E_{i},I_{i},1-(S_{i}+E_{i}+I_{i}))\geq 0\\ \max(S_{i},E_{i},I_{i},1-(S_{i}+E_{i}+I_{i}))\leq 1\end{array} ;\forall i\in[1,d]\right. \tag{30}\]_Then the solution is non-negative since:_  
\[min(S_{i},E_{i},I_{i},R_{i})\geq 0,max(S_{i},E_{i},I_{i},R_{i})\leq 1;i\in\mathbb{Z}_{ 0+}\]  
Proof.