## 2 The \(\mu\)-SEIR Model
### Boundaries  
The subsequent two results are direct from the combination of Equations (1)-(4) and they are respectively concerned with the fact that the total population is constant and the infection spreads, strictly increasing along consecutive samples, under certain constraints of the parameters and the susceptible subpopulation:  
**Proposition 1**. : _The total population \(N_{i}=S_{i}+E_{i}+I_{i}+R_{i}\) is constant for any integer \(i\in\mathbf{Z_{0^{+}}}\)._  
Proof. : When summing up Equations (1)-(4), one gets directly that  
\[N_{i+1}=N_{i}=N_{0}\]  
for any integer \(i\in\mathbf{Z_{0^{+}}}\). The subsequent result gives explicit conditions for the infection to expand at each two consecutive samples provided that the susceptible exceed a certain minimum threshold. This fact intuitively agrees with the known property of the significant verified pandemic spreading in its initial phase when the susceptible levels were very high. **Proposition 2**. : _Assume that \(S_{i}>max\big{\{}\frac{\mu}{\mu_{E}},\frac{\gamma(1-\gamma)}{\mu_{B}\eta_{1}} \big{\}}\). Then \(E_{i+1}>E_{i}\) and \(I_{i+1}>I_{i}\ \forall i>1\)._  
Proof. : Define \(\Delta E_{i}=E_{i+1}-E_{i}\) and \(\Delta I_{i}=I_{i+1}-I_{i}\). From Equation (2) we obtain that \(\Delta E_{i}=\beta_{I}I_{i}S_{i}+(\beta_{E}S_{i}-\mu)E_{i}\geq\beta_{I}I_{i}S_ {i}+(\beta_{E}S_{i}-\mu)E_{i}\) which, being \(S_{i}>\frac{\mu}{\beta_{E}}\) implies that \(\Delta E_{i}>0\) and \(E_{i+1}>E_{i}\). Now, from (3) we obtain that \(\Delta I_{i}=\mu E_{i}-\gamma I_{i}\). Then \(\Delta I_{i+1}=\mu E_{i+1}-\gamma I_{i+1}=\mu(E_{i}+\Delta E_{i})-\gamma(I_{i }+\Delta I_{i})=(1-\gamma)\Delta I_{i}+\mu\Delta E_{i}\) Which would be positive for all i+1 if \(\Delta E_{i}>0\), which it has already be proven, and \(\Delta I_{i}\geq 0\). If \(\Delta I_{i}<0\), \(\Delta I_{i+1}>0\) would be true only if \(\Delta E_{i}>\frac{\gamma-1}{\mu}\Delta I_{i}\). From Equations (2) and (3) we get that  
\[\beta_{I}S_{i}I_{i}+E_{i}(\beta_{E}S_{i}-\mu)>\frac{\gamma-1}{\mu}(\mu E_{i}- \gamma I_{i}) \tag{5}\]  
\[E_{i}(\beta_{E}S_{i}+1-\gamma-\mu)>I_{i}\bigg{(}\frac{\gamma(1-\gamma)}{\mu} -\beta_{I}S_{i}\bigg{)} \tag{6}\]Given that \(E_{i}>0\), \(I_{i}>0\)\(\forall i\in\mathbf{Z_{0+}}\), these inequalities will be always true if  
\[\left\{ \begin{array}{rcl}S_{i}&>&\frac{\gamma+\mu-1}{\beta_{E}}\\ S_{i}&>&\frac{\gamma(1-\gamma)}{\mu\beta_{I}}\end{array} \right. \tag{7}\]  
Then, if \(S_{i}>\max\{\frac{\gamma(1-\gamma)}{\mu\beta_{I}},\frac{\mu+\gamma-1}{\beta_{ E}},\frac{\mu}{\beta_{E}}\}=\max\{\frac{\gamma(1-\gamma)}{\mu\beta_{I}},\frac{\mu}{ \beta_{E}}\}\) the proposition holds. The vector sequence solution of dimension four is non-negative for all samples for any given non-negative initial conditions of all the subpopulations under some reasonable extra constraints on the initial conditions and parameters (so as to keep jointly both stability and non-negativity of the solution) as proved in the subsequent result:  
**Theorem 1**. : _Assume that \(\mu,\gamma\in(0,1]\), and \(\beta_{1}/\beta_{E}\leq\gamma/\mu\) with \(\beta_{I}>0\) and \(\beta_{E}>0\). Assume also that the initial conditions \((S_{0},E_{0},I_{0},R_{0})\) are subject to_  
\[\left\{ \begin{array}{rcl}0&\leq&S_{0}\leq\frac{\mu}{2\beta_{E}}\\ 0&\leq&E_{0}\leq\frac{1}{2\beta_{E}}\\ 0&\leq&I_{0}\leq\frac{1}{2\beta_{I}}\\ 0&\leq&R_{0}\end{array} \right. \tag{8}\]  
_Then, the vector sequence solution is non-negative for all samples. In addition, the sequences of all the subpopulations and that of the total population are bounded._  
Proof. : The proof follows by complete induction. Assume that there exists an integer \(k\geq 0\) such that \(min(S_{i},E_{i},I_{i},R_{i})\geq 0\), \(0\leq S_{i}\leq\frac{\mu}{2\beta_{E}}\), \(0\leq E_{i}\leq\frac{1}{2\beta_{E}}\) and \(0\leq I_{i}\leq\frac{1}{2\beta_{I}}\) for any \(i=0,1,2,\ldots,k\). Such an integer \(k\geq 0\) always exists since above constraints hold from the given hypotheses on the initial conditions (at least), i.e., for \(k=0\).