## SEIR Epidemic Model with Eventual Vaccination of Newborns
### Basic Reproduction Number and Stability of the Equilibrium Points
: _The components of the endemic equilibrium point may be rewritten, equivalently to (6), as follows:_  
\[S^{*}(q)=\frac{(1-q)A}{\mu R_{0}(q)};\ \ E^{*}(q)=\frac{(1-q)A}{\mu+\sigma} \bigg{(}1-\frac{1}{R_{0}(q)}\bigg{)}\] \[\Gamma(q)=\frac{\mu}{\beta}(R_{0}(q)-1);\ \ \ R^{*}(q)=\frac{qA}{\mu}+\frac{ \gamma}{\beta}(R_{0}(q)-1) \tag{15}\]  
_It is reachable if and only if \(R_{0}(q)\geq 1\) and it is coincident with the disease-free equilibrium point if and only if \(R_{0}(q)=1\)._  
**Theorem 3**. : _The endemic equilibrium point is globally asymptotically stable for any \(q\in[0\,\ 1\ ]\) if and only if the transmission rate satisfies \(\beta\in(\beta_{c}(q)\,\infty)\), equivalently, if and only if the basic reproduction number exceeds unity._  
Proof. : The Jacobian matrix at the endemic equilibrium point is:  
\[\mathbf{J}_{EQ^{*}}(q)=\left(\frac{\partial x^{T}}{\partial x}\right)_{EQ^{*} }=\left[ \begin{array}{cccc}-\mu R_{0}(q)&0&-\frac{(\mu+\sigma)(\mu+\gamma+ \alpha)}{\sigma}&0\\ \mu(R_{0}(q)-1)&-(\mu+\sigma)&\frac{(\mu+\sigma)(\mu+\gamma+\alpha)}{\sigma}&0 \\ 0&\sigma&-(\mu+\gamma+\alpha)&0\\ 0&0&\gamma&-\mu\end{array} \right] \tag{16}\]  
One of the eigenvalues of (16) is \(\lambda_{1}=-\mu\). Instead of calculating the other three eigenvalues of Equation (16), which is a difficult task, the Routh-Hurwitz stability criterion \(p(\lambda)=\sum_{i=0}^{3}a_{3-i}\lambda^{i}=0\) is applied to the characteristic equation of the third order remaining submatrix [31], where  
\[ \begin{array}{c}a_{0}=1;\ a_{1}=\mu(2+R_{0}(q))+\gamma+\alpha+\sigma;a_{2}= \mu R_{0}(q)(2\mu+\gamma+\alpha+\sigma);\\ a_{3}=\mu(R_{0}(q)-1)(\mu+\gamma+\alpha)(\mu+\sigma).\end{array}  \tag{17}\]  
The Routh-Hurwitz criterion is applied by first calculating the following determinants:  
\[ \begin{array}{c}\Delta_{0}=1\ ;\ \Delta_{1}=a_{1}=\mu(2+R_{0}(q))+\gamma+ \alpha+\sigma;\Delta_{2}=\left|\begin{array}{cc}a_{1}&a_{0}\\ a_{3}&a_{2}\end{array}\right|=a_{1}a_{2}-a_{3};\\ \Delta_{3}=\left|\begin{array}{ccc}a_{1}&a_{0}&0\\ a_{3}&a_{2}&a_{1}\\ 0&0&a_{3}\end{array}\right|=a_{3}\Delta_{2}.\end{array}  \tag{18}\]  
The number of characteristic roots of \(p(\lambda)=0\) with positive real part is the number of sign changes in the set \(\{\Delta_{0}\,\ \ \ \Delta_{1}\,\ \ \ \Delta_{2}/\Delta_{1}\,\ \ \ \Delta_{3}/\Delta_{2}\}\). Since \(\Delta_{0}>0\) and \(\Delta_{1}>0\), then there is no characteristic root with positive real part if and only if \(\Delta_{2}/\Delta_{1}>0\,\ \ \ \Delta_{3}/\Delta_{2}>0\ \ \text{if and only if}\)\(R_{0}(q)>1\), and \(\Delta_{2}/\Delta_{1}>0\) if and only if  
\[a_{1}a_{2}-a_{3}=\mu[\mu(2+R_{0}(q))R_{0}(q)\left(2\mu+\gamma+\alpha+\sigma \right)+(\gamma+\alpha+\sigma)(R_{0}(q)(\mu+\gamma+\alpha)+\mu+\sigma)]>0 \tag{19}\]  
which always holds. Then, (16) is a stability matrix if and only if \(R_{0}(q)>1\). Thus, the endemic equilibrium point is locally asymptotically stable if and only if \(R_{0}(q)>1\). It is now proven that such a stability property is global by using contradiction arguments. Assume that the solution is oscillatory periodic with some arbitrary oscillation period \(T>0\). The period is supposed arbitrary since it is being rebutted by contradiction arguments in the following that some such oscillation period exits so that no limit oscillation exists as a result. Define \(\hat{y}(\delta)=\lim\limits_{n\to\infty}y(nT+\tau)\) and \(\hat{y}_{M}(\delta)=\lim\limits_{n\to\infty}\max\limits_{\tau\in[0,\ \delta]}y(nT+\tau)\) for the four subpopulations \(y=S,\ E,\ I,\ R\). Note that all such claimed maximum values are positive realnumbers. From Equation (2) in the proof of Theorem 1, one gets for \(t=nT+\delta,n\in\mathbf{Z}_{0+}=\{z\in\mathbf{Z}:\ z\geq 0\}\), and \(\delta\in[0\,\ T)\) that the limits below exist for any such a \(\delta\in[0\,\ T)\):  
\[