## SEIR Epidemic Model with Eventual Vaccination of Newborns
### Basic Reproduction Number and Stability of the Equilibrium Points  
One of the most important parameters in epidemiology is the basic reproduction number \(R_{0}\), which is defined as the average number of secondary cases caused by the entry of an infectious individual into a totally susceptible population [1] (Chap. 2). There are several techniques for calculating the basic reproduction number, and in this work, the next generation method will be shown [16], where \(R_{0}\) is the spectral radius of the next generation matrix. So, to build this matrix, first, one defines the transmission and transition function matrices \(\mathbf{F}\) and \(\mathbf{-V}\) defined from the second and third equations of (1) by:  
\[\mathbf{F}=\left( \begin{array}{c}\beta SI\\ 0\end{array} \right)\! ;\ \mathbf{V}=\left( \begin{array}{c}(\mu+\sigma)E\\ -\sigma E+(\mu+\gamma+\alpha)I\end{array} \right) \tag{8}\]  
which are evaluated at the disease-free equilibrium point to yield the matrix of dynamics of the linearized system around the disease-free equilibrium point, that is, the Jacobian matrix of the infective subsystem \((E,I)\) at such a point,  
\[\mathbf{j}_{EQ^{0}}=\mathbf{F}-\mathbf{V}=\left(\frac{\partial F}{\partial x} \right)_{EQ^{0}}-\left(\frac{\partial V}{\partial x}\right)_{EQ^{0}}=-\mathbf{ V}\!\left(\mathbf{I}_{2}-\mathbf{V}^{-1}\mathbf{F}\right) \tag{9}\]  
Ref. [38], where \(\mathbf{I}_{2}\) is the identity matrix of second order. Since \(-\mathbf{V}=\mathbf{j}_{EQ^{0}}\mathbf{j}_{\beta=0}\) is a stability matrix since the linearized system around the disease-free equilibrium point is globally asymptotically stable in the absence of disease, that is, if the transmission rate is zero, one follows from Equation (9) that \(\mathbf{j}_{EQ^{0}}\) is a stability matrix if and only if the spectral radius of \(\mathbf{V}^{-1}\mathbf{F}\) is less than one or, equivalently, that of  
\[\mathbf{G}=\mathbf{F}\mathbf{V}^{-1}=\left[ \begin{array}{cc}\frac{A\sigma(1 -q)\beta}{\mu(\mu+\sigma)(\mu+\gamma+\alpha)}&\frac{A(1-q)\beta}{\mu(\mu+ \gamma+\alpha)}\\ 0&0\end{array} \right] \tag{10}\]  
that is  
\[R_{0}(q)=\frac{A\sigma(1-q)\beta}{\mu(\mu+\sigma)(\mu+\gamma+\alpha)} \tag{11}\]  
referred to as the basic reproduction number, is less than one, that is, if and only if \(\beta<\beta_{c}(q)\) and this Jacobian matrix is critically stable for \(\beta=\beta_{c}(q)\) and unstable for \(\beta>\beta_{c}(q)\). Comparing this result with Proposition 1, and since the eigenvalues of the Jacobian matrix Equation (9) of the infective pair \((E,I)\) are continuous functions of the transmission rate, yields the result that the disease-free equilibrium point is asymptotically stable if and only if the endemic one is unreachable, which happens for a sufficiently small transmission rate as follows:  
**Theorem 2**. : _The disease-free equilibrium point is the only reachable equilibrium point and it is globally asymptotically stable if and only if the transmission rate is small enough to satisfy \(\beta\in[0\,,\beta_{c}(q))\). It is coincident with the endemic equilibrium point and critically stable if \(\beta=\beta_{c}(q)\) and unstable if \(\beta>\beta_{c}(q)\)._  
Proof. : One has from the above discussion that for sufficiently close to the disease-free equilibrium point, \(E(t)\to 0\) and \(I(t)\to 0\) as \(t\to\infty\) if \(\beta<\beta_{c}(q)\). On the other hand, note from Proposition 1 that, if \(\beta<\beta_{c}(q)\), then the exposed subpopulation is negative which contradicts Theorem 1(i). Therefore, the endemic equilibrium is not attainable if the reproduction number is less than one so that the only global stable attractor is the disease-free equilibrium point. On the other hand, all the subpopulations are bounded for all time and any \(\beta\geq 0\) from Theorem 1.