## SEIR Epidemic Model with Eventual Vaccination of Newborns
### Basic Reproduction Number and Stability of the Equilibrium Points
\begin{split}\hat{S}(\delta)&=\lim_{n\to\infty}S(nT+ \delta)=(1-q)A\int_{0}^{\delta}e^{-\int_{t}^{\delta}(\beta t(s)+\mu)ds}d\tau \leq(1-q)A\delta;\\ \hat{E}(\delta)&=\lim_{n\to\infty}E(nT+\delta)\leq \beta\delta\hat{S}_{M}(\delta)\hat{I}_{M}(\delta);\hat{I}(\delta)=\lim_{n\to \infty}I(nT+\delta)\leq\sigma\delta\hat{E}_{M}(\delta);\\ \hat{R}(\delta)&=\lim_{n\to\infty}R(nT+\delta)\leq \delta\Big{(}\gamma I_{M}(\delta)+qA\Big{)}\end{split}  \tag{20}\]  
after using the mean value theorem for integrals since all the involved integrals in Equation (2) are continuous functions of time. Note that \(\hat{y}:[0\,\ T)\to\mathbf{R}_{0+}\) will define the periodic limit oscillation of the subpopulation \(z\) in the event that such an oscillation exists. By recombining the three first above equations, one gets for any given \(q\in(0\,\ 1]\), any \(\delta\in(0\,\ T]\), and any supposed arbitrary oscillation period \(T>0\) that, since \(\hat{E}(\tau)=\hat{E}_{M}(\delta)\) for some \(\tau\in(0\,\ \delta]\) and any \(\delta\in(0\,\ T]\):  
\[\hat{E}(\tau)\leq\beta A\sigma(1-q)\delta^{3}\hat{E}_{M}(\delta)\Leftrightarrow 1 \leq\beta A\sigma(1-q)\delta^{3}\]  
which is violated if  
\[\delta\in\left(0\,\ \min\left(T\,\ \frac{1}{\sqrt[3]{\beta A\sigma(1-q)}} \right)\right)\]  
Hence, a contradiction follows such that the exposed subpopulation does not have a limit oscillation of any potential oscillation period, since a chaotic behavior cannot exist either since the model has no small time-varying parameter. Similarly, one gets a similar conclusion on no existence of limit oscillations of any period of the infectious subpopulation since the oscillation condition is violated for a range of sufficiently small inter-period time instants for any arbitrary claimed oscillation period according to:  
\[\hat{I}_{M}(\delta)\leq\beta\sigma(1-q)A\delta^{3}\hat{I}_{M}(\delta)\]  
and similar conclusions can be deduced for the susceptible and the recovered individuals. Therefore, no limit cyclic exists surrounding the reachable endemic equilibrium point when the disease-free equilibrium point is unstable, that is, if the basic reproduction number exceeds unity. Furthermore, all solutions are bounded from Theorem 1. As a result, the endemic equilibrium point is globally asymptotically stable if the basic reproduction number exceeds unity, equivalently, if and only if \(\beta\in(\beta_{c}(q)\,\infty)\). \(\square\)  
**Remark 4**. : _It can be of interest to give some considerations which are based on known general results of the qualitative theory of differential equations which indicate that there is a unique global asymptotically stable attractor which is always one of the equilibrium points [43, 44, 45]. The global attractor is the disease-free equilibrium if the reproduction number is less than or equal to one and the endemic equilibrium point if such a parameter exceeds unity. In this way, there are no stable oscillations surrounding neither one of the equilibrium points nor jointly both of them. If the reproduction number is unity, then the asymptotic behavior is similar to that arising if it is less than unity so that the disease-free equilibrium point is a global asymptotically stable attractor. Thus, the constraints implying local asymptotic stability of any of the equilibrium points lead to their global asymptotic stability. Those considerations are deduced from the subsequent linked arguments:_  
1. _If_ \(R_{0}(q)<1\)_, then the disease-free equilibrium point is locally asymptotically stable and the endemic one is not reachable. A limit cycle surrounding it in any phase plane of two subpopulations, if any, would be unstable from the combined topology of equilibrium points and limit cycles. Therefore, it would be irrelevant for stability considerations. As a result, the disease-free equilibrium point is globally asymptotically stable (Theorem_ 2_)._
2. _If_ \(R_{0}(q)>1\)_, then the endemic equilibrium point is locally asymptotically stable and the disease-free equilibrium point is unstable.