## 1 Introduction
The local asymptotic stability is seen to also be global by combining properties of non-negativity of the solution with the values of Poincare index for individual or groups of singular related to the discussion on the existence of limit cycles. Concretely, the local asymptotic stability is proven to also be global on the basis of the fact that the stable attainable attractor is the unique disease-free equilibrium point if reproduction number is less than unity (in which case the endemic equilibrium point is not attainable). Contrarily, if the disease-free equilibrium point is unstable, in the case of a reproduction number exceeding unity, then it is shown that it cannot be surrounded by a limit cycle in any of the phase planes; the only stable and attainable attractor being then the endemic equilibrium point. Another alternative vaccination strategy, which is also discussed in the paper under the same basic model, consists of a periodic impulsive vaccination. Again, the vaccination effort can be accommodated to the current stocks of susceptible subpopulation. The stability properties of the periodic steady-state are performed via [PERSON]'s theory, with an explicit calculation of the eigenvalues of the monodromy matrix which are the Floquet multipliers. This method allows us to establish that, under a sufficiently small transmission rate, the local asymptotic stability is guaranteed according to a formula which makes explicit a trade-off between the vaccination interval between consecutive vaccination pulses and the susceptible population fraction to be vaccinated by a defining a reproduction-like number. To perform the above summarized study, this work is split into five sections which are organized as follows. Section 2 aims at establishing and discussing the SEIR model under eventual vaccination of newborns. The general formulation of the proposed SEIR model is given. Later on, the mathematical analysis of the model properties is discussed in detail in terms of local and global asymptotic stability concerns of both the disease-free and endemic equilibrium points, which are explicitly calculated as dependent of the vaccination of newborns levels. It is proven that the endemic equilibrium point is not reachable when the disease-free one is locally asymptotically stable, which happens as the basic reproduction number is less than unity. It is also proven that when the basic reproduction number exceeds unity then the endemic equilibrium point is reachable and locally asymptotically stable while the disease-free one is unstable. The reproduction number is improved with the levels of vaccination, which means that the reproduction number is kept below unity for wider ranges of potential values of the transmission rate as the vaccination effort increases. It is also proven that the local asymptotic stability of the equilibrium points also imply both respective global asymptotic stability properties. In the formulation of Section 2, the absence of vaccination of newborns might be understood as a particular case of the corresponding vaccination effort. On the other hand,in Section 3, an impulsive periodic vaccination strategy is proposed and studied, and the resulting model properties are formulated for the delay-free case. In this case, it is necessary to work with a resulting dynamic system of periodic coefficients, and, for this purpose, the Floquet theory is used to formulate the stability properties. Some numerical results are displayed and discussed in Section 4 for the COVID-19 pandemic, which are then performed to visualize the solutions for the two studied vaccination strategies. First, an endemic disease is simulated using the SEIR model, and then the mentioned vaccination strategies which are applied with the aim of eradicating the disease or, at least, softening its impact.