## SEIR Model with Impulsive Vaccination
### Periodic Solutions and Stability Results
Therefore, a third-order differential system being built with the first three relations of Equation (35) might be compacted into a third-order differential system of the form \(\dot{x}(t)=\mathbf{A}(t)x(t)\) with \(\mathbf{A}(t)=\mathbf{A}(t+T)\) being a third-order periodic matrix function of dynamics for all \(t\geq 0\). Its entries follow directly from inspection of Equation (35). Such a system has to be solved on a time interval \([0\,,\ T]\). A fundamental matrix \(X(t)=X(t+T)\) of the third-order differential system, that is being nonsingular for all time since its columns are particular linearly independent solutions to initial conditions satisfying \(X(0)=\mathbf{I}_{3}\), is defined for all \(t=nT\) as \(X(T)\), the monodromy matrix, being of the form:  
\[X(T)=Diag\left((1-p)e^{-\mu T},\quad\exp\left(-\frac{1}{2}\int_{0}^{T}P_{+}(t) dt\right),\quad\exp\left(-\frac{1}{2}\int_{0}^{T}P_{-}(t)dt\right)\right) \tag{36}\]  
where  
\[P_{+}(t)=\alpha+\sigma+\gamma+2\mu+\sqrt{\left(\alpha+\gamma-\sigma\right)^{2 }+4\beta\sigma\hat{S}(t)}\;;\;\forall t\in[0\,,\ T] \tag{37}\]  
\[P_{-}(t)=\alpha+\sigma+\gamma+2\mu-\sqrt{\left(\alpha+\gamma-\sigma\right)^{2 }+4\beta\sigma\hat{S}(t)}\;;\;\forall t\in[0\,,\ T]. \tag{38}\]  
The eigenvalues of the monodromy matrix, that is, its diagonal entries in this case are the Floquet multipliers [29; 30], and the matrix \(\mathbf{A}(t)\) is a stability matrix if the moduli of their Floquet multipliers are less than one. One gets by inspection of Equations (36)-(38) that the two first eigenvalues have a modulus less than one since \(\mu>0\), \(0\leq p\leq 1\) and \(P_{+}(t)>0\); \(\forall t\in[0\,,\ T]\) and the third eigenvalue has a modulus less than one if and only if  
\[T(\alpha+\sigma+\gamma+2\mu)>\int_{0}^{T}\sqrt{\left(\alpha+\gamma-\sigma\right) ^{2}+4\beta\sigma\hat{S}(t)dt} \tag{39}\]  
The condition Equation (39) implies that the periodic solution is locally asymptotically stable in the sense that the solution of the incremental system Equation (35) converges asymptotically to zero for sufficiently small initial conditions. Property (iv) has been proven. To prove Property (v), note that a sufficient condition for Equation (39) to hold is that  
\[\left(\alpha+\sigma+\gamma+2\mu\right)^{2}>\left(\alpha+\gamma-\sigma\right)^ {2}+4\beta\sigma\max_{t\in[0\,,\ T]}\hat{S}(t)\]  
which holds if, equivalently, Equation (26) holds. Property (v) has been proven. **Remark 6**. : _Theorem 4 (v) gives a testable trade-off between the sizes of the critical transmission rate \(\beta_{c}\) and those of the impulsive period (\(p\) and \(T\)) to guarantee the local stability of the periodic solution. In particular, for small periods for the impulsive vaccination actions, the critical transmission rate value compatible with the local asymptotic stability of the disease-free solution is high. That means that, in the main range of values of interest, the tolerance to the transmission rate guarantees that such a stability tolerance increases as the impulsive vaccination period decreases as intuitively expected. Note that it is not difficult to extend the performed study to the combination of the two combined vaccination strategies of vaccination of newborns and periodic impulsive vaccination. On the other hand, the generalization to impulsive vaccination controls under time-varying inter-impulse periods are an open design tool for related future work based on the adaptive sampling rate designs [43; 44]._