## SEIR Model with Impulsive Vaccination
### Periodic Solutions and Stability Results  
The disease-free periodic solution and its local stability property are characterized in the following through the ad hoc use of Floquet theory. For this purpose, the following result is stated and proven. **Theorem 4**: _The following properties hold:_  
1. _Any solution of Equation (_21_) is non-negative for all time for any given set of non-negative initial conditions._
2. _There exists a non-negative periodic disease-free solution of Equation (_21_) for each given set of non-negative finite initial conditions of the form_ \[\hat{E}(t)=\hat{I}(t)=0\] \[\hat{S}(t)=\frac{\hat{A}}{\mu}\Big{[}1-\frac{p}{1-(1  
Then, any solution of Equation (21) for any given non-negative initial conditions is non-negative for all time. Property (i) has been proven. To prove Property (ii), first equalize to zero all the time-derivatives in Equation (21) to get \(E(t)=I(t)=0\), and that the solution of the first differential equation is:  
\[S(t)=\frac{A}{\mu}\big{(}1-e^{-\mu(t-(n-1)T)}\big{)}+e^{-\mu(t-(n-1)T)}S[(n-1) T];\ \forall t\in[(n-1)T,\ nT),\forall n\in\mathbf{Z}_{+} \tag{27}\]  
In particular, the above equation leads to  
\[S(nT^{-})=e^{-\mu T}S[(n-1)T]+\frac{A}{\mu}\big{(}1-e^{-\mu T}\big{)} \tag{28}\]  
\[S(nT)=(1-p)S(nT^{-})=(1-p)\bigg{(}e^{-\mu T}S[(n-1)T]+\frac{A}{\mu}\big{(}1-e^ {-\mu T}\big{)}\bigg{)} \tag{29}\]  
at the impulsive time instants. Denoting \(S_{n}=S(nT)\) so that the stroboscopic mapping \(F:S_{n-1}\to S_{n}\); \(\forall n\in\mathbf{Z}_{+}\) defined by  
\[S_{n}=F(S_{n-1})=(1-p)\big{(}c+(S_{n-1}-c)e^{-\mu T}\big{)} \tag{30}\]  
with \(c=A/\mu\), has a unique fixed point satisfying \(S_{*}=F(S_{*})\) defined by the first relationship of Equation (22). In this way, the delay-free periodic solution of the susceptible, whose right and left limit values at the impulsive time instants are the two equations of Equation (22) is:  
\[\hat{S}(t)=\frac{A}{\mu}\Bigg{[}1-\Bigg{(}1-\frac{(1-p)\Big{(}1-e^{-\mu T} \Big{)}}{1-(1-p)e^{-\mu T}}\Bigg{)}e^{-\mu(t-(n-1)T)}\Bigg{]};\ \forall t\in[(n-1)T,\ nT),\forall n\in\mathbf{Z}_{+} \tag{31}\]  
The periodic disease-free solution of the recovered is given by \(\hat{R}(t)=\hat{N}(t)-\hat{S}(t)\), where \(\hat{N}(t)\) is the periodic solution of the total population which satisfies under zero infection the differential equation:  
\[\hat{N}(t)=A-\mu\hat{N}(t) \tag{32}\]  
whose transient solution is:  
\[\hat{N}(t)=\frac{A}{\mu}+\bigg{(}1-\frac{A}{\mu}\bigg{)}e^{-\mu t}. \tag{33}\]  
Taking limits as \(t\rightarrow\infty\) and subtracting Equation (31) from Equation (33) leads to the claimed steady-state periodic recovered oscillatory solution \(\hat{S}(t)=\underset{t\rightarrow\infty}{\lim}\big{(}\hat{N}(t)\big{)}-\hat{ R}(t)=A/\mu-\hat{R}(t)\). Property (ii) has been proven. Property (iii) is a direct consequence of Equations (22) and (23) for the disease-free solution. To prove Property (iv), the periodic oscillation is firstly perturbed with a small perturbation \((s(t)\,\ e(t)\,\ i(t)\,\ r(t))\), that is,  
\[S(t)=\hat{S}(t)+s(t);\ E(t)=\hat{E}(t)+e(t);\ I(t)=\hat{I}(t)+i(t);\ R(t)=\hat{ R}(t)+r(t) \tag{34}\]  
which results in the following modification of the differential system Equation (21) after neglecting the contribution to perturbation terms of orders higher than one:  
\[ \begin{array}{l}\dot{s}(t)=-\mu s(t)-\beta\hat{S}(t)i(t)-p\sum_{n=0}^{ \infty}s(nT^{-})\delta(t-nT);\ \dot{e}(t)=\beta\hat{S}(t)i(t)-(\mu+\sigma)e(t);\\ \dot{i}(t)=\sigma e(t)-(\mu+\gamma+\alpha)i(t);\ \dot{r}(t)=\gamma i(t)-\mu r(t)+p \sum_{n=0}^{\infty}s(nT^{-})\delta(t-nT)\end{array}  \tag{35}\]  
The periodic solution is stable if the perturbation vanishes through time. The function \(\hat{S}(t)\) is given by Equation (31) and it is a periodic coefficient of period \(T\) in Equation (35). The last equation is not needed for analysis if one considers the relationships \(\hat{R}(t)=\hat{N}(t)-\hat{S}(t)=A/\mu-\hat{S}(t)\) for the disease-free solution.