## SEIR Model with Impulsive Vaccination  
### The Epidemic Model Under Periodic Impulsive Vaccination  
Consider a modification of Equation (1) as follows with \(q=0\) (no vaccination of newborns) and under a periodic impulsive vaccination of gain \(p\in[0\,\ 1]\) and period \(T\) at the time instants \(t=nT\) for \(n\in\mathbf{Z}_{0+}\):  
\[\begin{array}{c}\dot{S}(t)=A-(\beta I+\mu)S(t)-pS(nT^{-})\delta(t-nT)\\ \\ \dot{E}(t)=\beta S(t)I(t)-(\mu+\sigma)E(t)\\ \\ \dot{I}(t)=\sigma E(t)-(\mu+\gamma+\alpha)I(t)\\ \\ \dot{R}=\gamma I-\mu R+pS(nT^{-})\delta(t-nT)\end{array} \tag{21}\]  
for all \(t\in[nT\,\ (n+1)T)\); \(\forall n\in\mathbf{Z}_{0+}\) with initial conditions \(S(0)\geq 0\), \(E(0)\geq 0\), \(I(0)\geq 0\), and \(R(0)\geq 0\) for \(n=0\), where \(\delta(t)\) is the Dirac distribution which reflects in the model dynamics the presence of a very intensive vaccination (ideally of impulsive infinity gain) concentrated in very short time intervals (ideally instantaneously administered at the time instants \(t_{n}=nT\)). The notation \(S(nT^{-})\) stands for the left limit of \(S(t)\) at \(t_{n}=nT\) while the right limit (i.e., the value as the impulse at \(t_{n}\) is accounted for) is simply denoted as \(S(nT)=S(nT^{+})=(1-p)S(nT^{-})\). The removed susceptible individuals at the impulsive time instants become recovered, which is reflected by the solution of the above fourth equation at the impulsive time instants, that is, \(R(nT)=R(nT^{+})=R(nT^{-})+pS(nT^{-})\). Note that \(p=0\) describes the absence of impulsive vaccination while \(p=1\) describes the total vaccination of the susceptible at the time instants \(t_{n}=nT\), since no susceptible individual could be contacted by an infectious one because of the total removal of the first ones from the environment under study. It can be pointed out that impulsive vaccination is equivalent to intervention measures like isolation or confinement since a number of susceptible are transferred to the recovered subpopulation. The above-mentioned equivalence is at the level of transitions between subpopulations although no immunity is expected from such a state transition from susceptible to recovered. In this sense, the total recovered subpopulation is understood in a very wide sense as those individuals who are not susceptible to being infected, rather than just as those which are in fact immune, while those being immune are also included in the recovered subpopulation.  
Note that Equation (3) is still valid for the dynamics of the total population. That means that there are no finite jumps of the total population at the impulsive time instants (in particular, \(N_{*}=N_{*}^{-}=A/\mu\) at the disease-free equilibrium point) and there are no jumps either of the exposed or the infectious subpopulations at the disease-free equilibrium point.  
By equalizing to zero the time-derivatives of the four subpopulations in Equation (21) while zeroing the exposed and infectious values, one gets the subsequent disease-free steady-state:  
\[S_{*}=\lim_{n\to\infty}S(nT)=\frac{A(1-p)\big{(}1-e^{-\mu T}\big{)}}{\mu(1- (1-p)e^{-\mu T}\big{)}}\ ;\ S_{*}^{-}=\lim_{n\to\infty}S(nT^{-})=\frac{S_{*}}{1-p}= \frac{A\big{(}1-e^{-\mu T}\big{)}}{\mu(1-(1-p)e^{-\mu T})} \tag{22}\]\[R_{*}=\lim_{n\to\infty}R(nT)=N_{*}-S_{*}=\frac{Ap}{\mu\left(1-(1-p)e^{-\mu t}\right) }\ ;\ R_{*}^{-}=\lim_{n\to\infty}R(nT^{-})=N_{*}-S_{*}^{-}=\frac{Ape^{-\mu T}}{\mu \left(1-(1-p)e^{-\mu t}\right)} \tag{23}\]  
with \(E_{*}=E_{*}^{-}=I_{*}=I_{*}^{-}=0\).  
**Remark 5**: _Note that if \(q=p=0\) (i.e., in the absence of both newborns and impulsive vaccination efforts) then the disease-free equilibrium point discussed in the former section coincides with the above disease-free steady state. Note also that Equations (22) and (23) imply that the disease-free steady state is a periodic oscillation rather than an equilibrium point for nonzero \(p\)._