The solution trajectories of the SVEIRS differential model (1.1)–(1.5) are given below.
Equation (1.1) yields
Equation (1.2) yields
Equation (1.3) is already in integral form. Equation (1.4) yields
Equation (1.5) yields
The asymptotic values of the partial populations can be calculated from (5.1)–(5.4) as time tends to infinity as follows provided that the involved righthandside integrals exist:
Remark 5.1.
If any of the above righthandside integrals with upperlimit does not exist, then the corresponding limit of the involved partial population as time tends to infinity does not exist and, then, the limit value has to be replaced by the existing limit superior as time tends to infinity. Note that if Theorems 4.2 and 4.3 hold then all the limit values at infinity of the partial populations exist since the total population is uniformly bounded and all the partial populations are nonnegative for all time. A conclusion of this feature is that under positivity and boundedness of the solutions, all the partial populations of the impulsivefree SVEIRS model (1.1)–(1.5) have finite limits as time tends to infinity. As a result, all the trajectory solutions converge asymptotically either to the diseasefree equilibrium point or to the endemic equilibrium point.
The considerations in Remark 5.1 are formally expressed as the subsequent important result by taking also into account the uniqueness of the infected population at any endemic equilibrium points, the uniqueness of the vaccinated population at such points (which follows from (3.12) and which implies the uniqueness of the total population at such an equilibrium endemic points (see Remark 5.1).
Theorem 5.2.
The following two properties hold.

(i)
The endemic equilibrium point is unique.

(ii)
Assume that Theorems 4.2 and 4.3 hold. Then, any solution trajectory of the SVEIRS impulsivefree vaccination model (1.1)–(1.5) generated for finite initial conditions converges asymptotically either to the diseasefree equilibrium point or to the endemic equilibrium point as time tends to infinity.
Now, assume that the solution trajectory converges to some locally asymptotically stable equilibrium point . The above equations ensure the existence of integrands , being nonnegative, and such that for some , for any initial conditions in some small neighbourhood of the equilibrium, such that
which leads to
Any equilibrium point also satisfies the following constraints from (5.1)–(5.4), and (1.3) after performing the replacements (for the initial time instant), for the final time instant, for any given , and taking limits as , , , , , and . This leads to the following implicit relations being independent of and being applicable for any (diseasefree or endemic) equilibrium point:
Equation (1.2) yields
Equation (1.4) yields
which is identical to (2.5), that is, either or . Equation (1.5) yields
which is identical to (2.6).
Remark 5.3.
Some fast observations useful for the model interpretation follow by simple inspection of (5.1)–(5.4) and (5.10).

(1)
If or , then as is impossible.

(2)
as occurs if is eventually a function of time, rather than a real constant, subject to for some , for almost all except possibly at a set of zero measure in the case that the total population does not extinguish.

(3)
can converge to zero exponentially with time, for instance, to the diseasefree equilibrium point, while being a function of exponential order and, in such a case, also converges to zero exponentially while being of exponential order and satisfying from (1.3).

(4)
as requires from (5.4) the two above corresponding conditions for the infective and vaccinated populations to converge to zero. In such a case, the convergence of the recovered population to zero is also at an exponential rate. An alternative condition for a convergence to zero of the recovered population, perhaps at a rate slower than exponential, is the convergence to zero of the function
(see (5.4) with alternating sign on any two consecutive appropriate time intervals of finite lengths).
5.1. Incorporation of Impulsive Vaccination to the SVEIRS Model
Impulsive vaccination involving cullingtype or removal actions on the susceptible population has being investigated recently in [18, 19]. It has also being investigated in [20] in the context of a very general SEIR model. Assume that the differential system (1.1)–(1.5) is used for modelling in open real intervals for some given real sequence of time intervals with , for all and an impulsive vaccination is used at time instants , in the sequence which leads to a susceptible culling (or partial removal of susceptible from the habitat) and corresponding vaccinated increase as follows for some given real sequence , where :
so that . The following simple result follows trivially.
Theorem 5.4.
Let and be arbitrary except that the second one has all its elements in for some . Then, there is no nonzero equilibrium point of the impulsive SVEIRS model (1.1)–(1.5), (5.14). If as , then the equilibrium points of the impulsive model are the same as those of the SVEIRS model (1.1)–(1.5).
Equations (5.1), (5.2) yield to the following recursive expressions:
The solutions inbetween two consecutive impulsive vaccinations are obtained by slightly modifying (5.15)–(5.18) by replacing by zero and by for . The following result about the oscillatory behaviour of the vaccinated population holds.
Theorem 5.5.
Assume that is a piecewise continuous function rather than a constant one with eventual bounded step discontinuities at the sequence of impulsive time instants. Define real sequences at impulsive time instants with general terms
Then, the vaccinated population is an oscillating function if there is a culling sequence of time instants for some given real sequence with , of impulsive gains if any two consecutive impulsive time instants satisfy some of the two conditions below.
Condition 1.
and with a regular piecewise continuous vaccination and .
Condition 2.
and with a regular piecewise continuous vaccination and .
Proof.
One gets directly by using lower and upperbounds in (5.18) via and for
so that as and also for some sufficiently large ,
if , and
if . Thus, and imply that and , and and imply that and for some and some sufficiently large intervals inbetween consecutive impulses and via the use of an admissible regular piecewise continuous vaccination and .
It turns out that Theorem 5.5 might be generalized by grouping a set of consecutive impulsive time instants such that and for some positive integers , .
A further property now described is that of the impulsivefree infection permanence in the sense that for sufficiently large initial conditions of the infected populations the infected population exceeds a, initial conditions dependent, positive lowerbound for all time if no impulsive vaccination is injected under any regular vaccination.
Theorem 5.6.
Assume that the SVEIRS model is positive (in the sense that no partial population reaches a negative value for any time under nonnegative initial conditions) and, furthermore, that the susceptible and vaccinated populations remain positive for all time. Assume also that the disease transmission constant is subject to the subsequent constraint
for some positive real constants dependent on the delay
Then, the infection is permanent for all time if it strictly exceeds zero along the initialization interval.
Proof.
Note from the solution trajectories of the susceptible and vaccinated populations (5.1) and (5.2) that for some and for some real constants and , if and . Then,
so that, from (5.3a), since the SVEIR model is positive
The two last inequalities following from the fact that the former one stands, for all (an "adhoc" complete induction reasoning will lead to an identical conclusion), and, furthermore,
Thus, the infection is permanent for all time if it is permanent for the initialization time interval .
It is also obvious the following simplification of Theorem 5.6.
Theorem 5.7.
Theorem 5.6 still holds under (5.24) and the "adhoc" modified inequality (5.23) if either the susceptible or the vaccinated (but not both) population reaches zero in finite time or tends to zero asymptotically.
Proof.
It is similar to that of Theorem 5.6 by zeroing either and and removing the inverses from the corresponding conditions and proofs.
Close to the above results is the asymptotic permanence of the infection under sufficiently large disease constant.
Theorem 5.8.
The infection is asymptotically permanent for a positive initialization of the infected population on its initialization interval if the disease transmission constant is large enough.
Proof.
Note from (5.26) that for any given small , there is a sufficiently large finite such that for any and any , one gets
if the disease transmission constant is large enough to satisfy
If either or (but not both) is zero, then its inverse is removed from the above condition.
The above results suggest that the infection removal require periodic culling (or partial removal) of the susceptible population through impulsive vaccination so that both populations can become extinguished according to (5.8).