On the Existence of Equilibrium Points, Boundedness, Oscillating Behavior and Positivity of a SVEIRS Epidemic Model under Constant and Impulsive Vaccination

1 Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain 2 Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain 3 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA 4 Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 5 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain


Introduction
Important control problems nowadays related to Life Sciences are the control of ecological models like, for instance, those of population evolution Beverton-Holt model, Hassell model, Ricker model, etc. 1-5 via the online adjustment of the species environment carrying capacity, that of the population growth or that of the regulated harvesting quota as well as the disease propagation via vaccination control. In a set of papers, several variants and generalizations of the Beverton-Holt model standard time-invariant, time-varying parameterized, generalized model or modified generalized model have been investigated at the levels of stability, cycle-oscillatory behavior, permanence, and control through the manipulation of the carrying capacity see, e.g., 1-5 . The design of related control actions has been proved to be important in those papers at the levels, for instance, of aquaculture exploitation or plague fighting. On the other hand, the literature about epidemic mathematical models is exhaustive in many books and papers. A nonexhaustive list of references is given in this manuscript, compare 6-14 see also the references listed therein . The sets of models include the most basic ones, 6, 7 . i SI-models where not removed-by-immunity population is assumed. In other words, only susceptible and infected populations are assumed. ii SIR-models, which include susceptible, infected, and removed-by-immunity populations. iii SEIR-models where the infected populations is split into two ones i.e., the "infected" which incubate the disease but do not still have any disease symptoms and the "infectious" or "infective" which do exhibit the external disease symptoms . The three above models have two possible major variants, namely, the so-called "pseudomass action models," where the total population is not taken into account as a relevant disease contagious factor or disease transmission power, and the so-called "true-mass action models", where the total population is more realistically considered as being an inverse factor of the disease transmission rates. There are other many variants of the above models, for instance, including vaccination of different kinds: constant 8 , impulsive 12 , discretetime, and so forth, incorporating point or distributed delays 12, 13 , oscillatory behaviors 14 , and so forth. On the other hand, variants of such models become considerably simpler for the disease transmission among plants 6,7 . Some generalizations involve the use of a mixed regular continuous-time/impulsive vaccination control strategies for generalized time-varying epidemic model which is subject to point and distributed time-varying delays, 12, 13, 15-17 . Other well-known types of epidemic models are the so-called SVEIRS epidemic models which incorporate the dynamics of a vaccinated population, and the "infected" population without external symptoms of the SEIR-type models is replaced with an "exposed" population subject to a certain dynamics, 18,19 . Thus, in the context of SVEIRS models, the infected and infectious populations of the SEIR models are joined in a single "infected" population I t while there is an exposed population E t present in the model. In this paper, we focus on the existence and some properties of disease-free and endemic equilibrium points of a SVEIRS model subject to an eventual constant regular vaccination rather than to an impulsive vaccination type. Some issues about boundedness and positivity of the model are also investigated. The following impulsive-free SVEIRS epidemic model, of a modified true-mass action type, with regular constant vaccination is being firstly considereḋ where S, V, E, I, and R are, respectively, the susceptible, vaccinated, exposed, infected or infective or infectious , and recovered populations, N t is the total population being the sum of the above ones, and V c ∈ 0, 1 is a constant vaccination action. There are potential latent and immune periods denoted by τ and ω, respectively, which are internal delays in the dynamic system 1.1 -1.5 , and b is the natural birth rate and death rate of the population. The parameter ν < b takes into account a vaccination action on newborns which decreases the incremental susceptible population through time, γ 1 is the average rate for vaccines to obtain immunity and move into recovered population, and β disease transmission constant and δβ are, respectively, average numbers for contacts of an infective with a susceptible and an infective with a vaccinated individual per unit of time, 18 . The periodic impulsive, rather than regular, vaccination action proposed in 18, 19 , can be got from 1.1 -1.5 with V c 0 while adding either corresponding "culling" action, or, alternatively, a less drastic "partial removal of susceptible from the habitat" action. This implies in practical terms to put in quarantine a part of the susceptible population so as to minimize the effects of the disease propagation what corresponds with a population decrease of the susceptible in the habitat under study and a parallel increase of the vaccinated populations at times being an integer multiple of some prefixed period T > 0. This paper investigate through Sections 2-4 the existence and uniqueness of the delay-free and endemic equilibrium points as well as the positivity and boundedness of the state-trajectory solutions under arbitrary nonnegative initial conditions and optional constant vaccination. Some generalized extensions concerning this model are given in Section 5 by using aperiodic impulsive vaccination with time-varying associated gains, in general, and investigating the state-trajectory solution properties. This impulsive vaccinations strategy will be performed as follows at a sequence of time instants {t k } k∈Z 0 ran in general at a nonperiodic "in-between" sampling interval sequence:

1.6
Examples are provided in Section 6. It has to be pointed out that other variants of epidemic models have been recently investigated as follows. In 20 , a mixed regular and impulsive vaccination action is proposed for a SEIR epidemic mode model which involves also mixed point and distributed delays. In 21 , an impulsive vaccination strategy is discussed for a SVEIR epidemic model whose latent period is a point delay while the existence of an immune period is not assumed. In 22 , a latent period is introduced in the susceptible population of a SIR epidemic model with saturated incidence rate. The disease-free equilibrium point results to be locally asymptotically stable if the reproduction number is less than unity while the endemic equilibrium point is locally asymptotically stable if such a number exceeds unity.

The Disease-Free Equilibrium Point
The potential existence of a disease-free equilibrium point is now discussed which asymptotically removes the disease if ν < b.
which imply the following further constraints: Two particular disease-free equilibrium points are Advances in Difference Equations

5
The disease-free equilibrium point satisfies the constraints

2.7
The proof follows directly from the above equations.
Note also that if ν 0, as in the particular case of impulsive-free SVEIRS model obtained from that discussed in 18, 19 , then the disease-free equilibrium satisfies E * V * I * R * 0, N * S * 1. In such a case, the model can be ran out with population normalized to unity.
Proof. Note from Theorem 2.1 that the disease-free equilibrium point satisfies from simple direct calculations that Advances in Difference Equations Note that the exposed population at the equilibrium defined by 1.3 can be equivalently described by a differential equation obtained by applying the Leibniz differentiation rule under the integral symbol to yielḋ The local asymptotic stability of the disease-free equilibrium point is guaranteed by that of the linearized incremental system about it. The linearized model about the equilibrium becomes to be defined from 1.1 , 1.2 , 2.10 and 1.4 , 1.5 by the state vector x t : S t , V t , E t , I t , R t T which satisfies the differential systeṁ where, after using the identities in Theorem 2.1 related to the equilibrium point and provided that Assertion 1 holds, one gets

2.12
Advances in Difference Equations 7 where ε β > ε β is a real constant, and , 2.14 and the matrices A * τ and A * ω are entry-wise defined by

2.15
with all the remaining entries being zero. The following inequalities apply for equivalent norms of either vectors or vector-induced norms of matrices M of dimension or, respectively, order n: Thus, one gets from the above inequalities 2.16 that

2.18
Note from 2.18 that √ 5 γ b α e b τ−ω ≤ b − b 0 for a given b and any given positive real constant b 0 < b if γ b α and τ − ω , if positive, are small enough such that, equivalently,

Advances in Difference Equations
Thus, one gets from 2.17 -2.19 that On the other hand, we can use from L'Hopital rule the following limit relations in the entries 1, 4 and 2, 4 of the matrix A * 0 : if the remaining parameters remain finite and then N * S * 1 and E * I * V * R * 0 from Theorem 2.1. By continuity with respect to parameters, for any sufficiently large M ∈ and, one gets from 2.14 ,

2.23
and for b being large enough such that it satisfies with b a being some existing real positive constant, depending on the vaccination constant V c , such that ν 1−V c ≥ β ε 1 / 1 η , it follows from inspection of 2.22 , 2.23 that A * 0 ∞ ≤ γ b α e bτ . Using again 2.16 , 2.17 , it follows that the following close constraint to 2.19 for large enough b: Advances in Difference Equations

2.27
On the other hand, note that the linearized system 2.11 -2.17 is asymptotically stable if and only if which is guaranteed under the two conditions below: is negative, and, furthermore, the following constraint holds which guarantees the above stability Condition 2 via 2.26 , 2.27 , and 2.13

2.30
The following result is proven from Theorem 2.1, by taking into account the above asymptotic stability conditions for the linearized incremental system about the disease-free equilibrium point, which imply that of the nonlinear one 1.1 -1.5 about the equilibrium point, and the related former discussion. Note that the statement of Theorem 2.3 guarantees the local stability of the disease-free equilibrium point under its existence condition of Theorem 2.1 requiring ν < b.

The Existence of Endemic Equilibrium Points, Uniqueness Issues, and Some Related Characterizations
The existence of endemic equilibrium points which keep alive the disease propagation is now discussed. It is proven that there is a unique equilibrium point with physical meaning since all the partial populations are nonnegative.
Thus, there is at least one endemic equilibrium point at which the susceptible, vaccinated, infected, exposed, and recovered populations are positive and the vaccinated population is zero if and only if V c 0 (i.e., in the absence of vaccination action). Furthermore, such an equilibrium point satisfies the constraints ii If the disease transmission constant is small enough satisfying β < β : ηe bτ γ b α / 1 δ for V c > 0, and β < ηe bτ γ b α for V c 0, then there is no reachable endemic equilibrium point.
Proof. The endemic equilibrium point is calculated as follows: since, otherwise, the above disease-free equilibrium point would be considered .

11
S * > 0 since, otherwise, the following contradiction would follow: , what is only possible in the disease-free equilibrium if the total population is extinguished, is a contradiction at the endemic point Remark 3.2. Note that if ω 0, then it follows from 1.3 and 2.4 that E t E * 0, for all t ∈ R 0 so that the SVEIRS model 1.1 -1.5 becomes a simpler SVIRS one without specification of the exposed population dynamics. Remark 3.3. Note that, under the constraints in Theorem 3.1 ii for β, if there is no reachable endemic equilibrium point because β < β, then the solution trajectory of 1.1 -1.5 can only either converge to the disease-free equilibrium point provided that it is at least locally asymptotically stable or to be bounded converging or not to an oscillatory solution or to diverge to an unbounded total population depending on the values of the parameterization of the model 1.1 -1.5 . Note that the endemic-free disease transmission constant upperbound β increases as η, τ and γ b α increase and also as δ decreases.
If V c > 0, then it follows from Theorem 3.1 that there exist positive constants α S , α V , α E , α I , and α R satisfying α −1 1 such that the endemic equilibrium points, if any, satisfy the constraints so that one gets from 3.3 -3.7 that if min S * , V * ≥ 1, otherwise, then only the upper-bounding constraint holds in 3.10 .9 is equivalent, since R * > 0 at the endemic equilibrium point, to 3.14 or, alternatively, with η < 1 hold. On the other hand, 3.12 is equivalent to where β 0 : β/e bτ γ b α so that 3.17 is of the form specifically as follows: Now, the same reasoning as that used for the susceptible endemic equilibrium component is applied to 3.18 to conclude that, since there is a nonzero vaccinated population for at most two endemic equilibrium points with min S * , V * ≥ 1, then which is obtained from 3.2 , and either or, a < 0, d > 0 in 3.18 . The uniqueness of the endemic equilibrium point with all partial populations being nonnegative for all time is now proven as follows. First, define auxiliary variables Thus, since N * α S S * , 3.14 can be rewritten as follows: If such an equation has two positive real roots for the susceptible equilibrium implicitly depending on N * , then either bα I η < 0 or bα I N * B < 0 what is impossible and leads to a contradiction. Then, there is a unique nonnegative susceptible population S * ≥ 0 at the two potentially existing endemic equilibrium points provided that the total population N * at the endemic equilibrium point is unique. Furthermore, simple inspection of the above equation implies strict positivity S * > 0. On the other hand, it follows from Theorem 3.1, 3.5 , that δV * / 1 ηV * e bτ γ b α /β − S * / 1 ηS * , which has a unique solution in V * for a given S * . Since there is a unique S * > 0, then there is a unique V * > 0 as a result. From 2.5 in the proof of Theorem 2.1, there is also a unique population at the infected population endemic equilibrium I * > 0, then unique related exposed and recovered equilibrium populations E * > 0 and R * > 0 from 2.4 and 2.6 , respectively. Thus, there is a unique endemic equilibrium point with all the partial populations being nonnegative. The above discussion concerning the existence of a unique endemic equilibrium point with all the partial populations being nonnegative is summarized as follows.
Theorem 3.4. Assume that V c ∈ 0, 1 and that γ b α e bτ η/ 1 δ ≤ β ≤ γ b α e bτ 1 η / 1 δ (the upper-bounding condition does not hold if min S * , V * < 1 so that N * α S S * α V V * α E E * α I I * α R R * for some positive constants α S , α V , α E , α I and α R . If N * is unique at the endemic equilibrium then, there is a unique endemic equilibrium point with all the corresponding partial populations being positive, and the following parametrical constraints hold:

3.23
The constants α S , α I , and α V satisfy either 3.15 , or 3.16 , and the constraint This result will be combined with some issues concerning the existence of limits of all the partial population at infinite time to conclude that there is a unique total population at the endemic equilibrium points so that, from Theorem 3.4, there is a unique endemic equilibrium point see Remark 5.1 and Theorem 5.2 in Section 5 .

About Infection Propagation and the Properties of Uniform Boundedness of the Total Population and Positivity of All the Partial Populations
This section discuses briefly the monotone increase of the infected population and the boundedness of the total population as well as the positivity of the model.

Theorem 4.1. If the infection propagates through t − τ, t with the infected population being monotone increasing, then
and if, furthermore, I t > I t − τ for t ∈ t * − τ, t * , then Now, rewrite 1.3 in differential equivalent form by using Leibnitz's rule as follows: i assume furthermore that ψ : e ντ β 1 δ 1 − e − b − ν τ /η b − ν e −bτ < 1, then the total population is uniformly bounded for all time, irrespective of the susceptible and vaccinated populations, for any bounded initial conditions and ii assume that the disease transmission constant is large enough satisfying β ≥ 1/ 1 δ sup t∈R 0 bη 1 η / ηe −bω I t − ω − 1 η e −bτ I t − τ subject to η/ 1 η > e b ω − τ and ω < τ, then N : R 0 → R 0 is monotone decreasing and of negative exponential order so that the total population exponentially extinguishes as a result.
Proof. Consider the SVEIRS model in differential form described by 1.1 , 1.2 , 1.4 , 1.5 , and 4.4 . Summing up the five equations, one gets directlẏ and Property i follows since ψ < 1. Two cases are now discussed separately related to the proof of Property ii . a Note that if the solution trajectory is positive subject to min S t , V t ≥ 1 equivalently if max S −1 t , V −1 t ≤ 1 , then Advances in Difference Equations so that one gets from 4.6 with identically zero upper-bound in 4.10 holds for some t ∈ R 0 if and only if N t I t 0 since b > ν and so that 4.10 still holds and the same conclusion arises. Thus, Property ii is proven.
A brief discussion about positivity is summarized in the next result. Proof. First note that all the partial populations are defined by continuous-time differentiable functions from 1.1 -1.5 . Then, if any partial population is negative, it is zero at some previous time instant. Assume that S σ ≥ 0 for σ < t and S t 0 at some time instant t. Then, from 1.1 Thus, S t ≥ 0. As a result, S t cannot reach negative values at any time instant. Assume that V σ ≥ 0 for σ < t and V t 0 at some time instant t. Then,V t νV c N t ≥ 0 from 1.2 so that V t ≥ 0. As a result, V t cannot reach negative values at any time. E t ≥ 0 for any time instant t from 1.3 . Assume that I σ ≥ 0 for σ < t and I t 0 at some time instant t. Then,İ t ≥ 0 from 1.4 . As a result, I t cannot reach negative values at any time. Finally, assume that R σ ≥ 0 for σ < t and R t 0 at some time instant t. Thus,Ṙ t γ 1 V t γ I t −I t−ω e −bω ≥ 0 from 1.5 if V t ≥ max γ/γ 1 I t− ω e −bω −I t , 0 . Thus,

17
if V t ≥ max γ/γ 1 I t − ω e −bω − I t , 0 when R t 0, then all the partial populations are uniformly bounded, since they are nonnegative and the total population N t is uniformly bounded from Theorem 4.2 i .
It is discussed in the next section that if the two above theorems related to positivity and boundedness hold, then the solution trajectories converge to either the disease-free equilibrium point or to the endemic equilibrium point.

Solution Trajectory of the SVEIRS Model
The solution trajectories of the SVEIRS differential model 1.1 -1.5 are given below.

Equation 1.3 is already in integral form. Equation 1.4 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 right-hand-side integrals exist:

5.5
Remark 5.1. If any of the above right-hand-side integrals with upper-limit ∞ 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. Now, assume that the solution trajectory converges to some locally asymptotically stable equilibrium point S * , V * , E * , I * , R * T . The above equations ensure the existence of integrands f S ∈ L 1 R 0 , f V ∈ L 1 R 0 , f I ∈ L 1 R 0 , being nonnegative, and f R t O e bt such that f R t e −bt ≤ K R t for some K R ∈ L 1 R 0 , for any initial conditions in some small neighbourhood of the equilibrium, such that

5.6
Advances in Difference Equations 19 which leads to

5.7
Any equilibrium point also satisfies the following constraints from 5.1 -5.4 , and 1.3 after performing the replacements 0 → kT for the initial time instant , t → k 1 T for the final time instant, for any given T ∈ R 0 , and taking limits as This leads to the following implicit relations being independent of T and being applicable for any disease-free or endemic equilibrium point:

Equation 1.4 yields
which is identical to 2.5 , that is, either I * 0 or 1

5.12
which is identical to 2.6 . 1 If b > 0 or V c < 1, then S t → 0 as t → ∞ is impossible.
2 V t → 0 as t → ∞ occurs if V c is eventually a function of time, rather than a real constant, subject to V c t ≤ KN −1 t e − t 0 γ 1 b δβI σ / 1 ηV σ dσ for some K ∈ R , for almost all t ∈ R 0 except possibly at a set of zero measure in the case that the total population does not extinguish.
3 I t can converge to zero exponentially with time, for instance, to the disease-free equilibrium point, while being a function of exponential order − γ b α and, in such a case, E t also converges to zero exponentially while being of exponential order − γ α and satisfying E t ≤ K ωe γ b α ω 1 δ /η e − γ α t from 1.3 .
4 R t → 0 as t → ∞ 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 .

Incorporation of Impulsive Vaccination to the SVEIRS Model
Impulsive vaccination involving culling-type 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 t k , t k 1 intervals for some given real sequence of time intervals {T k } k∈Z 0 with T k ≥ T min > 0, for all k ∈ Z 0 and an impulsive vaccination is used at time instants t k t k−1 T k−1 k−1 j 0 T j , t 0 ≥ 0 in the sequence {t k } k∈Z 0 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 {θ k } k∈Z 0 , where θ k ∈ 0, 1 : The following simple result follows trivially.

Advances in Difference Equations
The solutions in-between two consecutive impulsive vaccinations are obtained by slightly modifying 5.15 -5.18 by replacing θ k by zero and t k 1 by t k ρ < t k 1 for ρ ∈ 0, T k . The following result about the oscillatory behaviour of the vaccinated population holds.
Theorem 5.5. Assume that V c : R 0 → 0, 1 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

5.19
Then, the vaccinated population is an oscillating function if there is a culling sequence {t k } k∈Z 0 of time instants for some given real sequence {θ k } k∈Z 0 with θ k ∈ 0, 1 , of impulsive gains if any two consecutive impulsive time instants satisfy some of the two conditions below. Condition 1. V c1 t k ≤ 1 and 0 ≤ V c2 t k 1 ≤ 1 with a regular piecewise continuous vaccination V c t k ∈ V c1 t k , 1 and V c t k 1 ∈ 0, V c2 t k 1 .
Condition 2. 0 ≤ V c1 t k 1 ≤ 1 and V c2 t k ≤ 1 with a regular piecewise continuous vaccination V c t k ∈ 0, V c1 t k and V c t k 1 ∈ V c2 t k 1 , 1 .
Proof. One gets directly by using lower and upper-bounds in 5.18 via θ k 1 0 and θ k 1 1 Advances in Difference Equations 23 so that as T k → ∞ and also for some sufficiently large Thus, V c1 t k ≤ 1 and 0 ≤ V c2 t k 1 ≤ 1 imply that V t k 1 > V t k and V t k 2 < V t k 1 , and 0 ≤ V c1 t k ≤ 1 and V c2 t k 1 ≤ 1 imply that V t k 1 < V t k and V t k 2 > V t k 1 for some θ k , θ k 1 ∈ 0, 1 and some sufficiently large intervals in-between consecutive impulses T k and T k 1 via the use of an admissible regular piecewise continuous vaccination V c t k ∈ V c1 t k , 1 and V c t k 1 ∈ 0, V c2 t k 1 .
It turns out that Theorem 5.5 might be generalized by grouping a set of consecutive impulsive time instants such that V t k i 1 > V t k and V t k i j 1 < V t k i 1 for some positive integers i i k , j j i, k .
A further property now described is that of the impulsive-free infection permanence in the sense that for sufficiently large initial conditions of the infected populations the infected population exceeds a, initial conditions dependent, positive lower-bound 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 τ g β1 g β1 τ ≥ 1, g β2 g β2 τ ≥ max g β1 , g − e − γ b α τ 1 − e − γ b α τ , g g τ ≥ 1.

5.24
Then, the infection is permanent for all time if it strictly exceeds zero along the initialization interval. Proof. It is similar to that of Theorem 5.6 by zeroing either u S and u V 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 ε ∈ R 0 , there is a sufficiently large finite T T t, ε ∈ R 0 such that for any σ ≥ T and any t ∈ R 0 , one gets if the disease transmission constant is large enough to satisfy

5.29
If either u S or u V 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 .

Disease-Free Equilibrium Point
Consider now β 0.15 days −1 , ν 0.95b days −1 satisfying ν < b. The two particular cases corresponding to V c 0 and V c 1 in Section 2 will be studied separately. Thus, the following simulations have been obtained for the SVEIR system 1.1 -1.5 and V c 0.
As it can be appreciated in Figure 1, the vaccinated, exposed, infected, and removedby-immunity or "immune" or "recovered" population converge to zero and cannot be distinguished in the figure. This situation corresponds to the case when the disease naturally vanishes. On the other hand, the susceptible population presents a different dynamics, reaching a maximum and then converging to a nonzero endemic equilibrium point. Figure 2 shows a zoom on the equilibrium point reached by the model in Figure 1. It can be noted that the vaccinated, exposed, infectious, and removed populations are zero while the susceptible one converges to a number of 20 individuals. Furthermore, those values correspond to the ones stated in Theorem 2.1 for V c 0, since all the populations vanish except the susceptible which converges to S * b/ b − ν 20. If V c 1, then the solution trajectories converge to the equilibrium point as depicted in Figure 3.
In this case, only the exposed and infected tend to zero while the remaining populations tend to the values calculated in Theorem 2.1 when V c 1 14.07, E * I * 0. In the next example, the existence of an endemic equilibrium point is studied through a numerical simulation.

Endemic Equilibrium Point
In order to study the endemic equilibrium point, the value of β is changed now to a new value β 1.5 days −1 and ν 0.9b days −1 satisfying the condition β ≥ ηe bτ γ b α stated in Theorem 3.1 i for V c 0. Thus, the system's trajectories are showed in Figure 4.
A zoom on Figure 4 will show the equilibrium point of the system as represented in Figure 5. Figure 5 reveals that there is an endemic equilibrium point, associated to nonzero populations of the exposed and infectious, whose coordinates in view of 5.

6.2
As it can be verified in Figure 5, all the calculated values correspond to the ones obtained in the simulation example. Note that an endemic equilibrium point exists since the exposed and infected are different from zero. The endemic equilibrium point is lost when impulsive vaccination is applied on the system as the next section illustrates.

Effect of the Impulsive Vaccination on the Endemic Equilibrium Point
In this last simulation, the impulsive vaccination considered in Section 5 is introduced in the system. Thus, Equations 5.8 and 5.9 are used, at certain time instants, to remove population from susceptible while increasing the vaccinated. The impulsive vaccination is used once a week i.e., every 7 days with θ k θ 0.8. The following Figures 6 and 7 show the effect of the impulsive vaccination on the population of susceptible and vaccinated while Figures 8, 9, and 10 show the effect of impulses on the exposed, infected, and immune, respectively. Notice that the evolution of infected, exposed, and immune is continuous since the impulsive vaccination does not influence directly those populations. It can be appreciated in Figure 6 that, as expected, the impulsive vaccination reduces the susceptible population in a 20%, that is, such a population is reduced drastically during the first applications of the impulse. On the other hand, Figure 7 depicts the steady-state behavior of the susceptible and vaccinated. In agreement with Theorems 5.4 and 5.5 in Section 5, the endemic equilibrium point is now lost while an oscillatory periodic indeed steady-state behavior is obtained. Figures 8, 9, and 10 show the effect of impulse vaccination on the exposed, infected, and immune populations, respectively. It can be observed that the impulse vaccination reduces the maximum peak of the exposed and infected populations while increases the maximum peak of the immune population. However, the solution trajectory solutions are very similar to the nonimpulsive case after the transient. The main reason for this feature relies on the fact that the susceptible population tends to a very small value during the transient and, afterwards, the influence of its variations on the exposed, susceptible, and immune populations is less relevant. Hence, the improvement on the epidemic model state-trajectory solution is concentrated on the transient. In this way, the theoretical results of the manuscript have been illustrated through simulation examples.