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

- M. De la Sen
^{1, 2}Email author, - Ravi P. Agarwal
^{3, 4}, - A. Ibeas
^{5}and - S. Alonso-Quesada
^{1, 2}

**2011**:748608

https://doi.org/10.1155/2011/748608

© M. De la Sen et al. 2011

**Received: **17 January 2011

**Accepted: **23 February 2011

**Published: **13 March 2011

## Abstract

This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed.

## Keywords

## 1. 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).

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.

## 2. The Disease-Free Equilibrium Point

The potential existence of a disease-free equilibrium point is now discussed which asymptotically removes the disease if .

Theorem 2.1.

Two particular disease-free equilibrium points are , if , and , , , if .

If , then there is no disease-free equilibrium points.

Proof.

The proof follows directly from the above equations.

Remark 2.2.

Note that if , then . Note also that if , as in the particular case of impulsive-free SVEIRS model obtained from that discussed in [18, 19], then the disease-free equilibrium satisfies , . In such a case, the model can be ran out with population normalized to unity.

Assertion 1.

Proof.

since what also yields , that is, .

which is guaranteed under the two conditions below:

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.

Theorem 2.3.

Assume that . Then, there is a sufficiently large such that the disease-free equilibrium point is locally asymptotically stable for any constant vaccination and a sufficiently small amount , a sufficiently small delay and a sufficiently small difference delay (this being applicable if ) such that (2.30) holds.

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 .

## 3. 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.

Theorem 3.1.

Assume that . Then, the following properties hold.

- (i)Assume that for and for . 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 (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 for , and for , then there is no reachable endemic equilibrium point.

Proof.

(since, otherwise, the above disease-free equilibrium point would be considered).

Remark 3.2.

Note that if , then it follows from (1.3) and (2.4) that , for all 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 , if there is no reachable endemic equilibrium point because , 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 upper-bound increases as and increase and also as decreases.

If such an equation has two positive real roots for the susceptible equilibrium (implicitly depending on , then either or what is impossible and leads to a contradiction. Then, there is a unique nonnegative susceptible population at the two potentially existing endemic equilibrium points provided that the total population at the endemic equilibrium point is unique. Furthermore, simple inspection of the above equation implies strict positivity . On the other hand, it follows from Theorem 3.1, (3.5), that , which has a unique solution in for a given . Since there is a unique , then there is a unique 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 , then unique related exposed and recovered equilibrium populations and 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.

The constants , and satisfy either (3.15), or (3.16), and the constraint so that in (3.18).

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).

## 4. 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.

Proof.

Theorem 4.2.

Assume that . Then, the following properties hold provided that the SVEIR epidemic model (1.1)–(1.5) has nonnegative solution trajectories of all the partial populations for all time:

Proof.

- (a)

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.

Theorem 4.3.

Assume that . Then, the SVEIRS epidemic model (1.1)–(1.5) is positive in the sense that no partial population is negative at any time if its initial conditions are nonnegative and the vaccinated population exceeds a certain minimum measurable threshold in the event that the recovered population is zero as follows: if . The susceptible, vaccinated, exposed, and infected populations are nonnegative for all time irrespective of the above constraint. If, in addition, Theorem 4.2(i) holds, then all the partial populations of the SVEIRS model are uniformly bounded for all time.

Proof.

Thus, . As a result, cannot reach negative values at any time instant. Assume that for and at some time instant . Then, from (1.2) so that . As a result, cannot reach negative values at any time. for any time instant from (1.3). Assume that for and at some time instant . Then, from (1.4). As a result, cannot reach negative values at any time. Finally, assume that for and at some time instant . Thus, from (1.5) if . Thus, if when , then all the partial populations are uniformly bounded, since they are nonnegative and the total population 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.

## 5. Solution Trajectory of the SVEIRS Model

The solution trajectories of the SVEIRS differential model (1.1)–(1.5) are given below.

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 4.2 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 impulsive-free 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 disease-free 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 impulsive-free vaccination model (1.1)–(1.5) generated for finite initial conditions converges asymptotically either to the disease-free equilibrium point or to the endemic equilibrium point as time tends to infinity.

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)
- (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 disease-free 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

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).

The solutions in-between 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.

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.

if . Thus, and imply that and , and and imply that and for some and some sufficiently large intervals in-between 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 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.

Then, the infection is permanent for all time if it strictly exceeds zero along the initialization interval.

Proof.

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 "ad-hoc" 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.

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).

## 6. Simulation Results

This section contains some simulation examples illustrating the theoretical results introduced in Sections 2 and 5 concerning the existence and location of disease-free and endemic equilibrium points under regular and impulsive vaccination as well as the eventual oscillatory behavior. The objective of these examples is to numerically verify the theoretical expressions obtained there. The parameters of the model are:
(days)^{-1},
(days)^{-1},
days,
(days)^{-1},
,
days,
,
(days)^{-1}, and
days. The initial conditions are
,
,
,
, and
. Firstly, the disease-free case is considered in Section 6.1 while the endemic case will be treated in Sections 6.2 and 6.3.

### 6.1. Disease-Free Equilibrium Point

Consider now
(days)^{-1},
(days)^{-1} satisfying
. The two particular cases corresponding to
and
in Section 2 will be studied separately. Thus, the following simulations have been obtained for the SVEIR system (1.1)–(1.5) and
.

In the next example, the existence of an endemic equilibrium point is studied through a numerical simulation.

### 6.2. Endemic Equilibrium Point

^{-1}and (days)

^{-1}satisfying the condition stated in Theorem 3.1(i) for . Thus, the system's trajectories are showed in Figure 4.

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.

### 6.3. Effect of the Impulsive Vaccination on the Endemic Equilibrium Point

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.

## Declarations

### Acknowledgments

The authors thank the Spanish Ministry of Education for the Grant no. DPI2009-07197. They are also grateful to the Basque Government by its support through Grants nos. IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK SPE07UN04 and SPE09UN12. The authors are also grateful to the reviewers for their suggestions to improve the former version of the manuscript.

## Authors’ Affiliations

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