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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 746951, 13 pages
A Delayed Epidemic Model with Pulse Vaccination
1Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
2School of Sciences, Jimei University, Xiamen 361021, China
Received 14 November 2007; Revised 7 January 2008; Accepted 18 February 2008
Academic Editor: Leonid Berezansky
Copyright © 2008 Chunjin Wei and Lansun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate is proposed. We analyze the dynamical behaviors of this model and point out that there exists an infection-free periodic solution which is globally attractive if , , and the disease is permanent. Our results indicate that a short period of pulse or a large pulse vaccination rate is the sufficient condition for the eradication of the disease. The main feature of this paper is to introduce time delay and impulse into SEIRS model and give pulse vaccination strategies.
Infectious diseases are usually caused by pathogenic microorganisms, such as bacteria, viruses, parasites, or fungi; the diseases can be spread directly or indirectly. The severe and sudden epidemics of infectious diseases have a great influence on the human life and socioeconomy, which compel scientists to design and implement more effective control and preparedness pro- grams. Pulse vaccination is an effective method to use in attempts to control infectious diseases.
In recent years, epidemic mathematical models of ordinary differential equations have been studied by many authors (e.g., [1–3]). In most of the research literatures, authors always assume that the disease incubation is negligible, therefore, once infected, each susceptible in- dividual becomes infectious instantaneously and later recovers with a temporary acquired im- munity. An epidemic model based on these assumptions is customarily called SIR (susceptible, infectious, recovered) model. However, many diseases incubate inside the hosts for a period of time before the hosts become infectious. We assume that a susceptible individual first goes through a latent period after infection before becoming infectious. The resulting model is called SEIRS (susceptible, exposed, infectious, recovered) model. The SEIRS infections disease model is a very important biologic model and has been studied by many authors (e.g., [4–6]).
Bilinear and standard incidence rates have been frequently used in classical epidemic models . Simple dynamics of these models seem related to such functions. These different incidence rates have been proposed by researchers. Anderson et al. pointed out that standard incidence is more suitable than bilinear incidence [8–10]. Levin et al. have adopted an incidence form like or which depends on different infective disease and environments . L. S. Chen and J. Chen  set forth transmission effect like the saturation effect as the infection rate. In this paper, we will adopt the infection rate because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters.
On the one hand, the newborns of the infectious may already be infected with the disease at birth such as hepatitis and phthisis, and so forth. This is called vertical transmission. On the other hand, some diseases may be spread from one individual to another via horizontal contacting transmission. Some epidemic models with vertical transmission were studied by many authors. However, only a few literatures  deal with the analysis of disease with pulse vaccination, vertical and horizontal transmissions.
Most of the research literature on these epidemic models are established by ODE, delayed ODE or impulsive ODE. However, impulsive equations with time delay are not many [14, 15]. In this paper, we establish a delayed SEIRS epidemic disease model with pulse vaccination and nonlinear incidence rate. We study their dynamic behaviors, establish sufficient condition for disease-eradication, as well as investigate the role of incubation in disease transmission. The main feature of this paper is to introduce time delay and pulse vaccination into epidemic model and obtain some important qualitative properties with valid pulse vaccination strategy.
The organization of this paper is as follows. In the next section, we introduce the delayed SEIRS model with pulse vaccination. To prove our main results, we also give several definitions, notations, and lemmas. In Section 3, we investigate the dynamic behavior of the model with nonlinear incidence and the sufficient condition is obtained for the global attractivity of infection-free periodic solution and the permanence of the model. In the final section, we try to interpret our mathematical results in terms of their ecological implication and also point out some future research directions.
2. Model Formulation and Preliminary
In the following model, we study a population that is partitioned into four classes, the susceptible, exposed, infectious, and recovered, with sizes denoted by S, E, I, and R, respectively, and we consider pulse vaccination strategy in the delayed SEIRS epidemic model with nonlinear incidence rate the following mathematical model is formulated:
Here, all coefficients are positive constants, denotes the influx or recruitment of the susceptible and the exposed. The death rate for disease and physical disease rate are and , respectively. is the recovery rate of infectious individual. is the proportion of those vaccinated successfully, which is called impulsive vaccination rate. is the latent period of the disease. Consider the death of exposed individuals during latent period of disease, that is, term. The disease is propagated both vertically and horizontally, is the number of newborns of infectious who transfer to the susceptible class, and is the number of newborns of the infectious who are infected vertically.
The total population size can be determined by the differential equationwhich is derived by adding all equations in system (2.1). So we have It follows that
Before going to any detail, we simplify model (2.1) and mainly discuss the following model:The initial condition of (2.4) is given aswhere . From biological considerations, we discuss system (2.4) in the closed setwhere denotes the nonnegative cone of including its lower dimensional faces. It is easy to show that is positively invariant with respect to (2.4).
Before starting our main results, we give the
Lemma 2.1 (See ). Consider the following delay differential equation: where and for The following hold:
(i)if then (ii)if then Lemma 2.2. Consider the following impulsive differential equations: where Then there exists a unique positive periodic solution of (2.8): which is globally asymptotically stable, where
3. Global Attractivity of Infection-Free Periodic Solution
In this section, we study the existence of the infection-free periodic solution of system (2.4), in which infectious individuals are entirely absent from the population permanently, that is, for all . Under this condition, system (2.4) becomes the following impulsive system without delay: From the third and sixth equations of system (3.1), we have . Further, if it follows that from the second and sixth equations of system (2.1). In the following, we show that the susceptible population and recovered population oscillate with period , in synchronization with the periodic impulsive vaccination. Consider the following limit system of system (3.1): By Lemma 2.2, we know that the periodic solution of system (3.2), is globally asymptotically stable, where
Theorem 3.1. If then the infection-free periodic solution of system
globally attractive.Proof. Since we can choose small enough
such thatFrom the first equation of
it follows that Thus consider the following comparison impulsive
differential system:By (3.2), we know that the
periodic solution of system (3.5),is globally asymptotically
stable, where Let be the solution
of system (2.4)
with initial condition (2.5) and be the solution
of system (3.5) with initial value By the comparison theorem for impulsive differential
equations , there
exists an integer such thatthat is,Further, from the second
equation of system (2.4), we have that, for all Consider the following
comparison equation:From (3.4), we have thatAccording to Lemma 2.1, we
Set be the solution of system (2.4) with initial condition (2.5) and be the solution of (3.10) with initial condition By the comparison theorem in differential equation and the positivity of solution (with ), we have thatTherefore, for any (sufficiently small), there exists an such that for all By the fourth equation of system (2.4), we haveConsider the following comparison equation:
It is clear that by the comparison theorem, we have that there exists an integer such that for all Since is arbitrarily small, we haveIt follows from (3.12) and (3.14) that, there exists such thatFrom the second equation of system (2.1), we have It is easy to obtain that there exists an such that where So from the first equation of system (2.4), we have Consider the following comparison impulsive differential equations for and : By Lemma 2.2, we know that the periodic solution of system (3.19) iswhich is globally asymptotically stable, where By using the comparison theorem of impulsive differential equation , there exists an such thatLet then it follows from (3.8) and (3.22) that is globally attractive, that is,By the positivity of and sufficiently small it follows from (3.17) that From the restriction we have Therefore, the infection-free periodic solution is globally attractive. This completes the proof.Corollary 3.2. In system (2.4), the following states are true.
(i)If then infection-free periodic solution is globally attractive.(ii)If and then infection-free periodic solution is globally attractive, where (iii)If then infection-free periodic solution is globally attractive, where
Theorem 3.1 determines the global attractivity of system (2.4) in for the case From Corollary 3.2, we can see that a short pulse periodic (with ) or a large pulse vaccination rate (with ) is the sufficient condition for the global attractivity of infection-free periodic solution
In this section, it is noted that the disease is endemic if the infectious population persists above a certain threshold for sufficiently large time. The endemicity of the disease can be well captured and studied through the notion of uniform persistence and permanence.Definition 4.1. System (2.4) is said to be uniformly persistent if there exists an (independent of the initial data) such that every solution with initial conditions (2.5) of system (2.4) satisfiesDefinition 4.2. System (2.4) is said to be permanent if there exists a compact region such that every solution of system (2.4) with initial data (2.5) will eventually enter and remain in region
Theorem 4.3. If then there exists a positive constant such that each
positive solution of system
(2.4) satisfies for large enough.Proof. Note that the
second equation of system (2.4) can be rewritten as follows:DefineAccording to (4.3), we calculate
the derivative of along the
solution of (2.4):Since then and there
exists sufficiently small such
thatwhereWe claim that for any it is impossible that for all Otherwise, there is a such that for all It follows from
the first equation of (2.4) that we haveConsider the following
comparison impulsive system for :According to Lemma 2.2, we
obtain thatis the unique globally
asymptotically stable positive periodic solution, where
By the comparison theorem for impulsive differential equation , we know that there exists such that the following inequality holds for :ThusFrom (4.6), we have By (4.5) and (4.12), we haveLetWe will show that for all . Otherwise, there is a such that for and . However, the second equation of systems (2.4) and (4.12) imply thatThis is a contradiction, thus, for all . As a consequence, (4.13) leads towhich implies that as This contradicts with Hence, for any the inequality cannot hold for all Next, we are left to consider two cases:
(1) for large enough;(2) oscillates about for large enough.It is clear that if for large enough, then our aim is obtained. So we only need consider the case
LetIn the following, we will show that for large enough, let and satisfy and for where is sufficiently large such that for we can conclude that is uniformly continuous since the positive solution of (2.4) is ultimately bounded and is not affected by impulsive effects. Hence there exists a constant and is independent of the choice if such that for If our aim is obtained. If , since , and , it is obvious that for . If by the second equation of (2.4), we obtain for The same arguments can be continued, we can obtain for . Since the interval is arbitrarily chosen, we can conclude that for large enough. Based on the above discussions, the choice of is independent of the positive solution of (2.4), and we have proved that any positive solution of (2.4) satisfies for all sufficiently large . This completes the proof.Theorem 4.4. If then the system (2.4) is permanent.Proof. Suppose that be any solution of (2.4). From the first equation of (2.4), we haveSimilarly, we havewhereBy Theorem 4.3, the third equation of (2.4) becomesIt is easy to obtain that
SetBy Theorem 3.1 and above discussions, we know that the set is a global attractor in and, of course, every solution of system (2.4) with initial condition (2.5) will eventually enter and remain in region . Hence system (2.4) is permanent. This completes the proof. Denote Corollary 4.5. The following results are true.
(1)If then system (2.4) is permanent.(2)If then system (2.4) is permanent.
In this paper, we introduce the delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate of the form As a result, it is observed that nonlinear incidence, the latent period of disease, pulse vaccination rate, and pulse vaccination period bring effects on the dynamics of our model. Theorems 3.1 and 4.4 show that or implies that the disease will be eradicated, whereas or implies that the disease will be epidemic. Our results indicate that a short pulse time or a large pulse vaccinate rate will lead to eradication of the disease. In this paper, we only discuss and but for closed interval the dynamical behaviors of system (2.4) have not been studied, that is, the threshold parameter for the reproducing number between the eradication and the permanence of the disease has not been studied, which will be left in the future research.
This work was supported by National Natural Science Foundation of China (10771179).
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