# A Delayed Epidemic Model with Pulse Vaccination

Authors: Chunjin Wei; Lansun Chen
Journal: Discrete Dynamics in Nature and Society
 (2008-02-21)

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## Abstract

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 ifR1<1,  R2>1, 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.


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## Body

## 1. Introduction

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 [7]. 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 βSqIp or βSqIp/N
which depends on different infective disease and environments
[11]. L. S. Chen and J. Chen
[12] set forth
transmission effect like the saturation effect βS(t)/(1+aS(t)) as the
infection rate. In this paper, we will adopt the infection rate βS(t)/(1+aS(t))  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 [13] 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 Section3, 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 rateβ(S/(1+aS))I, the following mathematical model is
formulated:
(2.1)S′(t)=A−βS(t)I(t)1+aS(t)−μS(t)−(1−p)μI(t)+αR(t),t≠nT,E′(t)=βS(t)I(t)1+aS(t)−βe−μτS(t−τ)I(t−τ)1+aS(t−τ)−μE(t)+(1−p)μI(t),t≠nT,I′(t)=βe−μτS(t−τ)I(t−τ)1+aS(t−τ)−(r+d+μ)I(t),t≠nT,R′(t)=rI(t)−μR(t)−αR(t),t≠nT,Here, all coefficients are
positive constants,A denotes the
influx or recruitment of the susceptible and the exposed. The death rate for
disease and physical disease rate are d and μ, respectively. r is the recovery
rate of infectious individual. θ⁢(0<θ<1) is the
proportion of those vaccinated successfully, which is called impulsive
vaccination rate. τ is the latent
period of the disease. Consider the death of
exposed individuals during latent period of disease, that is, βe−μτ(S(t−τ)I(t−τ)/(1+aS(t−τ))) term. The
disease is propagated both vertically and horizontally,  pμI(0<p<1) is the number
of newborns of infectious who transfer to the susceptible class, and (1−p)μI is the number
of newborns of the infectious who are infected vertically.The total population sizeN(t)=S(t)+E(t)+I(t)+R(t) can be
determined by the differential equation(2.2)S(t+)=(1−θ)S(t),t=nT,n=1,2,…E(t+)=E(t),t=nT,n=1,2,…I(t+)=I(t),t=nT,n=1,2,…R(t+)=R(t)+θS(t),t=nT,n=1,2,….which is derived by adding all
equations in system (2.1). So we have A−(μ+d)N(t)≤N′(t)≤A−μN(t). It follows that(2.3)N′(t)=A−μN(t)−dI(t),Before going to any detail, we simplify model (2.1)
and mainly discuss the following model:(2.4)Aμ+d≤lim⁡t→∞inf⁡N(t)≤lim⁡t→∞sup⁡N(t)≤Aμ.The initial condition of (2.4) is given
as(2.5)S′(t)=A−βS(t)I(t)1+aS(t)−μS(t)−(1−p)μI(t)+αR(t),⁢t≠nT,I′(t)=βe−μτS(t−τ)I(t−τ)1+aS(t−τ)−(r+d+μ)I(t),t≠nT,R′(t)=rI(t)−μR(t)−αR(t),t≠nT,N′(t)=A−μN(t)−dI(t),t≠nT,S(t+)=(1−θ)S(t),t=nT,n=1,2,…I(t+)=I(t),t=nT,n=1,2,…R(t+)=R(t)+θS(t),t=nT,n=1,2,…N(t+)=N(t),t=nT,n=1,2,….where C+=C([−τ,0],R+4). From biological considerations, we discuss system
(2.4) in the
closed set(2.6)ϕ(ξ)=(ϕ1(ξ),ϕ2(ξ),ϕ3(ξ),ϕ4(ξ))∈C+,ϕi(0)>0,i=1,2,3,4,where R+4 denotes the
nonnegative cone of R4 including its
lower dimensional faces. It is easy to show that Ω is positively
invariant with respect to (2.4).Before starting our main results, we give the
following lemmas.Lemma 2.1 (See [16]).
Consider the following delay differential equation:(2.7)Ω={(S,I,R,N)∈R+4∣0≤S+I+R≤Aμ,⁢N≤Aμ},where a,b,τ>0 and x(t)>0 for t∈[−τ,0]. The following hold:
(i)
ifa<b, then lim⁡t→∞⁢x(t)=0,(ii)
ifa>b, then lim⁡t→∞⁢x(t)=+∞.Lemma 2.2.
Consider the following impulsive
differential equations:(2.8)x′(t)=ax(t−τ)−bx(t),where a>0,b>0,0<θ<1. Then there
exists a unique positive periodic solution of (2.8):(2.9)u′(t)=a−bu(t),t≠nT,u(t+)=(1−θ)u(t),t=nT,⁢n∈N,which is globally asymptotically
stable, where u*=a(1−θ)(1−e−bT)/b(1−(1−θ)e−bT).


## 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, I(t)=0 for all t≥0. Under this condition, system (2.4) becomes the
following impulsive system without delay: (3.1)ue(t)=ab+(u*−ab)e−b(t−kT),kT<t<(k+1)T,  From the third and sixth
equations of system (3.1), we have lim⁡t→∞N(t)=A/μ. Further, if I(t)=0, it follows that lim⁡t→∞E(t)=0 from the second
and sixth equations of system (2.1). In the
following, we show that the susceptible population S and recovered
population R oscillate with
period T, in synchronization with the periodic impulsive
vaccination. Consider the following limit system of system (3.1): (3.2)S′(t)=A−μS(t)+αR(t),⁢t≠nT,R′(t)=μR(t)−αR(t),t≠nT,N′(t)=A−μN(t),t≠nT,S(t+)=(1−θ)S(t),t=nT,n=1,2,…R(t+)=R(t)+θS(t),t=nT,n=1,2,…N(t+)=N(t),t=nT,n=1,2,….By Lemma 2.2, we know that the periodic solution of system (3.2),
(3.3)R(t)=Aμ−S(t),S′(t)=(α+μ)(Aμ−S(t)),t≠nT,S(t+)=(1−θ)S(t),t=nT,⁢n∈N.By Lemma , we know that the periodic solution of system (),
(3.3)Se(t)=Aμ+(S*−Aμ)e−(α+μ)(t−nT),nT<t≤(n+1)T,is globally asymptotically
stable, where S*=A(1−θ)(1−e−(α+μ)T)/μ(1−(1−θ)e−(α+μ)T).DenoteR1=βe−μτδ/(1+aδ)(r+d+μ), where δ=A(1−e−(α+μ)T)/μ(1−(1−θ)e−(α+μ)T).Theorem 3.1.
IfR1<1, then the infection-free periodic solution (Se(t),0,A/μ−Se(t),A/μ) of system
(2.4) is
globally attractive.Proof.
SinceR1<1, we can choose ε>0 small enough
such that(3.4)βe−μτ(δ+ε)1+a(δ+ε)<r+d+μ.From the first equation of
system (2.4),
it follows that S′(t)≤(α+μ)((A/μ)−S(t)). Thus consider the following comparison impulsive
differential system:(3.5)z′(t)=(α+μ)(Aμ−z(t)),t≠nT,z(t+)=(1−θ)z(t),t=nT,⁢n∈N.By (3.2), we know that the
periodic solution of system (3.5),(3.6)ze(t)=Se(t)=Aμ+(S*−Aμ)e−(α+μ)(t−nT),nT<t≤(n+1)T,is globally asymptotically
stable, where S*=A(1−θ)(1−e−(α+μ)T)/μ(1−(1−θ)e−(α+μ)T). Let (S(t),I(t),R(t),N(t)) be the solution
of system (2.4)
with initial condition (2.5) and S(0+)=S0>0,z(t) be the solution
of system (3.5) with initial value z(0+)=S0. By the comparison theorem for impulsive differential
equations [17], there
exists an integer n1>0 such that(3.7)S(t)<z(t)<ze(t)+ε,nT<t≤(n+1)T,⁢n>n1,that is,(3.8)S(t)<ze(t)+ε≤A(1−e−(α+μ)T)μ(1−(1−θ)e−(α+μ)T)+ε≐η.Further,  from the second
equation of system (2.4), we have that, for all t>nT+τ,n>n1,(3.9)I′(t)≤βηe−μτ1+aηI(t−τ)−(r+d+μ)I(t).Consider the following
comparison equation:(3.10)y′(t)=βηe−μτ1+aηy(t−τ)−(r+d+μ)y(t).From (3.4), we have that(3.11)βηe−μτ1+aη<(r+d+μ).According to Lemma 2.1, we
obtain that lim⁡t→∞y(t)=0.
Set(S(t),I(t),R(t),N(t)) be the solution
of system (2.4)
with initial condition (2.5) and I(ξ)=ϕ(ξ)>0(ξ∈[−τ,0]),y(t) be the solution
of (3.10) with initial condition y(ξ)=ϕ(ξ)>0(ξ∈[−τ,0]). By the
comparison theorem in differential equation and the positivity of solution
(with I(t)≥0),  we have
that(3.12)lim⁡t→∞I(t)=0.Therefore, for any ε1>0 (sufficiently
small), there exists an n2(n2T>n1T+τ) such that 0<I(t)<ε1 for all t>n2T. By the fourth equation of system (2.4), we have(3.13)N′(t)>A−μN(t)−dε1fort>n2T.Consider the following
comparison equation: z1′(t)=(A−dε1)−μz1(t).
It is clear thatlim⁡t→∞z1(t)=(A−dε1)/μ; by the
comparison theorem, we have that there exists an integer n3>n2 such that for
all t>n3T,N(t)≥(A−dε1)/μ−ε1.  Since ε1 is arbitrarily
small, we have(3.14)lim⁡t→∞N(t)=Aμ.It follows from (3.12) and (3.14)
that, there exists n4>n3 such that(3.15)I(t)<ε1,N(t)>Aμ−ε1fort>n4T.From the second equation of
system (2.1), we have (3.16)E′(t)≤(Aβε1μ+aA+(1−p)με1)−μE(t)fort>n4T.  It is easy to obtain that there
exists an n5>n4 such
that
(3.17)E(t)<δ1+ε1fort>n5T,where δ1=(Aβε1+(μ+aA)(1−p)με1)/μ(μ+Aa). So from the
first equation of system (2.4), we have (3.18)S′(t)≥(A+αAμ−(1−p)με1−αδ1−3αε1)−(βε1+μ+α)S(t).Consider the following
comparison impulsive differential equations for t>n5T and n>n5: (3.19)u′(t)=(A+αAμ−(1−p)με1−αδ1−3αε1)−(βε1+μ+α)u(t),t≠nT,u(t+)=(1−θ)u(t),t=nT,n∈N.By Lemma 2.2, we know that the
periodic solution of system (3.19) is(3.20)ue(t)=Θ+(u*−Θ)e−(α+μ+βε1)(t−nT),nT<t≤(n+1)T,which is globally asymptotically
stable, where  (3.21)Θ=A+Aα/μ−αδ1−3αε1−(1−p)με1α+μ+βε1,u*=Θ(1−θ)(1−e−(α+μ+βε1)T)1−(1−θ)e−(α+μ+βε1)T.By using the comparison theorem
of impulsive differential equation [17], there exists an n6>n5 such
that(3.22)S(t)>ue(t)−ε1,nT<t≤(n+1)T,n>n6.Let ε1→0, then it follows from (3.8) and (3.22) that (3.23)Se(t)=Aμ(1−θe−(α+μ)(t−nT)1−(1−θ)e−(α+μ)T),nT<t≤(n+1)T,is globally attractive, that is,(3.24)lim⁡t→∞S(t)=Se(t).By the positivity of E(t) and
sufficiently small ε1, it follows from (3.17) that (3.25)lim⁡t→∞E(t)=0.  From the restriction N(t)=S(t)+E(t)+I(t)+R(t), we have
limt→∞⁡R(t)=A/μ−Se(t).
Therefore, the infection-free
periodic solution (Se(t),0,A/μ−Se(t),A/μ) is globally
attractive. This completes the proof.Corollary 3.2.
In system (2.4), the following
states are true.
(i)
IfAβe−μτ<(r+d+μ)(μ+Aa),then infection-free periodic solution(Se(t),0,A/μ−Se(t),A/μ)is globally attractive.(ii)
IfAβe−μτ>(r+d+μ)(μ+Aa)andT<T*,then infection-free periodic solution(Se(t),0,A/μ−Se(t),A/μ)is globally
attractive, whereT*=1/(α+μ)ln⁡(1+(θ(r+d+μ)μ)/(Aβe−μτ−aA(r+d+μ)−(r+d+μ)μ).(iii)
Ifθ>θ*,then infection-free periodic solution(Se(t),0,A/μ−Se(t),A/μ)is globally attractive, whereθ*=(r+dμ)e−(α+μ)T−(r+d+μ)μ+(Aβe−μτ−a(r+d+μ)A)(1−e−(α+μ)T)/μ(r+d+μ)e−(α+μ)T.Theorem3.1 determines the global attractivity of
system (2.4) in Ω for the case R1<1. From Corollary 3.2, we can see that a short pulse
periodic (with T) or a large
pulse vaccination rate (with θ) is the
sufficient condition for the global attractivity of infection-free periodic
solution (Se(t),0,A/μ−Se(t),A/μ).


## 4. Permanence

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 m>0 (independent of
the initial data) such that every solution (S(t),I(t),R(t),N(t)) with initial
conditions (2.5) of system (2.4) satisfies(4.1)lim⁡t→∞inf⁡S(t)≥m,lim⁡t→∞inf⁡I(t)≥m,lim⁡t→∞inf⁡R(t)≥m,lim⁡t→∞inf⁡N(t)≥m.Definition 4.2.
System (2.4) is said to be
permanent if there exists a compact region Ω0∈intΩ such that every
solution of system (2.4) with initial data (2.5) will eventually enter and
remain in region Ω0.Denote(4.2)R2=(βe−μτ/(r+d+μ)−a)A(1−θ)(1−e−μT)μ(1−(1−θ)e−μT),I*=Aμ(R2−1)Aβ+(1−p)μ2R2.Theorem 4.3.
IfR2>1, then there exists a positive constant m such that each
positive solution (S(t),I(t),R(t),N(t)) of system
(2.4) satisfies I(t)≥m for t large enough.Proof.
Note that the
second equation of system (2.4) can be rewritten as follows:(4.3)I′(t)=I(t)(βe−μτS(t)1+aS(t)−(r+d+μ))−βe−μτddt∫t−τtS(u)I(u)1+aS(u)du.Define(4.4)V(t)=I(t)+βe−μτ∫t−τtS(u)1+aS(u)I(u)du.According to (4.3), we calculate
the derivative of V along the
solution of (2.4):(4.5)V′(t)=I(t)(βe−μτS(t)1+aS(t)−(r+d+μ))=(r+d+μ)I(t)(βe−μτS(t)(r+d+μ)(1+aS(t))−1).Since R2>1, then I*>0 and there
exists sufficiently small ε>0 such
that(4.6)βe−μτσ(r+d+μ)(1+aσ)>1,where(4.7)σ=(A−(1−p)μI*)(1−θ)(1−e−(βI*+μ)T)(βI*+μ)(1−(1−θ)e−(βI*+μ)T)−ε>0.We claim that for any t0>0, it is impossible that I(t)<I* for all t≥t0. Otherwise, there is a t0>0 such that I(t)<I* for all t≥t0. It follows from
the first equation of (2.4) that we have(4.8)S′(t)>(A−(1−p)μI*)−(βI*+μ)S(t).Consider the following
comparison impulsive system for t≥t0:(4.9)v′(t)=(A−(1−p)μI*)−(βI*+μ)v(t),t≠nT,v(t+)=(1−θ)v(t),t=nT,⁢n∈N.According to Lemma 2.2, we
obtain that(4.10)ve(t)=A−(1−p)μI*βI*+μ+(v*−A−(1−p)μI*βI*+μ)e−(βI*+μ)(t−nT),nT<t≤(n+1)T,is the unique globally
asymptotically stable positive periodic solution, where v*=(A−(1−p)μI*)(1−θ)(1−e−(βI*+μ)T)/(βI*+μ)(1−(1−θ)e−(βI*+μ)T).
By the comparison theorem for impulsive differential
equation [17], we know
that there exists t1>t0+τ such that the
following inequality holds for t>t1:(4.11)S(t)>ve(t)−ε.Thus(4.12)S(t)>v*−ε≐σ>0fort≥t1.From (4.6), we have βe−μτσ/(r+d+μ)(1+aσ)>1. By (4.5) and
(4.12), we have(4.13)V′(t)>(r+d+μ)I(t)(βe−μτσ(r+d+μ)(1+aσ)−1)fort≥t1.Let(4.14)Il=min⁡t∈[t1,t1+τ]I(t).We will show that I(t)≥Il for all t≥t1. Otherwise, there is a T0>0 such that I(t)≥Il for t∈[t1,t1+τ+T0],⁢I(t1+τ+T0)=Il, and I′(t1+τ+T0)≤0. However, the second equation of
systems (2.4) and (4.12) imply that(4.15)I′(t1+τ+T0)≥(r+d+μ)Il(βe−μτσ(r+d+μ)(1+aσ)−1)>0.This is a contradiction, thus, I(t)≥Il for all t≥t1. As a consequence, (4.13)
leads to(4.16)V′(t)>(r+d+μ)Il(βe−μτσ(r+d+μ)(1+aσ)−1)>0fort≥t1,which implies that V(t)→∞ as t→∞. This
contradicts with V(t)≤(A/μ)(1+Aτβe−μτ/μ). Hence, for any t0>0, the inequality I(t)<I* cannot hold for
all t≥t0. Next, we are left to consider two cases:
(1)
I(t)≥I* for t large enough;(2)
I(t) oscillates
about I* for t large enough.It is clear that if I(t)≥I* for t large enough,
then our aim is obtained. So we only need consider the case (2).
Let(4.17)m=min⁡{I*2,I*e−(r+d+μ)τ}.In the following, we will show
that I(t)≥m for t large enough,
let t*>0 and ι>0 satisfy I(t*)=I(ι+t*)=I*, and I(t)<I* for t*<t<t*+ι, where t* is sufficiently
large such that S(t)>σ for t*<t<t*+ι, we can conclude
that I(t) is uniformly
continuous since the positive solution of (2.4)
is ultimately bounded and I(t) is not affected
by impulsive effects. Hence there exists a constant T1⁢(0<T1<τ, and T1 is independent
of the choice if t*) such that I(t)>I*/2 for t*≤t≤t*+T1. If ι≤T1,our aim is obtained. If
T1<ι≤τ, since I′(t)>−(r+d+μ)I(t), and I(t*)=I*, it is obvious that I(t)≥I*e−(r+d+μ)τ for t*<t<t*+ι. If ι>τ; by the second equation of (2.4), we obtain I(t)≥I*e−(r+d+μ)τ for t*<t<t*+τ. The same
arguments can be continued, we can obtain I(t)≥I*e−(r+d+μ)τ for t*+τ<t<t*+ι. Since the interval [t*,t*+ι] is arbitrarily
chosen, we can conclude that I(t)≥m for t large enough.
Based on the above discussions, the choice of m is independent
of the positive solution of (2.4), and we have proved that
any positive solution of (2.4) satisfies I(t)≥m for all
sufficiently large t. This completes the proof.Theorem 4.4.
IfR2>1, then the system (2.4) is permanent.Proof.
Suppose that(S(t),I(t),R(t),N(t)) be any solution
of (2.4). From
the first equation of (2.4), we have(4.18)S′(t)≥pA−(βAμ+μ)S(t).Similarly, we have(4.19)lim⁡t→∞S(t)≥q,where(4.20)q=pAμβA+μ2(1−θ)(1−e−(β(A/μ)+μ)T)1−(1−θ)e−(β(A/μ)+μ)T−ε.By Theorem 4.3, the third
equation of (2.4) becomes(4.21)R′(t)≥rm−(μ+α)R(t).It is easy to obtain that(4.22)R(t)≥rmμ+α−ε≐ω.
Set(4.23)Ω0={(S,I,R,N)∣q≤S,⁢m≤I,⁢ω≤R,⁢S+I+R≤Aμ,Aμ+d−ε≤N≤Aμ}.By Theorem 3.1 and above
discussions, we know that the set Ω0 is a global
attractor in Ω and, of course,
every solution of system (2.4) with initial condition (2.5) will eventually enter
and remain in region Ω0. Hence system (2.4) is permanent. This
completes the proof.
Denote (4.24)T*=−1μln⁡11−θ(1−θ(βe−μτ/(r+d+μ)−a)A(βe−μτ/(r+d+μ)−a)A−μ),θ*=1−μeμTA(βe−μτ/(r+d+μ)−a)(eμT−1)+μ.Corollary 4.5.
The following results are true.
(1)
IfT>T*,then system(2.4)is permanent.(2)
Ifθ<θ*,then system(2.4)is permanent.


## 5. Discussion

In this paper,
we introduce the delayed SEIRS epidemic model with pulse vaccination and
nonlinear incidence rate of the formβ(S(t)I(t)/(1+aS(t))). 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 R1<1,θ>θ*,
or T<T* implies that
the disease will be eradicated, whereas R2>1,θ<θ*,  or T>T* 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 R1<1 and R2>1, but for closed interval [R1,R2], 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.



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*Source: 746951-2008-03-23.xml*