# Dynamical Analysis of SIR Epidemic Models with Distributed Delay

Authors: Wencai Zhao; Tongqian Zhang; Zhengbo Chang; Xinzhu Meng; Yulin Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction numberR is got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.


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## Body

## 1. Introduction

Infectious diseases have always been the problem that people have to face. Emerging diseases pose a continual threat to public health such as SARS and avian influenza. It is very necessary to establish and study mathematical models which can reflect the spread of the infectious diseases. A famous model in which the population is partitioned into three classes, the susceptible, infectious and recovered, with sizes denoted byS, I, and R, respectively, that could be used to describe an influenza epidemic was developed early in the 20th century by Kermack and McKendrick [1]. This model known as the susceptible-infectious-recovered (SIR) model is as follows:
(1)S′(t)=Λ-βS(t)I(t)-μS(t),I′(t)=βS(t)I(t)-rI(t)-μI(t),R′(t)=rI(t)-μR(t),
where S(t) denotes the number of members of a population susceptible to the disease, I(t) denotes the total population of infectives with some virus at time t,R(t) denotes the number of members who have been removed from the possibility of infection through a temporal immunity. In the model, parameters μ,β,r, and Λ are positive constants, where μ represents the death rates of susceptibles, infectives, and recovered. β is the contact rate and r is the recovery rate from the infected compartment. Λ is the recruitment rate of the susceptible population. The SIR infectious disease model is a basic but important biologic model and has been studied by many authors [1–17]. Many diseases have the incubation period, such as rabies; the incubation period in rabies ranges from about two weeks to several months, and rarely even to years [18–21]. Diseases with incubation period always lead to time delay in the epidemic models, so delay differential equation is widely used in the epidemic mathematical model, such as [2, 7, 8, 10, 12–16, 22]. Based on model (1), a new epidemic model with distributed delay is as follows:
(2)S′(t)=Λ-βI(t)∫-∞t‍F(t-τ)S(τ)dτ-μS(t),I′(t)=βI(t)∫-∞t‍F(t-τ)S(τ)dτ-rI(t)-μI(t),R′(t)=rI(t)-μR(t),
where the function F(t)=ae-at,a>0,∫0+∞‍F(τ)dτ=1.As is known to all, one of strategies to control infectious diseases is vaccination. Then a number of epidemic models in ecology can be formulated as dynamical systems of differential equations with vaccination [23–26]. Based on ODE, systems with sudden perturbations lead to impulsive differential equations. The theory of impulsive differential equations has been studied intensively and systematically in [27–34]. Compared to continuous constant vaccination, pulse vaccination seems more reasonable in the real world. Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as a strategy for the elimination of childhood viral infections such as measles hepatitis, parotitis, smallpox, and phthisis. Under the pulse vaccination strategy (PVS) [9, 11, 17, 28, 35–41], what we are interested is how large a fraction of the population should we keep vaccinated in order to prevent the agent from establishing; that is, it is very important for us to investigate the conditions under which a given agent can invade a partially vaccinated population. Thus, we also need to consider the following epidemic model with distributed delay and pulse vaccination strategy at fixed moments, which is more realistic as
(3)S′(t)=Λ-βI(t)∫-∞t‍F(t-τ)S(τ)dτ-μS(t),t≠nT,I′(t)=βI(t)∫-∞t‍F(t-τ)S(τ)dτ-rI(t)-μI(t),t≠nT,R′(t)=rI(t)-μR(t),t≠nT,S(t+)=(1-δ)S(t),t=nT,I(t+)=I(t),t=nT,R(t+)=R(t)+δS(t),t=nT,
where δ (with 0<δ<1) is the proportion of those vaccinated successfully to all of the susceptibles and T is a fixed positive constant and denotes the period of the impulsive effect, n={1,2,…}.The organization of this paper is as follows. In Section2, we will show the boundedness of the SIR system and give the transformation of models and some lemmas. In Section 3, we will analyze the local stability of equilibrium of system (7) and give the basic reproduction number. In Section 4.1, we will prove the existence and globally asymptotical stability of the periodic solution of the “infection-free” model. In Section 4.2, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. Finally, we give numerical analysis and biological conclusions to show our main results.


## 2. Transformation of Models and Prerequisites

For system (2), the total population size N(t)=S(t)+I(t)+R(t) satisfies N′(t)=Λ-μN(t) and limt→∞N(t)=Λ/μ. Hence it is sufficient to consider system (2) with respect to D={(S,I,R)∈R+3∣S(t)+I(t)+R(t)≤(Λ/μ)+ϵ}. Note that the variable R does not appear in the first and second equations of system (2); hence, we only need to consider the subsystem of (2) as follows:
(4)S′(t)=Λ-βI(t)∫-∞t‍F(t-τ)S(τ)dτ-μS(t),I′(t)=βI(t)∫-∞t‍F(t-τ)S(τ)dτ-rI(t)-μI(t).
In order to study the system, we can use the chain transformation Z(t)=∫-∞t‍F(t-τ)S(τ)dτ. Since
(5)∫-∞t‍F(t-τ)dτ=limA→-∞∫At‍ae-a(t-τ)dτ=1
and ∫-∞t‍F(t-τ)S(τ)dτ is convergent, then
(6)ΔZ(t)=∫-∞t+‍F(t-τ)S(τ)dτ-∫-∞t‍F(t-τ)S(τ)dτ=0.
Hence, system (4) becomes
(7)S′(t)=Λ-βI(t)Z(t)-μS(t),I′(t)=βI(t)Z(t)-rI(t)-μI(t),Z′(t)=a(S(t)-Z(t)).
We understand the relationship between the two systems as follows. If (S,I):[0,+∞)→R2 is the solution of system (4) corresponding to continuous and bounded initial function S(0)=S0,I(0)=I0, then (S,I,Z):[0,+∞)→R3 is a solution of system (7) with S(0)=S0,I(0)=I0,Z(0)=Z0, and Z(0)=∫-∞0‍F(-τ)S(τ)dτ=∫-∞0‍aeaτS(τ)dτ. Conversely, if (S,I,Z) is any solution of system (7) defined on the entire real line and bounded on (-∞,0], then Z is given by Z(t)=∫-∞t‍F(t-τ)S(τ)dτ, and so (S,I) satisfies system (4).System (7) will be analyzed with the following initial conditions S(0)=S0>0, I(0)=I0>0, Z(0)=Z0>0.By the mean value theorem of integrals, there existsς∈(-∞,t), such that
(8)Z(t)=∫-∞t‍F(t-τ)S(τ)dτ=S(ς)∫-∞t‍F(t-τ)dτ.
We know that S(t)<(Λ/μ)+ɛ and ∫-∞t‍F(t-τ)dτ=1; hence we get Z(t)<(Λ/μ)+ɛ.By the same transformation, from system (3), we have
(9)S′(t)=Λ-βI(t)Z(t)-μS(t),t≠nT,I′(t)=βI(t)Z(t)-rI(t)-μI(t),t≠nT,Z′(t)=a(S(t)-Z(t)),t≠nT,S(t+)=(1-δ)S(t),t=nT,I(t+)=I(t),t=nT,Z(t+)=Z(t),t=nT.Motivated by the application of systems (9) to population dynamics (refer to [27]), we assume that solutions of systems (9) satisfy the initial conditions ϕ∈Ch+, and ϕ(0)≥0 lie in Ch+={ϕ=(ϕ1(s),ϕ2(s),ϕ3(s))∈Ch:ϕi(0)≥0(i=1,2,3)}, where ϕi(s) is positive, bounded, and continuous function for s∈[0,+∞).We would like to have the following definitions first.Definition 1.
LetR+=[0,∞). The map V:R+×R+3→R+ is said to belong to class V0 if  (i)
Vis continuous in (kT,(k+1)T]×R+3,k∈N, and for each x∈R+3, lim(t,z)→(kT+,x)V(t,z)=V(kT+,x) exist,(ii)
V is locally Lipschitzian in x.Also we have the following lemmas.Lemma 2 (see [31]).
LetV:R+×R+3→R+, and V∈V0. Assume that
(10)D+V(t,z(t))≤(≥)g(t,V(t,z(t))),t≠nT,V(t,z(t+))≤(≥)Ψn(V(t,z(t))),t=nT,
where g:R+×R+→R is continuous in (nT,(n+1)T]×R+ and for each x∈R+, n∈N, lim(t,y)→((nT)+,x)g(t,y)=g((nT)+,x) exist; Ψn:R+→R+ is nondecreasing. Let r(t)=r(t,0,u0) be the maximal (minimal) solution of the scalar impulsive differential equation
(11)u′=g(t,u),t≠nT,u(t+)=Ψn(u(t)),t=nT,u(0+)=u0,
existing on [0,∞). Then V(0+,z0)≤(≥)u0 implies that V(t,z(t))≤(≥)r(t),t≥0, where z(t)=z(t,0,z0) is any solution of (9) existing on [0,∞).Lemma 3.
Consider the following system:(12)x′(t)=Λ-μx(t),t≠nT,y′(t)=a(x(t)-y(t)),t≠nT,x(t+)=(1-δ)x(t),t=nT,y(t+)=y(t),t=nT.
Then system (12) has a unique positive T-periodic solution x*(t),y*(t) as
(13)x*(t)=Λμ(1-δe-μ(t-nT)1-(1-δ)e-μT),y*(t)=Λμ(1+aδe-a(t-nT)(1-e-μT)-e-μ(t-nT)(1-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),x*(0+)=Λμ(1-δ1-(1-δ)e-μT),y*(0+)=Λμ(1-aδ(e-μT-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),
and for each solution, x(t)→x*(t),y(t)→x*(t) as t→∞.Proof.
Solving the first equation of system (12), we have
(14)x(t)=x(nT+)e-μ(t-nT)+Λμ(1-e-μ(t-nT)),nT<t≤(n+1)T.
Substituting x(t) into the second equation of (12), we integrate both sides in interval (nT,(n+1)T], and we get
(15)y(t)=y(nT+)e-a(t-nT)+aa-μ(x(nT+)-Λμ)(e-μ(t-nT)-e-a(t-nT))+Λμ(1-e-a(t-nT)).
By using stroboscopic map of difference equation, we have
(16)x((n+1)T+)=(1-δ)x((n+1)T)=(1-δ)x(nT+)e-μT+Λμ(1-δ)(1-e-μT),y((n+1)T+)=y((n+1)T)=y(nT+)e-aT+aa-μ(x(nT+)-Λμ)×(e-μT-e-aT)+Λμ(1-e-aT).
The fixed point of the above mapping is
(17)x*=Λμ(1-δ1-(1-δ)e-μT),y*=Λμ(1-aδ(e-μT-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)).
Therefore, we can get the following T-periodic solution (x*(t),y*(t)) of system (12):
(18)x*(t)=Λμ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,y*(t)=Λμ(1+aδ(1-e-μT)e-a(t-nT)-(1-e-aT)e-μ(t-nT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),nT<t≤(n+1)T.
Next, we will prove the attractivity of periodic solutions. Let (x(t),y(t)) be an any solution of the system (12); then, for t∈(nT,(n+1)T], we have
(19)x(t)=x*(t)+(x(nT+)-x*)e-μ(t-nT),y(t)=y*(t)+(y(nT+)-y*)e-a(t-nT)+aa-μ(x(nT+)-x*)(e-μ(t-nT)-e-a(t-nT)).
On the one hand, by the recurrence formula, we have
(20)x((n+1)T+)=(1-δ)e-μTx(nT+)+Λμ(1-δ)(1-e-μT).
Thus
(21)x(nT+)=(1-δ)ne-nμTx(0+)+[1+(1-δ)e-μT+⋯+(1-δ)n-1e-(n-1)μT]×Λμ(1-δ)(1-e-μT)=(1-δ)ne-nμTx(0+)+Λμ(1-δ)(1-e-μT)1-(1-δ)ne-nμT1-(1-δ)e-μT⟶x*(n⟶∞),
and then we have x(t)→x*(t)(n→∞).
On the other hand,(22)y((n+1)T+)=y((n+1)T)=y(nT+)e-aT+aa-μ(x(nT+)-Λμ)×(e-μT-e-aT)+Λμ(1-e-aT),
and limn→∞x(nT+)=x*, for n large enough we have the following approximate recursive formula:
(23)y((n+1)T+)=y(nT+)e-aT+aa-μ(x*-Λμ)×(e-μT-e-aT)+Λμ(1-e-aT)=y(nT+)e-aT+aδΛ(e-aT-e-μT)μ(a-μ)(1-(1-δ)e-μT)+Λμ(1-e-aT).
Thus, for any m∈N+, we get
(24)y((m+N)T+)=y(NT+)e-maT+[1+e-aT+⋯+e-(m-1)aT]×(aδΛ(e-aT-e-μT)μ(a-μ)(1-(1-δ)e-μT)+Λμ(1-e-aT))=y(NT+)e-maT+1-e-maT1-e-aT×(aδΛ(e-aT-e-μT)μ(a-μ)(1-(1-δ)e-μT)+Λμ(1-e-aT))⟶Λμ(1-aδ(e-μT-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT))(m⟶∞).
Then we have limn→∞y(nT+)=y*, thus y(t)→y*(t)(t→∞). The proof is completed.


## 3. The Stability of Equilibrium of System (7) and the Basic Reproduction Number

In this section, we will consider the local stability of equilibrium of system (7) and give the basic reproduction number. Obviously, the system (7) has a disease-free equilibrium E0(Λ/μ,0,Λ/μ) and an endemic equilibrium E*(S*,I*,Z*), where S*=(μ+r)/β, I*=(βΛ-μ(μ+r))/β(μ+r), Z*=(μ+r)/β. Let R=βΛ/μ(μ+r); we have the following theorem.Theorem 4.
System (7) always has a disease-free equilibrium E0. If and only if R>1, system (7) has an endemic equilibrium E*.


### 3.1. Local Stability of Disease-Free EquilibriumE0

We calculate the Jacobian matrix of system (7) evaluated at E0; one gets the following matrix:
(25)J(E0)=(-μ-βΛμ00βΛμ-μ-r0a0-a).
Obviously, E0 is locally asymptotically stable if β(Λ/μ)-μ-r<0 which implies that R<1, and unstable if R>1. Then R=βΛ/μ(μ+r) can be used as the basic reproductive number. Thus, we obtain the following result.Theorem 5.
IfR<1, then disease-free equilibrium E0 of system (7) is locally asymptotically stable and unstable if R>1.


### 3.2. Local Stability of Endemic EquilibriumE*

About the local stability of endemic equilibriumE*, we have the following theorem.Theorem 6.
IfR>1 and a≥r or 1<R<(r+μ)/(r-a), the equilibrium E* of system (7) is locally asymptotically stable.Proof.
The Jacobian matrix of system (7) evaluated at E* is
(26)J(E*)=(-μ-βZ*-βI*00βI*a0-a).
Let λi(i=1,2,3) be its eigenvalues with Reλ1≤Reλ2≤Reλ3. After a simple calculation, it follows that
(27)detJ(E*)=-aβ2I*Z*<0.
For detJ(E*)=λ1λ2λ3<0, there are two cases as follows:  (i)
Reλi<0 for i=1,2,3;(ii)
Reλ1<0≤Reλ2≤Reλ3.
Now we prove that the case (ii) is not true. Note thatdetJ(E*)=-aβ2I*Z*<0; one gets trJ(E*)<0, that is, λ1+λ2+λ3<0. If the case (ii) is true, we have that Re(λ1+λ2)<0 and Re(λ1+λ3)<0. The second additive compound matrix [6] of J(E*) (see the Appendix) is as follows:
(28)J[2](E*)=(-μβI*βI*0-μ-a-βZ*-a0-a),detJ[2](E*)=-a((βI*+μ)(μ+a)-β2I*Z*)=-aμ(R(a-r)+(μ+r)),
where S*=(μ+r)/β,I*=(βΛ-μ(μ+r))/β(μ+r), and Z*=(μ+r)/β is used. Notice that R>1, and two cases will happen as follows:  (a)
ifa≥r, then detJ[2](E*)<0;(b)
ifa<r, but 1<R<(r+μ)/(r-a), then detJ[2](E*)<0.
According to the property of the second additive compound matrix [6], the eigenvalues of J[2](E*) are λi+λj,1≤i<j≤3. Then, we have
(29)(λ1+λ2)(λ1+λ3)(λ2+λ3)<0.
Notice that Re(λ1+λ2)<0 and Re(λ1+λ3)<0, then we get Re(λ2+λ3)<0, which contradicts with case (ii). Therefore, Reλi<0 for i=1,2,3. So E* is locally asymptotically stable for R>1 and a≥r or 1<R<(r+μ)/(r-a). This completes the proof.


### 3.3. Analysis atR=1

In this section, we consider the stability of system (7) under R=1 using the center manifold theory, as described in [42, Theorem 4.1]. To apply this method, the following simplification and change of variables are made first. Let S=x1,I=x2,Z=x3, and the system (7) becomes
(30)x˙1=Λ-βx2x3-μx1=f1,x˙2=βx2x3-rx2-μx2=f2,x˙3=a(x1-x3)=f3,
with R=1 corresponding to β=β*=μ(μ+r)/Λ. The virus-free equilibrium is E0(Λ/μ,0,Λ/μ). The linearization matrix of system (7) around the infection-free equilibrium E0 when β=β* is (31)J(f)=(-μ-β*Λμ00β*Λμ-(μ+r)0a0-a)=(-μ-(μ+r)0000a0-a).
The matrix J(f) has eigenvalues (0,-μ,-a)T, which meets the requirement of a simple zero eigenvalue and others having negative real part. A right eigenvector η corresponding to the zero eigenvalue is η=(-(μ+r)/μ,1,-(μ+r)/μ)T and the left eigenvector satisfying ξη=1 is ξ=(0,1,0). For the system (7), we can get
(32)a=∑i,j,k=13‍ξkηiηj∂2fk∂xi∂xj(Λμ,0,Λμ)=-2Λ(μ+r)2<0,b=∑i,k=13‍ξkηi∂2fk∂xi∂β(Λμ,0,Λμ)=Λμ>0.
Thus, a<0,b>0, by item (iv) of Theorem 4.1 in [42]; we can give the following result.Theorem 7.
The disease-free equilibriumE0 for system (7) is locally asymptotically stable for R near 1.


## 3.1. Local Stability of Disease-Free EquilibriumE0

We calculate the Jacobian matrix of system (7) evaluated at E0; one gets the following matrix:
(25)J(E0)=(-μ-βΛμ00βΛμ-μ-r0a0-a).
Obviously, E0 is locally asymptotically stable if β(Λ/μ)-μ-r<0 which implies that R<1, and unstable if R>1. Then R=βΛ/μ(μ+r) can be used as the basic reproductive number. Thus, we obtain the following result.Theorem 5.
IfR<1, then disease-free equilibrium E0 of system (7) is locally asymptotically stable and unstable if R>1.


## 3.2. Local Stability of Endemic EquilibriumE*

About the local stability of endemic equilibriumE*, we have the following theorem.Theorem 6.
IfR>1 and a≥r or 1<R<(r+μ)/(r-a), the equilibrium E* of system (7) is locally asymptotically stable.Proof.
The Jacobian matrix of system (7) evaluated at E* is
(26)J(E*)=(-μ-βZ*-βI*00βI*a0-a).
Let λi(i=1,2,3) be its eigenvalues with Reλ1≤Reλ2≤Reλ3. After a simple calculation, it follows that
(27)detJ(E*)=-aβ2I*Z*<0.
For detJ(E*)=λ1λ2λ3<0, there are two cases as follows:  (i)
Reλi<0 for i=1,2,3;(ii)
Reλ1<0≤Reλ2≤Reλ3.
Now we prove that the case (ii) is not true. Note thatdetJ(E*)=-aβ2I*Z*<0; one gets trJ(E*)<0, that is, λ1+λ2+λ3<0. If the case (ii) is true, we have that Re(λ1+λ2)<0 and Re(λ1+λ3)<0. The second additive compound matrix [6] of J(E*) (see the Appendix) is as follows:
(28)J[2](E*)=(-μβI*βI*0-μ-a-βZ*-a0-a),detJ[2](E*)=-a((βI*+μ)(μ+a)-β2I*Z*)=-aμ(R(a-r)+(μ+r)),
where S*=(μ+r)/β,I*=(βΛ-μ(μ+r))/β(μ+r), and Z*=(μ+r)/β is used. Notice that R>1, and two cases will happen as follows:  (a)
ifa≥r, then detJ[2](E*)<0;(b)
ifa<r, but 1<R<(r+μ)/(r-a), then detJ[2](E*)<0.
According to the property of the second additive compound matrix [6], the eigenvalues of J[2](E*) are λi+λj,1≤i<j≤3. Then, we have
(29)(λ1+λ2)(λ1+λ3)(λ2+λ3)<0.
Notice that Re(λ1+λ2)<0 and Re(λ1+λ3)<0, then we get Re(λ2+λ3)<0, which contradicts with case (ii). Therefore, Reλi<0 for i=1,2,3. So E* is locally asymptotically stable for R>1 and a≥r or 1<R<(r+μ)/(r-a). This completes the proof.


## 3.3. Analysis atR=1

In this section, we consider the stability of system (7) under R=1 using the center manifold theory, as described in [42, Theorem 4.1]. To apply this method, the following simplification and change of variables are made first. Let S=x1,I=x2,Z=x3, and the system (7) becomes
(30)x˙1=Λ-βx2x3-μx1=f1,x˙2=βx2x3-rx2-μx2=f2,x˙3=a(x1-x3)=f3,
with R=1 corresponding to β=β*=μ(μ+r)/Λ. The virus-free equilibrium is E0(Λ/μ,0,Λ/μ). The linearization matrix of system (7) around the infection-free equilibrium E0 when β=β* is (31)J(f)=(-μ-β*Λμ00β*Λμ-(μ+r)0a0-a)=(-μ-(μ+r)0000a0-a).
The matrix J(f) has eigenvalues (0,-μ,-a)T, which meets the requirement of a simple zero eigenvalue and others having negative real part. A right eigenvector η corresponding to the zero eigenvalue is η=(-(μ+r)/μ,1,-(μ+r)/μ)T and the left eigenvector satisfying ξη=1 is ξ=(0,1,0). For the system (7), we can get
(32)a=∑i,j,k=13‍ξkηiηj∂2fk∂xi∂xj(Λμ,0,Λμ)=-2Λ(μ+r)2<0,b=∑i,k=13‍ξkηi∂2fk∂xi∂β(Λμ,0,Λμ)=Λμ>0.
Thus, a<0,b>0, by item (iv) of Theorem 4.1 in [42]; we can give the following result.Theorem 7.
The disease-free equilibriumE0 for system (7) is locally asymptotically stable for R near 1.


## 4. Disease Impulsive Control for System (9)

### 4.1. The Existence and Globally Asymptotical Stability of the “Infection-Free” Periodic Solution of System (9)

We demonstrate the expression of the infectives-free solution of the system (9) firstly, in which the infectives are entirely absent from the population permanently. Consider the infectives-free subsystem of system (9) in the form
(33)S′(t)=Λ-μS(t),t≠nT,Z′(t)=a(S(t)-Z(t)),t≠nT,S(t+)=(1-δ)S(t),t=nT,Z(t+)=Z(t),t=nT.
By Lemma 3, system (33) has a unique positive T-periodic solution (S*(t),Z*(t)) given by
(34)S*(t)=Λμ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,Z*(t)=Λμ(1+aδe-a(t-nT)(1-e-μT)-e-μ(t-nT)(1-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),nT<t≤(n+1)T,S*(0+)=Λμ(1-δ1-(1-δ)e-μT),Z*(0+)=Λμ(1-aδ(e-μT-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),
and for each solution, S(t)→S*(t),Z(t)→Z*(t) as t→∞. Hence, we have Theorem 8.Theorem 8.
The system (9) has an “infection-free” periodic solution (S*(t),0,Z*(t)) for t∈(nT,(n+1)T],n∈N.In next section, we will prove that “infection-free” periodic solution(S*(t),0,Z*(t)) is globally asymptotically stable.Theorem 9.
Let(S(t),I(t),Z(t)) be any solution of (9), and then (S*(t),0,Z*(t)) is globally asymptotically stable provided ℜ<1, where ℜ=((βΛ/μ)(T-δ(1-e-μT)/μ(1-(1-δ)e-μT)))/(μ+r)T.Proof.
Firstly, we will prove the local stability. The local stability of periodic solution(S*(t),0,Z*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. This may be written as
(35)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0≤t<T,
where Φ satisfies
(36)dΦ(t)dt=(-μ-βZ*(t)00βZ*(t)-μ-r0a0-a)Φ(t),
and Φ(0)=E, the identity matrix. Hence, the fundamental solution matrix is (37)Φ(t)=(e-μtA00e∫0t‍(βZ*(s)-μ-r)ds0C0e-at).
There is no need to calculate the exact form of A,C as it is not required in the analysis that follows. The linearization of the fourth, fifth, and sixth equations of system (9) becomes
(38)(u(nT+)v(nT+)w(nT+))=(1-δ00010001)(u(nT)v(nT)w(nT)).
The stability of the periodic solution (S*(t),0,Z*(t)) is determined by the eigenvalues of
(39)M=(1-δ00010001)Φ(T),
which are
(40)λ1=(1-δ)e-μT<1,λ2=e-aT<1,λ3=e∫0T‍(βZ*(t)-μ-r)dt.
According to Floquet theory (see [43]), (S*(t),0,Z*(t)) is locally stable if |λ3|<1.
Denote(41)ℜ=(βΛ/μ)(T-δ(1-e-μT)/μ(1-(1-δ)e-μT))(μ+r)T,
For ℜ<1, we have
(42)∫0T‍(βZ*(t)-μ-r)dt<0,
which leads to |λ3|<1. Thus the periodic solution (S*(t),0,Z*(t)) is locally stable.
In the following, we prove the global attractivity. Sinceℜ<1 holds, we have
(43)βΛμ(T-δ(1-e-μT)μ(1-(1-δ)e-μT))-(μ+r)T<0.
We can choose a ɛ>0 small enough such that
(44)η=βΛμ((1+2ɛ)T-δ(1-e-μT)μ(1-(1-δ)e-μT))-(μ+r)T<0.
Note that S′(t)≤Λ-μS(t), by impulsive differential inequalities, we have
(45)S(t)<S*(t)+ɛ
for all t large enough. For simplification, we may assume that (45) holds for all t≥0. From the third equation of system (9) and (45), we get
(46)Z′(t)≤a(S*(t)+ɛ-Z(t)),t≠nT,Z(t+)=Z(t),t=nT.
By impulsive differential inequalities, we have
(47)Z(t)<P*(t)+ɛ,
where P*(t) is the periodic solution of the following equation:
(48)P′(t)=a(S*(t)+ɛ-P(t)),t≠nT,P(t+)=P(t),t=nT,P*(t)=Λμ(1+aδe-a(t-nT)(1-e-μT)-e-μ(t-nT)(1-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT))+ɛ,nT<t≤(n+1)T.
From the second and fifth equations of the system (9), we have
(49)I′(t)≤βI(t)(P*(t)+ɛ)-μI(t)-rI(t),t≠nT,I(t+)=I(t),t=nT,
which leads to
(50)I((n+1)T)≤I(nT+)exp∫nT(n+1)T‍(β(P*(t)+ɛ)-(μ+r))dt=I(nT)exp∫nT(n+1)T‍(β(P*(t)+ɛ)-(μ+r))dt=I(nT)eη.
Hence I(nT)≤I(0)enη and I(nT)→0 as n→+∞. Therefore, I(t)→0 as t→+∞, since I(t)≤I(nT) for nT<t≤(n+1)T.
Next, we prove thatS(t)→S*(t),Z(t)→Z*(t) as t→+∞. For all ɛ>0, there must exist a T1>0 such that 0≤I(t)<ɛ for t>T1. Without loss of generality, we may assume that 0≤I(t)<ɛ for all t≥0, and then from system (9), we have
(51)Λ-βΛɛμ-μS(t)≤S′(t)≤Λ-μS(t).
Then we have q(t)<S(t)<S*(t)+ɛ1 and q(t)→q*(t) as t→+∞, where q(t) is the solution of
(52)q′(t)=Λ-βΛɛμ-μq(t),t≠nT,q(t+)=(1-δ)q(t),t=nT,q(0+)=S(0+),q*(t)=Λ(μ-βɛ)μ2(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T.
Therefore, for any ɛ1>0, there exists a T2>0 such that q*(t)-ɛ1<S(t)<S*(t)+ɛ1. Let ɛ→0; we have S*(t)-ɛ1≤S(t)≤S*(t)+ɛ1 for t large enough, which implies S(t)→S*(t) as t→+∞.
Similarly,Z(t)→Z*(t) as t→+∞ can be analyzed by the same method as the above, so we omit it. This completes the proof.


### 4.2. Permanence of System (9)

Theorem 10.
Ifℜ>1, then system (9) is permanent, that is, there exist three positive constants m1,  m2, and m3 such that S(t)≥m1,I(t)≥m2, and Z(t)≥m3 for t large enough.Proof.
Suppose that theX(t)=(S(t),I(t),Z(t)) is any positive solution of system (9). From system (9), we can get
(53)S′(t)≥Λ-βΛ2μ2-μS(t),t≠nT,S(t+)=(1-δ)S(t),t=nT.
Consider the following impulsive differential equation:
(54)G′(t)=Λ-βΛ2μ2-μG(t),t≠nT,G(t+)=(1-δ)G(t),t=nT,G(0+)=S(0+).
Then, we have S(t)≥G(t)>G*(t)-ɛ, where
(55)G*(t)=Λ-βΛ2/μ2μ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,G0*=(Λ-βΛ2/μ2)(1-δ)(1-e-μT)μ(1-(1-δ)e-μT).
So we have
(56)S(t)≥G(t)>G*(t)-ɛ>G0*-ɛ=m1>0
for t large enough. From (56) and the third and sixth equations of the system (9), we have
(57)Z′(t)>a(m1-Z(t)),t≠nT,Z(t+)=Z(t),t=nT.
Consider the following system:
(58)H′(t)=a(m1-H(t)),t≠nT,H(t+)=Z(t),t=nT.
Obviously, H(t)→m1 as t→∞, Then there exists ɛ>0 such that
(59)Z(t)>m1-ɛ=m3>0.
Next, we prove that there exists a constantm2>0 such that I(t)≥m2 for t large enough. We will do it in the following two steps.
Step (I). Sinceℜ>1, we can choose m4,ɛ>0 small enough such that
(60)ω=β((Λ-βm4μ-2ɛ)T-δ(Λ-βm4)(1-e-μT)μ2(1-(1-δ)e-μT))-(μ+r)T>0.
We will prove that I(t)<(μ/Λ)m4 cannot hold for all t≥0. Otherwise,
(61)S′(t)≥Λ-βm4-μS(t),t≠nT,S(t+)=(1-δ)S(t),t=nT.
So we have S(t)≥m(t)>m*(t)-ɛ for t large enough, where m(t) is the solution of
(62)m′(t)=Λ-βm4-μm(t),t≠nT,m(t+)=(1-δ)m(t),t=nT,(63)m*(t)=Λ-βm4μ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,m*=Λ-βm4μ(1-δ1-(1-δ)e-μT).
From the third and sixth equations of the system (9), we have
(64)Z′(t)≥a(m*(t)-ɛ-Z(t)),t≠nT,Z(t+)=Z(t),t=nT,
and Z(t)>h*(t)-ɛ, where
(65)h*(t)=Λ-βm4μ+Λ-βm4μaδ×((1-e-μT)e-a(t-nT)-(1-e-aT)e-μ(t-nT)(a-μ)(1-e-aT)(1-(1-δ)e-μT))-ɛ.
So we have
(66)I′(t)≥(β(h*(t)-ɛ)-μ-r)I(t),t≠nT,I(t+)=I(t),t=nT.
Integrating (66) on (nT,(n+1)T], we have
(67)I(n+1)T≥I(nT+)exp∫nT(n+1)T‍(β(h*(t)-ɛ)-μ-r)dt=I(nT)eω.
Then I(nT)≥I(0)enω→∞ as n→∞, which is a contradiction to the boundedness of I(t). Hence, there exists a t1>0 such that I(t1)≥(μ/Λ)m4. For the sake of simplification, we let (μ/Λ)m4 be m4.
Step (II). IfI(t)≥m4 for all t≥t1, and we let m2=m4, then our aim is obtained. Otherwise, let t*=inft>t1{t∣I(t)<m4}, there are two possible case for t*.
Case (I).t*=n1T,n1∈Z+. Then I(t)≥m4 for t∈[t1,t*) and I(t*)=m4. Choose n2,n3∈Z+, such that
(68)n2T>1μln1+m*ɛ,e-(n2+1)(μ+r)Ten3ω>1.
Let T′=(n2+n3)T, we claim that there exists a t2∈(t*,t*+T′] such that I(t2)>m4. Otherwise, consider (62) with m(n1T+)=S(n1T+) for t∈(nT,(n+1)T] and n1≤n≤n1+n2+n3. We have
(69)|m(t)-m*(t)|≤(1+m*)e-μ(t-t*)<ɛ
for (n1+n2)T<t≤(n1+n2+n3)T. So as in the above step (I), we have
(70)I(t*+T′)≥I(t*+n2T)en3ω.
From system (9), we get
(71)I′(t)≥(-μ-r)I(t),t≠nT,I(t+)=I(t),t=nT,
for t∈[t*,t*+n2T]. Integrating (71) on [t*,t*+n2T], we have
(72)I(t*+n2T)≥m4e(-μ-r)n2T.
Thus, we have
(73)I(t*+T′)≥m4e(-μ-r)n2Ten3ω>m4,
which is a contradiction.
Lett~=inft>t*{t∣I(t)>m4}, and then for t∈(t*,t~),I(t)≤m4 and I(t~)=m4, since I(t) is continuous and I(t+)=I(t) when t=nT. For t∈(t*,t~), suppose t~∈(t*+(k-1)T,t*+kT],k∈Z+,k≤n2+n3; from (71), we have
(74)I(t)≥I(t*)e(-μ-r)(t-t*)≥m4e(n2+n3)(-μ-r)T.
Let m2′=m4e(n2+n3)(-μ-r)T, hence; we have I(t)≥m2′ for t∈(t*,t~). For t>t~, the same arguments can be continued since I(t~)≥m4.
Case (II).t*≠nT,n∈Z+. Then I(t)≥m4 for t∈[t1,t*] and I(t*)=m4; suppose t*∈(n4T,(n4+1)T),n4∈Z+. There are two possible cases for t∈(t*,(n4+1)T),n4∈Z+.
Case (IIa).I(t)≤m4 for all t∈(t*,(n4+1)T). We claim that there must be a t2′∈[(n4+1)T,(n4+1)T+T′] such that I(t2′)>m4. Otherwise, consider (62) with m((n4+1)T+)=S((n4+1)T+); we have
(75)|m(t)-m*(t)|≤(1+m*)e-μ(t-(n4+1)T)
for t∈(nT,(n+1)T] and n4+1≤n≤n4+1+n2+n3. By a similar argument as in step II case (I), we get
(76)I((n4+1+n2+n3)T)≥I((n4+1+n2)T)en3ω.
Since I(t)≤m4 for t∈(t*,(n4+1)T), (71) holds on [t*,(n4+1+n2)T], so we have
(77)I((n4+1+n2)T)≥m4e(n2+1)(-μ-r)T.
Thus
(78)I((n4+1+n2+n3)T)≥m4e(n2+1)(-μ-r)Ten3ω>m4,
which is a contradiction. Let t¯=inft>t*{t∣I(t)>m4}, and then for ∈(t*,t¯),I(t)≤m4 and I(t¯)=m4. For t∈(t*,t¯), suppose t¯∈(n4T+(k′-1)T,n4T+k′T],k′∈Z+,k′≤1+n2+n3, we have
(79)I(t)≥m4e(1+n2+n3)(-μ-r)T.
Let m2=m4e(1+n2+n3)(-μ-r)T<m2′, so I(t)≥m2 for t∈(t*,t¯). For t>t¯, the same arguments can be continued since I(t¯)≥m4.
Case (IIb). There exists at∈(t*,(n4+1)T) such that I(t)>m4. Let t**=inft>t*{t∣I(t)>m4}, and then for t∈(t*,t**),I(t)≤m4 and I(t**)=m4. For t∈(t*,t**), integrating (71) on (t*,t**), we have
(80)I(t)≥I(t*)e(-μ-r)(t-t*)≥m4e(-μ-r)T>m2.
So, I(t)≥m2 for t∈(t*,t**). Since I(t**)≥m4, for t>t**, the same arguments can be continued.
Hence,I(t)≥m2 for all t≥t1. The proof is completed.


## 4.1. The Existence and Globally Asymptotical Stability of the “Infection-Free” Periodic Solution of System (9)

We demonstrate the expression of the infectives-free solution of the system (9) firstly, in which the infectives are entirely absent from the population permanently. Consider the infectives-free subsystem of system (9) in the form
(33)S′(t)=Λ-μS(t),t≠nT,Z′(t)=a(S(t)-Z(t)),t≠nT,S(t+)=(1-δ)S(t),t=nT,Z(t+)=Z(t),t=nT.
By Lemma 3, system (33) has a unique positive T-periodic solution (S*(t),Z*(t)) given by
(34)S*(t)=Λμ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,Z*(t)=Λμ(1+aδe-a(t-nT)(1-e-μT)-e-μ(t-nT)(1-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),nT<t≤(n+1)T,S*(0+)=Λμ(1-δ1-(1-δ)e-μT),Z*(0+)=Λμ(1-aδ(e-μT-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT)),
and for each solution, S(t)→S*(t),Z(t)→Z*(t) as t→∞. Hence, we have Theorem 8.Theorem 8.
The system (9) has an “infection-free” periodic solution (S*(t),0,Z*(t)) for t∈(nT,(n+1)T],n∈N.In next section, we will prove that “infection-free” periodic solution(S*(t),0,Z*(t)) is globally asymptotically stable.Theorem 9.
Let(S(t),I(t),Z(t)) be any solution of (9), and then (S*(t),0,Z*(t)) is globally asymptotically stable provided ℜ<1, where ℜ=((βΛ/μ)(T-δ(1-e-μT)/μ(1-(1-δ)e-μT)))/(μ+r)T.Proof.
Firstly, we will prove the local stability. The local stability of periodic solution(S*(t),0,Z*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. This may be written as
(35)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0≤t<T,
where Φ satisfies
(36)dΦ(t)dt=(-μ-βZ*(t)00βZ*(t)-μ-r0a0-a)Φ(t),
and Φ(0)=E, the identity matrix. Hence, the fundamental solution matrix is (37)Φ(t)=(e-μtA00e∫0t‍(βZ*(s)-μ-r)ds0C0e-at).
There is no need to calculate the exact form of A,C as it is not required in the analysis that follows. The linearization of the fourth, fifth, and sixth equations of system (9) becomes
(38)(u(nT+)v(nT+)w(nT+))=(1-δ00010001)(u(nT)v(nT)w(nT)).
The stability of the periodic solution (S*(t),0,Z*(t)) is determined by the eigenvalues of
(39)M=(1-δ00010001)Φ(T),
which are
(40)λ1=(1-δ)e-μT<1,λ2=e-aT<1,λ3=e∫0T‍(βZ*(t)-μ-r)dt.
According to Floquet theory (see [43]), (S*(t),0,Z*(t)) is locally stable if |λ3|<1.
Denote(41)ℜ=(βΛ/μ)(T-δ(1-e-μT)/μ(1-(1-δ)e-μT))(μ+r)T,
For ℜ<1, we have
(42)∫0T‍(βZ*(t)-μ-r)dt<0,
which leads to |λ3|<1. Thus the periodic solution (S*(t),0,Z*(t)) is locally stable.
In the following, we prove the global attractivity. Sinceℜ<1 holds, we have
(43)βΛμ(T-δ(1-e-μT)μ(1-(1-δ)e-μT))-(μ+r)T<0.
We can choose a ɛ>0 small enough such that
(44)η=βΛμ((1+2ɛ)T-δ(1-e-μT)μ(1-(1-δ)e-μT))-(μ+r)T<0.
Note that S′(t)≤Λ-μS(t), by impulsive differential inequalities, we have
(45)S(t)<S*(t)+ɛ
for all t large enough. For simplification, we may assume that (45) holds for all t≥0. From the third equation of system (9) and (45), we get
(46)Z′(t)≤a(S*(t)+ɛ-Z(t)),t≠nT,Z(t+)=Z(t),t=nT.
By impulsive differential inequalities, we have
(47)Z(t)<P*(t)+ɛ,
where P*(t) is the periodic solution of the following equation:
(48)P′(t)=a(S*(t)+ɛ-P(t)),t≠nT,P(t+)=P(t),t=nT,P*(t)=Λμ(1+aδe-a(t-nT)(1-e-μT)-e-μ(t-nT)(1-e-aT)(a-μ)(1-e-aT)(1-(1-δ)e-μT))+ɛ,nT<t≤(n+1)T.
From the second and fifth equations of the system (9), we have
(49)I′(t)≤βI(t)(P*(t)+ɛ)-μI(t)-rI(t),t≠nT,I(t+)=I(t),t=nT,
which leads to
(50)I((n+1)T)≤I(nT+)exp∫nT(n+1)T‍(β(P*(t)+ɛ)-(μ+r))dt=I(nT)exp∫nT(n+1)T‍(β(P*(t)+ɛ)-(μ+r))dt=I(nT)eη.
Hence I(nT)≤I(0)enη and I(nT)→0 as n→+∞. Therefore, I(t)→0 as t→+∞, since I(t)≤I(nT) for nT<t≤(n+1)T.
Next, we prove thatS(t)→S*(t),Z(t)→Z*(t) as t→+∞. For all ɛ>0, there must exist a T1>0 such that 0≤I(t)<ɛ for t>T1. Without loss of generality, we may assume that 0≤I(t)<ɛ for all t≥0, and then from system (9), we have
(51)Λ-βΛɛμ-μS(t)≤S′(t)≤Λ-μS(t).
Then we have q(t)<S(t)<S*(t)+ɛ1 and q(t)→q*(t) as t→+∞, where q(t) is the solution of
(52)q′(t)=Λ-βΛɛμ-μq(t),t≠nT,q(t+)=(1-δ)q(t),t=nT,q(0+)=S(0+),q*(t)=Λ(μ-βɛ)μ2(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T.
Therefore, for any ɛ1>0, there exists a T2>0 such that q*(t)-ɛ1<S(t)<S*(t)+ɛ1. Let ɛ→0; we have S*(t)-ɛ1≤S(t)≤S*(t)+ɛ1 for t large enough, which implies S(t)→S*(t) as t→+∞.
Similarly,Z(t)→Z*(t) as t→+∞ can be analyzed by the same method as the above, so we omit it. This completes the proof.


## 4.2. Permanence of System (9)

Theorem 10.
Ifℜ>1, then system (9) is permanent, that is, there exist three positive constants m1,  m2, and m3 such that S(t)≥m1,I(t)≥m2, and Z(t)≥m3 for t large enough.Proof.
Suppose that theX(t)=(S(t),I(t),Z(t)) is any positive solution of system (9). From system (9), we can get
(53)S′(t)≥Λ-βΛ2μ2-μS(t),t≠nT,S(t+)=(1-δ)S(t),t=nT.
Consider the following impulsive differential equation:
(54)G′(t)=Λ-βΛ2μ2-μG(t),t≠nT,G(t+)=(1-δ)G(t),t=nT,G(0+)=S(0+).
Then, we have S(t)≥G(t)>G*(t)-ɛ, where
(55)G*(t)=Λ-βΛ2/μ2μ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,G0*=(Λ-βΛ2/μ2)(1-δ)(1-e-μT)μ(1-(1-δ)e-μT).
So we have
(56)S(t)≥G(t)>G*(t)-ɛ>G0*-ɛ=m1>0
for t large enough. From (56) and the third and sixth equations of the system (9), we have
(57)Z′(t)>a(m1-Z(t)),t≠nT,Z(t+)=Z(t),t=nT.
Consider the following system:
(58)H′(t)=a(m1-H(t)),t≠nT,H(t+)=Z(t),t=nT.
Obviously, H(t)→m1 as t→∞, Then there exists ɛ>0 such that
(59)Z(t)>m1-ɛ=m3>0.
Next, we prove that there exists a constantm2>0 such that I(t)≥m2 for t large enough. We will do it in the following two steps.
Step (I). Sinceℜ>1, we can choose m4,ɛ>0 small enough such that
(60)ω=β((Λ-βm4μ-2ɛ)T-δ(Λ-βm4)(1-e-μT)μ2(1-(1-δ)e-μT))-(μ+r)T>0.
We will prove that I(t)<(μ/Λ)m4 cannot hold for all t≥0. Otherwise,
(61)S′(t)≥Λ-βm4-μS(t),t≠nT,S(t+)=(1-δ)S(t),t=nT.
So we have S(t)≥m(t)>m*(t)-ɛ for t large enough, where m(t) is the solution of
(62)m′(t)=Λ-βm4-μm(t),t≠nT,m(t+)=(1-δ)m(t),t=nT,(63)m*(t)=Λ-βm4μ(1-δe-μ(t-nT)1-(1-δ)e-μT),nT<t≤(n+1)T,m*=Λ-βm4μ(1-δ1-(1-δ)e-μT).
From the third and sixth equations of the system (9), we have
(64)Z′(t)≥a(m*(t)-ɛ-Z(t)),t≠nT,Z(t+)=Z(t),t=nT,
and Z(t)>h*(t)-ɛ, where
(65)h*(t)=Λ-βm4μ+Λ-βm4μaδ×((1-e-μT)e-a(t-nT)-(1-e-aT)e-μ(t-nT)(a-μ)(1-e-aT)(1-(1-δ)e-μT))-ɛ.
So we have
(66)I′(t)≥(β(h*(t)-ɛ)-μ-r)I(t),t≠nT,I(t+)=I(t),t=nT.
Integrating (66) on (nT,(n+1)T], we have
(67)I(n+1)T≥I(nT+)exp∫nT(n+1)T‍(β(h*(t)-ɛ)-μ-r)dt=I(nT)eω.
Then I(nT)≥I(0)enω→∞ as n→∞, which is a contradiction to the boundedness of I(t). Hence, there exists a t1>0 such that I(t1)≥(μ/Λ)m4. For the sake of simplification, we let (μ/Λ)m4 be m4.
Step (II). IfI(t)≥m4 for all t≥t1, and we let m2=m4, then our aim is obtained. Otherwise, let t*=inft>t1{t∣I(t)<m4}, there are two possible case for t*.
Case (I).t*=n1T,n1∈Z+. Then I(t)≥m4 for t∈[t1,t*) and I(t*)=m4. Choose n2,n3∈Z+, such that
(68)n2T>1μln1+m*ɛ,e-(n2+1)(μ+r)Ten3ω>1.
Let T′=(n2+n3)T, we claim that there exists a t2∈(t*,t*+T′] such that I(t2)>m4. Otherwise, consider (62) with m(n1T+)=S(n1T+) for t∈(nT,(n+1)T] and n1≤n≤n1+n2+n3. We have
(69)|m(t)-m*(t)|≤(1+m*)e-μ(t-t*)<ɛ
for (n1+n2)T<t≤(n1+n2+n3)T. So as in the above step (I), we have
(70)I(t*+T′)≥I(t*+n2T)en3ω.
From system (9), we get
(71)I′(t)≥(-μ-r)I(t),t≠nT,I(t+)=I(t),t=nT,
for t∈[t*,t*+n2T]. Integrating (71) on [t*,t*+n2T], we have
(72)I(t*+n2T)≥m4e(-μ-r)n2T.
Thus, we have
(73)I(t*+T′)≥m4e(-μ-r)n2Ten3ω>m4,
which is a contradiction.
Lett~=inft>t*{t∣I(t)>m4}, and then for t∈(t*,t~),I(t)≤m4 and I(t~)=m4, since I(t) is continuous and I(t+)=I(t) when t=nT. For t∈(t*,t~), suppose t~∈(t*+(k-1)T,t*+kT],k∈Z+,k≤n2+n3; from (71), we have
(74)I(t)≥I(t*)e(-μ-r)(t-t*)≥m4e(n2+n3)(-μ-r)T.
Let m2′=m4e(n2+n3)(-μ-r)T, hence; we have I(t)≥m2′ for t∈(t*,t~). For t>t~, the same arguments can be continued since I(t~)≥m4.
Case (II).t*≠nT,n∈Z+. Then I(t)≥m4 for t∈[t1,t*] and I(t*)=m4; suppose t*∈(n4T,(n4+1)T),n4∈Z+. There are two possible cases for t∈(t*,(n4+1)T),n4∈Z+.
Case (IIa).I(t)≤m4 for all t∈(t*,(n4+1)T). We claim that there must be a t2′∈[(n4+1)T,(n4+1)T+T′] such that I(t2′)>m4. Otherwise, consider (62) with m((n4+1)T+)=S((n4+1)T+); we have
(75)|m(t)-m*(t)|≤(1+m*)e-μ(t-(n4+1)T)
for t∈(nT,(n+1)T] and n4+1≤n≤n4+1+n2+n3. By a similar argument as in step II case (I), we get
(76)I((n4+1+n2+n3)T)≥I((n4+1+n2)T)en3ω.
Since I(t)≤m4 for t∈(t*,(n4+1)T), (71) holds on [t*,(n4+1+n2)T], so we have
(77)I((n4+1+n2)T)≥m4e(n2+1)(-μ-r)T.
Thus
(78)I((n4+1+n2+n3)T)≥m4e(n2+1)(-μ-r)Ten3ω>m4,
which is a contradiction. Let t¯=inft>t*{t∣I(t)>m4}, and then for ∈(t*,t¯),I(t)≤m4 and I(t¯)=m4. For t∈(t*,t¯), suppose t¯∈(n4T+(k′-1)T,n4T+k′T],k′∈Z+,k′≤1+n2+n3, we have
(79)I(t)≥m4e(1+n2+n3)(-μ-r)T.
Let m2=m4e(1+n2+n3)(-μ-r)T<m2′, so I(t)≥m2 for t∈(t*,t¯). For t>t¯, the same arguments can be continued since I(t¯)≥m4.
Case (IIb). There exists at∈(t*,(n4+1)T) such that I(t)>m4. Let t**=inft>t*{t∣I(t)>m4}, and then for t∈(t*,t**),I(t)≤m4 and I(t**)=m4. For t∈(t*,t**), integrating (71) on (t*,t**), we have
(80)I(t)≥I(t*)e(-μ-r)(t-t*)≥m4e(-μ-r)T>m2.
So, I(t)≥m2 for t∈(t*,t**). Since I(t**)≥m4, for t>t**, the same arguments can be continued.
Hence,I(t)≥m2 for all t≥t1. The proof is completed.


## 5. Numerical Analysis and Conclusion

To verify the theoretical results obtained in this paper, we will give some numerical simulations. We consider the hypothetical set of parameter values asΛ=0.2, μ=0.2, β=0.1, r=0.3, and a=0.1 with (S0,I0,Z0)=(0.7,0.3,0.3). By calculation, we know that R=0.2<1, and according to Theorem 5, and we know that the disease-free equilibrium E0(1,0,1) of system (7) is locally asymptotically stable for this case (see Figure 1). We set the hypothetical set of parameter values as Λ=1.8, μ=0.2, β=0.08, r=0.3, a=0.1 with (S0,I0,Z0)=(0.7,0.3,0.3). By calculation, we know that R=1.44>1, and according to Theorem 6, we know that the endemic equilibrium E*(6.25,1.1,6.25) of system (7) is locally asymptotically stable for this case (see Figure 2).Illustration of basic behavior of system (7) on the threshold values R=0.2<1, where Λ=0.2,μ=0.2,β=0.1,r=0.3,a=0.1, the initial value is (S0,I0,Z0)=(0.7,0.3,0.3), and E0=(1,0,1).
(a)
Time series ofS(t),I(t),Z(t)
(b)
Phase portrait ofS(t),I(t)
(c)
Phase portrait ofS(t),Z(t)
(d)
Phase portrait ofI(t),Z(t)
(e)
Phase portrait ofS(t),I(t),Z(t)Illustration of basic behavior of system (7) on the threshold values R=1.44>1, where Λ=1.8,μ=0.2,β=0.08,r=0.3,a=0.1, the initial value is (S0,I0,Z0)=(0.7,0.3,0.3), and E*=(6.25,1.1,6.25).
(a)
Time series ofS(t),I(t),Z(t)
(b)
Phase portrait ofS(t),I(t)
(c)
Phase portrait ofS(t),Z(t)
(d)
Phase portrait ofI(t),Z(t)
(e)
Phase portrait ofS(t),I(t),Z(t)On the other hand, we consider the hypothetical set of parameter values asΛ=1,μ=0.2,β=0.5,r=0.6,a=3, and T=1 with (S0,I0,Z0)=(0.7,0.3,0.3). If δ=0.8, by calculation, we know that ℜ=0.4155<1. According to Theorem 9, we know that the “infection-free” periodic solution of system (9) is globally asymptotically stable for this case (see Figure 3). We can explain this in the epidemiology that if we take such a strategy by improving vaccination proportion of susceptible persons in practice, as a result, the infectious population vanishes; that is, diseases eliminate. Figures 3(b), 3(c), 3(d), and 3(e) show the “infection-free” periodic solution under ℜ=0.4155<1. In contrast, if we decrease vaccination proportion of susceptible persons to 0.2, the diseases will be permanent (in this case ℜ=1.4831.) (see Figure 4). Comparing Figure 1 with Figure 3 and Figure 2 with Figure 4, we find that the impulse vaccination proportion of susceptible persons has played a very important role in the actual epidemic prevention.Illustration of basic behavior of system (9) on the threshold values ℜ=0.4155<1, where Λ=1,μ=0.2,β=0.5,r=0.6,a=3,T=1, and δ=0.8.
(a)
Time series ofS(t),I(t),Z(t)
(b)
Phase portrait ofS(t),I(t)
(c)
Phase portrait ofS(t),Z(t)
(d)
Phase portrait ofI(t),Z(t)
(e)
Phase portrait ofS(t),I(t),Z(t)Illustration of basic behavior of system (9) on the threshold values ℜ=1.4831>1, where Λ=1,μ=0.2,β=0.5,r=0.6,a=3,T=1, and δ=0.2.
(a)
Time series ofS(t),I(t),Z(t)
(b)
Phase portrait ofS(t),I(t)
(c)
Phase portrait ofS(t),Z(t)
(d)
Phase portrait ofI(t),Z(t)
(e)
Phase portrait ofS(t),I(t),Z(t)In this paper, a SIR epidemic model with distributed delay is proposed. The dynamics behavior of the model without vaccination or under impulsive vaccination is studied, respectively. By using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed and by using Floquet's theorem, small-amplitude perturbation skills and comparison theorem the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model under impulsive vaccination are obtained. Lastly, we give some numerical simulation to illustrate our main conclusions. We think our mathematical results would be helpful in diseases control.



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*Source: 154387-2013-07-22.xml*