# An SEIV Epidemic Model for Childhood Diseases with Partial Permanent Immunity

Authors: Mei Bai; Lishun Ren
Journal: Computational and Mathematical Methods in Medicine
 (2015)

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## Abstract

An SEIV epidemic model for childhood disease with partial permanent immunity is
studied. The basic reproduction numberR0 has been worked out. The local and global asymptotical
stability analysis of the equilibria are performed, respectively. Furthermore, if we take the treated
rate τ as the bifurcation parameter, periodic orbits will bifurcate from endemic equilibrium when
τ passes through a critical value. Finally, some numerical simulations are given to support our
analytic results.


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## Body

## 1. Introduction

It is primarily important for health administrators to protect children from disease that can be prevented by vaccination. Although preventive vaccines have reduced the incidence of infectious diseases among children, childhood disease is an important public health problem. We often use mathematical models to realize the transmission dynamics of childhood diseases and to estimate control programs [1–4]. Recently, many scholars study the SEIV epidemic models [5, 6]. In those models, let S(t), I(t), and V(t), respectively, represent the number of susceptible individuals at time t, infective individuals at time t, and vaccinated individuals at time t. At the earliest, most researches on these types of models assume that the disease incubation is negligible, so that each susceptible individual, once infected, instantaneously turns into infectious and later recovers obtaining a permanent immunity. Soon afterwards, the models become more general. Researchers assume that a susceptible individual first goes through a latent period after infection before becoming infectious (we called E represents exposed individuals but not yet infectious).In [7], the authors discussed the following model: (1)S˙(t)=A(1-p)+ωV-μS-βSI,V˙(t)=Ap-ωV-μV+τI,E˙(t)=βSI-μE-σE,I˙(t)=σE-μI-τI,where all parameters are positive. Parameter A represents the number of additional populations of childhood; ω represents the rate at which vaccine wanes; μ represents the natural death rate; β represents the rate at which susceptible individuals become infected by those who are infectious; p represents the fraction of recruited individuals who are vaccinated; τ represents the rate at which infected individuals are treated; and σ represents the rate at which exposed individuals become infectious.In model (1), βSI is called incidence rate which plays an important role in the transmission dynamics. In addition, incidence rate can determine the tendency of epidemics. At the earliest, in the classical epidemic disease model, scholars made much focus on the bilinear incidence [8, 9]. In 1945, Wilson and Worcester discussed the nonlinear incidence rate [10, 11]. Later, the incidence function grows into more general nonlinear forms. In [12], the authors have considered a SEIV model with nonlinear incidence rate βSI(1+αI). The paper discussed the basic reproduction of the system and bifurcation phenomenon. And this incidence function is more in line with actual situation. One of the strategies to control infectious diseases is vaccination in [13, 14].And under the above circumstance, in [15], the authors have studied the following model: (2)S˙(t)=A(1-p)+ωV-μS-βSI(1+αI),V˙(t)=Ap-ωV-μV+τI,E˙(t)=βSI(1+αI)-μE-σE,I˙(t)=σE-μI-τI.In [15] they have supplied a framework of discussing the transmission dynamics of the epidemic model where the preventive vaccine may lose efficacy over time. And it has showed that if the vaccination coverage level is below the threshold, the disease will persist within the population. In addition, if the vaccination coverage level exceeds a certain threshold value, the disease can be eradicated from the population through constructing a proper Lyapunov function by using global stability analysis of the model.In the process of treatment, some patients can not be cured; therefore we should consider the disease-caused death on the basic of the above models and make the parameterϵ be the rate at which infectious individuals lose their life due to disease during the process of treatment. Moreover, about some diseases, some cured patients can not obtain a permanent immunity. Thus, this paper also considers the SEIV epidemic models for childhood disease with partial permanent immunity based on above models and denotes τ as the rate of I transforming to S. Namely, when τ=0, all recoverers obtain permanent immunity. When τ=1, all recoverers become susceptible individuals. When 0<τ<1, partial infective individuals become susceptible individuals and the number is τI. So model (2) is transformed to model (3). Model (3) is described as follows: (3)S˙(t)=(1-p)A+ωV+τI-μS-βSI(1+αI),V˙(t)=pA-ωV-μV,E˙(t)=βSI(1+αI)-μE-σE,I˙(t)=σE-μI-τI-ϵI.Assume the initial values are satisfied with the following:(4)S(0)≥0,V(0)≥0,E(0)≥0,I(0)≥0.System (3) which we present will be analyzed to decide the optimal vaccine coverage level needed to control the disease. The rest of this paper is organized as follows. In Section 2, we calculate the basic reproduction number R0, which determines the spread of infection. In Section 3, the local stability of equilibria is analyzed. We discuss the bifurcation phenomenon and illustrate that when the treated rate τ crosses through a critical value, system (3) undergoes Hopf bifurcation at the positive equilibrium in Section 4. By constructing the Lyapunov function and a generalization of the Poincaré-Bendixson criterion, we discuss the global stability of disease-free equilibrium and endemic equilibrium, respectively, in Section 5. Some numerical examples are presented to illustrate theoretical analysis in Section 6. In Section 7 we discuss our findings.


## 2. The Basic Reproduction Number

In the following, we will calculate the basic reproduction number of system (3). The basic reproduction number, denoted by R0, is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [16]. Obviously, system (3) always has a disease-free equilibrium P0(A[μ(1-p)+ω]/μ(μ+ω),pA/(μ+ω),0,0); that is, E=I=0. And {(S,E,I,V)∣S>0, E≥0,I≥0,V>0} is a positively invariant set of system (3). Adding up the four equations in system (3), we can obtain (5)ddtS+E+I+V=A-μ(S+E+I+V)-ϵI=μ[Aμ-(S+E+I+V)]-ϵI≤μ[Aμ-(S+E+I+V)].And limsup⁡t→∞⁡(S+E+I+V)≤A/μ. Therefore, the set (6)Ω=(S,E,I,V)∣S>0,E≥0,I≥0,V>0,AμmmcS+E+I+V≤Aμis positively invariant for system (3). Next we will discuss the dynamic characteristic of system (3) on Ω. Set x=(E,I,S,V)T; then system (3) can be rewritten as (7)dxdt=Φ(x)-Ψ(x),where (8)Φ(x)=βSI(1+αI)000,Ψ(x)=μE+σEμI+τI+ϵI-σE-A(1-p)-ωV-τI+μS+βSI(1+αI)-pA+ωV+μV.Define (9)F=∂Φi∂xj(x0),V=∂Ψi∂xj(x0)with1≤i,j≤2.We have (10)F=0βA[μ(1-p)+ω]μ(μ+ω)00,V=μ+σ0-σμ+τ+ϵ.It is easy to get (11)V-1=1μ+σ0σ(μ+σ)(μ+τ+ϵ)1μ+τ+ϵ.FV-1 develops a meaningful definition of R0 and is the expected number of new infections for system (3). ρ(FV-1)=σβA[μ(1-p)+ω]/μ(μ+ω)(μ+σ)(μ+τ+ϵ) is the spectral radius of matrix FV-1. Thus by [16] (12)R0=σβA[μ(1-p)+ω]μ(μ+ω)(μ+σ)(μ+τ+ϵ).Set (13)R0*=1αμM·[2αμMβ[M-(μ+ω)στ]-β[M-στ(μ+ω)]],R1*=β[M-τσ(ω+μ)]αμM,where M=(μ+ω)(μ+σ)(μ+τ+ϵ).Lemma 1.
Assume thatR0, R0*, and R1* are defined as (12),  (13), and M>στ(μ+ω); then (i)
R0*≤1;(ii)
R1*<1⇔R1*<R0*.Proof.
(i) From the definition ofR0*, we know (14)R0*-1=2αμMβ[M-(μ+ω)στ]mmcαμMβ[M-(μ+ω)στ]-βM-στμ+ω-αμMαμM-1,for M>στ(μ+ω) and α,β,μ,M>0; then (15)β[M-στ(μ+ω)]≥0,αμM≥0,β[M-στ(μ+ω)]+αμM≥2αμMβ[M-(μ+ω)στ];thus (16)R0*-1≤0;that is (17)R0*≤1.
(ii) From the definition ofR1*, we know (18)R1*<1⟺R1*-1=β[M-τσ(ω+μ)]-αμMαμM<0⟺β[M-τσ(ω+μ)]<αμM,R1*<R0*⟺R1*-R0*=2βM-τσω+μ-2αμMβM-μ+ωσταμM<0⟺β[M-τσ(ω+μ)]<αμMβ[M-(μ+ω)στ]⟺β[M-τσ(ω+μ)]<αμM;this completes the proof.


## 3. Local Stability of Equilibria

In the following, we will discuss the local stability of the equilibriaP0 and P*.Theorem 2 (see [16]).
The disease-free equilibriumP0 is locally asymptotically stable if R0<1; it is unstable if R0>1.Theorem 3.
(i) SupposeR1*<1. When R0<R0* system (3) has no real equilibria; when R0*<R0<1 there are two endemic equilibria, P1* and P2*, and when R0≥1 there is only one endemic equilibrium P*.
(ii) SupposeR1*>1. When R0<R0* system (3) has no real equilibria; when R0*<R0<1 there is no endemic equilibria, and when R0≥1 there is only one unique endemic equilibrium P*.Proof.
Through the following system, we can calculate the endemic equilibriaP*(S*,V*,E*,I*): (19)A(1-p)+ωV*+τI*-μS*-βS*I*(1+αI*)=0,Ap-ωV*-μV*=0,βS*I*(1+αI*)-μE*-σE*=0,σE*-μI*-τI*-ϵI*=0.We can get (20)E*=(μ+τ+ϵσ)I*,S*=(μ+σ)(μ+τ+ϵ)σβ(1+αI*),V*=pAμ+ω.I* is satisfied with the following equation and is positive: (21)k1I*2+k2I*+k3=0,where (22)k1=αβ[(μ+ω)στ-M],k2=M(μαR0-β)+σβτ(μ+ω),k3=μM(R0-1).
We have(23)k1<0,k2>0⟺R0>R1*;k3>0⟺R0>1;it indicates the case of equilibria for system (3). More specifically, when k3>0 system (3) has only one endemic equilibrium; when k3<0, k2>0, and (k2)2-4k1k3>0 it has two endemic equilibria; otherwise it has no endemic equilibria by the Descartes rule of signs. And, for k3<0, k2>0, and (k2)2-4k1k3=0, that is, when R0=R0*, we notice there exists a bifurcation point. Actually, the formula k22-4k1k3 can be represented with respect to R0 so that(24)k22-4k1k3=μMαR0+βM-στμ+ω2-4μMαβ[M-στ(μ+ω)].Hence, (k2)2-4k1k3≥0 when R0≥R0*. Considering all the analysis results, (i) and (ii) can be obtained easily.Theorem 4.
ForR0>1, the endemic equilibrium P* of system (3) is locally asymptotically stable satisfying c3>0 and c1c2-c3>0, where c1, c2, and c3 are shown in the following proof.Proof.
System (3) has only one endemic equilibrium P* for R0>1. At the equilibrium P*=(S*,V*,E*,I*) the matrix of the linearized system (3) is(25)J(P*)=-μ-βI*(1+αI*)ω0τ-βS*(1+2αI*)0-ω-μ00βI*(1+αI*)0-μ-σβS*(1+2αI*)00σ-τ-μ-ϵ.The characteristic equation is (26)(λ+ω+μ)(λ3+c1λ2+c2λ+c3)=0,where (27)c1=(μ+E)+(2μ+τ+σ+ϵ)=3μ+τ+σ+ϵ+E,c2=(μ+E)(2μ+τ+σ+ϵ)+(τ+μ+ϵ)(μ+σ)-σF,c3=(μ+E)(τ+μ+ϵ)(μ+σ)+στE-μσF,E=βI*(1+αI*),F=βS*(1+2αI*).It is easy to get (28)c1c2-c3=(2μ+σ+E)τ2+μ+E5μ+2σ+2ϵ+Emmmc+(μ+σ)(4μ+σ+2ϵ+E)mmmc-σF-(μ+E)(μ+σ)-σEτ+(μ+E)(σ+ϵ+E+3μ)(2μ+σ+ϵ)+(μ+σ)(3μ+σ+ϵ+E)(μ+ϵ)-σF(3μ+σ+ϵ+E)-(μ+E)(μ+σ)(μ+ϵ)+μσF=Dτ2+Gτ+H.It is clear that c1>0. By the Hurwitz criterion, epidemic equilibrium P* is locally asymptotically stable for c3>0 and c1c2-c3>0.


## 4. Bifurcation Analysis

From Theorem3 we can see that R0=1 is a bifurcation value. Actually, the disease-free equilibrium changes its stability when being across R0=1. Next, we investigate the nature of the bifurcation concerning the disease-free equilibrium P0(A[μ(1-p)+ω]/μ(μ+ω),pA/(μ+ω),0,0) when R0=1. In other words, we will discuss under what conditions system (3) can undergo a forward or a backward bifurcation. And we need the results in [17, 18]. In order to introduce it, consider the following equation which has a parameter ϕ: (29)dxdt=f(x,ϕ);f:Rn×Rn,f∈C2(Rn×Rn).Without loss of generality, for all values of the parameter ϕ, assume 0 is an equilibrium for system (29); that is, (30)f(0,ϕ)≡0,∀ϕ=0.Lemma 5 (see [17]).
Suppose the following.(A1)
Q=Dxf(0,0) is the linearization matrix of system (29) around the equilibrium x=0 with ϕ evaluated at 0. 0 is a simple eigenvalue of Q and all other eigenvalues of Q have negative real parts.(A2)
MatrixQ has a (nonnegative) right eigenvector V and a left eigenvector W with respect to the zero eigenvalue.Definefk as the kth component of f, and (31)a=∑k,i,j=1n‍wkvivj∂2fk∂xi∂xj(0,0),b=∑k,i=1n‍wkvi∂2fk∂xi∂ϕ(0,0).And a and b totally decide the local dynamic of system (29) around x=0.(i)
Considera>0, b>0. If ϕ<0, with |ϕ|≪1, x=0 is locally asymptotically stable and there exists a positive unstable equilibrium; if 0<ϕ≤1, x=0 is unstable and there exists a negative and locally asymptotically stable equilibrium.(ii)
Considera<0, b<0. If ϕ<0, with |ϕ|≪1, x=0 is unstable; if 0<ϕ≤1, x=0 is locally asymptotically stable and there exists a positive unstable equilibrium.(iii)
Considera>0, b<0. If ϕ<0, with |ϕ|≪1, x=0 is unstable and there exists a locally asymptotically stable negative equilibrium; if 0<ϕ≤1, x=0 is stable and a positive unstable equilibrium emerges.(iv)
Considera<0, b>0. If ϕ changes from negative to positive,   x=0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.Remark 6.
The requirement thatV is nonnegative is unnecessary by [17].It seems that a transcritical bifurcation occurs atϕ=0: more specifically, the bifurcation at ϕ=0 is forward when a<0 and b>0; the bifurcation at ϕ=0 is backward when a>0 and b>0.Next considerϕ=0 as the bifurcation parameter, so that R0<1 for ϕ<0 and R0>1 for ϕ>0 and so that x0 is a disease-free equilibrium for system (29) of all values of ϕ.Take into account the following system:(32)dxdt=f(x,ϕ),where f is continuously differentiable at least twice in both x and ϕ. The disease-free equilibrium is the line (x0;ϕ). And the disease-free equilibrium changes its local stability at the point (x0;ϕ) [16].Next we will exhibit that there exist nontrivial equilibria near the bifurcation point(x0;0).LetS=x1, V=x2, E=x3, and I=x4; then system (3) becomes (33)x1˙t=A(1-p)+ωx2(t)+τx4(t)-μx1(t)-βx1(t)x4(t)(1+αx4(t))≔f1,x2˙(t)=Ap-ωx2(t)-μx2(t)≔f2,x3˙(t)=βx1(t)x4(t)(1+αx4(t))-μx3(t)-σx3(t)≔f3,x4˙(t)=σx3(t)-μx4(t)-τx4(t)-ϵx4(t)≔f4.We will show that system (33) may exhibit a backward bifurcation when R0=1 by applying Lemma 5. Think of the disease-free equilibrium P0(A[μ(1-p)+ω]/μ(μ+ω),pA/(μ+ω),0,0) and notice that the condition R0=1 can be seen as β=β*=μ(μ+τ+ϵ)(μ+σ)(μ+ω)/σA[(1-p)μ+ω] in terms of the parameter β.Calculate the eigenvalues of the following matrix:(34)J(P0,β*)=-μω0τ-β*A[μ(1-p)+ω]μ(μ+ω)0-ω-μ0000-μ-σβ*A[μ(1-p)+ω]μ(μ+ω)00σ-τ-μ-ϵ;we can obtain λ1=-μ, λ2=-μ-ω, λ3=-2μ-σ-τ, and λ4=0.The matrixJ(P0,β*) has a simple eigenvalue of 0; and all others have negative real parts. Thus, we can make use of the center manifold theory. The disease-free equilibrium P0 is a nonhyperbolic equilibrium when β=β* (i.e., when R0=1). This completes the verification with respect to (A1) of Lemma 5.Now we setV=(v1,v2,v3,v4)T as a right eigenvector associated with the zero eigenvalue λ4=0. It is calculated by (35)-μω0τ-β*A[μ(1-p)+ω]μ(μ+ω)0-ω-μ0000-μ-σβ*A[μ(1-p)+ω]μ(μ+ω)00σ-τ-μ-ϵv1v2v3v4=0.Expanding (35), we can have (36)-μv1+ωv2+τv4-β*A[μ(1-p)+ω]μ(μ+ω)v4=0,(-ω-μ)v2=0,(-μ-σ)v3+β*A[μ(1-p)+ω]μ(μ+ω)v4=0,σv3-(τ+μ+σ)v4=0.Expanding (36), we have (37)V=(μ+σ)[στ(μ+ω)-M]μM,0,1,στ+μ+ϵT.And the left eigenvector W=(w1,w2,w3,w4) satisfying W·V=1 is obtained by (38)-μw1=0,ωw1-(ω+μ)w2=0,-(μ+σ)w3+σw4=0,τw1-β*A[μ(1-p)+ω]μ(μ+ω)w1+β*A[μ(1-p)+ω]μ(μ+ω)w3-(τ+μ+ϵ)w4=0.From (38), the left eigenvector W turns out to be (39)W=0,0,(μ+τ+ϵ)2μ+σ+τ+ϵ,(μ+σ)(μ+τ+ϵ)σ(2μ+σ+τ+ϵ)T.Computing the following formulas, we get (40)∂2f1∂x1∂x4=∂2f1∂x4∂x1=-β,∂2f1∂x42=-2αβA[μ(1-p)+ω]μ(μ+ω),∂2f3∂x1∂x4=∂2f3∂x4∂x1=β,∂2f3∂x42=2αβA[μ(1-p)+ω]μ(μ+ω),∂2f1∂x4∂β=∂2f1∂β∂x4=-A[μ(1-p)+ω]μ(μ+ω),∂2f3∂x4∂β=∂2f3∂β∂x4=-A[μ(1-p)+ω]μ(μ+ω),and all the other second-order partial derivatives are equal to zero.So, we evaluatea and b as follows: (41)a=∑k,i,j=14‍wkvivj∂2fk∂xi∂xj(P0,β*),b=∑k,i=14‍wkvi∂2fk∂xi∂β(P0,β*).From system (33), and the terms (∂2fk/∂xi∂xj)(P0,β*) and (∂2fk/∂xi∂β)(P0,β*) which are nonzero, the following are deduced: (42)a=2w1v1v4∂2f1∂x1∂x4(P0,β*)+w1v42∂2f1∂x42(P0,β*)+2w3v1v4∂2f3∂x1∂x4(P0,β*)+w3v42∂2f3∂x42(P0,β*),b=2w1v4∂2f1∂x4∂β(P0,β*)+2w3v4∂2f3∂x4∂β(P0,β*).By (37) and (39), we obtain (43)a=2σβμ(2μ+σ+τ+ϵ)(μ+σ)[στ(μ+ω)-M]+(μ+σ)σαA[μ(1-p)+ω],b=2σA[μ(1-p)+ω]μ(2μ+σ+τ+ϵ)(μ+ω).Obviouslyb is always positive. Therefore the sign of the coefficient a determines the local dynamics around the disease-free equilibrium for β=β* by Lemma 5.Remark 7.
Setα0=(M-στμ+ω)/σA[μ(1-p)+ω]. The coefficient a is positive if and only if α>α0. Under this circumstance, the direction of the bifurcation for system (3) at R0=1 is backward. Considering condition (12), we get that the condition α>α0 is equivalent to the condition R1*<1 at the bifurcation, that is, when R0=1.Theorem 8.
LetR0=1. System (3) shows a backward bifurcation when R1*<1 and a forward bifurcation when R1*>1.Furthermore, taking the treated rateτ as the bifurcation parameter, we can get the following.Theorem 9.
LetR0>1. When τ passes through a critical value, system (3) undergoes Hopf bifurcation at the positive equilibrium P*.Proof.
If system (3) shows Hopf bifurcation, there must exist τ=τ*, which satisfies the following conditions: (44)ih(τ*)≡c1(τ*)c2(τ*)-c3(τ*)=0,(45)iiddτReλττ=τ*≠0.From (28) and (44), we can calculate the critical value τ*.
Forτ=τ*, we have (46)c1c2=c3.From (26) and (46), we have (47)(λ2+c2)(λ+c1)=0,which has three roots: (48)λ1=ic2,λ2=-ic2,λ3=-c1.For all τ, the roots are all in the following general forms: (49)λ1(τ)=α1(τ)+iα2(τ),λ2(τ)=α1(τ)-iα2(τ),λ3(τ)=-c1.Next, we prove the transversality condition (50)ddτReλjττ=τ*≠0,j=1,2.We substitute λj(τ)=α1(τ)+iα2(τ) into (47) and calculate the derivative, getting (51)J1(τ)α1′(τ)-G1(τ)α2′(τ)+K1(τ)=0,G1(τ)α1′(τ)+J1(τ)α2′(τ)+H1(τ)=0,where (52)G1=6α1(τ)α2(τ)+2c1(τ)α2(τ),H1=2α1(τ)α2(τ)c1′(τ)+c2′(τ)α2(τ),J1=3α12(τ)+2c1(τ)α1(τ)+c2(τ)-3α22(τ),K1=α12(τ)c1′(τ)+c2′(τ)α1(τ)+c3′(τ)-c1′(τ)α22(τ).For(53)G1(τ*)H1(τ*)+J1(τ*)K1(τ*)≠0,we obtain (54)ddτReλjττ=τ*=G1H1+J1K1J12+G12τ=τ*≠0.
Hence, the transversality condition is confirmed. This verifies the result.


## 5. Global Stability of Equilibria

Theorem 10.
LetR0(1+αA/μ)<1. If R1*>1 the disease-free equilibrium P0 is globally asymptotically stable in Ω; if R1*<1 the disease-free equilibrium P0 is globally asymptotically stable in Ω when R0<R0*.Proof.
WhenR0(1+αA/μ)<1, R0<1. If R1*>1, P0 is the only equilibrium of  (3) which is located in Ω. From the first equation of  (3), we obtain dS/dt≤(1-p)A+(A/μ)(ω+τ)-(μ+ω)S. A solution of the equation dy/dt=(1-p)A+(A/μ)(ω+τ)-(μ+ω)y is a upper solution of S(t). Due to that y→A[μ(1-p)+(ω+τ)]/μ(μ+ω) when t→∞, we can easily get that, for a small enough ε>0 which is sufficiently small, there exists a t0 such that S(t)≤y(t)≤A[μ(1-p)+(ω+τ)]/μ(μ+ω)+ε as t>t0.
Considering the Lyapunov functionL=σE+(μ+σ)I, thus (55)L′=σE′+(μ+σ)I′=I[σβS(1+αI)-(μ+σ)(μ+τ+ϵ)].
ForR0(1+αA/μ)<1, we can choose ε small enough such that (1+αA/μ)R0-1+(1+αA/μ)εσβ/(μ+σ)(μ+τ+ϵ)<0. Thus,(56)L′≤σβAμ1-p+ω+τμμ+ω+εmmc·1+αAμ-(μ+σ)(μ+τ+ϵ)I=(μ+σ)(μ+τ+ϵ)·[(1+αAμ)R0-1+(1+αA/μ)εσβ(μ+σ)(μ+τ+ϵ)]I≤0,and L′=0 if and only if I=0. The singleton P0 is the maximum positive invariant set in {(S,V,E,I)∈Ω,L′=0}. The global stability of P0 for every solution follows from LaSalle’s Invariance Principle.
IfR1*<1, system (3) has two endemic equilibrium when R0*<R0<1. Furthermore, system (3) shows a backward bifurcation. That means we should require that R0 becomes much smaller than unity (less than a critical value R0*) so that the disease can be eliminated. Thus, when R1*<1 the disease-free equilibrium P0 is globally asymptotically stable in Ω when R0<R0*.
For system (3), we discuss global stability of the endemic equilibrium P* for R0>1. Due to S+V+E+I→A/μ when t→∞, we can determine V(t) by S(t), E(t), and I(t). So system (3) can be changed into the following limit system:(57)S˙t=A(1-p)+ω(Aμ-S(t)-E(t)-I(t))+τI-μS-βSI(1+αI),E˙(t)=βSI(1+αI)-μE-σE,I˙(t)=σE-μI-τI-ϵI.
We need the following results [15] to obtain the result we want.
Consider the system as follows:(58)dxdt=f(t,x),(59)dydt=g(y),where f and g are locally Lipschitz in x∈Rn and continuous. And for all positive t values its solutions exist. If f(t,x)→g(x) when t→∞ locally uniformly for x∈Rn, then system (58) is defined as asymptotically autonomous with limit system (59).Lemma 11.
SetP is a locally asymptotically stable equilibrium of (59) and x is the ω-limit set of a forward bounded solution x(t) of (58). If x includes a point y0 such that the solution of (59) with y(0)=y0 converges to P when t→∞, then ω=P; that is, x(t)→P when t→∞.Corollary 12.
If solutions of system (58) are bounded and the equilibrium P of the limit system (59) is globally asymptotically stable, then any solution x(t) of system (58) satisfies x(t)→P when t→∞.Next, we obtain sufficient conditions that endemic equilibriumP* is globally asymptotically stable for R0>1 by the geometrical approach [9]. Firstly, we briefly introduce this geometrical approach.Let aC1 function x→f(x)∈Rn be in an open set D∈Rn. Consider the differential equation (60)dxdt=f(x).Denote x(0,x0)=x0 by x(t,x0) which is the solution to (60). We establish the following two assumptions. (H1)
There exists a compact absorbing setK⊂D.(H2)
Equation (60) has a unique equilibrium x- in D.If the equilibriumx- is locally stable, it is globally stable in D and all trajectories in D converge to x-. For n≥2, we mean a condition satisfied by f which rules out the existence of nonconstant periodic solutions of (60) by Bendixson’s criterion. The classical Bendixson’s condition divf(x)<0 for n=2 is robust under C1 local perturbations of f. About higher-dimensional systems, the C1 robust properties have been discussed.If there exists a neighborhoodU of x0 and T>0 such that U∩x(t,U) is empty for all t>T, then a point x0∈D is called wandering for (59). For example, all limit points and equilibria are nonwandering. We will introduce the global stability principle in [19] which is suited for autonomous systems.Lemma 13 (see [19]).
Assume that(H1) and (H2) hold. And suppose that (60) satisfies Bendixson’s criterion that is robust under C1 local perturbations of f at all nonequilibrium nonwandering points for (60). Then, x- is globally stable in D provided it is stable.To have the robustness required by Lemma13, we show the following Bendixson criterion [19]. Let x→P(x) be a matrix-valued function that is C1 for x∈D. Assume that P-1(x) exists and is continuous for x∈K, which is the compact absorbing set. Define a quantity q2- as (61)q2-=limt→∞sup⁡sup⁡x∈K1t∫0tμ(B(x(s,x0)))ds,where (62)B=PfP-1+P∂f[2]∂xP-1.By substituting the derivative in the direction of f into each entry p of P, the matrix Pf is obtained. μ(B) is the Lozinski1˘ measure of B in terms of a vector norm · in RN: (63)μ(B)=limh→0+I+hB-1h.IfD is simply connected, the condition q2-<0 excludes the existence of any orbit that attracts a simple closed rectifiable curve that is invariant for (62), such as homoclinic orbits, heteroclinic cycles, and periodic orbits in [19]. And it is robust under C1 local perturbations of f near any nonequilibrium point that is nonwandering. In particular, the following lemma is proved in [19].Lemma 14.
Assume thatD is simply connected and that the hypotheses (H1) and (H2) hold. Then, if  q2-<0, the unique equilibrium x- of (62) is globally stable in D.Next, we will obtain the main result.Theorem 15.
IfR0>1, system (3) admits a unique endemic equilibrium P*. It is globally asymptotically stable in terms of solutions of (3) initiating in the interior of Ω, provided that inequality (77) or (78), and c1>0, c1c2-c3>0 are satisfied.Proof.
The Jacobian of system (57) is as follows: (64)J=-μ-ω-βI(1+αI)-ω-ω+τ-βS(1+2αI)βI(1+αI)-μ-σβS(1+2αI)0σ-τ-μ-ϵ.
From Theorem3(ii), we obtain that there exists the endemic equilibrium P* and it is unique due to R0>1. We will analyse the stability of P* by the method in [9]. Due to Lemma 14, the global stability of P* requires the following sufficient conditions: (i) there must exist a compact absorbing set in the interior of Ω (i.e., condition (H1)); (ii) P* in the interior of Ω is unique (i.e., condition (H2)); and (iii) the requirement q2-<0.System (3) satisfies (H1)-(H2) under the assumption R0>1. Actually, P0 is unstable when R0>1. As P0 is unstable and P0∈∂Ω, we can obtain the uniform persistence.AsΩ is bounded, the uniform persistence implies that there exist a compact absorbing set in the interion of Ω for system (3) (see [20]). Therefore, (H1) is verified. Also, P* is the only equilibrium in the interior of Ω, so that P* is unique; that is, (H2) is verified, too.Next we will look for conditions which satisfied (H3). Consider the Jacobian matrix (64) and get the second additive compound matrix J[2](S,E,I):(65)J[2]=-2μ-ω-σ-βI(1+αI)βS(1+2αI)ω-τ-βS(1+2αI)σ-2μ-ω-τ-ϵ-βI(1+αI)-ω0βI(1+αI)-2μ-σ-τ-ϵ.Letp(x)=P(S,E,I)=diag{E/I,E/I,E/I}. Then PfP-1=diag{E′/E-I′/I,E′/E-I′/I,E′/E-I′/I}. Thus, the matrix B=PfP-1+PJ[2]P-1 can be written in block form as (66)B=B11B12B21B22,where(67)B11=E′E-I′I-βI(1+αI)-2μ-ω-σ,B12=[βS(1+2αI),βS(1+2αI)+ω-τ],B21=σ,0T,B22=E′E-I′I-βI(1+αI)-2μ-ω-τ-ϵ-ωβI(1+αI)E′E-I′I-2μ-σ-τ-ϵ.Set(k-,l-,m-) be the vectors in R3. We choose a standard in R3 as (k-,l-,m-)=max⁡{k-,l-+m-} and set μ be the Lozinski1˘ measure in term of this standard. Applying the technique in [21], the following can be obtained:(68)μ(B)≤sup⁡{g1,g2},where (69)g1=μ1(B11)+B12,g2=μ1(B22)+B21.We can obtain (70)μ1(B11)=B11=E′E-I′I-βI(1+αI)-2μ-ω-σ,B21=σ,μ1B22=max⁡⁡E′E-I′I-2μ-ω-τ-ϵ,mmmmccE′E-I′I-2μ-σ-τ-ϵ-ω=E′E-I′I-2μ-ω-τ-ϵ,B12=max⁡{βS(1+2αI),βS(1+2αI)+ω-τ}=ω-τ+βS1+2αI,ω-τ>0,βS1+2αI,ω-τ<0.From the second and third equations of system (57), we can obtain (71)E′E=βSI(1+αI)E-(μ+σ),I′I=σEI-(μ+τ+ϵ).Thus, we obtain (72)g1=μ1(B11)+B12=E′E-I′I-βI(1+αI)-2μ-σ+βS1+2αI-τ,ω>τ,E′E-I′I-βI1+αI-2μ-ω-σ+βS1+2αI,ω<τ,g2=μ1(B22)+B21=E′E-I′I-2μ-ω-τ-ϵ+σ.We can get (73)g1≤E′E-μ-σ+ϵ-βI(1+αI)+βS1+2αI,ω>τ,E′E-μ-ω-σ+ϵ-βI(1+αI)+βS1+2αI,ω<τ,g2≤E′E-μ-ω+σ.Hence (74)μ(B)≤E′E+max⁡-μ-ω+σ,mmmmmm-μ-σ+ϵmmmmmm-βI(1+αI)mmmmmm+βS(1+2αI),ω>τ,E′E+max⁡-μ-ω+σ,mmmmmm-μ-ω-σ+ϵmmmmmm-βI(1+αI)mmmmmm+βS(1+2αI),ω<τ.Considering c≤S, I≤A/μ, where c is the constant of uniform persistence; it is obvious that (75)μ(B)≤E′E-d,where (76)d=min⁡-μ-ω+σ,Aμmmmc-μ-σ+ϵ-βc(1+αc)mmmc+βAμ(1+2αAμ),ω>τ,min⁡-μ-ω+σ,Aμmmmc-μ-ω-σ+ϵ-βc(1+αc)mmmc+βAμ(1+2αAμ),ω<τ.And if (77)μ+ω<σ,βc(1+αc)+μ+σ-ϵ<βAμ(1+2αAμ),ω>τ,or (78)μ+ω<σ,βc(1+αc)+μ+ω+σ-ϵ<βAμ(1+2αAμ),ω<τ,holds, then d>0.For each(S(t),E(t),I(t))∈Ω, we obtain (79)1t∫0tμ(B)ds≤1t∫0t1μ(B)ds+1t∫t1tμ(B)ds≤1t∫0t1μ(B)ds+1tlog⁡E(t)E(t1)-d≤-d2<0.Due to Theorem4 and Lemma 14, if R0>1, then the endemic equilibrium P* of system (3) is globally stable in Ω.


## 6. Numerical Simulations

Next, we show some numerical examples to support our analytic results.Example 1.
We take parametersp=0.9, A=1000000, ω=0.000005, μ=2, τ=40, β=0.00009, α=0.000005, σ=0.006, and ϵ=0.08. Then we can obtain M=168.8254>στ(μ+ω)=0.4800 and R0=0.00031987<1 which satisfies Theorem 2 and R0(1+αA/μ)=0.0011<1 which satisfies Theorem 10. Therefore, system (3) has a disease-free equilibrium P0(50001,450000,0,0) and it is globally asymptotically stable (Figure 1).Figure 1
The global stability of disease-free equilibrium.Example 2.
We take parametersp=0.6, A=1000000, β=0.00005, α=0.000005, μ=0.02, ω=0.05, σ=45.6, and ϵ=23. Under these parameters, due to Theorem 9, we calculate the critical value τ*=39.9918. If we take τ=20<τ*, we can get M=137.3801>στ(μ+ω)=63.8400, R0=48.1293, R1*=267.6519, and R0*=0.0032. Therefore R0*<1<R0<R1*, due to Theorem 3 system (3) has a disease-free equilibrium P0(41429000,8571400,0,0) and an endemic equilibrium P*(432250,187060,198280,8571400). And we can calculate d=45.53>0 which guarantees q2-<0. From Theorem 15, we can get that the endemic equilibrium is globally asymptotically stable, which is demonstrated by Figure 2. If we take τ=40>τ*, then the endemic equilibrium P* becomes unstable and a periodic orbits bifurcates from P*, which is demonstrated by Figure 3.Figure 2
The global stability of endemic equilibrium.Figure 3
The periodic solution of system (3).


## 7. Discussion

In this paper, considering disease-caused death and partial permanent immunity, we modified the SEIV epidemic model in [15]. Applying the method of [16], we calculated the basic reproduction number R0 and found that when R0=1 and R1*<1 system (3) shows backward bifurcation. If R1*<1, system (3) has a unique endemic equilibrium when R0≥1 and has two endemic equilibria when R0<1. If R1*>1, system (3) has a unique endemic equilibrium when R0≥1 and has no endemic equilibrium when R0<1. Also system (3) always has a disease-free equilibrium P0. Local and global asymptotic stability of the disease-free equilibrium are determined by R0<1 and R0(1+αA/μ)<1, respectively. Also we have studied the local and global asymptotic stability of the endemic equilibrium. Moreover, taking the disease-caused death rate τ as bifurcation parameter, we discussed the Hopf bifurcation of system (3). We found that when R0>1, there is always a critical value τ*, such that system (3) exhibits Hopf bifurcation at P* when τ passes through τ*.From the sense of epidemiology, whenR0(1+αA/μ)<1, if R1*>1 holds or R1*<1, R0<R0* hold; system (3) has one disease-free equilibrium which is globally stable. Namely, the disease will be eradicated. And when R0>1 and inequality (77) or (78) holds, system (3) has a unique endemic equilibrium P* which is global asymptotically stable. Under this circumstance, the infectious disease becomes endemic disease. If R0>1, system (3) has a unique endemic equilibrium P* and we found that when the rate τ becomes sufficiently large the disease will break out periodically. And differentiating the bifurcation coefficient a partially with respect to p, we can get ∂a/∂p=-σαAμ(μ+σ)<0, which means that vaccinating more susceptible populations decreases the likelihood of the occurrence of backward bifurcation [15].



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