# Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

Authors: Hai-Feng Huo; Li-Xiang Feng
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

An epidemic model with incomplete treatment and vaccination for the newborns and susceptibles
is constructed. We establish that the global dynamics are completely determined by the basic
reproduction numberR0. If R0≤1, then the disease-free equilibrium is globally asymptotically
stable. If R0>1, the endemic equilibrium is globally asymptotically stable. Some numerical
simulations are also given to explain our conclusions.


---

## Body

## 1. Introduction

Epidemiological models describing a directly transmitted viral or bacterial agent in a closed population and consisting of susceptibles (S), infectives (I), and recovers (R) were considered by Kermack and Mckendrick (1927). For some diseases, such as influenza and tuberculosis, on adequate contact with an infectious individual, a susceptible becomes exposed for a while, that is, infected but not yet infectious. Thus it is realistic to introduce a latent compartment (usually denoted by E), leading to an SEIR model [1]. Such type of models has been widely discussed in recent decades [2–8].Vaccination is important for the elimination of infectious disease. Usually, the vaccination process are different schedules for different disease and vaccines. For some disease, such as hepatitis B virus infection [9], doses should be taken by vaccinees several times and there must be some fixed time intervals between two doses. Considering the time for vaccines to obtain immunity and possibility to be infected before vaccination, Liu et al. [10] studied the vaccination effects via two SVIR models according to continuous vaccination strategy and pulse vaccination strategy (PVS), respectively. Li et al. [11] considered that the vaccine is available for both the susceptibles and the newborns, and the immunity of the vaccinated individuals is temporary and that the efficiency of vaccine is not complete.In [12–15], it is assumed that the treated individuals have partial immunity and can be infected through contacts with infectious individuals. Yang et al. [15] incorporated the incomplete treatment into TB epidemic model with treatment, it is assumed that the being treated individuals are kept at an isolated environment, therefore, individuals in treatment compartment are not able to infect others or be infected. Since individual's symptoms of TB may disappear after being treated, but a few of tubercle bacillus may still be left in the body of the treated individual [16, 17], then the treated individual may still be a TB carrier and become latent or may enter the infectious compartment for treatment failure [18].Motivated by these works, in this paper, we consider an SVEIT epidemiological model with varying infectivity. The model assumes that the vaccine is available for both the susceptibles and the newborns and the immunity of the vaccinated individuals is temporary, and that the efficiency of vaccine is not complete. And we also incorporate the incomplete treatment into the epidemic model.The organization of this paper is as follows. In the next section, the epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is formulated. In Section3, the basic reproduction number and the existence of equilibria are investigated. The global stability of the disease-free and endemic equilibria are proved in Section 4, and some numerical simulations are given in Section 5. In the last section, we give some brief discussions.


## 2. The Model Formulation

In this section, we introduce an epidemic model with incomplete treatment and vaccination. The total population is partitioned into five compartments: the susceptible compartment (S), the vaccinated compartment (V), the latent compartment (E), the infectious compartment (I), and the treatment compartment (T). The population flow among those compartments is shown in the following diagram (Figure 1).Figure 1
The transfer diagram for the model (2.1).The schematic diagram leads to the following system of ordinary differential equations:(2.1)S′=qμA-(μ+p)S-βSI+ɛV,V′=(1-q)μA+pS-βσVI-(μ+ɛ)V,E′=βI(S+σV)-(μ+γ)E+(1-k)δT,I′=γE-(μ+α1+ξ)I+kδT,T′=ξI-(μ+α2+δ)T.
Here, μA is the birth rate of the population; μ is the natural death rate of the population; q(0<q<1) is the fraction of the unvaccinated newborns, 1-q is the fraction of the vaccinated newborns; p is the vaccinating rate coefficient for the susceptible individuals; ɛ is the rate coefficient of losing the immunity from the vaccination. β and σβ(0<σ<1) are the transmission coefficient of the infection for the susceptible and vaccinated individuals from the infectious individuals, where 0<σ<1 shows that the efficiency of the vaccine is not complete (100%); γ is the rate coefficient of transfer from the latent compartment to the infectious one; ξ is the percapita treatment rate for the infectious individuals; δ is the rate coefficient at which a treated individual leaves compartment T; α1 and α2 are the disease-induced death rate coefficients for individuals in compartments I and T, respectively; k(0≤k≤1) is the fraction of the drug-resistant individuals in the treated individuals. k reflects the failure of treatment, where k=0 means that all the treated individuals will become latent, while k=1 means that the treatment fails and all the treated individuals will still be infectious.It is important to show positivity and boundedness for the system (2.1) as they represent populations. Firstly, we present the positivity of the solutions. System (2.1) can be put into the matrix form(2.2)X′=G(X),
where X=(S,V,E,I,T)T∈R5 and G(X) is given by(2.3)G(X)=(G1(X)G2(X)G3(X)G4(X)G5(X))=(qμA-(μ+p)S-βSI+ɛV(1-q)μA+pS-βσVI-(μ+ɛ)VβI(S+σV)-(μ+γ)E+(1-k)δTγE-(μ+α1+ξ)I+kδTξI-(μ+α2+δ)T).
It is easy to check that(2.4)Gi(X)|Xi(t)=0,Xt∈C+≥0,i=1,2,3,4,5.
Due to Lemma  2 in [19], any solution of (2.1) is such that X(t)∈R+5 for all t≥0.Summing equations in (2.1) yields(2.5)(S+V+E+I+T)′=μ[A-(S+V+E+I+T)]-α1I-α2T≤μ[A-(S+V+E+I+T)],
then it follows that limt→+∞sup[S(t)+V(t)+E(t)+I(t)+T(t)]≤A, so the set(2.6)Ω={(S,V,E,I,T)∈R+5:S+V+E+I+T≤A}
is positively invariant for (2.1). Therefore, we will consider the global stability of (2.1) on the set Ω.


## 3. The Basic Reproduction Number and Existence of Equilibria

The model has a disease-free equilibriumP0(S0,V0,0,0,0), where(3.1)S0=A(ɛ+qμ)p+μ+ɛ,V0=A[p+(1-q)μ]p+μ+ɛ.
In the following, the basic reproduction number of system (2.1) will be obtained by the next generation matrix method formulated in [20].Letx=(E,I,T,S,V)T, then system (2.1) can be written as(3.2)dxdt=F(x)-V(x),
where(3.3)F(x)=(βI(S+σV)0000),V(x)=((μ+γ)E-(1-k)δT(μ+α1+ξ)I-γE-kδT(μ+α2+δ)T-ξIβSI+(μ+p)S-qμA-ɛVβσVI+(μ+ɛ)V-(1-q)μA-pS).
The Jacobian matrices of ℱ(x) and 𝒱(x) at the disease-free equilibrium P0 are, respectively,(3.4)DF(P0)=(F3×3000),DV(P0)=(V3×3000βS00μ+p-ɛ0βσV00-pμ+ɛ),
where(3.5)F=(0β(S0+σV0)0000000),V=(μ+γ0-(1-k)δ-γμ+α1+ξ-kδ0-ξμ+α2+δ).
The model reproduction number, denoted by R0 is thus given by(3.6)R0=ρ(FV-1)=βγ(μ+α2+δ)(S0+σV0)(μ+γ)(μ+α1+ξ)(μ+α2+δ)-δξ[(1-k)γ+(μ+γ)k].The endemic equilibriumP*(S*,V*,E*,I*,T*) of system (2.1) is determined by equations(3.7)qμA-(μ+p)S-βSI+ɛV=0,(1-q)μA+pS-βσVI-(μ+ɛ)V=0,βI(S+σV)-(μ+γ)E+(1-k)δT=0,γE-(μ+α1+ξ)I+kδT=0,ξI-(μ+α2+δ)T=0.
The first two equations in (3.7) lead to(3.8)S=μA[q(μ+ɛ+βσI)+ɛ(1-q)](p+μ+βI)(μ+ɛ+βσI)-pɛ,V=μA[(1-q)(p+μ+βI)+pq](p+μ+βI)(μ+ɛ+βσI)-pɛ.
From the last equation in (3.7), we have(3.9)T=ξμ+α2+δI.
Substituting (3.9) into the fourth equation in (3.7) gives(3.10)E=(μ+α1+ξ)(μ+α2+δ)-kδξ(μ+α2+δ)γI.
For I≠0, substituting (3.9), (3.10) into the third equation in (3.7) gives(3.11)S+σV=1β(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγ(μ+α2+δ)γ.
Substituting (3.8) into (3.11) yields(3.12)H(I)∶=[q(μ+ɛ+βσI)+ɛ(1-q)]+σ[(1-q)(p+μ+βI)+pq](p+μ+βI)(μ+ɛ+βσI)-pɛ-(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγβγ(μ+α2+δ)μA=βσI+(ɛ+qμ)+σ[p+(1-q)μ](p+μ+βI)(μ+ɛ+βσI)-pɛ-(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγβγ(μ+α2+δ)μA=0.
Direct calculation shows(3.13)H′(I)β=-(βσI)2+2βσI(ɛ+qμ)+σ[p+(1-q)μ]+σ0[(p+μ+βI)(μ+ɛ+βσI)-pɛ]2<0,
where(3.14)σ0=(p+μ)[p+(1-q)μ]σ2+σ[p(ɛ+qμ)+ɛ(p+(1-q)μ)]+(μ+ɛ)(ɛ+qμ),
then function H(I) is decreasing for I>0. Since (p+μ+βI)(μ+ɛ+βσI)-pɛ>βI(βσI+ɛ+σp+μ), and it follows from σ≤1 that βσI+(ɛ+qμ)+σ[p+(1-q)μ]<βσI+ɛ+σp+μ, then(3.15)H(I)<1βI-(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγβγ(μ+α2+δ)μA.
Thus,(3.16)H(0)=(ɛ+qμ)+σ[p+(1-q)μ]μ(p+μ+ɛ)-(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγβγ(μ+α2+δ)μA=(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγβγ(μ+α2+δ)μA(R0-1),H(A)<1βA-(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-kδξ]-(1-k)δξγβγ(μ+α2+δ)μA=(μ+α2+δ)[γμ-(μ+γ)(μ+α1+ξ)]+δξ(γ+μk)βAγμ(μ+α2+δ)=-(μ+α2+δ)[μ(μ+α1+ξ)+γ(α1+ξ)]+δξ(γ+μk)βAγμ(μ+α2+δ)=-(μ+α2+δ)[μ(μ+α1)+γα1]+ξ(μ+γ)(μ+α2)+μδξ(1-k)βAγμ(μ+α2+δ).<0.
Therefore, by the monotonicity of function H(I), for (3.12) there exists a unique positive root in the interval (0,A) when R0>1; there is no positive root in the interval (0,A) when R0≤1. We summarize this result in Theorem 3.1.Theorem 3.1.
For system (2.1), there is always the disease-free equilibrium P0(S0,V0,0,0,0). When R0>1, besides the disease-free equilibrium P0, system (2.1) also has a unique endemic equilibrium P*(S*,V*,E*,I*,T*), where
(3.17)S*=μA[q(μ+ɛ+βσI*)+ɛ(1-q)](p+μ+βI*)(μ+ɛ+βσI*)-pɛ,V*=μA[(1-q)(p+μ+βI*)+pq](p+μ+βI*)(μ+ɛ+βσI*)-pɛ,E*=(μ+α1+ξ)(μ+α2+δ)-kδξ(μ+α2+δ)γI*,T*=ξμ+α2+δI*,
and I* is the unique positive root of equation H(I)=0.


## 4. Global Stability of Equilibria

Theorem 4.1.
For system (2.1), the disease-free equilibrium P0 is globally stable if R0≤1; the endemic equilibrium P* is globally stable if R0>1.


### 4.1. The Proof Global Stability of the Disease-Free Equilibrium

For the disease-free equilibriumP0(S0,V0,0,0,0), S0 and V0 satisfy equations(4.1)qμA-(μ+p)S-βSI+ɛV=0,(1-q)μA+pS-βσVI-(μ+ɛ)V=0,
then (2.1) can be rewritten as follows:(4.2)S′=S[qμA(1S-1S0)+ɛ(VS-V0S0)-βI],V′=V[(1-q)μA(1V-1V0)+p(VS-V0S0)-βσI],E′=βI[(S0+σV0)+(S-S0)+σ(V-V0)]-(μ+γ)E+(1-k)δT,I′=γE-(μ+α1+ξ)I+kδT,T′=ξI-(μ+α2+δ)T.
Define the Lyapunov function(4.3)V1=(S-S0-S0ln⁡SS0)+(V-V0-V0ln⁡VV0)+E+μ+γγI+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)T.
The derivative of V1 is given by(4.4)V1′=(S-S0)[qμA(1S-1S0)+ɛ(VS-V0S0)-βI]+(V-V0)[(1-q)μA(1V-1V0)+p(VS-V0S0)-βσI]+βI[(S0+σV0)+(S-S0)+σ(V-V0)]-(μ+γ)E+(1-k)δT+μ+γγ[γE-(μ+α1+ξ)I+kδT]+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)[ξI-(μ+α2+δ)T]=(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-δξ[(1-k)γ+(μ+γ)k]γ(μ+α2+δ)(R0-1)I+F(S,I),
where(4.5)F(S,I)=μqA(S-S0)(1S-1S0)+(1-q)μA(V-V0)(1V-1V0)+ɛ(S-S0)(VS-V0S0)+p(V-V0)(SV-S0V0).
Denote x=S/S0,y=V/V0; then(4.6)F(S,I)=μqA(x-1)(1x-1)+(1-q)μA(y-1)(1y-1)+ɛV0(x-1)(yx-1)+pS0(y-1)(xy-1)=:F¯(x,y).
Applying (4.1) to function F¯(x,y) yields(4.7)F¯(x,y)=(2μqA+2μ(1-q)A+ɛV0+pS0)-μS0x-μqA1x-μV0y-μ(1-q)A1y-ɛV0yx-pS0xy.
By [11], we have F¯(x,y)≤0 for x,y>0 and F¯(x,y)=0 if and only if x=y=1. Since (μ+γ)(μ+α1+ξ)(μ+α2+δ)>δξ[(1-k)γ+(μ+γ)k] and R0≤1, then V1′≤0. It follows from LaSalle invariance principle [21] that the disease-free equilibrium P0 is globally asymptotically stable when R0≤1.


### 4.2. The Proof Global Stability of the Endemic Equilibrium

For the endemic equilibriumP*(S*,V*,E*,I*,T*), S*,V*,E*,I*, and T* satisfy equations(4.8)qμA-(μ+p)S-βSI+ɛV=0,(1-q)μA+pS-βσVI-(μ+ɛ)V=0,βI(S+σV)-(μ+γ)E+(1-k)δT=0,γE-(μ+α1+ξ)I+kδT=0,ξI-(μ+α2+δ)T=0,
which will be used many times in the following inference.By applying (4.8) and denoting(4.9)SS*=x,VV*=y,EE*=z,II*=u,TT*=w,
we have(4.10)x′=x[qμAS*(1x-1)-βI*(u-1)+ɛV*S*(yx-1)],y′=y[(1-q)μAV*(1y-1)+pS*V*(xy-1)-βσI*(u-1)],z′=zβI*E*[S*(xuz-1)+σV*(yuz-1)]+z(1-k)δT*E*(wz-1),u′=u[γE*I*(zu-1)+kδT*I*(wu-1)],w′=wξI*T*(uw-1).
Define the Lyapunov function(4.11)V2=S*(x-1-ln⁡x)+V*(y-1-ln⁡y)+E*(z-1-ln⁡z)+μ+γγI*(u-1-ln⁡u)+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)T*(w-1-ln⁡w).The derivative ofV2 is given by
(4.12)V2′=S*x-1xx′+V*y-1yy′+E*z-1zz′+μ+γγI*u-1uu′+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)T*w-1ww′=(x-1)[qμA(1x-1)-βS*I*(u-1)+ɛV*(yx-1)]+(y-1)[(1-q)μA(1y-1)+pS*(xy-1)-βσV*I*(u-1)]+βI*(z-1)[S*(xuz-1)+σV*(yuz-1)]+(z-1)(1-k)δT*(wz-1)+μ+γγ(u-1)[γE*(zu-1)+kδT*(wu-1)]+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)ξI*(w-1)(uw-1)=qμA(x-1)(1x-1)-βS*I*(x-1)(u-1)+ɛV*(x-1)(yx-1)+(1-q)μA(y-1)(1y-1)+pS*(y-1)(xy-1)-βσV*I*(y-1)(u-1)+βS*I*(z-1)(xuz-1)+βσV*I*(z-1)(yuz-1)+(1-k)δT*(z-1)(wz-1)+(μ+γ)E*(u-1)(zu-1)+μ+γγkδT*(u-1)(wu-1)+[(μ+γ)k+γ(1-k)]δT*γ×(w-1)(uw-1)=qμA(2-x-1x)+(1-q)μA(2-y-1y)-βS*I*(xu-x-u+1)+ɛV*(y-x-yx+1)+pS*(x-y-xy+1)-βσV*I*(yu-y-u+1)+βS*I*(xu-z-xuz+1)+βσV*I*(yu-z-yuz+1)+(1-k)δT*(w-z-wz+1)+(μ+γ)E*(z-u-zu+1)+μ+γγkδT*(w-u-wu+1)+[(μ+γ)k+γ(1-k)]δT*γ(u-w-uw+1)=2qμA+2(1-q)μA+ɛV*+pS*+(1-k)δT*+(μ+γ)E*+μ+γγkδT*+(μ+γ)k+γ(1-k)γδT*-μS*x-qμA1x-μV*y-(1-q)μA1y+z[-βS*I*-βσV*I*-(1-k)δT*+(μ+γ)E*]+u[βS*I*+βσV*I*-(μ+γ)E*-μ+γγkδT*+(μ+γ)k+γ(1-k)γδT*]+w[(1-k)δT*+μ+γγkδT*-(μ+γ)k+γ(1-k)γδT*]-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-(μ+γ)E*zu-(1-k)δT*wz-μ+γγkδT*wu-(μ+γ)k+γ(1-k)γδT*uw=[2qμA+2(1-q)μA+ɛV*+pS*+βI*(S*+σV*)+3(1-k)δT*+2μ+γγkδT*]-μS*x-qμA1x-μV*y-(1-q)μA1y-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-βI*(S*+σV*)zu-(1-k)δT*zu-(1-k)δT*wz-μ+γγkδT*wu-[(μ+γ)k+γ(1-k)]γδT*uw=[2qμA+2(1-q)μA+ɛV*+pS*+βI*(S*+σV*)]-μS*x-qμA1x-μV*y-(1-q)μA1y-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-βI*(S*+σV*)zu+(1-k)δT*(3-zu-wz-uw)+μ+γγkδT*(2-wu-uw)=:F¯(x,y,z,u)+(1-k)δT*(3-zu-wz-uw)+μ+γγkδT*(2-wu-uw).
Following [11], βI*(S*+σV*)=(μ+γ)E* and(4.13)F(x,y,z,u)=[2qμA+2(1-q)μA+ɛV*+pS*+(μ+γ)E*]-μS*x-qμA1x-μV*y-(1-q)μA1y-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-(μ+γ)E*zu≤0,
we have F¯(x,y,z,u)≤0 for x,y,z,u>0 and F¯(x,y,z,u)=0 if and only if x=y=1 and z=u. Therefore, V2′≤0 for x,y,z,u,w>0 and V2′=0 if and only if x=y=1,z=u=w, and the maximum invariant set of system (2.1) on the set {(x,y,z,u,w):V2′=0} is the singleton (1,1,1,1,1). Thus, for system (2.1), the endemic equilibrium P* is globally asymptotically stable if R0>1 by LaSalle invariance principle [21].


## 4.1. The Proof Global Stability of the Disease-Free Equilibrium

For the disease-free equilibriumP0(S0,V0,0,0,0), S0 and V0 satisfy equations(4.1)qμA-(μ+p)S-βSI+ɛV=0,(1-q)μA+pS-βσVI-(μ+ɛ)V=0,
then (2.1) can be rewritten as follows:(4.2)S′=S[qμA(1S-1S0)+ɛ(VS-V0S0)-βI],V′=V[(1-q)μA(1V-1V0)+p(VS-V0S0)-βσI],E′=βI[(S0+σV0)+(S-S0)+σ(V-V0)]-(μ+γ)E+(1-k)δT,I′=γE-(μ+α1+ξ)I+kδT,T′=ξI-(μ+α2+δ)T.
Define the Lyapunov function(4.3)V1=(S-S0-S0ln⁡SS0)+(V-V0-V0ln⁡VV0)+E+μ+γγI+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)T.
The derivative of V1 is given by(4.4)V1′=(S-S0)[qμA(1S-1S0)+ɛ(VS-V0S0)-βI]+(V-V0)[(1-q)μA(1V-1V0)+p(VS-V0S0)-βσI]+βI[(S0+σV0)+(S-S0)+σ(V-V0)]-(μ+γ)E+(1-k)δT+μ+γγ[γE-(μ+α1+ξ)I+kδT]+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)[ξI-(μ+α2+δ)T]=(μ+γ)[(μ+α1+ξ)(μ+α2+δ)-δξ[(1-k)γ+(μ+γ)k]γ(μ+α2+δ)(R0-1)I+F(S,I),
where(4.5)F(S,I)=μqA(S-S0)(1S-1S0)+(1-q)μA(V-V0)(1V-1V0)+ɛ(S-S0)(VS-V0S0)+p(V-V0)(SV-S0V0).
Denote x=S/S0,y=V/V0; then(4.6)F(S,I)=μqA(x-1)(1x-1)+(1-q)μA(y-1)(1y-1)+ɛV0(x-1)(yx-1)+pS0(y-1)(xy-1)=:F¯(x,y).
Applying (4.1) to function F¯(x,y) yields(4.7)F¯(x,y)=(2μqA+2μ(1-q)A+ɛV0+pS0)-μS0x-μqA1x-μV0y-μ(1-q)A1y-ɛV0yx-pS0xy.
By [11], we have F¯(x,y)≤0 for x,y>0 and F¯(x,y)=0 if and only if x=y=1. Since (μ+γ)(μ+α1+ξ)(μ+α2+δ)>δξ[(1-k)γ+(μ+γ)k] and R0≤1, then V1′≤0. It follows from LaSalle invariance principle [21] that the disease-free equilibrium P0 is globally asymptotically stable when R0≤1.


## 4.2. The Proof Global Stability of the Endemic Equilibrium

For the endemic equilibriumP*(S*,V*,E*,I*,T*), S*,V*,E*,I*, and T* satisfy equations(4.8)qμA-(μ+p)S-βSI+ɛV=0,(1-q)μA+pS-βσVI-(μ+ɛ)V=0,βI(S+σV)-(μ+γ)E+(1-k)δT=0,γE-(μ+α1+ξ)I+kδT=0,ξI-(μ+α2+δ)T=0,
which will be used many times in the following inference.By applying (4.8) and denoting(4.9)SS*=x,VV*=y,EE*=z,II*=u,TT*=w,
we have(4.10)x′=x[qμAS*(1x-1)-βI*(u-1)+ɛV*S*(yx-1)],y′=y[(1-q)μAV*(1y-1)+pS*V*(xy-1)-βσI*(u-1)],z′=zβI*E*[S*(xuz-1)+σV*(yuz-1)]+z(1-k)δT*E*(wz-1),u′=u[γE*I*(zu-1)+kδT*I*(wu-1)],w′=wξI*T*(uw-1).
Define the Lyapunov function(4.11)V2=S*(x-1-ln⁡x)+V*(y-1-ln⁡y)+E*(z-1-ln⁡z)+μ+γγI*(u-1-ln⁡u)+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)T*(w-1-ln⁡w).The derivative ofV2 is given by
(4.12)V2′=S*x-1xx′+V*y-1yy′+E*z-1zz′+μ+γγI*u-1uu′+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)T*w-1ww′=(x-1)[qμA(1x-1)-βS*I*(u-1)+ɛV*(yx-1)]+(y-1)[(1-q)μA(1y-1)+pS*(xy-1)-βσV*I*(u-1)]+βI*(z-1)[S*(xuz-1)+σV*(yuz-1)]+(z-1)(1-k)δT*(wz-1)+μ+γγ(u-1)[γE*(zu-1)+kδT*(wu-1)]+δ[(μ+γ)k+γ(1-k)]γ(μ+α2+δ)ξI*(w-1)(uw-1)=qμA(x-1)(1x-1)-βS*I*(x-1)(u-1)+ɛV*(x-1)(yx-1)+(1-q)μA(y-1)(1y-1)+pS*(y-1)(xy-1)-βσV*I*(y-1)(u-1)+βS*I*(z-1)(xuz-1)+βσV*I*(z-1)(yuz-1)+(1-k)δT*(z-1)(wz-1)+(μ+γ)E*(u-1)(zu-1)+μ+γγkδT*(u-1)(wu-1)+[(μ+γ)k+γ(1-k)]δT*γ×(w-1)(uw-1)=qμA(2-x-1x)+(1-q)μA(2-y-1y)-βS*I*(xu-x-u+1)+ɛV*(y-x-yx+1)+pS*(x-y-xy+1)-βσV*I*(yu-y-u+1)+βS*I*(xu-z-xuz+1)+βσV*I*(yu-z-yuz+1)+(1-k)δT*(w-z-wz+1)+(μ+γ)E*(z-u-zu+1)+μ+γγkδT*(w-u-wu+1)+[(μ+γ)k+γ(1-k)]δT*γ(u-w-uw+1)=2qμA+2(1-q)μA+ɛV*+pS*+(1-k)δT*+(μ+γ)E*+μ+γγkδT*+(μ+γ)k+γ(1-k)γδT*-μS*x-qμA1x-μV*y-(1-q)μA1y+z[-βS*I*-βσV*I*-(1-k)δT*+(μ+γ)E*]+u[βS*I*+βσV*I*-(μ+γ)E*-μ+γγkδT*+(μ+γ)k+γ(1-k)γδT*]+w[(1-k)δT*+μ+γγkδT*-(μ+γ)k+γ(1-k)γδT*]-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-(μ+γ)E*zu-(1-k)δT*wz-μ+γγkδT*wu-(μ+γ)k+γ(1-k)γδT*uw=[2qμA+2(1-q)μA+ɛV*+pS*+βI*(S*+σV*)+3(1-k)δT*+2μ+γγkδT*]-μS*x-qμA1x-μV*y-(1-q)μA1y-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-βI*(S*+σV*)zu-(1-k)δT*zu-(1-k)δT*wz-μ+γγkδT*wu-[(μ+γ)k+γ(1-k)]γδT*uw=[2qμA+2(1-q)μA+ɛV*+pS*+βI*(S*+σV*)]-μS*x-qμA1x-μV*y-(1-q)μA1y-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-βI*(S*+σV*)zu+(1-k)δT*(3-zu-wz-uw)+μ+γγkδT*(2-wu-uw)=:F¯(x,y,z,u)+(1-k)δT*(3-zu-wz-uw)+μ+γγkδT*(2-wu-uw).
Following [11], βI*(S*+σV*)=(μ+γ)E* and(4.13)F(x,y,z,u)=[2qμA+2(1-q)μA+ɛV*+pS*+(μ+γ)E*]-μS*x-qμA1x-μV*y-(1-q)μA1y-ɛV*yx-pS*xy-βS*I*xuz-βσV*I*yuz-(μ+γ)E*zu≤0,
we have F¯(x,y,z,u)≤0 for x,y,z,u>0 and F¯(x,y,z,u)=0 if and only if x=y=1 and z=u. Therefore, V2′≤0 for x,y,z,u,w>0 and V2′=0 if and only if x=y=1,z=u=w, and the maximum invariant set of system (2.1) on the set {(x,y,z,u,w):V2′=0} is the singleton (1,1,1,1,1). Thus, for system (2.1), the endemic equilibrium P* is globally asymptotically stable if R0>1 by LaSalle invariance principle [21].


## 5. Numerical Simulation

In this section, some numerical results of system (2.1) are presented for supporting the analytic results obtained above. All of the parameter values are estimated. The model parameters are taken as: A=55.496, μ=0.0143, q=0.6, β=0.05, σ=0.2, p=0.05, ɛ=0.03, γ=0.00368, δ=1.5, α1=0.3, α2=0.05. First, we choose ξ=0.9, k=0.3, numerical simulation gives R0=0.8809<1, then the disease-free equilibrium P0 is globally asymptotically stable (Figure 2). Second, we choose ξ=0.8, k=0.36, numerical simulation gives R0=1, then the disease-free equilibrium P0 is globally asymptotically stable (Figure 3). At last, we choose ξ=0.6, k=0.5, numerical simulation gives R0=1.2938>1, then the endemic equilibrium P* is globally asymptotically stable (Figure 4).Figure 2
WhenR0<1, the disease-free equilibrium P0 is globally asymptotically stable.Figure 3
WhenR0=1, the disease-free equilibrium P0 is globally asymptotically stable.Figure 4
WhenR0>1, the endemic equilibrium P* is globally asymptotically stable.


## 6. Discussion

We have formulated an epidemic model with incomplete treatment and vaccination and investigated their dynamical behaviors. By means of the next-generation matrix, we obtained their basic reproduction number,R0, which play a crucial role. By constructing Lyapunov function, we proved the global stability of their equilibria: when the basic reproduction number is less than or equal to one, all solutions converge to the disease-free equilibrium, that is, the disease dies out eventually; when the basic reproduction number exceeds one, the unique endemic equilibrium is globally stable, that is, the disease will persist in the population and the number of infected individuals tends to a positive constant.For system (2.1), k reflects the failure of treatment. Direct calculation shows that ∂R0/∂k>0, then decreasing the treatment failure coefficient is helpful to reduce epidemic infection. The realization of decreasing k mainly depends on decreasing the appearance of drug resistance. On the other hand, ∂R0/∂ξ<0 implies that increasing ξ has positive effect on epidemic control, since increasing ξ is to shorten the period of the infectious compartment in the nonisolated environment, that is, to start treating as soon as possible. This measure is effective to control the spread of the epidemic. Figure 5 shows the relation among the basic reproduction number R0, the treatment failure coefficient k, and the per-capita treatment rate ξ.Figure 5
The relationship amongR0, k and ξ.



---

*Source: 530267-2012-03-13.xml*