# Stability Analysis of a Vector-Borne Disease with Variable Human Population

Authors: Muhammad Ozair; Abid Ali Lashari; Il Hyo Jung; Young Il Seo; Byul Nim Kim
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A mathematical model of a vector-borne disease involving variable human population is analyzed. The varying population size includes a term for disease-related deaths. Equilibria and stability are determined for the system of ordinary differential equations. IfR0≤1, the disease-“free” equilibrium is globally asymptotically stable and the disease always dies out. If R0>1, a unique “endemic” equilibrium is globally asymptotically stable in the interior of feasible region and the disease persists at the “endemic” level. Our theoretical results are sustained by numerical simulations.


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## Body

## 1. Introduction

Vector-borne diseases such as malaria, dengue fever, plague, and West Nile fever are infectious diseases caused by the influx of viruses, bacteria, protozoa, or rickettsia which are primarily transmitted by disease-transmitting biological agents, called vectors. A vector-borne disease is transmitted by a pathogenic microorganism from an infected host to another organism results to form from an infection by blood-feeding arthropods [1].Vector-borne diseases, in particular, mosquito-borne disease, are transmitted to humans by blood-sucker mosquitoes, which have been a big problem for the public health in the world. The literature dealing with the mathematical theory on vector-borne diseases is quite extensive. Many mathematical models concerning the emergence and reemergence of the vector-host infectious disease have been proposed and analyzed in the literature [2, 3].By direct transmission models, we mean that the infection moves from person to person directly, with no environmental source, intermediate vector, or host. In a vector-host model, direct transmission may take place by transfusion-related transmission, transplantation-related transmission, and needle-stick-related transmission [4]. Some models have been developed to study the dynamics of a vector-borne disease that considers a direct mode of transmission in human host population [5–7].Mathematical modeling has proven to play an important role in gaining some insights into the transmission dynamics of infectious diseases and suggest control strategies. Appropriate mathematical models can provide a qualitative assessment for the problem. Some mathematical models discussed in [8–10] provide, best understanding about the dynamics and control of infectious diseases. Immense literature on the use of mathematical models for communicable diseases is available [11, 12]. The assumption of constant population size in epidemiological models is usually valid when we study the diseases of short duration with limited effects on mortality. It may not be valid when dealing with endemic diseases such as malaria, which has a high mortality rate. Ngwa and Shu [13] assumed density-dependent death rates in both vector and human populations, so that the total populations are varying with time that includes disease-related deaths. Esteva and Vargas [14] analyzed the effect of variable host population size and disease-induced death rate. Recently Ozair et. al analyzed vector-host disease model with nonlinear incidence [15].In this paper, based on the ideas posed in [6, 14], we develop and analyze a vector-host disease model considering a direct mode of transmission as well as a variable human population. The aim of this paper is to establish stability properties of equilibria and the threshold parameter R0 that completely determines the existence of endemic or disease-free equilibrium. If R0⩽1, the disease-free equilibrium is globally asymptotically stable. If R0>1, a unique endemic equilibrium exists and is globally asymptotically stable under parametric restrictions. However, in numerical simulations it is shown that the disease still can be “endemic” even if the conditions are violated.The rest of the paper is organized as follows. In Section2, we present a formulation of the extended mathematical model. The dimensionless formulation of proposed model is carried out in Section 3. Section 4 devotes existence and uniqueness of “endemic” equilibria. In Section 5, we use Lyapunov function theory to show global stability of disease-“free" equilibrium and geometric approach to prove global stability of “endemic” equilibrium. Discussions and simulations are done in Section 6.


## 2. Model Formulation

The human population is partitioned into subclasses of individuals who are susceptible, infectious, and recovered, with sizes denoted bySh(t), Ih(t), and Rh(t), respectively. The vector population is subdivided into susceptible and infectious vectors, with sizes denoted by Sv(t) and Iv(t), respectively. The mosquito population does not have an immune class, since their infective period ends with their death. Thus, Nh(t)=Sh(t)+Ih(t)+Rh(t) and Nv(t)=Sv(t)+Iv(t) are, respectively, the total human and vector populations at time t. The model is given by the following system of differential equations:
(1)dSh(t)dt=b1Nh-β1ShIhNh-β2ShIvNv-μhSh,dIh(t)dt=β1ShIhNh+β2ShIvNv-μhIh-γhIh-δhIh,dRh(t)dt=γhIh-μhRh,dSv(t)dt=μvNv-β3SvIhNh-μvSv,dIv(t)dt=β3SvIhNh-μvIv.In model (1), b1 is the recruitment rate of humans into the population which is assumed to be susceptible. Susceptible hosts get infected via two routes of transmission, through a contact with an infected individual and through being bitten by an infectious vector. We denote the infection rate of susceptible individuals which results from effective contact with infectious individuals by β1 and β2 is the infection rate of susceptible humans resulting due to the biting of infected vectors. The incidence of new infections via direct and indirect route of transmission is given by the standard incidence form β1(ShIh/Nh) and β2(ShIv/Nh), respectively. The term μh is the natural mortality rate of humans. We assume that infectious individuals acquire permanent immunity by the rate γh. The infectious humans suffer from disease-induced death at a rate δh. The recruitment and natural death rate of vector population is assumed to be μv. The susceptible vectors become infectious as a result of biting effect of infectious humans at a rate β3, so that the incidence of newly infected vectors is again given by standard incidence form β3(SvIh/Nh). The total human population is governed by the following equation:
(2)dNhdt=b1Nh-μhNh-δhIh.


## 3. Dimensionless Formulation

Denotesh=Sh/Nh, ih=Ih/Nh, rh=Rh/Nh, sv=Sv/Nv, and iv=Iv/Nv. It is easy to verify that sh, ih, rh, sv, and iv satisfy the following system (see the Appendix details for):
(3)dsh(t)dt=b1(1-sh)-β1shih-β2shiv+δhshih,dih(t)dt=β1shih+β2shiv-(b1+γh+δh)ih+δhih2,drh(t)dt=γhih-b1rh+δhihrh,dsv(t)dt=μv(1-sv)-β3svih,div(t)dt=β3svih-μviv,
where solutions are restricted to sh+ih+rh=1 and sv+iv=1. Before analyzing the unnormalized model (1) and (2), we consider the normalized model (3) by scaling, and so we can study the following reduced system that describes the dynamics of the proportion of individuals in each class
(4)dsh(t)dt=b1(1-sh)-β1shih-β2shiv+δhshih,dih(t)dt=β1shih+β2shiv-(b1+γh+δh)ih+δhih2,div(t)dt=β3(1-iv)ih-μviv,
determining rh from
(5)drh(t)dt=γhih-b1rh+δhihrh,
or from rh=1-sh-ih and sv from sv=1-iv, respectively. The correlation between normalized and unnormalized models is explained in the Appendix. Throughout this work, we study the reduced system (4) in the closed, positively invariant set Γ={(sh,ih,iv)∈R+3,0≤sh+ih≤1,0≤iv≤1}, where R+3 denotes the nonnegative cone of R3 with its lower dimensional faces.


## 4. Existence of Equilibria

We seek the conditions for the existence and stability of the disease-“free” equilibrium (DFE)E0(sh0,0,0) and the “endemic” proportion equilibrium E*(sh*,ih*,iv*). Obviously, E0(1,0,0)∈Γ is the DFE of (4), which exists for all positive parameters. The Jacobian matrix of (4) at an arbitrary point E(sh,ih,iv) takes the following form:(6)J(E)=(-b1-β1ih-β2iv+δhih-(β1-δh)sh-β2shβ1ih+β2ivβ1sh-(b1+γh+δh)+2δhihβ2sh0β3(1-iv)-β3ih-μv).To analyze the stability of DFE, we calculate the characteristic equation of J(E) at E=E0 as follows:
(7)(λ+b1)(λ2+λ(μv+b1+γh+δh-β1)+μv(b1+γh+δh)(1-R0)λ2),
where
(8)R0=β1b1+γh+δh+β2β3μv(b1+γh+δh).
By Routh Hurwitz criteria [16], all roots of (7) have negative real parts if and only if R0<1. So, E0 is locally asymptotically stable for R0<1. If R0>1, the characteristic equation (7) has positive eigenvalue, and  E0 is thus unstable. We established the following theorem.Theorem 1.
The disease-free equilibrium is locally asymptotically stable wheneverR0<1 and unstable for R0>1.In order to find the “endemic” equilibrium of (4), we set the right hand side of (4) equal to zero and get
(9)iv*=β3ih*μv+β3ih*,sh*=b1(μv+β3ih*)(μv+β3ih*)(b1+(β1-δh)ih*)+β2β3ih*,
where ih* is a positive solution of the equation
(10)f(ih*)=A3ih*3+A2ih*2+A1ih*+A0=0,
where
(11)A3=β3δh(β1-δh),A2=β2β3δh+(μvδh-β3(b1+γh+δh))(β1-δh)+b1β3δh,A1=b1(μvδh+β1β3)-(b1+γh+δh)(β2β3+b1β3+μv(β1-δh)),A0=(b1+γh+δh)b1μv(R0-1).
From right hand side of (5), we have γhih*=(b1-δhih*)rh*>0 and second equation of (9) β1μv+β2β3-μvδh>β3δhih*, which means that
(12)0<ih*<min{1,b1δh,(β1μv+β2β3μvδh-1)μvβ3}.
If (β1μv+β2β3)/μvδh≤1, there is no positive ih*, and therefore the only equilibrium point in Γ is E0. Note that this is a special case of R0<1.Assume thatR0>1.(1)
Ifβ1>δh, then A3>0, we have f(-∞)<0, f(∞)>0 and f(0)=A0>0. Further,f(1)<0(ifb1/δh≥1) and f(b1/δh)<0. Thus, there exists unique ih* such that f(ih*)=0 (see Figure 1).(2)
Ifβ1=δh, then A3=0 and f(ih*)=A2ih*2+A1ih*+A0, where A2=β2β3δh+b1β3δh>0. We observe thatf(-∞)>0, f(∞)>0 and f(0)=A0>0. Moreover, f(1)<0(ifb1/δh≥1) and f(b1/δh)<0. Therefore, there exists unique ih* such that f(ih*)=0 (see Figure 2).(3)
Ifβ1<δh, then A3<0, we have f(-∞)>0, f(∞)<0 and still f(0)=A0>0, f(1)<0(ifb1/δh≥1), f(b1/δh)<0. In this case, we can say that there is only one root or  three roots in the interval (0,1)ifb1/δh≥1 or (0,b1/δh)ifb1/δh<1.We know thatf(ih*)=0 has three real roots if and only if
(13)q24+p327≤0,where
(14)p=A1A3-(A2)23(A3)2,q=A0A3-A1A23(A3)2+2(A2)327(A3)3,
or
(15)R^1=18A0A1A2A3-4A0(A2)3-4(A1)3A3+(A1)2(A2)227(A0)2(A3)2≥1.Figure 1
β
1
>
δ
h.Figure 2
β
1
=
δ
h.IfR^1<1, there is unique ih* such that f(ih*)=0 in the feasible interval.IfR^1>1, there are three different real roots for f(ih*)=0 say ih1*,ih2*,ih3*(ih1*<ih2*<ih3*). Note that, differentiating with respect to ih*, we obtain
(16)f′(ih*)=3A3ih*2+2A2ih*+A1.The three different real roots forf(ih*)=0 are in the feasible interval if and only if the following inequalities are satisfied:
(17)0<-A23A3<1,f′(0)=A1<0,f′(1)=3A3+2A2+A1<0(ifb1δh≥1),f′(b1δh)=3A3(b1δh)2+2A2(b1δh)+A1<0(ifb1δh<1).
If R^1=1, there are three real roots for f(ih*)=0, in which at least two are identical. Similarly, if inequalities (17) are satisfied, then there are three real roots for f(ih*)=0 in the feasible interval, say ih1*,ih2*,ih3*(ih1*=ih2*).Assume thatR0=1.(1)
Ifβ1=δh, then A3=0 and (10) reduces to ih*(A2ih*+A1)=0, which implies that ih*=0 or ih*=-A1/A2, which is positive but it lies outside the interval (0,1)ifb1/δh≥1 or (0,b1/δh)ifb1/δh<1.(2)
Ifβ1>δh, then A3>0, we have ih*(A3ih*2+A2ih*+A1)=0, which implies that ih*=0 or ih* is the solution of the equation
(18)g(ih*)=A3ih*2+A2ih*+A1=0,where g(-∞)>0, g(∞)>0, and g(0)=A1<0. Moreover, g(1)<0(ifb1/δh≥1) and g(b1/δh)<0ifb1/δh<1. Therefore, there exists no ih* such that g(ih*)=0 in the interval (0,1)ifb1/δh≥1 or (0,b1/δh)ifb1/δh<1.In summary, regarding the existence and the number of the “endemic” equilibria, we have the following.Theorem 2.
Suppose thatβ1≥δh. There is always a disease-“free” equilibrium for system (4); if R0>1, then there is a unique “endemic” equilibrium E*(sh*,ih*,iv*) with coordinates satisfying (9) and (10) besides the disease-“free” equilibrium.


## 5. Global Dynamics

### 5.1. Global Stability of the Disease-“Free” Equilibrium

In this subsection, we analyze the global behavior of the equilibria for system (4). The following theorem provides the global property of the disease-free equilibrium E0 of the system.Theorem 3.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable in the interior of Γ.Proof.
To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function:(19)L(t)=μvih(t)+β2iv(t).
Calculating the time derivative of L along (4), we obtain
(20)L′(t)=μvih′(t)+β2iv′(t)=μv[δhih2β1shih+β2shiv-(b1+γh+δh)ih+δhih2]+β2(β3svih-μviv)=μv[ih2δhih2β1(1-ih)ih+β2(1-ih)iv-(b1+γh+δh)ih+δhih2]+β2[β3(1-iv)ih-μviv]=μv[β1ih-β1ih2+β2iv-β2ihiv-(b1+γh+δh)ih+δhih2]+β2(β3ih-β3ivih-μviv)=μvβ1ih-μvβ1ih2+μvβ2iv-μvβ2ihiv-μv(b1+γh+δh)ih+μvδhih2+β2β3ih-β2β3ivih-β2μviv=-μv(b1+γh+δh)(1-R0)ih-μv(β1-δh)ih2-μvβ2ihiv-β2β3ivih.
Thus, L′(t) is negative if R0≤1 and L′=0 if and only if ih=0. Consequently, the largest compact invariant set in {(Sh,Ih,Iv)∈Γ,L′=0}, when R0≤1, is the singelton {E0}. Hence, LaSalle’s invariance principle [16] implies that “E0" is globally asymptotically stable in Γ. This completes the proof.


### 5.2. Global Stability of “Endemic” Equilibrium

Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibriumE* in the feasible region Γ. We have omitted the detailed introduction of this approach, and we refer the interested readers to see [17]. We summarize this approach below.Consider aC1 map f:x↦f(x) from an open set D⊂Rn to Rn such that each solution x(t,x0) to the differential equation
(21)x′=f(x).
is uniquely determined by the initial value x(0,x0). We have the following assumptions:(H1)D is simply connected;(H2) there exists a compact absorbing set K⊂D;(H3) (21) has unique equilibrium x- in D.LetP:x↦P(x) be a nonsingular (n2)×(n2) matrix-valued function which is C1 in D and a vector norm |·| on RN, where N=(n2).Letμ be the Lozinskiĭ measure with respect to the |·|. Define a quantity q-2 as
(22)q-2=limsupt→∞supx0∈K1t∫0t‍μ(B(x(s,x0)))ds,
where B=PfP-1+PJ[2]P-1, the matrix Pf is obtained by replacing each entry p of P by its derivative in the direction of f, (pij)f, and J[2] is the second additive compound matrix of the Jacobian matrix J of (21). The following result has been established in Li and Muldowney [17].Theorem 4.
Suppose that(H1), (H2), and (H3) hold, the unique endemic equilibrium E* is globally stable in Γ if  q-2<0.ObviouslyΓ is simply connected and E* is a unique endemic equilibrium for R0>1 in Γ. To apply the result of the above theorem for global stability of endemic equilibrium E*, we first prove the uniform persistence of (4) when the threshold parameter R0>1, by applying the acyclicity Theorem (see [18]).Definition 5 (see [19]).
The system (4) is uniformly persistent, that is, there exists c>0 (independent of initial conditions), such that liminft→∞sh≥c,liminft→∞ih≥c,liminft→∞iv≥c.
LetX be a locally compact metric space with metric d and let Γ be a closed nonempty subset of X with boundary Γ and interior Γ∘. Clearly, Γ∘ is a closed subset of Γ. Let Φt be a dynamical system defined on Γ. A set B in X is said to be invariant if Φ(B,t) = B. Define M∂:={x∈Γ:Φ(t,x)∈Γ,forallt≥0}.Lemma 6 (see [18]).
Assume that(a)
Φt has a global attractor;(b)
there existsM={M1,…,Mk} of pair-wise disjoint, compact and isolated invariant set on ∂Γ such that
(1)
⋃x∈M∂‍ω(x)⊆⋃j=1k‍Mj;(2)
no subsets ofM form a cycle on ∂Γ;(3)
eachMj is also isolated in Γ;(4)
Ws(Mj)⋂‍Γ∘=ϕ for each 1≤j≤k, where Ws(Mj) is stable manifold of Mj. Then Φt is uniformly persistent with respect to Γ.Proof.
We haveΓ={(sh,ih,iv)∈R+3,0≤sh+ih≤1,0≤iv≤1}, Γ∘={(sh,ih,iv)∈R+3sh,ih>0}, ∂Γ=Γ/Γ∘. Obviously M∂=∂Γ. Since Γ is bounded and positively invariant, so there exists a compact set M in which all solutions of system (4) initiated in Γ ultimately enter and remain forever. On sh-axis we have sh′=b1(1-sh) which means sh→1 as t→∞. Thus, E0 is the only omega limit point on ∂Γ, that is, ω(x)=E0 for all x∈M∂. Furthermore, M=E0 is a covering of Ω=⋃x∈M∂‍ω(x), because all solutions initiated on the sh-axis converge to E0. Also E0 is isolated and acyclic. This verifies that hypothesis (1) and (2) hold. When R0>1, the disease-“free” equilibrium (DFE) E0 is unstable from Theorem 1 and also Ws(M)=∂Γ. Hypothesis (3) and (4) hold. Therefore, there always exists a global attractor due to ultimate boundedness of solutions.The boundedness ofΓ and the above lemma imply that (4) has a compact absorbing set K⊂Γ [19]. Now we shall prove that the quantity q-2<0. We choose a suitable vector norm |·| in R3 and a 3×3 matrix-valued function
(23)P(x)=(1000ihiv000ihiv).
Obviously, P is C1 and nonsingular in the interior of Ω. Linearizing system (4) about an endemic equilibrium E* gives the following Jacobian matrix:(24)J(E*)=(-b1sh-(β1-δh)sh-β2shβ1ih+β2ivβ1sh-(b1+γh+δh)+2δhihβ2sh0β3(1-iv)-β3ih-μv).The second additive compound matrix of J(E*) is given by
(25)J[2]=(M11β2shβ2shβ3(1-iv)M22-(β1-δh)sh0β1ih+β2ivM33),
where
(26)M11=-b1sh+β1sh-(b1+γh+δh)+2δhih,M22=-b1sh-β3ih-μv,M33=β1sh-(b1+γh+δh)+2δhih-β3ih-μv.
The matrix B=PfP-1+PJ[2]P-1 can be written in block form as
(27)B=(B11B12B21B22),
with
(28)B11=-b1sh+β1sh-(b1+γh+δh)+2δhih,B12=(β2shivih,β2shivih),B21=((ihiv)β3(1-iv)0),B22=(Q11Q12Q21Q22),
where
(29)Q11=-b1sh-β3ih-μv,Q12=-(β1-δh)sh,Q21=β1ih+β2iv,Q22=β1sh-(b1+γh+δh)+2δhih-β3ih-μv,ivih(ihiv)f=ih′ih-iv′iv.
Consider the norm in R3 as |(u,v,w)|=max(|u|,|v|+|w|) where (u,v,w) denotes the vector in R3. The Lozinskiĭ, measure with respect to this norm is defined as μ(B)≤sup(g1,g2), where
(30)g1=μ1(B11)+|B12|,g2=μ1(B22)+|B21|.
From system (4), we can write
(31)ih′ih=β1sh+β2shivih-(b1+γh+δh)+δhih,iv′iv=β3(1-iv)ihiv-μv.
Since B11 is a scalar, its Lozinskiĭ measure with respect to any vector norm in R1 will be equal to B11. Thus
(32)B11=-b1sh+β1sh-(b1+γh+δh)+2δhih,|B12|=β2shivih,
and g1 will become
(33)g1=-b1sh+β1sh-(b1+γh+δh)+2δhih+β2shivih=ih′ih-b1sh+δhih≤ih′ih-b1+δhih.
Also |B21|=(ih/iv)β3(1-iv), |B12| and |B21| are the operator norms of B12 and B21 which are mapping from R2, to R and from R to R2 respectively, and R2 is endowed with the l1 norm. μ1(B22) is the Lozinskiĭ measure of 2×2 matrix B22 with respect to l1 norm in R2. 								(34)μ(B22)=Sup{ivih(ihiv)f-b1sh-β3ih-μv+β1ih+β2iv,ivih(ihiv)f+(β1-δh)sh+β1sh-(b1+γh+δh)+2δh-β3ih-μvihivih}≤ih′ih-iv′iv-b1+δhih-β3ih-μv,
if β1≤γh/2. Hence
(35)g2≤ih′ih-iv′iv-b1+δhih-β3ih-μv+(ihiv)β3(1-iv)=ih′ih-b1+δhih-β3ih.
Thus,
(36)μ(B)=Sup{g1,g2}≤ih′ih-b1+δh≤ih′ih-β-1,
where β-1=min(γh/2,b1/2). Since (4) is uniformly persistent when R0>1, so for T>0 such that t>T implies ih(t)≥c, iv(t)≥c and (1/t)logih(t)<(β-1/2) for all (sh(0),ih(0),iv(0))∈K. Thus,
(37)1t∫0t‍μ(B)dt<logih(t)t-β-1<-β-12,
for all (sh(0),ih(0),iv(0))∈K, which further implies that q-2<0. Therefore, all the conditions of Theorem 4 are satisfied. Hence, unique endemic equilibrium E* is globally stable in Γ.


## 5.1. Global Stability of the Disease-“Free” Equilibrium

In this subsection, we analyze the global behavior of the equilibria for system (4). The following theorem provides the global property of the disease-free equilibrium E0 of the system.Theorem 3.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable in the interior of Γ.Proof.
To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function:(19)L(t)=μvih(t)+β2iv(t).
Calculating the time derivative of L along (4), we obtain
(20)L′(t)=μvih′(t)+β2iv′(t)=μv[δhih2β1shih+β2shiv-(b1+γh+δh)ih+δhih2]+β2(β3svih-μviv)=μv[ih2δhih2β1(1-ih)ih+β2(1-ih)iv-(b1+γh+δh)ih+δhih2]+β2[β3(1-iv)ih-μviv]=μv[β1ih-β1ih2+β2iv-β2ihiv-(b1+γh+δh)ih+δhih2]+β2(β3ih-β3ivih-μviv)=μvβ1ih-μvβ1ih2+μvβ2iv-μvβ2ihiv-μv(b1+γh+δh)ih+μvδhih2+β2β3ih-β2β3ivih-β2μviv=-μv(b1+γh+δh)(1-R0)ih-μv(β1-δh)ih2-μvβ2ihiv-β2β3ivih.
Thus, L′(t) is negative if R0≤1 and L′=0 if and only if ih=0. Consequently, the largest compact invariant set in {(Sh,Ih,Iv)∈Γ,L′=0}, when R0≤1, is the singelton {E0}. Hence, LaSalle’s invariance principle [16] implies that “E0" is globally asymptotically stable in Γ. This completes the proof.


## 5.2. Global Stability of “Endemic” Equilibrium

Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibriumE* in the feasible region Γ. We have omitted the detailed introduction of this approach, and we refer the interested readers to see [17]. We summarize this approach below.Consider aC1 map f:x↦f(x) from an open set D⊂Rn to Rn such that each solution x(t,x0) to the differential equation
(21)x′=f(x).
is uniquely determined by the initial value x(0,x0). We have the following assumptions:(H1)D is simply connected;(H2) there exists a compact absorbing set K⊂D;(H3) (21) has unique equilibrium x- in D.LetP:x↦P(x) be a nonsingular (n2)×(n2) matrix-valued function which is C1 in D and a vector norm |·| on RN, where N=(n2).Letμ be the Lozinskiĭ measure with respect to the |·|. Define a quantity q-2 as
(22)q-2=limsupt→∞supx0∈K1t∫0t‍μ(B(x(s,x0)))ds,
where B=PfP-1+PJ[2]P-1, the matrix Pf is obtained by replacing each entry p of P by its derivative in the direction of f, (pij)f, and J[2] is the second additive compound matrix of the Jacobian matrix J of (21). The following result has been established in Li and Muldowney [17].Theorem 4.
Suppose that(H1), (H2), and (H3) hold, the unique endemic equilibrium E* is globally stable in Γ if  q-2<0.ObviouslyΓ is simply connected and E* is a unique endemic equilibrium for R0>1 in Γ. To apply the result of the above theorem for global stability of endemic equilibrium E*, we first prove the uniform persistence of (4) when the threshold parameter R0>1, by applying the acyclicity Theorem (see [18]).Definition 5 (see [19]).
The system (4) is uniformly persistent, that is, there exists c>0 (independent of initial conditions), such that liminft→∞sh≥c,liminft→∞ih≥c,liminft→∞iv≥c.
LetX be a locally compact metric space with metric d and let Γ be a closed nonempty subset of X with boundary Γ and interior Γ∘. Clearly, Γ∘ is a closed subset of Γ. Let Φt be a dynamical system defined on Γ. A set B in X is said to be invariant if Φ(B,t) = B. Define M∂:={x∈Γ:Φ(t,x)∈Γ,forallt≥0}.Lemma 6 (see [18]).
Assume that(a)
Φt has a global attractor;(b)
there existsM={M1,…,Mk} of pair-wise disjoint, compact and isolated invariant set on ∂Γ such that
(1)
⋃x∈M∂‍ω(x)⊆⋃j=1k‍Mj;(2)
no subsets ofM form a cycle on ∂Γ;(3)
eachMj is also isolated in Γ;(4)
Ws(Mj)⋂‍Γ∘=ϕ for each 1≤j≤k, where Ws(Mj) is stable manifold of Mj. Then Φt is uniformly persistent with respect to Γ.Proof.
We haveΓ={(sh,ih,iv)∈R+3,0≤sh+ih≤1,0≤iv≤1}, Γ∘={(sh,ih,iv)∈R+3sh,ih>0}, ∂Γ=Γ/Γ∘. Obviously M∂=∂Γ. Since Γ is bounded and positively invariant, so there exists a compact set M in which all solutions of system (4) initiated in Γ ultimately enter and remain forever. On sh-axis we have sh′=b1(1-sh) which means sh→1 as t→∞. Thus, E0 is the only omega limit point on ∂Γ, that is, ω(x)=E0 for all x∈M∂. Furthermore, M=E0 is a covering of Ω=⋃x∈M∂‍ω(x), because all solutions initiated on the sh-axis converge to E0. Also E0 is isolated and acyclic. This verifies that hypothesis (1) and (2) hold. When R0>1, the disease-“free” equilibrium (DFE) E0 is unstable from Theorem 1 and also Ws(M)=∂Γ. Hypothesis (3) and (4) hold. Therefore, there always exists a global attractor due to ultimate boundedness of solutions.The boundedness ofΓ and the above lemma imply that (4) has a compact absorbing set K⊂Γ [19]. Now we shall prove that the quantity q-2<0. We choose a suitable vector norm |·| in R3 and a 3×3 matrix-valued function
(23)P(x)=(1000ihiv000ihiv).
Obviously, P is C1 and nonsingular in the interior of Ω. Linearizing system (4) about an endemic equilibrium E* gives the following Jacobian matrix:(24)J(E*)=(-b1sh-(β1-δh)sh-β2shβ1ih+β2ivβ1sh-(b1+γh+δh)+2δhihβ2sh0β3(1-iv)-β3ih-μv).The second additive compound matrix of J(E*) is given by
(25)J[2]=(M11β2shβ2shβ3(1-iv)M22-(β1-δh)sh0β1ih+β2ivM33),
where
(26)M11=-b1sh+β1sh-(b1+γh+δh)+2δhih,M22=-b1sh-β3ih-μv,M33=β1sh-(b1+γh+δh)+2δhih-β3ih-μv.
The matrix B=PfP-1+PJ[2]P-1 can be written in block form as
(27)B=(B11B12B21B22),
with
(28)B11=-b1sh+β1sh-(b1+γh+δh)+2δhih,B12=(β2shivih,β2shivih),B21=((ihiv)β3(1-iv)0),B22=(Q11Q12Q21Q22),
where
(29)Q11=-b1sh-β3ih-μv,Q12=-(β1-δh)sh,Q21=β1ih+β2iv,Q22=β1sh-(b1+γh+δh)+2δhih-β3ih-μv,ivih(ihiv)f=ih′ih-iv′iv.
Consider the norm in R3 as |(u,v,w)|=max(|u|,|v|+|w|) where (u,v,w) denotes the vector in R3. The Lozinskiĭ, measure with respect to this norm is defined as μ(B)≤sup(g1,g2), where
(30)g1=μ1(B11)+|B12|,g2=μ1(B22)+|B21|.
From system (4), we can write
(31)ih′ih=β1sh+β2shivih-(b1+γh+δh)+δhih,iv′iv=β3(1-iv)ihiv-μv.
Since B11 is a scalar, its Lozinskiĭ measure with respect to any vector norm in R1 will be equal to B11. Thus
(32)B11=-b1sh+β1sh-(b1+γh+δh)+2δhih,|B12|=β2shivih,
and g1 will become
(33)g1=-b1sh+β1sh-(b1+γh+δh)+2δhih+β2shivih=ih′ih-b1sh+δhih≤ih′ih-b1+δhih.
Also |B21|=(ih/iv)β3(1-iv), |B12| and |B21| are the operator norms of B12 and B21 which are mapping from R2, to R and from R to R2 respectively, and R2 is endowed with the l1 norm. μ1(B22) is the Lozinskiĭ measure of 2×2 matrix B22 with respect to l1 norm in R2. 								(34)μ(B22)=Sup{ivih(ihiv)f-b1sh-β3ih-μv+β1ih+β2iv,ivih(ihiv)f+(β1-δh)sh+β1sh-(b1+γh+δh)+2δh-β3ih-μvihivih}≤ih′ih-iv′iv-b1+δhih-β3ih-μv,
if β1≤γh/2. Hence
(35)g2≤ih′ih-iv′iv-b1+δhih-β3ih-μv+(ihiv)β3(1-iv)=ih′ih-b1+δhih-β3ih.
Thus,
(36)μ(B)=Sup{g1,g2}≤ih′ih-b1+δh≤ih′ih-β-1,
where β-1=min(γh/2,b1/2). Since (4) is uniformly persistent when R0>1, so for T>0 such that t>T implies ih(t)≥c, iv(t)≥c and (1/t)logih(t)<(β-1/2) for all (sh(0),ih(0),iv(0))∈K. Thus,
(37)1t∫0t‍μ(B)dt<logih(t)t-β-1<-β-12,
for all (sh(0),ih(0),iv(0))∈K, which further implies that q-2<0. Therefore, all the conditions of Theorem 4 are satisfied. Hence, unique endemic equilibrium E* is globally stable in Γ.


## 6. Discussions and Simulations

This paper deals with a vector-host disease model which allows a direct mode of transmission and varying human population. It concerns diseases with long duration and substantial mortality rate (e.g., malaria). Our main results are concerned with the global dynamics of transformed proportionate system. We have constructed Lyapunov function to show the global stability of disease-“free” equilibrium and the geometric approach is used to prove the global stability of “endemic” equilibrium. The epidemiological correlations between the two systems (normalized and unnormalized) have also been discussed. The dynamical behavior of the proportionate model is determined by the basic reproduction number of the disease. The model has a globally asymptotically stable disease-“free” equilibrium wheneverR0≤1 (Figures 3 and 4). When R0>1, the disease persists at an “endemic” level (Figures 5 and 6) if β1<min(b1/2,γh/2). Figures 7, 8, 9, and 10 describe numerically “endemic” level of infectious individuals and infectious vectors under the condition β1<min(b1/2,γh/2). We here question that what are the dynamics of the proportionate system (4) even if the condition β1<min(b1/2,γh/2) is not satisfied? We see numerically that if δh,γh/2<β1<b1/2 or γh/2<β1=δh<b1/2 then infectious individuals and infectious vectors will also approach to endemic level for different initial conditions (Figures 11, 12, 13, and 14). It is also numerically shown that the same is true for the case δh,b1/2<β1<γh/2 or b1/2<β1=δh<γh/2 (Figures 15, 16, 17, and 18). This implies that the condition β1<min(b1/2,γh/2) is weak for the global stability of unique “endemic” equilibrium.Figure 3
b
1
=
1; β1=0.02; β2=0.4; β3=0.6; γh=0.3; μv=0.2; β1=δh=0.02; R0=0.92.Figure 4
b
1
=
1; β1=0.02; β2=0.4; β3=0.6; γh=0.3; μv=0.2; β1>δh=0.01; R0=0.93.Figure 5
b
1
=
2; β1=0.025; β2=0.65; β3=0.75; γh=0.000045; μv=0.2; β1>δh=0.000025; R0=1.23.Figure 6
b
1
=
2; β1=0.0025; β2=0.65; β3=0.75; γh=0.000045; μv=0.2; β1=δh=0.0025; R0=1.22.Figure 7
b
1
=
2; β1=0.4; β2=0.85; β3=0.75; γh=0.85; μv=0.2; δh=0.0000001<β1<b1/2=1, γh/2=0.425; R0=1.26.Figure 8
b
1
=
2; β1=0.4; β2=0.85; β3=0.75; γh=0.85; μv=0.2; δh=0.0000001<β1<b1/2=1, γh/2=0.425; R0=1.26.Figure 9
b
1
=
2; β1=0.04; β2=0.85; β3=0.65; γh=0.85; μv=0.1; 0.04=δh=β1<b1/2=1, γh/2=0.425; R0=1.92.Figure 10
b
1
=
2; β1=0.04; β2=0.85; β3=0.65; γh=0.85; μv=0.1; 0.04=δh=β1<b1/2=1, γh/2=0.425; R0=1.92.Figure 11
b
1
=
1; β1=0.01; β2=0.85; β3=0.95; γh=0.015; μv=0.25; 0.009=δh, γh/2=0.0075<β1<b1/2=0.5; R0=3.16.Figure 12
b
1
=
1; β1=0.01; β2=0.85; β3=0.95; γh=0.015; μv=0.25; 0.009=δh, γh/2=0.0075<β1<b1/2=0.5; R0=3.16.Figure 13
b
1
=
1; β1=0.01; β2=0.85; β3=0.95; γh=0.015; μv=0.25; γh/2=0.0075<0.01=δh=β1<b1/2=0.5; R0=3.16.Figure 14
b
1
=
1; β1=0.01; β2=0.85; β3=0.95; γh=0.015; μv=0.25; γh/2=0.0075<0.01=δh=β1<b1/2=0.5; R0=3.16.Figure 15
b
1
=
0.78; β1=0.4; β2=0.65; β3=0.55; γh=0.8; μv=0.15; 0.35=δh, b1/2=0.39<β1<γh/2=0.0075; R0=1.44.Figure 16
b
1
=
0.78; β1=0.4; β2=0.65; β3=0.55; γh=0.8; μv=0.15; 0.35=δh, b1/2=0.39<β1<γh/2=0.0075; R0=1.44.Figure 17
b
1
=
0.78; β1=0.4; β2=0.65; β3=0.55; γh=0.8; μv=0.15; b1/2=0.39<0.4=δh=β1<γh/2=0.0075; R0=1.40.Figure 18
b
1
=
0.78; β1=0.4; β2=0.65; β3=0.55; γh=0.8; μv=0.15; b1/2=0.39<0.4=δh=β1<γh/2=0.0075; R0=1.40.



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*Source: 293293-2013-04-08.xml*