research article global dynamics of a host-vector-predator...

Research ArticleGlobal Dynamics of a Host-Vector-PredatorMathematical Model

Fengyan Zhou1,2 and Hongxing Yao1,3

1 Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China2Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 31200, China3 School of Finance and Economics, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Correspondence should be addressed to Hongxing Yao; [email protected]

Received 18 April 2014; Accepted 2 July 2014; Published 22 July 2014

Academic Editor: Peter G L Leach

Copyright © 2014 F. Zhou and H. Yao.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A mathematical model which links predator-vector(prey) and host-vector theory is proposed to examine the indirect effect ofpredators on vector-host dynamics. The equilibria and the basic reproduction number R

0are obtained. By constructing Lyapunov

functional and using LaSalle’s invariance principle, global stability of both the disease-free and disease equilibria are obtained.Analytical results show thatR

0provides threshold conditions on determining the uniform persistence and extinction of the disease,

and predator density at any time should keep larger or equal to its equilibrium level for successful disease eradication. Finally, takingthe predation rate as parameter, we provide numerical simulations for the impact of predators on vector-host disease control. It isillustrated that predators have a considerable influence on disease suppression by reducing the density of the vector population.

1. Introduction

Host-vector diseases are infectious diseases caused by aninfectious microbe transmitted by a blood (or sap-) suckingarthropod called vectors, which carry the diseasewithout get-ting it themselves. For instance, human and animal diseasesare such as malaria, dengue fever, West Nile virus, Chagasdisease, sleeping sickness, and Lyme disease. Host-vectordisease also affects other living organisms, such as plants.Examples of vector-borne infections in crops include tobaccomosaic virus (TMV), tomato spotted wilt virus (TSWV),tomato yellow leaf curl virus (TYLCV), cucumber mosaicvirus (CMV), and potato virus Y (PVY) [1]. Vector-borneinfections in trees include the pine wilt disease and the redring disease in palms [2].

Recently, the frequent occurrence of natural disastersand environmental degradation around the world createsconditions suitable for breeding of vectors so that vector-borne diseases are constantly emerging. It is estimated thatalmost three-fourths of the new infectious diseases in recentyears belong to vector-borne infectious disease. Vector-bornediseases remain a serious global threat to humans, livestock,and crops and cause great economic losses in agriculture and

forestry; thus; control of such diseases is of great economicand public health concern.

The present mode of controlling vector-borne diseaseincludes using bednets, spraying insecticides [3], and sterileinsect technique (SIT) [4–6]. Effective vaccines and treatmentcontrol have not been adopted for most of the diseasesexcept that malaria, which is preventable and curable whentreatment and prevention measures are taken properly. Onepotential approach to control vector-borne disease is to intro-duce biological enemies (biocontrol agents) of the vector.Compared to insecticidemethod and sterile insect technique,introducing biological enemies (biocontrol agents) of thevector is more safe and cost-efficient. Moreover, it will leadto ecological balance and environment protection. Biologicalcontrol of vectors has been successfully applied in controllinga variety of disease pathogens, including crop diseases suchas the tomato leaf curl virus in India and the cassava mosaicvirus in sub-Saharan Africa [7, 8], as well as tree diseasessuch as pine wilt disease in Japan and China [9, 10]. Forhuman diseases, Entomogenous fungi are used as promisingbiopesticides for tick and sleeping sickness control [11, 12].Predators have been introduced as biological control agentsof vectors for various diseases such as malaria, dengue fever,

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 245650, 10 pageshttp://dx.doi.org/10.1155/2014/245650

2 Journal of Applied Mathematics

tick disease, and Lyme disease [13–20]. Several recent studiessuggested that predators led to a decline in local cases ofdengue fever in Vietnam and Thailand [21, 22] and malariain India [23, 24].

Mathematical models provide a powerful tool to under-stand the dynamics of disease spreading through a populationand in decision-making process regarding disease predic-tion and disease control. Since Ross [25] first proposed amalaria model in 1911, many authors have been attracted tomathematical modeling, and very large good results on thesubject have been presented. For example, the authors in [26]studied the global dynamics and back bifurcation of a vector-borne disease model with horizontal transmission in hostpopulation, which includes exposed classes both in host andvector populations and extended models studied in [27, 28].

However, compared with biological control of herbivo-rous pests, which has long been established as amajor compo-nent of pestmanagement programmers and is aimed to directdecrease pest densities by pest enemies [29, 30], biologicalcontrol of vectors has been seldom investigated based onmathematical models. The main reason is that modeling thebiological control of vectors is a complex interaction, mainlyincluding the virus-host interaction, vector-host interaction,and vector- (prey-) enemy interaction, which is more com-plicated than the only pest-enemy interaction [31, 32] ofbiological pest control.

Up to know, only several authors have studied thebiological control of vector to reduce the disease incidenceby mathematical models. For examples, Moore et al. [33]first proposed a host-vector-predator model to study theeffect of predator on the transmission and control of vector-borne disease.The efficacy of three types of biocontrol agents:predator/parasitoid, competitor, and pathogen of the vector,is compared in [34] to reduce disease incidence. Zhou andYao [35] improved the model proposed in [33], study thedisease control threshold, and limit cycles with persistenceof disease or without disease. However, both papers [33, 35]focus on the constant total host populations, and the hostsand vectors are simply divided into susceptible and infectiveones.

Motivated by the above considerations, in this paper weconsider a new host-vector-predator model, where both totalhost and vector population are time-dependent populationsize and the total host population is divided into foursubpopulations of susceptible hosts, exposed hosts, infectioushosts, and recovered hosts. Furthermore, the total vectorpopulation is divided into three subpopulations of susceptiblehosts, exposed hosts, and infectious hosts. The aim is toexplore the global dynamics of the proposed host-vector-predator model and the impacts of predator on host-vectordisease control.

The rest of the paper is organized as follows. In Section 2,we present a formulation of the mathematical model. Theequilibria and the basic reproduction number are givenin Section 3. In Section 4, global stability analysis of theequilibria is investigated. In Section 5, we use some numer-ical simulations to explore the impact of predators on theprevalence of vector-borne disease. Conclusions are given inSection 6.

2. Model Formulation

The basic model for the transmission dynamics of vector-borne disease with predator control is given by the followingdeterministic system of nonlinear differential equations:

𝑑𝑆ℎ (𝑡)

𝑑𝑡= 𝑏1− 𝛽1𝑆ℎ (𝑡) 𝐼V (𝑡) − 𝜇ℎ𝑆ℎ (𝑡) ,

𝑑𝐸ℎ (𝑡)

𝑑𝑡= 𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − (𝛼ℎ + 𝜇ℎ) 𝐸ℎ (𝑡) ,

𝑑𝐼ℎ (𝑡)

𝑑𝑡= 𝛼ℎ𝐸ℎ(𝑡) − (𝛾

ℎ+ 𝛿ℎ+ 𝜇ℎ) 𝐼ℎ(𝑡) ,

𝑑𝑅ℎ(𝑡)

𝑑𝑡= 𝛾ℎ𝐼ℎ(𝑡) − 𝜇

ℎ𝑅ℎ(𝑡) ,

𝑑𝑆V (𝑡)

𝑑𝑡= 𝑏2− 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝜇V𝑆V (𝑡) − ℎ𝑆V (𝑡) 𝑃V (𝑡) ,

𝑑𝐸V (𝑡)

𝑑𝑡= 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡)−𝛼V𝐸V (𝑡)−𝜇V𝐸V (𝑡)−ℎ𝐸V (𝑡) 𝑃V (𝑡) ,

𝑑𝐼V (𝑡)

𝑑𝑡= 𝛼V𝐸V (𝑡) − 𝜇V𝐼V (𝑡) − ℎ𝐼V (𝑡) 𝑃V (𝑡) ,

𝑑𝑃V (𝑡)

𝑑𝑡= 𝜀ℎ (𝑆V (𝑡) + 𝐸V (𝑡) + 𝐼V (𝑡)) 𝑃V (𝑡) − 𝑒𝑃V (𝑡) ,

(1)

with initial conditions

𝑆ℎ (0) ≥ 0, 𝐸

ℎ (0) ≥ 0, 𝐼ℎ (0) ≥ 0,

𝑅ℎ(0) ≥ 0, 𝑆V (0) ≥ 0, 𝐸V (0) ≥ 0,

𝐼V (0) ≥ 0, 𝑃V (0) ≥ 0,

(2)

where the total host population at time 𝑡 denoted by𝑁ℎ(𝑡) is

divided into four subpopulations of susceptible hosts 𝑆ℎ(𝑡),

exposed hosts 𝐸ℎ(𝑡), infectious hosts 𝐼

ℎ(𝑡), and recovered

hosts 𝑅ℎ(𝑡), so that 𝑁

ℎ(V) = 𝑆

ℎ(𝑡) + 𝐸

ℎ(𝑡) + 𝐼

ℎ(𝑡) + 𝑅

ℎ(𝑡).

The total vector population at time 𝑡 denoted by 𝑁V(𝑡) issplit into susceptible vectors 𝑆V(𝑡), exposed vectors 𝐸V(𝑡) andinfectious vectors 𝐼V(𝑡), so that𝑁V(𝑡) = 𝑆V(𝑡)+𝐸V(𝑡)+𝐼V(𝑡).Thepredator of the vector at time 𝑡 is represented by 𝑃V(𝑡). 𝑏1 and𝑏2are, respectively, the recruitment rate of hosts and vectors.

Parameters 𝛽1and 𝛽

2are, respectively, the rate of biting from

susceptible hosts to infected hosts and susceptible vectors toinfected vectors. 𝜇

ℎand 𝜇V are, respectively, the natural death

rate of infected hosts and vectors. Exposed hosts and vectorsdevelop symptoms of the disease and move to the infectiousclass at rates 𝛼

ℎand 𝛼V, respectively. 𝛾ℎ is the natural recovery

rate from the infected hosts. 𝛿ℎis the disease-caused death

rate of infected hosts. ℎ and 𝜀 are, respectively, the predationrates and conversation rate of the predator to the vector. 𝑒is the mortality of the predator including naturally, beingpreyed both by humans and other animals.

Journal of Applied Mathematics 3

Adding the host equations in (1), we have

𝑑𝑁ℎ(𝑡)

𝑑𝑡= 𝑏1− 𝜇ℎ𝑁ℎ(𝑡) − 𝛿

ℎ𝐼ℎ(𝑡) , (3)

and from the last four equations of system (1), we have

𝑑𝑁V (𝑡)

𝑑𝑡= 𝑏2− 𝜇V𝑁V (𝑡) − ℎ𝑁V (𝑡) 𝑃V (𝑡) ,

𝑑𝑃V (𝑡)

𝑑𝑡= 𝜀ℎ𝑁V (𝑡) 𝑃V (𝑡) − 𝑒𝑃V (𝑡) .

(4)

By (3) and (4), we have

𝑑𝑁ℎ(𝑡)

𝑑𝑡≤ 𝑏1− 𝜇ℎ𝑁ℎ(𝑡) ,

𝑑𝑁V (𝑡)

𝑑𝑡≤ 𝑏2− 𝜇V𝑁V (𝑡) ,

𝑑𝑃V (𝑡)

𝑑𝑡= 𝜀ℎ𝑁V (𝑡) 𝑃V (𝑡) − 𝑒𝑃V (𝑡) .

(5)

Let

Γ = { (𝑆ℎ, 𝐸ℎ, 𝐼ℎ, 𝑅ℎ, 𝑆V, 𝐸V, 𝐼V, 𝑃V) ∈ 𝑅

+

8

| 0 ≤ 𝑆ℎ+ 𝐸ℎ+ 𝐼ℎ+ 𝑅ℎ≤𝑏1

𝜇ℎ

,

0 ≤ 𝑆V + 𝐸V + 𝐼V ≤𝑏2

𝜇V, 𝑃V ≥ 0} ;

(6)

then, it is easy to verify that Γ is positively an invariant ofsystem (1).

Since the fourth equation for 𝑅ℎ(𝑡) in system (1) does

not influence the dynamics behavior of system, we can omitthe equations for 𝑅

ℎ(𝑡). System (1) in the invariant space Γ

can be written as the following seven dimensional nonlinearsystems:

𝑑𝑆ℎ(𝑡)

𝑑𝑡= 𝑏1− 𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − 𝜇ℎ𝑆ℎ (𝑡) ,

𝑑𝐸ℎ(𝑡)

𝑑𝑡= 𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − (𝛼ℎ + 𝜇ℎ) 𝐸ℎ (𝑡) ,

𝑑𝐼ℎ(𝑡)

𝑑𝑡= 𝛼ℎ𝐸ℎ(𝑡) − 𝑚𝐼

ℎ(𝑡) ,

𝑑𝑆V (𝑡)

𝑑𝑡= 𝑏2− 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝜇V𝑆V (𝑡) − ℎ𝑆V (𝑡) 𝑃V (𝑡) ,

𝑑𝐸V (𝑡)

𝑑𝑡= 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡)−𝛼V𝐸V (𝑡)−𝜇V𝐸V (𝑡)−ℎ𝐸V (𝑡) 𝑃V (𝑡) ,

𝑑𝐼V (𝑡)

𝑑𝑡= 𝛼V𝐸V (𝑡) − 𝜇V𝐼V (𝑡) − ℎ𝐼V (𝑡) 𝑃V (𝑡) ,

𝑑𝑃V (𝑡)

𝑑𝑡= 𝜀ℎ (𝑆V (𝑡) + 𝐸V (𝑡) + 𝐼V (𝑡)) 𝑃V (𝑡) − 𝑒𝑃V (𝑡) ,

(7)

where𝑚 = 𝛾ℎ+ 𝛿ℎ+ 𝜇ℎ. Let

Γ = { (𝑆ℎ, 𝐸ℎ, 𝐼ℎ, 𝑆V, 𝐸V, 𝐼V, 𝑃V) ∈ 𝑅

+

7

| 0 ≤ 𝑆ℎ+ 𝐸ℎ+ 𝐼ℎ≤𝑏1

𝜇ℎ

,

0 ≤ 𝑆V + 𝐸V + 𝐼V ≤𝑏2

𝜇V, 𝑃V ≥ 0} .

(8)

Obviously, for system (7), the region Γ is positively invariant.

3. The Equilibria and the Basic ReproductionNumber

Lemma 1. The equilibria of system (7) are as follows.

(i) If the predator is present, that is, 𝑃V(𝑡) > 0 for any𝑡 ∈ [0, +∞), then there exist a disease-free equilib-rium 𝐸

1(𝑆0

ℎ, 0, 0, 𝑆

0

V , 0, 0, ��V) and a disease equilibrium𝐸2(𝑆ℎ, 𝐸ℎ, 𝐼ℎ, 𝑆V, 𝐸V, 𝐼V, ��V) if 𝑅0 is larger than one.

(ii) If the predator is absent, that is, 𝑃V(𝑡) ≡ 0 for any𝑡 ∈ [0, +∞), then there exist a disease-free equilibrium𝐸3((𝑏1/𝜇ℎ)0, 0, (𝑏

2/𝜇V), 0, 0, 0) and a disease equilib-

rium 𝐸4(_𝑆 ℎ,

_𝐸ℎ,

_𝐼 ℎ,

_𝑆 V,

_𝐸V,

_𝐼 V, 0) if 𝑅1 is larger than

one.

where

𝑆0

ℎ=𝑏1

𝜇ℎ

, 𝑆0

V = ��V,

𝑆ℎ=𝑄1𝑄2𝑄3(𝜇V + ℎ��V + 𝛽2𝐼ℎ)

𝛼ℎ𝛼V𝛽1𝛽2��V

,

𝐼ℎ=

𝑅2

0− 1

𝑄21𝑄22𝑄3𝑄4𝜇ℎ(𝜇ℎ𝛽2𝑄3+ 𝛼V𝛽1𝛽2��V)

,

𝐸ℎ=𝑄2

𝛼ℎ

𝐼ℎ, 𝑆V =

𝑏2

𝜇V + ℎ��V + 𝛽2𝐼ℎ,

𝐸V =𝛼V𝛽2��V𝐼ℎ (𝜇V + ℎ��V)

𝛼V𝑄3 (𝜇V + ℎ𝑃V + 𝛽2𝐼ℎ),

𝐼V =𝛼V𝛽2��V𝐼ℎ

𝑄3(𝜇V + ℎ��V + 𝛽2𝐼ℎ)

,

_𝑆 ℎ =

𝑄1𝑄2𝑄5𝜇V (𝜇V + 𝛽2

_𝐼 ℎ)

𝛼ℎ𝛼V𝛽1𝛽2𝑏2

,


_𝐼 ℎ =

𝑅2

1− 1

𝑄21𝑄22𝑄5𝜇ℎ[𝜇ℎ𝜇V𝛽2𝑄5 + 𝛼V𝛽1𝛽2𝑏2]

,

_𝐸ℎ =

𝑄2

𝛼ℎ

_𝐼 ℎ,

_𝑆 V =

𝑏2

𝜇V + 𝛽2_𝐼 ℎ

,

_𝐸V =

𝜇V_𝐼 V

𝛼V,

_𝐼 V =

𝑏2𝛼V𝛽2

_𝐼 ℎ

𝜇V𝑄5 (𝜇V + 𝛽2_𝐼 ℎ)

,

𝑄1= 𝛼ℎ+ 𝜇ℎ, 𝑄

2= 𝑚 = 𝛾

ℎ+ 𝛿ℎ+ 𝜇ℎ,

𝑄3= 𝛼V + 𝜇V + ℎ��V, 𝑄

4= (𝜇V + ℎ��V) ,

𝑄5= 𝛼V + 𝜇V, ��V =

𝑒

𝜀ℎ,

��V =𝑏2− 𝜇V��V

ℎ��Vif 𝑏2𝜇V>𝑒

𝜀ℎ.

(9)

Here (𝑏2/𝜇V) is the equilibrium level of total vector population

of system (7) without predators, while (𝑒/𝜀ℎ) is the equilibriumlevel of total vector population of system (7) with predators.(𝑏2/𝜇V) > (𝑒/𝜀ℎ) is the sufficient and necessary conditions to

ensure ��V > 0. Therefore, from the viewpoint of biologicalmeaning, if the predators are present, we always assume that(𝑏2/𝜇V) > (𝑒/𝜀ℎ).

𝑅0

= √(𝛼ℎ𝛼V𝛽1𝛽2𝑏1��V/𝜇ℎ𝑄1𝑄2𝑄3𝑄4) and 𝑅

1=

√(𝛼ℎ𝛼V𝑏1𝑏2𝛽1𝛽2/𝜇ℎ𝜇

2

V𝑄1𝑄2𝑄5), respectively, are the basicreproduction number of system (7) and system (7) withoutpredators by [36, 37].

Remark 2. It is not difficult to find that𝑅0is less than𝑅

1.That

is, predation results into a reduction in the basic reproductionnumber 𝑅

0of system (7).

4. Global Stability Analysis

The purpose of this section is to discuss the global stability ofthe disease-free and disease equilibria of system (7) to obtainthe control condition under which diseases can be eradicatedby predators preying on vectors. Before giving themain proof,we first give the following Lemma.

Lemma 3. For system (4), the unique positive equilibrium(��V, ��V) is globally asymptotically stable, where ��V and ��V aregiven in Lemma 1.

By constructing Lyapunov function 𝑉(𝑡) =

(𝜀/2)(𝑁V(𝑡) − ��V)2

+ ��V(𝑃V(𝑡) − ��V − ��V ln(𝑃V(𝑡)/��V))and using LaSalle’s invariance principle, it is not difficultto prove that the positive equilibrium (��V, ��V) is globallyasymptotically stable. Here we omit it.

Theorem 4. If 𝑅0≤ 1 and for any 𝑡 ∈ [0, +∞), 𝑃V(𝑡) ≥ ��V;

then, the disease-free equilibrium 𝐸1of system (7) is globally

asymptotically stable.

Proof. We construct the following Lyapunov functional:

𝐿 (𝑡) = 𝑤0(𝑆ℎ(𝑡) − 𝑆

0

ℎ− 𝑆0

ℎln𝑆ℎ(𝑡)

𝑆0ℎ

) + 𝑤1𝐸ℎ(𝑡)

+ 𝑤2𝐼ℎ (𝑡) + 𝑤3 (𝑆V (𝑡) − 𝑆

0

V − 𝑆0

V ln𝑆V (𝑡)

𝑆0V)

+ 𝑤4𝐸V (𝑡) + 𝑤5𝐼V (𝑡)

+ 𝑤6(𝑃V (𝑡) − ��V − ��V ln

𝑃V (𝑡)

��V) ,

(10)

where 𝑤0= 𝑤1= (𝛼

ℎ/𝑄1), 𝑤2= 1, 𝑤

3= 𝑤4=

(𝑏1𝛼ℎ𝛼V𝛽1/𝜇ℎ𝑄1𝑄3𝑄4), 𝑤5 = (𝑏

1𝛼ℎ𝛽1/𝜇ℎ𝑄1𝑄4), 𝑤6

=

(𝑤3/𝜀).Calculating the derivative of 𝐿(𝑡) along the solution of (7)

yields that

𝑑𝐿 (𝑡)

𝑑𝑡= 𝑤0(𝑆ℎ (𝑡) − 𝑆

0

ℎ

𝑆ℎ(𝑡)

) [𝑏1− 𝛽1𝑆ℎ (𝑡) 𝐼V (𝑡) − 𝜇ℎ𝑆ℎ (𝑡)]

+ 𝑤1[𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − (𝛼ℎ + 𝜇ℎ) 𝐸ℎ (𝑡)]

+ 𝑤2[𝛼ℎ𝐸ℎ (𝑡) − 𝑚𝐼ℎ (𝑡)]

+ 𝑤3(𝑆V (𝑡) − 𝑆

0

V

𝑆V (𝑡))

× [𝑏2− 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝜇V𝑆V (𝑡) − ℎ𝑆V (𝑡) 𝑃V (𝑡)]

+ 𝑤4[𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝛼V𝐸V (𝑡)

−𝜇V𝐸V (𝑡) − ℎ𝐸V (𝑡) 𝑃V (𝑡)]

+ 𝑤5[𝛼V𝐸V (𝑡) − 𝜇V𝑆V (𝑡) − ℎ𝐼V (𝑡) 𝑃V (𝑡)]

+ 𝑤6(𝑃V (𝑡) − ��V

𝑃V (𝑡))

× [𝜀ℎ (𝑆V (𝑡) + 𝐸V (𝑡) + 𝐼V (𝑡)) − 𝑒] 𝑃V (𝑡) .

(11)

By the equilibrium conditions 𝑏1= 𝜇ℎ𝑆0

ℎand 𝑏

2= 𝜇V𝑆

0

V +

ℎ𝑆0

V��V, it follows that

𝑑𝐿 (𝑡)

𝑑𝑡

= −𝜇ℎ𝑤0

(𝑆ℎ(𝑡) − 𝑆

0

ℎ)2

𝑆ℎ (𝑡)

− 𝜇V𝑤3(𝑆V(𝑡) − 𝑆

0

V)2

𝑆V (𝑡)

+ (𝑤1− 𝑤0) 𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) + (𝑤4 − 𝑤3) 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡)

+ (𝑤2𝛼ℎ− 𝑤1𝑄1) 𝐸ℎ(𝑡) + (𝑤

5𝛼V − 𝑤4𝑄3) 𝐸V (𝑡)


+ [𝑤3𝛽2��V − 𝑤2𝑄2] 𝐼ℎ (𝑡) + [𝑤0𝛽1

𝑏1

𝜇ℎ

− 𝑤5𝑄4] 𝐼V (𝑡)

+ 𝑤3ℎ(𝑆V (𝑡) − 𝑆

0

V)

𝑆V (𝑡)(𝑆0

V��V − 𝑆V (𝑡) 𝑃V (𝑡))

+ 𝑤4ℎ𝐸V (𝑡) (��V − 𝑃V (𝑡))

+ 𝑤5ℎ𝐼V (𝑡) (��V − 𝑃V (𝑡)) + 𝑤6𝜀ℎ𝐼V (𝑡) (𝑃V (𝑡) − ��V)

× (𝑁V (𝑡) − ��V) .

(12)

Since

𝑤3ℎ(𝑆V (𝑡) − 𝑆

0

V)

𝑆V (𝑡)(𝑆0

V��V − 𝑆V (𝑡) 𝑃V (𝑡))

+ 𝑤4ℎ𝐸V (𝑡) (��V − 𝑃V (𝑡))

= −𝑤3ℎ��V

(𝑆V (𝑡) − 𝑆0

V)2

𝑆V (𝑡)

+ 𝑤3ℎ (𝑆V (𝑡) − 𝑆

0

V) (��V − 𝑃V (𝑡)) ,

(13)

then using 𝑆0ℎ= 𝑏1/𝜇ℎ, 𝑆0

V = ��V we have

𝑑𝐿 (𝑡)

𝑑𝑡

= −𝜇ℎ𝑤0

(𝑆ℎ(𝑡) − (𝑏

1/𝜇ℎ))2

𝑆ℎ(𝑡)

− 𝑤3(𝜇V + ℎ��V)

(𝑆V(𝑡) − ��V)2

𝑆V (𝑡)

+ 𝑄2(𝑅2

0− 1) 𝐼

ℎ(𝑡) + [𝑤

3ℎ (𝑆V (𝑡) − ��V)

+𝑤4ℎ𝐸V (𝑡) + 𝑤5ℎ𝐼V (𝑡) ]

× (��V − 𝑃V (𝑡))

+ 𝑤6𝜀ℎ (𝑃V (𝑡) − ��V) (𝑁V (𝑡) − ��V) .

(14)

After some rearrangement, we have

𝑑𝐿 (𝑡)

𝑑𝑡

= −𝜇ℎ𝑤0

(𝑆ℎ(𝑡) − (𝑏

1/𝜇ℎ))2

𝑆ℎ(𝑡)

− 𝑤3(𝜇V + ℎ��V)

(𝑆V(𝑡) − ��V)2

𝑆V (𝑡)

+ 𝑄2(𝑅2

0− 1) 𝐼

ℎ (𝑡) + ℎ (𝑤5 − 𝑤3) 𝐼V (𝑡) (��V − 𝑃V) .

(15)

Since 𝑃V(𝑡) ≥ ��V, then we have 𝑁V(𝑡) ≤ ��V for any 𝑡 ∈[0, +∞). Otherwise, if 𝑁V(𝑡) > ��V for any 𝑡 ∈ [0, +∞), then𝜀ℎ𝑁V(𝑡) − 𝑒 > 0; thus, from the last equation of system (7),we have (𝑑𝑃V(𝑡)/𝑑𝑡) > 0. On the other hand, by Lemma 1,lim𝑡→+∞

𝑃V(𝑡) = ��V, so for any 𝑡 ∈ [0, +∞), 𝑃V(𝑡) < ��V. This

contradicts 𝑃V(𝑡) ≥ ��V. So𝑁V(𝑡) ≤ ��V when 𝑃V(𝑡) ≥ ��V for any𝑡 ∈ [0, +∞). Thus, (𝑑𝐿(𝑡)/𝑑𝑡) ≤ 0 if 𝑅

0≤ 1 and 𝑃V(𝑡) ≥ ��V

for any 𝑡 ∈ [0, +∞). Furthermore, (𝑑𝐿(𝑡)/𝑑𝑡) = 0 if and onlyif 𝑆ℎ= 𝑆0

ℎ, 𝑆V = 𝑆

0

V = ��V, 𝐸ℎ = 𝐸V = 𝐼ℎ= 𝐼V = 0 and

𝑃V = ��V. Consequently, the largest compact invariant set in{(𝑆ℎ(𝑡), 𝐸ℎ(𝑡), 𝐼ℎ(𝑡), 𝑆V(𝑡), 𝐸V(𝑡), 𝐼V(𝑡), 𝑃V(𝑡)) ∈ Γ : 𝑑𝐿/𝑑𝑡 = 0}

when 𝑅0≤ 1 is the singleton 𝐸

1. Then by Lyapunov-LaSalle

theorem, the equilibrium 𝐸1is globally stable if 𝑅

0≤ 1 and

𝑃V(𝑡) ≥ ��V for any 𝑡 ∈ [0, +∞).

Remark 5. From Theorem 4 we can see that when 𝑅0≤

1 the disease-free equilibrium 𝐸1of system (7) is globally

asymptotically stable if the predator population size 𝑃V(𝑡) isnot less than the predator equilibrium level ��V, which dependson the total equilibrium vector density ��V. That is, thepredator density threshold for successful disease eradicationis 𝑃V(𝑡) = ��V. If 𝑃V(𝑡) ≥ ��V and 𝑅0 ≤ 1 then disease can beeradicated; otherwise; pathogen persists though predators areintroduced and 𝑅

0≤ 1.

Similar to the proof of Theorem 4, we have the followingcorollary to show that the disease-free equilibrium 𝐸

3of

system (7) in absence of predators is global asymptoticallystable.

Corollary 6. If 𝑅1≤ 1 then the disease-free equilibrium 𝐸

3of

system (7) without predators is globally asymptotically stable.

A global stability result for the endemic equilibrium𝐸2of

the system (7) is given below.

Theorem 7. The endemic equilibrium state 𝐸2of system (7) is

globally asymptotically stable if

𝑅0> 1, 𝛼V =

𝛽2𝑆V𝐼ℎ

𝐸V, (16)

where 𝑆V, 𝐼ℎ and 𝐸V are the disease equilibrium value ofsusceptible vectors, infected hosts, and exposed vectors of system(7).

Proof. Consider the following Lyapunov functional:

𝑉 (𝑡)

= 𝑐1(𝑆ℎ(𝑡) − 𝑆

ℎln𝑆ℎ(𝑡)

𝑆ℎ

) + 𝑐2(𝐸ℎ(𝑡) − 𝐸

ℎln𝐸ℎ(𝑡)

𝐸ℎ

)

+ 𝑐3(𝐼ℎ(𝑡) − 𝐼

ℎln𝐼ℎ(𝑡)

𝐼ℎ

) + 𝑐4(𝑆V (𝑡) − 𝑆V ln

𝑆V (𝑡)

𝑆V)

+ 𝑐5(𝐸V (𝑡) − 𝐸V ln

𝐸V (𝑡)

𝐸V) + 𝑐6(𝐼V (𝑡) − 𝐼V ln

𝐼V (𝑡)

𝐼V)

+ 𝑐7(𝑃V (𝑡) − ��V ln

𝑃V (𝑡)

��V) ,

(17)


where 𝑐2= 𝑐1, 𝑐3= (𝑐1𝛽1𝑎1/𝑑1), 𝑐4= 𝑐5= (𝑐1𝛽1𝑎1/𝛽2𝑎2),

𝑐6= (𝑐5𝛽2𝑎2/𝑑2), 𝑐7= (𝑐4/𝜀), 𝑆ℎ𝐼V = 𝑎

1, 𝑆V𝐼ℎ = 𝑎

2, 𝑑1=

𝛼ℎ𝐸ℎ, 𝑑2= 𝛼V𝐸V.

Calculate the derivative of 𝑉(𝑡) along the solution of (1);this yields that

𝑑𝑉 (𝑡)

𝑑𝑡

= 𝑐1(1 −

𝑆ℎ

𝑆ℎ (𝑡)

) [𝑏1− 𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − 𝜇ℎ𝑆ℎ (𝑡)]

+ 𝑐2(1 −

𝐸ℎ

𝐸ℎ (𝑡)

) [𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − (𝛼ℎ + 𝜇ℎ) 𝐸ℎ (𝑡)]

+ 𝑐3(1 −

𝐼ℎ

𝐼ℎ (𝑡)

) [𝛼ℎ𝐸ℎ(𝑡) − 𝑚𝐼

ℎ(𝑡)]

+ 𝑐4(1 −

𝑆V

𝑆V (𝑡))

× [𝑏2− 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝜇V𝑆V (𝑡) − ℎ𝑆V (𝑡) 𝑃V (𝑡)]

+ 𝑐5(1 −

𝐸V

𝐸V (𝑡))

× [𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝛼V𝐸V (𝑡) − 𝜇V𝐸V (𝑡) − ℎ𝐸V (𝑡) 𝑃V (𝑡)]

+ 𝑐6(1 −

𝐼V

𝐼V (𝑡))

× [𝛼V𝐸V (𝑡) − 𝜇V𝐼V (𝑡) − ℎ𝐼V (𝑡) 𝑃V (𝑡)]

+ 𝑐7(1 −

��V

𝑃V (𝑡))

× [𝜀ℎ (𝑆V (𝑡) + 𝐸V (𝑡) + 𝐼V (𝑡)) 𝑃V (𝑡) − 𝑒𝑃V (𝑡)] .

(18)

System (7) satisfied the following relations at equilibriumpoint:

𝑏1= 𝛽1𝑆ℎ𝐼V + 𝜇ℎ𝑆ℎ,

(𝛼ℎ+ 𝜇ℎ) =

𝛽1𝑆ℎ𝐼V

𝐸ℎ

,

𝑚 =𝛼ℎ𝐸ℎ

𝐼ℎ

,

𝑏2= 𝛽2𝑆V𝐼ℎ + 𝜇V𝑆V + ℎ𝑆V��V,

(𝛼V + 𝜇V) =𝛽2𝑆V𝐼ℎ

𝐸V− ℎ��V,

𝜇V =𝛼V𝐸V

𝐼V− ℎ��V,

𝑒 = 𝜀ℎ��V, ��V = 𝑆V + 𝐸V + 𝐼V.

(19)

Substituting 𝑏1up to 𝑒 in the above equation, we obtain

𝑑𝑉 (𝑡)

𝑑𝑡

= −𝑐1𝜇ℎ


ℎ)2

𝑆ℎ(𝑡)

− 𝑐4𝜇V(𝑆V (𝑡) − 𝑆V)

2

𝑆V (𝑡)

+ 𝑐1(1 −

𝑆ℎ

𝑆ℎ(𝑡)) [𝛽1𝑆ℎ𝐼V − 𝛽1𝑆ℎ (𝑡) 𝐼V (𝑡)]

+ 𝑐2(1 −

𝐸ℎ

𝐸ℎ(𝑡)) [𝛽1𝑆ℎ(𝑡) 𝐼V (𝑡) − 𝐸ℎ

𝛽1𝑆ℎ𝐼V

𝐸ℎ

]

+ 𝑐3(1 −

𝐼ℎ

𝐼ℎ(𝑡)) [𝛼ℎ𝐸ℎ (𝑡) − 𝛼ℎ𝐸ℎ

𝐼ℎ (𝑡)

𝐼ℎ

]

+ 𝑐4(1 −

𝑆V

𝑆V (𝑡))

× [𝛽2𝑆V𝐼ℎ − 𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) + ℎ𝑆V��V − ℎ𝑆V (𝑡) 𝑃V (𝑡)]

+ 𝑐5(1 −

𝐸V

𝐸V (𝑡))

× [𝛽2𝑆V (𝑡) 𝐼ℎ (𝑡) − 𝛽2𝑆V𝐼ℎ

𝐸V (𝑡)

𝐸V

−ℎ𝐸V (𝑡) 𝑃V (𝑡) + ℎ𝐸V (𝑡) ��V]

+ 𝑐6(1 −

𝐼V

𝐼V (𝑡))

× [𝛼V𝐸V (𝑡) − 𝛼V𝐸V𝐼V

𝐼V (𝑡)− ℎ𝐼V (𝑡) 𝑃V (𝑡) + ℎ𝐼V (𝑡) ��V]

+ 𝑐7(1 −

��V

𝑃V (𝑡))

× [𝜀ℎ (𝑆V (𝑡) + 𝐸V (𝑡) + 𝐼V (𝑡) − 𝑆V − 𝐸V − 𝐼V)] 𝑃V (𝑡) .

(20)

Set

𝑆ℎ

𝑆ℎ

= 𝑥1,

𝐸ℎ

𝐸ℎ

= 𝑥2,

𝐼ℎ

𝐼ℎ

= 𝑥3,

𝑆V

𝑆V= 𝑥4,

𝐸V

𝐸V= 𝑥5,

𝐼V

𝐼V= 𝑥6,

𝑃V

��V= 𝑥7, 𝑆V��V = 𝑎3, 𝐸V��V = 𝑎4,

𝐼V��V = 𝑎5;

(21)


then, from the above equation, we obtain

𝑑𝑉 (𝑡)

𝑑𝑡

= −𝑐1𝜇ℎ


ℎ)2

𝑆ℎ(𝑡)


2

𝑆V (𝑡)

+ 𝑐1𝛽1𝑎1− 𝑐1𝛽1𝑎1𝑥1𝑥6− 𝑐1𝛽1𝑎1

1

𝑥1

+ 𝑐1𝛽1𝑎1𝑥6

+ 𝑐2𝛽1𝑎1+ 𝑐2𝛽1𝑎1𝑥1𝑥6− 𝑐2𝛽1𝑎1𝑥2− 𝑐2𝛽1𝑎1

𝑥1𝑥6

𝑥2

+ 𝑐3𝑑1+ 𝑐3𝑑1𝑥2− 𝑐3𝑑1𝑥3− 𝑐3𝑑1

𝑥2

𝑥3

+ 𝑐4𝛽2𝑎2+ 𝑐4𝛽2𝑎2𝑥3− 𝑐4𝛽2𝑎2𝑥3𝑥4− 𝑐4𝛽2𝑎2

1

𝑥4

+ 𝑐4ℎ𝑎3− 𝑐4ℎ𝑎3𝑥4𝑥7− 𝑐4ℎ𝑎3

1

𝑥4

+ 𝑐4ℎ𝑎3𝑥7

+ 𝑐5𝛽2𝑎2+ 𝑐5𝛽2𝑎2𝑥3𝑥4− 𝑐5𝛽2𝑎2𝑥5− 𝑐5𝛽2𝑎2

𝑥3𝑥4

𝑥5

− 𝑐5ℎ𝑎4𝑥5𝑥7+ 𝑐5ℎ𝑎4𝑥5+ 𝑐5ℎ𝑎4𝑥7− 𝑐5ℎ𝑎4

+ 𝑐6𝑑2+ 𝑐6𝑑2𝑥5− 𝑐6𝑑2

𝑥5

𝑥6

− 𝑐6𝑑2𝑥6

− 𝑐6ℎ𝑎5𝑥6𝑥7+ 𝑐6ℎ𝑎5𝑥6+ 𝑐6ℎ𝑎5𝑥7− 𝑐6ℎ𝑎5

+ 𝑐7𝜀ℎ𝑎3𝑥4𝑥7+ 𝑐7𝜀ℎ𝑎4𝑥5𝑥7+ 𝑐7𝜀ℎ𝑎5𝑥6𝑥7

− 𝑐7𝜀ℎ𝑎3𝑥7− 𝑐7𝜀ℎ𝑎4𝑥7− 𝑐7𝜀ℎ𝑎5𝑥7

− 𝑐7𝜀ℎ𝑎3𝑥4− 𝑐7𝜀ℎ𝑎4𝑥5− 𝑐7𝜀ℎ𝑎5𝑥6

+ 𝑐7𝜀ℎ𝑎3+ 𝑐7𝜀ℎ𝑎4+ 𝑐7𝜀ℎ𝑎5.

(22)

After some rearrangement, we have

𝑑𝑉 (𝑡)

𝑑𝑡

= −𝑐1𝜇ℎ


ℎ)2

𝑆ℎ(𝑡)


2

𝑆V (𝑡)

+ (𝑐1𝛽1𝑎1− 𝑐6𝑑2) 𝑥6+ (𝑐3𝑑1− 𝑐2𝛽1𝑎1) 𝑥2

+ (𝑐4𝛽2𝑎2− 𝑐3𝑑1) 𝑥3+ (𝑐6𝑑2− 𝑐5𝛽2𝑎2) 𝑥5

+ (𝑐2𝛽1𝑎1− 𝑐1𝛽1𝑎1) 𝑥1𝑥6+ (𝑐5𝛽2𝑎2− 𝑐4𝛽2𝑎2) 𝑥3𝑥4

+ (𝑐7𝜀 − 𝑐4) ℎ𝑎3𝑥4𝑥7+ (𝑐7𝜀 − 𝑐5) ℎ𝑎4𝑥5𝑥7

+ (𝑐7𝜀 − 𝑐6) ℎ𝑎5𝑥6𝑥7

+ (𝑐4𝑎3+ 𝑐5𝑎4+ 𝑐6𝑎5− 𝑐7𝜀𝑎3− 𝑐7𝜀𝑎4− 𝑐7𝜀𝑎5) ℎ𝑥7

+ (𝑐7𝜀ℎ𝑎4+ 𝑐7𝜀ℎ𝑎5− 𝑐5ℎ𝑎4− 𝑐6ℎ𝑎5)

+ (𝑐4ℎ𝑎3+ 𝑐7𝜀ℎ𝑎3− 𝑐4ℎ𝑎3

1

𝑥4

− 𝑐7𝜀ℎ𝑎3𝑥4)

+ 𝑐1𝛽1𝑎1− 𝑐1𝛽1𝑎1

1

𝑥1

− 𝑐2𝛽1𝑎1

𝑥1𝑥6

𝑥2

+ 𝑐2𝛽1𝑎1

− 𝑐3𝑑1

𝑥2

𝑥3

+ 𝑐3𝑑1+ 𝑐4𝛽2𝑎2− 𝑐4𝛽2𝑎2

1

𝑥4

− 𝑐5𝛽2𝑎2

𝑥3𝑥4

𝑥5

+ 𝑐5𝛽2𝑎2− 𝑐6𝑑2

𝑥5

𝑥6

+ 𝑐6𝑑2.

(23)

By some reduction, it follows that

𝑑𝑉 (𝑡)

𝑑𝑡

= −𝑐1𝜇ℎ


ℎ)2

𝑆ℎ(𝑡)


2

𝑆V (𝑡)

+ 𝑐4ℎ𝑎3(2 − 𝑥

4−1

𝑥4

)

+ 𝑐1𝛽1𝑎1(6 −

1

𝑥1

−𝑥1𝑥6

𝑥2

−𝑥2

𝑥3

−1

𝑥4

−𝑥3𝑥4

𝑥5

−𝑥5

𝑥6

) .

(24)

Since the arithmetic mean is greater than or is equal to thegeometric mean, then

2 − 𝑥4−1

𝑥4

≥ 0,

1

𝑥1

+𝑥1𝑥6

𝑥2

+𝑥2

𝑥3

+1

𝑥4

+𝑥3𝑥4

𝑥5

+𝑥5

𝑥6

≥ 6.

(25)

Thus, it follows from (24) that (𝑑𝑉(𝑡)/𝑑𝑡) ≤ 0 in Ω. Theequation (𝑑𝑉(𝑡)/𝑑𝑡) = 0 holds if and only if 𝑥

1= 𝑥2= 𝑥3=

𝑥4= 𝑥5= 𝑥6= 1; that is, 𝑆

ℎ= 𝑆ℎ, 𝐸ℎ= 𝐸ℎ, 𝐼ℎ= 𝐼ℎ, 𝑆V =

𝑆V, 𝐸V = 𝐸V, 𝐼V = 𝐼V, 𝑃V = ��V. Therefore, we prove the globalstability of the disease in Γ. The maximal compact invariantset in {𝑆

ℎ, 𝐸ℎ, 𝐼ℎ, 𝑆V, 𝐸V, 𝐼V, 𝑃V ∈ Γ : 𝑑𝑉(𝑡)/𝑑𝑡 = 0} is {𝐸2}when

𝑅0> 1 and 𝛼V = (𝛽3𝑆V𝐼ℎ/𝐸V). From the LaSalle’s invariance

principle, we finish the proof of Theorem 7.

Similar to the proof of Theorem 7, we have the followingcorollary to show that the disease equilibrium 𝐸

4of system

(7) in absence of predators is globally asymptotically stable.

Corollary 8. The endemic equilibrium state 𝐸4of system (7)

without predators is globally asymptotically stable if 𝑅1> 1.

Remark 9. From Theorems 4 and Theorem 7, Corollary 6and Corollary 8 we find that 𝑅

0= 1 and 𝑅

1= 1

provide threshold conditions on determining the uniformpersistence and extinction of the disease with and without


0 100 200 300 400 500 600

0

0.2

0.4

0.6

0.8

1

t

Popu

latio

n siz

e

Sh

Eh

Ih

S�

E�

I�

Figure 1: Global asymptotically stability of the unique disease equilibrium of system (7) without predators when 𝑅1> 1.

0

0.1

0.15

0.2

0.25

0.3

0.35

0.45

0.4

0.05

0

500 1000

t

Ih

h = 0

h = 0.1 h = 0.3

h = 0.5

(a)

0

500 10000

0.2

0.4

0.6

0.8

1I�

t

h = 0

h = 0.1 h = 0.3

h = 0.5

(b)

Figure 2: The population densities of the infected hosts and vectors of system (7) without and with predators.

predators, respectively.Moreover, we also find that the diseaseis persistent without predators if 𝑅

1> 1, then by introducing

predators, the disease will tend to be extinct if 𝑅0≤ 1.

5. Numerical Simulations

In this section, we examine the effects of predators on thetransmission dynamics of vector-borne diseases by somenumerical simulations.We choose the following set of param-eter values (it should be stated that these parameters arechosen for illustrative purpose only and may not necessarilybe realistic epidemiologically): 𝑏

1= 0.786098, 𝑏

2= 0.190028,

𝛽1= 0.961509, 𝛽

2= 0.535061, 𝜇

ℎ= 0.914518, 𝜇V =

0.0150508, 𝛼ℎ= 0.645664, 𝛼V = 0.19636, and 𝑚 = 1.120469;

By simple calculations, the basic reproduction number 𝑅1=

11.2826 > 1; then, by Corollary 8, the disease equilibrium ofsystem (7) without predators is globally asymptotically stable.That is, the disease persists without predators (see Figure 1).

Choose 𝑒 = 0.1, 𝜀 = 0.15; we consider the effectof predators on disease control by comparing equilibriumlevel of infected hosts and vectors with different values ofℎ in absence of the predators, that is, ℎ = 0; then, by theCorollary 8 disease persists (see the solid line of Figure 2). Byintroducing predators of the vector population, we find thatthe equilibrium infection levels have been reduced a bit if the


predation rate ℎ is equal to 0.1; however, the disease persists(see the dotted line of Figure 2). Increasing the predationrate such that ℎ = 0.3, we find that though the disease stillpersists, the equilibrium levels of infected hosts and vectorshave been greatly lessened (see the dot and dashed line ofFigure 2). By enhancing the predation rate ℎ such that ℎ =0.5 > ℎ

∗= 0.4755 (ℎ

∗ satisfies that 𝑅0= 1), then 𝑅

0=

0.9411 < 1; therefore, byTheorem 4 the vector-borne diseasecan be eradicated by introducing vector predators (see thedashed line of Figure 2).

Remark 10. From Figures 1 and 2, we find that predation hasa positive effect on vector-host disease control by reducingvector density. Furthermore, disease can be eradicated ifpredation rate ℎ is large enough such that 𝑅

0(ℎ) ≤ 1 and

the predator density 𝑃V(𝑡) satisfies 𝑃V(𝑡) ≥ ��V, where ��V is thepredator equilibrium level.

6. Conclusions

In this paper, we propose a host-vector-predator coupledmodel with variable host and vector population size toinvestigate the effect of predators on vector-borne diseasecontrol by analyzing the global stability of the disease-freeand disease equilibria. It is shown that the basic reproductionnumber 𝑅

0characterizes the disease transmission dynamics:

if 𝑅0≤ 1, then there exists only the disease-free equilibrium

which is globally asymptotically stable when predator densityat any time keeps larger or equal to its equilibrium level;that is, disease tends to be extinct when predators areintroduced, and if 𝑅

0> 1, then there is a disease equilibrium

which is globally asymptotically stable; that is, disease stillpersists though predators are introduced. As corollaries, theglobally stability of system (7) without predators is given. Toexamine the effect of predator on disease control, numericalsimulations are given by choosing to focus on parameter ℎ,which is predation rate. We conclude that predation leadsto decrease of equilibrium levels both for infected host andvector population; as ℎ increases, then the infected host andvector equilibrium population will be lessened. Furthermore,if ℎ > ℎ

∗(ℎ∗ satisfies that 𝑅

0= 1), then vector-borne

diseases can be eradicated.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Thework is supported by the National Natural Science Foun-dation of China (Grant nos. 70871056, 70871056, 71271103,and 71371087), the National Research Subject of EducationInformation Technology (Grant no. 126240641), and theInnovative Foundation for Doctoral Candidate of JiangsuProvince, China (Grant no. CXZZ13 0687).

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