Global Dynamics of a Host-Vector-Predator Mathematical Model

Abstract

A mathematical model which links predator-vector(prey) and host-vector theory is proposed to examine the indirect effect of predators on vector-host dynamics. The equilibria and the basic reproduction number R₀ are 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 that R₀ provides 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, taking the predation rate as parameter, we provide numerical simulations for the impact of predators on vector-host disease control. It is illustrated 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 an infectious microbe transmitted by a blood (or sap-) sucking arthropod called vectors, which carry the disease without getting it themselves. For instance, human and animal diseases are such as malaria, dengue fever, West Nile virus, Chagas disease, sleeping sickness, and Lyme disease. Host-vector disease also affects other living organisms, such as plants. Examples of vector-borne infections in crops include tobacco mosaic virus (TMV), tomato spotted wilt virus (TSWV), tomato yellow leaf curl virus (TYLCV), cucumber mosaic virus (CMV), and potato virus Y (PVY) [1]. Vector-borne infections in trees include the pine wilt disease and the red ring disease in palms [2].

Recently, the frequent occurrence of natural disasters and environmental degradation around the world creates conditions suitable for breeding of vectors so that vector-borne diseases are constantly emerging. It is estimated that almost three-fourths of the new infectious diseases in recent years belong to vector-borne infectious disease. Vector-borne diseases 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 economic and public health concern.

The present mode of controlling vector-borne disease includes using bednets, spraying insecticides [3], and sterile insect technique (SIT) [4–6]. Effective vaccines and treatment control have not been adopted for most of the diseases except that malaria, which is preventable and curable when treatment and prevention measures are taken properly. One potential approach to control vector-borne disease is to introduce biological enemies (biocontrol agents) of the vector. Compared to insecticide method and sterile insect technique, introducing biological enemies (biocontrol agents) of the vector is more safe and cost-efficient. Moreover, it will lead to ecological balance and environment protection. Biological control of vectors has been successfully applied in controlling a variety of disease pathogens, including crop diseases such as the tomato leaf curl virus in India and the cassava mosaic virus in sub-Saharan Africa [7, 8], as well as tree diseases such as pine wilt disease in Japan and China [9, 10]. For human diseases, Entomogenous fungi are used as promising biopesticides for tick and sleeping sickness control [11, 12]. Predators have been introduced as biological control agents of vectors for various diseases such as malaria, dengue fever, tick disease, and Lyme disease [13–20]. Several recent studies suggested that predators led to a decline in local cases of dengue fever in Vietnam and Thailand [21, 22] and malaria in India [23, 24].

Mathematical models provide a powerful tool to understand the dynamics of disease spreading through a population and in decision-making process regarding disease prediction and disease control. Since Ross [25] first proposed a malaria model in 1911, many authors have been attracted to mathematical modeling, and very large good results on the subject 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 host population, which includes exposed classes both in host and vector populations and extended models studied in [27, 28].

However, compared with biological control of herbivorous pests, which has long been established as a major component of pest management programmers and is aimed to direct decrease pest densities by pest enemies [29, 30], biological control of vectors has been seldom investigated based on mathematical models. The main reason is that modeling the biological control of vectors is a complex interaction, mainly including the virus-host interaction, vector-host interaction, and vector- (prey-) enemy interaction, which is more complicated than the only pest-enemy interaction [31, 32] of biological pest control.

Up to know, only several authors have studied the biological control of vector to reduce the disease incidence by mathematical models. For examples, Moore et al. [33] first proposed a host-vector-predator model to study the effect 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 and Yao [35] improved the model proposed in [33], study the disease control threshold, and limit cycles with persistence of disease or without disease. However, both papers [33, 35] focus on the constant total host populations, and the hosts and vectors are simply divided into susceptible and infective ones.

Motivated by the above considerations, in this paper we consider a new host-vector-predator model, where both total host and vector population are time-dependent population size and the total host population is divided into four subpopulations of susceptible hosts, exposed hosts, infectious hosts, and recovered hosts. Furthermore, the total vector population is divided into three subpopulations of susceptible hosts, exposed hosts, and infectious hosts. The aim is to explore the global dynamics of the proposed host-vector-predator model and the impacts of predator on host-vector disease control.

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

2. Model Formulation

The basic model for the transmission dynamics of vector-borne disease with predator control is given by the following deterministic system of nonlinear differential equations: 𝑑𝑆_ℎ(𝑡)𝑑𝑡=𝑏₁−𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝜇_ℎ𝑆_ℎ(𝑡),𝑑𝐸_ℎ(𝑡)𝑑𝑡=𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝛼_ℎ+𝜇_ℎ𝐸_ℎ(𝑡),𝑑𝐼_ℎ(𝑡)𝑑𝑡=𝛼_ℎ𝐸_ℎ(𝑡)−𝛾_ℎ+𝛿_ℎ+𝜇_ℎ𝐼_ℎ(𝑡),𝑑𝑅_ℎ(𝑡)𝑑𝑡=𝛾_ℎ𝐼_ℎ(𝑡)−𝜇_ℎ𝑅_ℎ(𝑡),𝑑𝑆_𝑣(𝑡)𝑑𝑡=𝑏₂−𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝜇_𝑣𝑆_𝑣(𝑡)−ℎ𝑆_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝐸_𝑣(𝑡)𝑑𝑡=𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐸_𝑣(𝑡)−ℎ𝐸_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝐼_𝑣(𝑡)𝑑𝑡=𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐼_𝑣(𝑡)−ℎ𝐼_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝑃_𝑣(𝑡)𝑆𝑑𝑡=𝜀ℎ_𝑣(𝑡)+𝐸_𝑣(𝑡)+𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)−𝑒𝑃_𝑣(𝑡), (1) with initial conditions 𝑆_ℎ(0)≥0,𝐸_ℎ(0)≥0,𝐼_ℎ(0)𝑅≥0,_ℎ(0)≥0,𝑆_𝑣(0)≥0,𝐸_𝑣(0)𝐼≥0,_𝑣(0)≥0,𝑃_𝑣(0)≥0, (2) where the total host population at time 𝑡 denoted by 𝑁_ℎ(𝑡) is divided into four subpopulations of susceptible hosts 𝑆_ℎ(𝑡), exposed hosts 𝐸_ℎ(𝑡), infectious hosts 𝐼_ℎ(𝑡), and recovered hosts 𝑅_ℎ(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 𝑁_𝑣(𝑡) is split into susceptible vectors 𝑆_𝑣(𝑡), exposed vectors 𝐸_𝑣(𝑡) and infectious vectors 𝐼_𝑣(𝑡), so that 𝑁_𝑣(𝑡)=𝑆_𝑣(𝑡)+𝐸_𝑣(𝑡)+𝐼_𝑣(𝑡). The predator of the vector at time 𝑡 is represented by 𝑃_𝑣(𝑡). 𝑏₁ and 𝑏₂ are, respectively, the recruitment rate of hosts and vectors. Parameters 𝛽₁ and 𝛽₂ are, respectively, the rate of biting from susceptible hosts to infected hosts and susceptible vectors to infected vectors. 𝜇_ℎ and 𝜇_𝑣 are, respectively, the natural death rate of infected hosts and vectors. Exposed hosts and vectors develop symptoms of the disease and move to the infectious class at rates 𝛼_ℎ and 𝛼_𝑣, respectively. 𝛾_ℎ is the natural recovery rate from the infected hosts. 𝛿_ℎ is the disease-caused death rate of infected hosts. ℎ and 𝜀 are, respectively, the predation rates and conversation rate of the predator to the vector. 𝑒 is the mortality of the predator including naturally, being preyed both by humans and other animals.

Adding the host equations in (1), we have 𝑑𝑁_ℎ(𝑡)𝑑𝑡=𝑏₁−𝜇_ℎ𝑁_ℎ(𝑡)−𝛿_ℎ𝐼_ℎ(𝑡), (3) and from the last four equations of system (1), we have𝑑𝑁_𝑣(𝑡)𝑑𝑡=𝑏₂−𝜇_𝑣𝑁_𝑣(𝑡)−ℎ𝑁_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝑃_𝑣(𝑡)𝑑𝑡=𝜀ℎ𝑁_𝑣(𝑡)𝑃_𝑣(𝑡)−𝑒𝑃_𝑣(𝑡). (4) By (3) and (4), we have 𝑑𝑁_ℎ(𝑡)𝑑𝑡≤𝑏₁−𝜇_ℎ𝑁_ℎ(𝑡),𝑑𝑁_𝑣(𝑡)𝑑𝑡≤𝑏₂−𝜇_𝑣𝑁_𝑣(𝑡),𝑑𝑃_𝑣(𝑡)𝑑𝑡=𝜀ℎ𝑁_𝑣(𝑡)𝑃_𝑣(𝑡)−𝑒𝑃_𝑣(𝑡). (5) Let Γ=𝑆_ℎ,𝐸_ℎ,𝐼_ℎ,𝑅_ℎ,𝑆_𝑣,𝐸_𝑣,𝐼_𝑣,𝑃_𝑣∈𝑅⁺₈∣0≤𝑆_ℎ+𝐸_ℎ+𝐼_ℎ+𝑅_ℎ≤𝑏₁𝜇_ℎ,0≤𝑆_𝑣+𝐸_𝑣+𝐼_𝑣≤𝑏₂𝜇_𝑣,𝑃_𝑣≥0; (6) then, it is easy to verify that Γ is positively an invariant of system (1).

Since the fourth equation for 𝑅_ℎ(𝑡) in system (1) does not influence the dynamics behavior of system, we can omit the equations for 𝑅_ℎ(𝑡). System (1) in the invariant space Γ can be written as the following seven dimensional nonlinear systems: 𝑑𝑆_ℎ(𝑡)𝑑𝑡=𝑏₁−𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝜇_ℎ𝑆_ℎ(𝑡),𝑑𝐸_ℎ(𝑡)𝑑𝑡=𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝛼_ℎ+𝜇_ℎ𝐸_ℎ(𝑡),𝑑𝐼_ℎ(𝑡)𝑑𝑡=𝛼_ℎ𝐸_ℎ(𝑡)−𝑚𝐼_ℎ(𝑡),𝑑𝑆_𝑣(𝑡)𝑑𝑡=𝑏₂−𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝜇_𝑣𝑆_𝑣(𝑡)−ℎ𝑆_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝐸_𝑣(𝑡)𝑑𝑡=𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐸_𝑣(𝑡)−ℎ𝐸_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝐼_𝑣(𝑡)𝑑𝑡=𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐼_𝑣(𝑡)−ℎ𝐼_𝑣(𝑡)𝑃_𝑣(𝑡),𝑑𝑃_𝑣(𝑡)𝑆𝑑𝑡=𝜀ℎ_𝑣(𝑡)+𝐸_𝑣(𝑡)+𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)−𝑒𝑃_𝑣(𝑡), (7) where 𝑚=𝛾_ℎ+𝛿_ℎ+𝜇_ℎ. Let Γ=𝑆_ℎ,𝐸_ℎ,𝐼_ℎ,𝑆_𝑣,𝐸_𝑣,𝐼_𝑣,𝑃_𝑣∈𝑅⁺₇∣0≤𝑆_ℎ+𝐸_ℎ+𝐼_ℎ≤𝑏₁𝜇_ℎ,0≤𝑆_𝑣+𝐸_𝑣+𝐼_𝑣≤𝑏₂𝜇_𝑣,𝑃_𝑣≥0. (8) Obviously, for system (7), the region Γ is positively invariant.

3. The Equilibria and the Basic Reproduction Number

Lemma 1. The equilibria of system (7) are as follows.(i)If the predator is present, that is, 𝑃_𝑣(𝑡)>0 for any 𝑡∈[0,+∞), then there exist a disease-free equilibrium 𝐸₁(𝑆⁰_ℎ,0,0,𝑆⁰_𝑣,0,0,𝑃_𝑣) and a disease equilibrium 𝐸₂(𝑆_ℎ,𝐸_ℎ,𝐼̃_ℎ,𝑆_𝑣,𝐸_𝑣,𝐼̃_𝑣,𝑃_𝑣) if 𝑅₀ is larger than one.(ii)If the predator is absent, that is, 𝑃_𝑣(𝑡)≡0 for any 𝑡∈[0,+∞), then there exist a disease-free equilibrium 𝐸₃((𝑏₁/𝜇_ℎ)0,0,(𝑏₂/𝜇_𝑣),0,0,0) and a disease equilibrium 𝐸₄(^⌢𝑆_ℎ,𝐸^⌢_ℎ,^⌢𝐼_ℎ,^⌢𝑆_𝑣,𝐸^⌢_𝑣,^⌢𝐼_𝑣,0) if 𝑅₁ is larger than one.where𝑆⁰_ℎ=𝑏₁𝜇_ℎ,𝑆⁰_𝑣=𝑁_𝑣,𝑆_ℎ=𝑄₁𝑄₂𝑄₃𝜇_𝑣+ℎ𝑃_𝑣+𝛽₂𝐼̃_ℎ𝛼_ℎ𝛼_𝑣𝛽₁𝛽₂𝑁_𝑣,𝐼̃_ℎ=𝑅²₀−1𝑄²₁𝑄²₂𝑄₃𝑄₄𝜇_ℎ𝜇_ℎ𝛽₂𝑄₃+𝛼_𝑣𝛽₁𝛽₂𝑁_𝑣,𝐸_ℎ=𝑄₂𝛼_ℎ𝐼̃_ℎ,𝑆_𝑣=𝑏₂𝜇_𝑣+ℎ𝑃_𝑣+𝛽₂𝐼̃_ℎ,𝐸_𝑣=𝛼_𝑣𝛽₂𝑁_𝑣𝐼̃_ℎ𝜇_𝑣+ℎ𝑃_𝑣𝛼_𝑣𝑄₃𝜇_𝑣+ℎ𝑃_𝑣+𝛽₂𝐼̃_ℎ,𝐼̃_𝑣=𝛼_𝑣𝛽₂𝑁_𝑣𝐼̃_ℎ𝑄₃𝜇_𝑣+ℎ𝑃_𝑣+𝛽₂𝐼̃_ℎ⌢,𝑆_ℎ=𝑄₁𝑄₂𝑄₅𝜇_𝑣𝜇_𝑣+𝛽₂^⌢𝐼_ℎ𝛼_ℎ𝛼_𝑣𝛽₁𝛽₂𝑏_2⌢,𝐼_ℎ=𝑅²₁−1𝑄²₁𝑄²₂𝑄₅𝜇_ℎ𝜇_ℎ𝜇_𝑣𝛽₂𝑄₅+𝛼_𝑣𝛽₁𝛽₂𝑏_2⌢,𝐸_ℎ=𝑄₂𝛼_ℎ^⌢𝐼_ℎ,^⌢𝑆_𝑣=𝑏₂𝜇_𝑣+𝛽₂^⌢𝐼_ℎ⌢,𝐸_𝑣=𝜇_𝑣^⌢𝐼_𝑣𝛼_𝑣,^⌢𝐼_𝑣=𝑏₂𝛼_𝑣𝛽₂^⌢𝐼_ℎ𝜇_𝑣𝑄₅𝜇_𝑣+𝛽₂^⌢𝐼_ℎ,𝑄₁=𝛼_ℎ+𝜇_ℎ,𝑄₂=𝑚=𝛾_ℎ+𝛿_ℎ+𝜇_ℎ,𝑄₃=𝛼_𝑣+𝜇_𝑣+ℎ𝑃_𝑣,𝑄₄=𝜇_𝑣+ℎ𝑃_𝑣,𝑄₅=𝛼_𝑣+𝜇_𝑣,𝑁_𝑣=𝑒,𝑃𝜀ℎ_𝑣=𝑏₂−𝜇_𝑣𝑁_𝑣ℎ𝑁_𝑣𝑏if²𝜇_𝑣>𝑒.𝜀ℎ (9) Here (𝑏₂/𝜇_𝑣) is the equilibrium level of total vector population of system (7) without predators, while (𝑒/𝜀ℎ) is the equilibrium level of total vector population of system (7) with predators. (𝑏₂/𝜇_𝑣)>(𝑒/𝜀ℎ) is the sufficient and necessary conditions to ensure 𝑃_𝑣>0. Therefore, from the viewpoint of biological meaning, if the predators are present, we always assume that (𝑏₂/𝜇_𝑣)>(𝑒/𝜀ℎ).

𝑅₀=(𝛼_ℎ𝛼_𝑣𝛽₁𝛽₂𝑏₁𝑁_𝑣/𝜇_ℎ𝑄₁𝑄₂𝑄₃𝑄₄) and 𝑅₁=(𝛼_ℎ𝛼_𝑣𝑏₁𝑏₂𝛽₁𝛽₂/𝜇_ℎ𝜇²_𝑣𝑄₁𝑄₂𝑄₅), respectively, are the basic reproduction number of system (7) and system (7) without predators by [36, 37].

Remark 2. It is not difficult to find that 𝑅₀ is less than 𝑅₁. That is, predation results into a reduction in the basic reproduction number 𝑅₀ of system (7).

4. Global Stability Analysis

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

Lemma 3. For system (4), the unique positive equilibrium (𝑁_𝑣,𝑃_𝑣) is globally asymptotically stable, where 𝑁_𝑣 and 𝑃_𝑣 are given in Lemma 1.

By constructing Lyapunov function 𝑉(𝑡)=(𝜀/2)(𝑁_𝑣(𝑡)−𝑁_𝑣)²+𝑁_𝑣(𝑃_𝑣(𝑡)−𝑃_𝑣−𝑃_𝑣ln(𝑃_𝑣(𝑡)/𝑃_𝑣)) and using LaSalle’s invariance principle, it is not difficult to prove that the positive equilibrium (𝑁_𝑣,𝑃_𝑣) is globally asymptotically stable. Here we omit it.

Theorem 4. If 𝑅₀≤1 and for any 𝑡∈[0,+∞),𝑃_𝑣(𝑡)≥𝑃_𝑣; then, the disease-free equilibrium 𝐸₁ of system (7) is globally asymptotically stable.

Proof. We construct the following Lyapunov functional: 𝐿(𝑡)=𝑤₀𝑆_ℎ(𝑡)−𝑆⁰_ℎ−𝑆⁰_ℎ𝑆ln^ℎ(𝑡)𝑆⁰_ℎ+𝑤₁𝐸_ℎ(𝑡)+𝑤₂𝐼_ℎ(𝑡)+𝑤₃𝑆_𝑣(𝑡)−𝑆⁰_𝑣−𝑆⁰_𝑣𝑆ln^𝑣(𝑡)𝑆⁰_𝑣+𝑤₄𝐸_𝑣(𝑡)+𝑤₅𝐼_𝑣(𝑡)+𝑤₆𝑃_𝑣(𝑡)−𝑃_𝑣−𝑃_𝑣𝑃ln^𝑣(𝑡)𝑃_𝑣, (10) where 𝑤₀=𝑤₁=(𝛼_ℎ/𝑄₁),𝑤₂ = 1,𝑤₃=𝑤₄=(𝑏₁𝛼_ℎ𝛼_𝑣𝛽₁/𝜇_ℎ𝑄₁𝑄₃𝑄₄),𝑤₅ = (𝑏₁𝛼_ℎ𝛽₁/𝜇_ℎ𝑄₁𝑄₄),𝑤₆=(𝑤₃/𝜀).
Calculating the derivative of 𝐿(𝑡) along the solution of (7) yields that 𝑑𝐿(𝑡)𝑑𝑡=𝑤₀𝑆_ℎ(𝑡)−𝑆⁰_ℎ𝑆_ℎ(𝑡)𝑏₁−𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝜇_ℎ𝑆_ℎ(𝑡)+𝑤₁𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝛼_ℎ+𝜇_ℎ𝐸_ℎ(𝑡)+𝑤₂𝛼_ℎ𝐸_ℎ(𝑡)−𝑚𝐼_ℎ(𝑡)+𝑤₃𝑆_𝑣(𝑡)−𝑆⁰_𝑣𝑆_𝑣(𝑡)×𝑏₂−𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝜇_𝑣𝑆_𝑣(𝑡)−ℎ𝑆_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑤₄𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐸_𝑣(𝑡)−ℎ𝐸_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑤₅𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝑆_𝑣(𝑡)−ℎ𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑤₆𝑃_𝑣(𝑡)−𝑃_𝑣𝑃_𝑣(𝑡)×𝜀ℎ𝑆_𝑣(𝑡)+𝐸_𝑣(𝑡)+𝐼_𝑣(𝑡)−𝑒𝑃_𝑣(𝑡). (11) By the equilibrium conditions  𝑏₁=𝜇_ℎ𝑆⁰_ℎ and  𝑏₂=𝜇_𝑣𝑆⁰_𝑣+ℎ𝑆⁰_𝑣𝑃_𝑣, it follows that 𝑑𝐿(𝑡)𝑑𝑡=−𝜇_ℎ𝑤₀𝑆_ℎ(𝑡)−𝑆⁰_ℎ²𝑆_ℎ(𝑡)−𝜇_𝑣𝑤₃𝑆_𝑣(𝑡)−𝑆⁰_𝑣²𝑆_𝑣(𝑡)+𝑤₁−𝑤₀𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)+𝑤₄−𝑤₃𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)+𝑤₂𝛼_ℎ−𝑤₁𝑄₁𝐸_ℎ(𝑡)+𝑤₅𝛼_𝑣−𝑤₄𝑄₃𝐸_𝑣(𝑡)+𝑤₃𝛽₂𝑁_𝑣−𝑤₂𝑄₂𝐼_ℎ(𝑡)+𝑤₀𝛽₁𝑏₁𝜇_ℎ−𝑤₅𝑄₄𝐼_𝑣(𝑡)+𝑤₃ℎ𝑆_𝑣(𝑡)−𝑆⁰_𝑣𝑆_𝑣(𝑡)𝑆⁰_𝑣𝑃_𝑣−𝑆_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑤₄ℎ𝐸_𝑣(𝑡)𝑃_𝑣−𝑃_𝑣(𝑡)+𝑤₅ℎ𝐼_𝑣(𝑡)𝑃_𝑣−𝑃_𝑣(𝑡)+𝑤₆𝜀ℎ𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)−𝑃_𝑣×𝑁_𝑣(𝑡)−𝑁_𝑣. (12) Since 𝑤₃ℎ𝑆_𝑣(𝑡)−𝑆⁰_𝑣𝑆_𝑣(𝑡)𝑆⁰_𝑣𝑃_𝑣−𝑆_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑤₄ℎ𝐸_𝑣(𝑡)𝑃_𝑣−𝑃_𝑣(𝑡)=−𝑤₃ℎ𝑃_𝑣𝑆_𝑣(𝑡)−𝑆⁰_𝑣²𝑆_𝑣(𝑡)+𝑤₃ℎ𝑆_𝑣(𝑡)−𝑆⁰_𝑣𝑃_𝑣−𝑃_𝑣(𝑡), (13) then using 𝑆⁰_ℎ=𝑏₁/𝜇_ℎ,𝑆⁰_𝑣=𝑁_𝑣 we have 𝑑𝐿(𝑡)𝑑𝑡=−𝜇_ℎ𝑤₀𝑆_ℎ(𝑡)−(𝑏₁/𝜇_ℎ)²𝑆_ℎ(𝑡)−𝑤₃𝜇_𝑣+ℎ𝑃_𝑣𝑆_𝑣(𝑡)−𝑁_𝑣²𝑆_𝑣(𝑡)+𝑄₂𝑅²₀−1𝐼_ℎ(𝑡)+𝑤₃ℎ𝑆_𝑣(𝑡)−𝑁_𝑣+𝑤₄ℎ𝐸_𝑣(𝑡)+𝑤₅ℎ𝐼_𝑣(𝑡×𝑃_𝑣)−𝑃_𝑣(𝑡)+𝑤₆𝜀ℎ𝑃_𝑣(𝑡)−𝑃_𝑣𝑁_𝑣(𝑡)−𝑁_𝑣. (14) After some rearrangement, we have 𝑑𝐿(𝑡)𝑑𝑡=−𝜇_ℎ𝑤₀𝑆_ℎ(𝑡)−(𝑏₁/𝜇_ℎ)²𝑆_ℎ(𝑡)−𝑤₃𝜇_𝑣+ℎ𝑃_𝑣𝑆_𝑣(𝑡)−𝑁_𝑣²𝑆_𝑣(𝑡)+𝑄₂𝑅²₀−1𝐼_ℎ(𝑡)+ℎ𝑤₅−𝑤₃𝐼_𝑣(𝑡)𝑃_𝑣−𝑃_𝑣. (15) Since 𝑃_𝑣(𝑡)≥𝑃_𝑣, then we have 𝑁_𝑣(𝑡)≤𝑁_𝑣 for any 𝑡∈[0,+∞). Otherwise, if 𝑁_𝑣(𝑡)>𝑁_𝑣 for any 𝑡∈[0,+∞), then 𝜀ℎ𝑁_𝑣(𝑡)−𝑒>0; thus, from the last equation of system (7), we have (𝑑𝑃_𝑣(𝑡)/𝑑𝑡)>0. On the other hand, by Lemma 1, lim_𝑡→+∞𝑃_𝑣(𝑡)=𝑃_𝑣, so for any 𝑡∈[0,+∞),𝑃_𝑣(𝑡)<𝑃_𝑣. This contradicts 𝑃_𝑣(𝑡)≥𝑃_𝑣. So 𝑁_𝑣(𝑡)≤𝑁_𝑣 when 𝑃_𝑣(𝑡)≥𝑃_𝑣 for any 𝑡∈[0,+∞). Thus, (𝑑𝐿(𝑡)/𝑑𝑡)≤0 if 𝑅₀≤1 and 𝑃_𝑣(𝑡)≥𝑃_𝑣 for any 𝑡∈[0,+∞). Furthermore, (𝑑𝐿(𝑡)/𝑑𝑡)=0 if and only if 𝑆_ℎ=𝑆⁰_ℎ,𝑆_𝑣=𝑆⁰_𝑣=𝑁_𝑣,𝐸_ℎ=𝐸_𝑣=𝐼_ℎ=𝐼_𝑣=0 and 𝑃_𝑣=𝑃_𝑣. Consequently, the largest compact invariant set in {(𝑆_ℎ(𝑡),𝐸_ℎ(𝑡),𝐼_ℎ(𝑡),𝑆_𝑣(𝑡),𝐸_𝑣(𝑡),𝐼_𝑣(𝑡),𝑃_𝑣(𝑡))∈Γ∶𝑑𝐿/𝑑𝑡=0} when 𝑅₀≤1 is the singleton 𝐸₁. Then by Lyapunov-LaSalle theorem, the equilibrium 𝐸₁ is globally stable if 𝑅₀≤1 and 𝑃_𝑣(𝑡)≥𝑃_𝑣 for any 𝑡∈[0,+∞).

Remark 5. From Theorem 4 we can see that when 𝑅₀≤1 the disease-free equilibrium 𝐸₁ of system (7) is globally asymptotically stable if the predator population size 𝑃_𝑣(𝑡) is not less than the predator equilibrium level 𝑃_𝑣, which depends on the total equilibrium vector density 𝑁_𝑣. That is, the predator density threshold for successful disease eradication is 𝑃_𝑣(𝑡)=𝑃_𝑣. If 𝑃_𝑣(𝑡)≥𝑃_𝑣 and 𝑅₀≤1 then disease can be eradicated; otherwise; pathogen persists though predators are introduced and 𝑅₀≤1.

Similar to the proof of Theorem 4, we have the following corollary to show that the disease-free equilibrium 𝐸₃ of system (7) in absence of predators is global asymptotically stable.

Corollary 6. If 𝑅₁≤1 then the disease-free equilibrium 𝐸₃ of system (7) without predators is globally asymptotically stable.

A global stability result for the endemic equilibrium 𝐸₂ of the system (7) is given below.

Theorem 7. The endemic equilibrium state 𝐸₂ of system (7) is globally asymptotically stable if 𝑅₀>1,𝛼_𝑣=𝛽₂𝑆_𝑣𝐼̃_ℎ𝐸_𝑣, (16) where 𝑆_𝑣, 𝐼̃_ℎ and 𝐸_𝑣 are the disease equilibrium value of susceptible vectors, infected hosts, and exposed vectors of system (7).

Proof. Consider the following Lyapunov functional: 𝑉(𝑡)=𝑐₁𝑆_ℎ(𝑡)−𝑆_ℎ𝑆ln^ℎ(𝑡)𝑆_ℎ+𝑐₂𝐸_ℎ(𝑡)−𝐸_ℎ𝐸ln^ℎ(𝑡)𝐸_ℎ+𝑐₃𝐼_ℎ(𝑡)−𝐼̃_ℎ𝐼ln^ℎ(𝑡)𝐼̃_ℎ+𝑐₄𝑆_𝑣(𝑡)−𝑆_𝑣𝑆ln^𝑣(𝑡)𝑆_𝑣+𝑐₅𝐸_𝑣(𝑡)−𝐸_𝑣𝐸ln^𝑣(𝑡)𝐸_𝑣+𝑐₆𝐼_𝑣(𝑡)−𝐼̃_𝑣𝐼ln^𝑣(𝑡)𝐼̃_𝑣+𝑐₇𝑃_𝑣(𝑡)−𝑃_𝑣𝑃ln^𝑣(𝑡)𝑃_𝑣, (17) where 𝑐₂=𝑐₁,𝑐₃=(𝑐₁𝛽₁𝑎₁/𝑑₁), 𝑐₄=𝑐₅=(𝑐₁𝛽₁𝑎₁/𝛽₂𝑎₂), 𝑐₆=(𝑐₅𝛽₂𝑎₂/𝑑₂),𝑐₇=(𝑐₄/𝜀), 𝑆_ℎ𝐼̃_𝑣=𝑎₁, 𝑆_𝑣𝐼̃_ℎ=𝑎₂, 𝑑₁=𝛼_ℎ𝐸_ℎ,𝑑₂=𝛼_𝑣𝐸_𝑣.
Calculate the derivative of 𝑉(𝑡) along the solution of (1); this yields that 𝑑𝑉(𝑡)𝑑𝑡=𝑐₁𝑆1−^ℎ𝑆_ℎ(𝑡)𝑏₁−𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝜇_ℎ𝑆_ℎ(𝑡)+𝑐₂𝐸1−^ℎ𝐸_ℎ(𝑡)𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝛼_ℎ+𝜇_ℎ𝐸_ℎ(𝑡)+𝑐₃𝐼̃1−^ℎ𝐼_ℎ(𝑡)𝛼_ℎ𝐸_ℎ(𝑡)−𝑚𝐼_ℎ(𝑡)+𝑐₄𝑆1−^𝑣𝑆_𝑣(𝑡)×𝑏₂−𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝜇_𝑣𝑆_𝑣(𝑡)−ℎ𝑆_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑐₅𝐸1−^𝑣𝐸_𝑣(𝑡)×𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐸_𝑣(𝑡)−ℎ𝐸_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑐₆𝐼̃1−^𝑣𝐼_𝑣(𝑡)×𝛼_𝑣𝐸_𝑣(𝑡)−𝜇_𝑣𝐼_𝑣(𝑡)−ℎ𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑐₇𝑃1−^𝑣𝑃_𝑣(𝑡)×𝜀ℎ𝑆_𝑣(𝑡)+𝐸_𝑣(𝑡)+𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)−𝑒𝑃_𝑣(𝑡). (18) System (7) satisfied the following relations at equilibrium point: 𝑏₁=𝛽₁𝑆_ℎ𝐼̃_𝑣+𝜇_ℎ𝑆_ℎ,𝛼_ℎ+𝜇_ℎ=𝛽₁𝑆_ℎ𝐼̃_𝑣𝐸_ℎ,𝛼𝑚=^ℎ𝐸_ℎ𝐼̃_ℎ,𝑏₂=𝛽₂𝑆_𝑣𝐼̃_ℎ+𝜇_𝑣𝑆_𝑣+ℎ𝑆_𝑣𝑃_𝑣,𝛼_𝑣+𝜇_𝑣=𝛽₂𝑆_𝑣𝐼̃_ℎ𝐸_𝑣−ℎ𝑃_𝑣,𝜇_𝑣=𝛼_𝑣𝐸_𝑣𝐼̃_𝑣−ℎ𝑃_𝑣,𝑒=𝜀ℎ𝑁_𝑣,𝑁_𝑣=𝑆_𝑣+𝐸_𝑣+𝐼̃_𝑣. (19) Substituting 𝑏₁ up to 𝑒 in the above equation, we obtain 𝑑𝑉(𝑡)𝑑𝑡=−𝑐₁𝜇_ℎ𝑆_ℎ(𝑡)−𝑆_ℎ²𝑆_ℎ(𝑡)−𝑐₄𝜇_𝑣𝑆_𝑣(𝑡)−𝑆_𝑣²𝑆_𝑣(𝑡)+𝑐₁𝑆1−^ℎ𝑆_ℎ(𝑡)𝛽₁𝑆_ℎ𝐼̃_𝑣−𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)+𝑐₂𝐸1−^ℎ𝐸_ℎ(𝑡)𝛽₁𝑆_ℎ(𝑡)𝐼_𝑣(𝑡)−𝐸_ℎ𝛽₁𝑆_ℎ𝐼̃_𝑣𝐸_ℎ+𝑐₃𝐼̃1−^ℎ𝐼_ℎ(𝑡)𝛼_ℎ𝐸_ℎ(𝑡)−𝛼_ℎ𝐸_ℎ𝐼_ℎ(𝑡)𝐼̃_ℎ+𝑐₄𝑆1−^𝑣𝑆_𝑣(𝑡)×𝛽₂𝑆_𝑣𝐼̃_ℎ−𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)+ℎ𝑆_𝑣𝑃_𝑣−ℎ𝑆_𝑣(𝑡)𝑃_𝑣(𝑡)+𝑐₅𝐸1−^𝑣𝐸_𝑣(𝑡)×𝛽₂𝑆_𝑣(𝑡)𝐼_ℎ(𝑡)−𝛽₂𝑆_𝑣𝐼̃_ℎ𝐸_𝑣(𝑡)𝐸_𝑣−ℎ𝐸_𝑣(𝑡)𝑃_𝑣(𝑡)+ℎ𝐸_𝑣(𝑡)𝑃_𝑣+𝑐₆𝐼̃1−^𝑣𝐼_𝑣(𝑡)×𝛼_𝑣𝐸_𝑣(𝑡)−𝛼_𝑣𝐸_𝑣𝐼_𝑣𝐼̃_𝑣(𝑡)−ℎ𝐼_𝑣(𝑡)𝑃_𝑣(𝑡)+ℎ𝐼_𝑣(𝑡)𝑃_𝑣+𝑐₇𝑃1−^𝑣𝑃_𝑣(𝑡)×𝜀ℎ𝑆_𝑣(𝑡)+𝐸_𝑣(𝑡)+𝐼_𝑣(𝑡)−𝑆_𝑣−𝐸_𝑣−𝐼̃_𝑣𝑃_𝑣(𝑡). (20) Set 𝑆_ℎ𝑆_ℎ=𝑥₁,𝐸_ℎ𝐸_ℎ=𝑥₂,𝐼_ℎ𝐼̃_ℎ=𝑥₃,𝑆_𝑣𝑆_𝑣=𝑥₄,𝐸_𝑣𝐸_𝑣=𝑥₅,𝐼_𝑣𝐼̃_𝑣=𝑥₆,𝑃_𝑣𝑃_𝑣=𝑥₇,𝑆_𝑣𝑃_𝑣=𝑎₃,𝐸_𝑣𝑃_𝑣=𝑎₄,𝐼̃_𝑣𝑃_𝑣=𝑎₅; (21) then, from the above equation, we obtain 𝑑𝑉(𝑡)𝑑𝑡=−𝑐₁𝜇_ℎ𝑆_ℎ(𝑡)−𝑆_ℎ²𝑆_ℎ(𝑡)−𝑐₄𝜇_𝑣𝑆_𝑣(𝑡)−𝑆_𝑣²𝑆_𝑣(𝑡)+𝑐₁𝛽₁𝑎₁−𝑐₁𝛽₁𝑎₁𝑥₁𝑥₆−𝑐₁𝛽₁𝑎₁1𝑥₁+𝑐₁𝛽₁𝑎₁𝑥₆+𝑐₂𝛽₁𝑎₁+𝑐₂𝛽₁𝑎₁𝑥₁𝑥₆−𝑐₂𝛽₁𝑎₁𝑥₂−𝑐₂𝛽₁𝑎₁𝑥₁𝑥₆𝑥₂+𝑐₃𝑑₁+𝑐₃𝑑₁𝑥₂−𝑐₃𝑑₁𝑥₃−𝑐₃𝑑₁𝑥₂𝑥₃+𝑐₄𝛽₂𝑎₂+𝑐₄𝛽₂𝑎₂𝑥₃−𝑐₄𝛽₂𝑎₂𝑥₃𝑥₄−𝑐₄𝛽₂𝑎₂1𝑥₄+𝑐₄ℎ𝑎₃−𝑐₄ℎ𝑎₃𝑥₄𝑥₇−𝑐₄ℎ𝑎₃1𝑥₄+𝑐₄ℎ𝑎₃𝑥₇+𝑐₅𝛽₂𝑎₂+𝑐₅𝛽₂𝑎₂𝑥₃𝑥₄−𝑐₅𝛽₂𝑎₂𝑥₅−𝑐₅𝛽₂𝑎₂𝑥₃𝑥₄𝑥₅−𝑐₅ℎ𝑎₄𝑥₅𝑥₇+𝑐₅ℎ𝑎₄𝑥₅+𝑐₅ℎ𝑎₄𝑥₇−𝑐₅ℎ𝑎₄+𝑐₆𝑑₂+𝑐₆𝑑₂𝑥₅−𝑐₆𝑑₂𝑥₅𝑥₆−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=−𝑐₁𝜇_ℎ𝑆_ℎ(𝑡)−𝑆_ℎ²𝑆_ℎ(𝑡)−𝑐₅𝜇_𝑣𝑆_𝑣(𝑡)−𝑆_𝑣²𝑆_𝑣(𝑡)+𝑐₁𝛽₁𝑎₁−𝑐₆𝑑₂𝑥₆+𝑐₃𝑑₁−𝑐₂𝛽₁𝑎₁𝑥₂+𝑐₄𝛽₂𝑎₂−𝑐₃𝑑₁𝑥₃+𝑐₆𝑑₂−𝑐₅𝛽₂𝑎₂𝑥₅+𝑐₂𝛽₁𝑎₁−𝑐₁𝛽₁𝑎₁𝑥₁𝑥₆+𝑐₅𝛽₂𝑎₂−𝑐₄𝛽₂𝑎₂𝑥₃𝑥₄+𝑐₇𝜀−𝑐₄ℎ𝑎₃𝑥₄𝑥₇+𝑐₇𝜀−𝑐₅ℎ𝑎₄𝑥₅𝑥₇+𝑐₇𝜀−𝑐₆ℎ𝑎₅𝑥₆𝑥₇+𝑐₄𝑎₃+𝑐₅𝑎₄+𝑐₆𝑎₅−𝑐₇𝜀𝑎₃−𝑐₇𝜀𝑎₄−𝑐₇𝜀𝑎₅ℎ𝑥₇+𝑐₇𝜀ℎ𝑎₄+𝑐₇𝜀ℎ𝑎₅−𝑐₅ℎ𝑎₄−𝑐₆ℎ𝑎₅+𝑐₄ℎ𝑎₃+𝑐₇𝜀ℎ𝑎₃−𝑐₄ℎ𝑎₃1𝑥₄−𝑐₇𝜀ℎ𝑎₃𝑥₄+𝑐₁𝛽₁𝑎₁−𝑐₁𝛽₁𝑎₁1𝑥₁−𝑐₂𝛽₁𝑎₁𝑥₁𝑥₆𝑥₂+𝑐₂𝛽₁𝑎₁−𝑐₃𝑑₁𝑥₂𝑥₃+𝑐₃𝑑₁+𝑐₄𝛽₂𝑎₂−𝑐₄𝛽₂𝑎₂1𝑥₄−𝑐₅𝛽₂𝑎₂𝑥₃𝑥₄𝑥₅+𝑐₅𝛽₂𝑎₂−𝑐₆𝑑₂𝑥₅𝑥₆+𝑐₆𝑑₂. (23) By some reduction, it follows that 𝑑𝑉(𝑡)𝑑𝑡=−𝑐₁𝜇_ℎ𝑆_ℎ(𝑡)−𝑆_ℎ²𝑆_ℎ(𝑡)−𝑐₅𝜇_𝑣𝑆_𝑣(𝑡)−𝑆_𝑣²𝑆_𝑣(𝑡)+𝑐₄ℎ𝑎₃2−𝑥₄−1𝑥₄+𝑐₁𝛽₁𝑎₁16−𝑥₁−𝑥₁𝑥₆𝑥₂−𝑥₂𝑥₃−1𝑥₄−𝑥₃𝑥₄𝑥₅−𝑥₅𝑥₆. (24) Since the arithmetic mean is greater than or is equal to the geometric mean, then 2−𝑥₄−1𝑥₄1≥0,𝑥₁+𝑥₁𝑥₆𝑥₂+𝑥₂𝑥₃+1𝑥₄+𝑥₃𝑥₄𝑥₅+𝑥₅𝑥₆≥6. (25) Thus, it follows from (24) that (𝑑𝑉(𝑡)/𝑑𝑡)≤0 in Ω. The equation (𝑑𝑉(𝑡)/𝑑𝑡)=0 holds if and only if 𝑥₁=𝑥₂=𝑥₃=𝑥₄=𝑥₅=𝑥₆=1; that is, 𝑆_ℎ=𝑆_ℎ,𝐸_ℎ=𝐸_ℎ,𝐼_ℎ=𝐼̃_ℎ,𝑆_𝑣=𝑆_𝑣,𝐸_𝑣=𝐸_𝑣,𝐼_𝑣=𝐼̃_𝑣,𝑃_𝑣=𝑃_𝑣. Therefore, we prove the global stability of the disease in Γ. The maximal compact invariant set in {𝑆_ℎ,𝐸_ℎ,𝐼_ℎ,𝑆_𝑣,𝐸_𝑣,𝐼_𝑣,𝑃_𝑣∈Γ∶𝑑𝑉(𝑡)/𝑑𝑡=0} is {𝐸₂} when 𝑅₀>1 and 𝛼_𝑣=(𝛽₃𝑆_𝑣𝐼̃_ℎ/𝐸_𝑣). From the LaSalle’s invariance principle, we finish the proof of Theorem 7.

Similar to the proof of Theorem 7, we have the following corollary to show that the disease equilibrium 𝐸₄ of system (7) in absence of predators is globally asymptotically stable.

Corollary 8. The endemic equilibrium state 𝐸₄ of system (7) without predators is globally asymptotically stable if 𝑅₁>1.

Remark 9. From Theorems 4 and Theorem 7, Corollary 6 and Corollary 8 we find that 𝑅₀=1 and 𝑅₁=1 provide threshold conditions on determining the uniform persistence and extinction of the disease with and without predators, respectively. Moreover, we also find that the disease is persistent without predators if 𝑅₁>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 the transmission dynamics of vector-borne diseases by some numerical simulations. We choose the following set of parameter values (it should be stated that these parameters are chosen for illustrative purpose only and may not necessarily be realistic epidemiologically): 𝑏₁=0.786098, 𝑏₂=0.190028, 𝛽₁=0.961509, 𝛽₂=0.535061, 𝜇_ℎ=0.914518, 𝜇_𝑣=0.0150508, 𝛼_ℎ=0.645664, 𝛼_𝑣=0.19636, and 𝑚=1.120469; By simple calculations, the basic reproduction number 𝑅₁=11.2826>1; then, by Corollary 8, the disease equilibrium of system (7) without predators is globally asymptotically stable. That is, the disease persists without predators (see Figure 1).

Figure 1 

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

Choose 𝑒=0.1, 𝜀=0.15; we consider the effect of predators on disease control by comparing equilibrium level of infected hosts and vectors with different values of ℎ in absence of the predators, that is, ℎ=0; then, by the Corollary 8 disease persists (see the solid line of Figure 2). By introducing predators of the vector population, we find that the 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 predation rate such that ℎ=0.3, we find that though the disease still persists, the equilibrium levels of infected hosts and vectors have been greatly lessened (see the dot and dashed line of Figure 2). By enhancing the predation rate ℎ such that ℎ=0.5>ℎ^∗=0.4755(ℎ^∗satisfiesthat𝑅₀=1), then 𝑅₀=0.9411<1; therefore, by Theorem 4 the vector-borne disease can be eradicated by introducing vector predators (see the dashed line of Figure 2).

(a)

(b)

Figure 2 

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

Remark 10. From Figures 1 and 2, we find that predation has a positive effect on vector-host disease control by reducing vector density. Furthermore, disease can be eradicated if predation rate ℎ is large enough such that 𝑅₀(ℎ)≤1 and the predator density 𝑃_𝑣(𝑡) satisfies 𝑃_𝑣(𝑡)≥𝑃_𝑣, where 𝑃_𝑣 is the predator equilibrium level.

6. Conclusions

In this paper, we propose a host-vector-predator coupled model with variable host and vector population size to investigate the effect of predators on vector-borne disease control by analyzing the global stability of the disease-free and disease equilibria. It is shown that the basic reproduction number 𝑅₀ characterizes the disease transmission dynamics: if 𝑅₀≤1, then there exists only the disease-free equilibrium which is globally asymptotically stable when predator density at any time keeps larger or equal to its equilibrium level; that is, disease tends to be extinct when predators are introduced, and if 𝑅₀>1, then there is a disease equilibrium which is globally asymptotically stable; that is, disease still persists though predators are introduced. As corollaries, the globally stability of system (7) without predators is given. To examine the effect of predator on disease control, numerical simulations are given by choosing to focus on parameter ℎ, which is predation rate. We conclude that predation leads to decrease of equilibrium levels both for infected host and vector population; as ℎ increases, then the infected host and vector equilibrium population will be lessened. Furthermore, if ℎ>ℎ^∗(ℎ^∗satisfiesthat𝑅₀=1), then vector-borne diseases can be eradicated.

Conflict of Interests

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

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant nos. 70871056, 71271103, and 71371087), the National Research Subject of Education Information Technology (Grant no. 126240641), and the Innovative Foundation for Doctoral Candidate of Jiangsu Province, China (Grant no. CXZZ13_0687).