# Dynamics and Biocontrol: The Indirect Effects of a Predator Population on a Host-Vector Disease Model

Authors: Fengyan Zhou; Hongxing Yao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A model of the interactions among a host population, an insect-vector population, which transmits virus from hosts to hosts, and a vector predator population is proposed based on virus-host, host-vector, and prey (vector)-enemy theories. The model is investigated to explore the indirect effect of natural enemies on host-virus dynamics by reducing the vector densities, which shows the basic reproduction numbersR01 (without predators) and R02 (with predators) that provide threshold conditions on determining the uniform persistence and extinction of the disease in a host population. When the model is absent from predator, the disease is persistent if R01>1; in such a case, by introducing predators of a vector, then the insect-transmitted disease will be controlled if R02<1. From the point of biological control, these results show that an additional predator population of the vector may suppress the spread of vector-borne diseases. In addition, there exist limit cycles with persistence of the disease or without disease in presence of predators. Finally, numerical simulations are conducted to support analytical results.


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## Body

## 1. Introduction

In recent years there has been an increasing attention of the importance of studying vector-host diseases, whereby diseases are not transmitted directly from hosts to hosts, but through a vector. For instance, human and animal diseases such as malaria, dengue fever, West Nile virus, and Lyme disease are transmitted by arthropod vectors. Also, plant virus diseases such as tobacco mosaic virus, African cassava mosaic virus, cucumber mosaic virus, potato virus Y, beet yellows virus, and rice dwarf virus are transmitted by arthropod vectors. Bailey [1] pointed out that the vector population greatly influenced the modeling and control of infectious diseases. Moreover, vector-host diseases cause serious public health threat and great economic losses of agricultural production. Consequently, some effective control methods should be considered to prevent diseases transmitted by insect-vectors.Generally, pesticide is the most used method of insect-transmitted diseases control. The abuse of pesticides exacerbates disease incidence by selecting insecticide resistant vectors [2] and heavily affects reproduction and predation behavior of the beneficial predators of the vector [3]. Moreover, the repeated use of pesticides will deteriorate environment and human health. Therefore, it is not reasonable to depend fully on insecticides for the control of vector populations.The potential for predation to reduce disease prevalence in a host population also has implications for the biological control of vector populations. For example, in [4–11] predators have been introduced as biological control agents of vectors for various diseases such as malaria, dengue fever, and Lyme disease. Several recent studies suggested that predators led to a decline in local cases of dengue fever in Vietnam and Thailand [12, 13] and malaria in India [14, 15].However, compared with biological control of herbivorous pests, which has long been established as a major component of pest management programmes and aims to direct decrease pest densities by pest enemies [16, 17], biological control of vectors has been seldom investigated. The main reason is that biological control of vectors is a complex interaction process, mainly including the virus-host interaction, vector-host interaction, and vector (prey)-enemy interaction, which is more complicated than the only pest-enemy interaction [18–23] of biological pest control. The goal of biological control agents of vectors is to reduce disease incidence through the indirect action of natural enemy that lowers the density of a target vector population and is not necessarily to eliminate the vector [24]. The identification of a vector population threshold density plays an important role in determining whether predation could sufficiently reduce the vector population density [25]. Therefore, from the mathematical viewpoint, it is also important to investigate how vector (prey)-predator dynamics may be influenced by the additional vector mortality.Previous results on vector-borne diseases mainly focused on the vector-host dynamics [26–34] or virus-host dynamics [35] without considering the vector-enemy dynamics. Only several authors have studied the biological control of vector to reduce the disease incidence. In [36, 37], different mathematical models are used to examine how Wolbachia affects the transmission of dengue based on the fact that Wolbachia reduces the lifespan of infected mosquito Aedes aegypti. Moore et al. [25] first proposed a host-vector-predator model and studied how the indirect effect of predators on vectors affects prevalence of vector-borne diseases. For plant virus diseases transmitted by arthropod vectors, Okamoto and Amarasekare [38] compared the efficacy of three types of biocontrol agents: predator/parasitoid, competitor, and pathogen of the vector (Whitefly, which transmits the African cassava mosaic virus from cassavas to cassavas) to reduce disease incidence. Both papers [25] and [38], however, focused on the indirect effect of natural enemies on the vector based on numerical analysis, explicit mathematical analysis has not been done in detail to obtain the threshold value determining the prevalence or extinction of disease, and more complex dynamics such as Hopf bifurcation in present of the vector predators have not been considered.Motivated by the above mentioned surveys, the main objective of this study is to investigate the effects of predators on the prevalence of vector-borne disease by analyzing the stability of disease-free and disease equilibria in present of predators. Moreover, disease control threshold and limit cycles with persistence of disease or without disease in present of predators will be studied.The rest of the paper is organized as follows. Section2, we mainly formulate our basic mathematical model of host-vector-predator interactions. The equilibria and the reproduction number are given in Section 3. In Section 4, the local stability of the equilibria for model (2) is analyzed. In Section 5, the global stability and Hopf bifurcation of model (2) are investigated; moreover, the control threshold is presented. In Section 6, we use some numerical simulations to explore the effect of predators on the persistence or eradication of virus transmitted by vectors, the stability results, and existence of limit cycles of our model with the predator population. Finally, concluding remarks are given in Section 7.


## 2. Model Formulation

In this section we mainly formulate a mathematical model to depict the interactions among a host population, a vector population, and a vector predator population.First, we will give the following assumptions.(1)
The total host populationK(t) is divided into susceptible (healthy) and infected hosts, respectively, with densities denoted by S(t) and I(t), respectively, and the total vector population is also partitioned into susceptible and infectious subclasses, with the densities denoted by Xu(t) and Xi(t), respectively.(2)
Virus transmission is induced by vectors feeding on infected hosts or infected vectors feeding on susceptible hosts, without considering the direct transmission between susceptible and infected hosts.(3)
The virus in vectors may not cause vectors’ death.(4)
The new born vectors are susceptible and unable to transmit virus from hosts to hosts.(5)
Vectors in both classes experience density-independent and density-dependent mortality in addition to death from predation.(6)
Vectors’ predators are beneficial insects which can prey on vectors and may not damage hosts and environment, denoted byP(t). The Holling type II functional [39] is used for interactions between vectors and predators.(7)
When there are no predators, we assume that the dynamics of the total vector populationX=Xu+Xi is governed by equation dX/dt=rX(1-X/M-mX/(S+I)), where r is the intrinsic growth rate, M is vector’s carrying capacity of the environment, and m denotes the extent to which vectors are limited by the host population, and we assume that m=0 if the total host population on which the vectors depend keeps constant.Our assumptions on the dynamical transfer of the host, the vector, and the predator population are demonstrated in Figure1.Figure 1
Flow chart of the host-vector-predator model.In Figure1, f(S,I)>0 is the recruited rate of the host (or the replanting rate for the plant host) and μ and d are the natural death rate and disease-induced mortality of the infected host I, respectively. b1(K) denotes the per capita rate of contacts on hosts by vectors. Generally, b1(K) is a monotonic increasing function on K. In this paper, we assume b1(K)=b1K. α and β are the disease transmission coefficients from infected hosts to uninfected vectors and from infected vectors to uninfected hosts, respectively; b, w, and r=b-w are the natural birth rate, the natural death rate and the intrinsic growth rate of vectors, respectively. The term hX/(1+aX) denotes the Holling type II functional response of the predator. h is the capturing rate (or the attacking rate) of predators, a is predator’s satiety rate, and γ denotes the conversation factor of the predator, which describes the conversion efficiency of the consumed vectors into individual predators. e is the natural mortality of the predator. All the parameters above are positive.Using the transfer diagram, the system which depicts the interactions among the host, the vector, and the predator population can be derived:(1)dSdt=f(S,I)-b1(K)βXiSK-μS,dIdt=b1(K)βXiSK-(μ+d)I,dXudt=b(Xu+Xi)-b1(K)αXuIK-wXu-rXuM(Xu+Xi)-rmXu(S+I)(Xu+Xi)-hXu1+a(Xu+Xi)P,dXidt=b1(K)αXuIK-wXi-rXiM(Xu+Xi)-rmXi(S+I)(Xu+Xi)-hXi1+a(Xu+Xi)P,dPdt=γh(Xu+Xi)1+a(Xu+Xi)P-eP.In this paper, if we further assume that the number of total host population keeps unchanged, that is,K is a constant [25, 35], then from assumption (7), the total vector population X without P can be governed by the logistic equation dX/dt=rX(1-X/M). Using assumptions that S+I=K, b1(K)=b1K, K is a constant, and dX/dt=rX(1-X/M), then (1) can be reduced as
(2)dIdt=b1βXi(K-I)-(μ+d)I,dXdt=rX(1-XM)-hX1+aXP,dXidt=b1α(X-Xi)I-wXi-rXiMX-hXi1+aXP,dPdt=γhX1+aXP-eP,
with initial conditions
(3)ψ={(I(0),X(0),Xi(0),P(0))∣0≤I(0)≤K,0≤Xi(0)≤X(0)≤M,P(0)≥0}.
It easily follows that model (2) can be studied in the positively invariant set:
(4)Ω={(I,X,Xi,P)∈R4+∣0≤I≤K,0≤Xi≤X≤M,P≥0}.
Without considering the predator, model (2) can be reduced to the following system:
(5)dIdt=b1βXi(K-I)-(μ+d)I,dXdt=rX(1-XM),dXidt=b1α(X-Xi)I-wXi-rXiMX,
and the positively invariant set of system (5) is
(6)T={(I,X,Xi)∈R3+∣0≤I≤K,0≤Xi≤X≤M}.
Compared with model (2.1) in [25], the main differences in our model are that (i) we assume that the vector population dynamics is governed by the logistic equation because the total host population on which vectors depend is a constant. (ii) Holling type II functional responses are used in our paper to describe the interaction between vectors and predators, which is more logistic and realistic because Holling type II functional responses introduce satiety effect in the predator population [38, 39]. (iii) The linear per capita rate of contacts on hosts by vectors is considered.The purpose of this paper is to investigate the effect of predators on the prevalence of disease among hosts by directly reducing the vector density. For this purpose, as preliminaries, we first give the equilibria and the reproduction numbers of (2) and (5) in the next section.


## 3. The Equilibria and the Reproduction Number

Lemma 1.
The equilibria of system (2) are as follows. (i)
If the equilibrium value of the predator is zero, then there exist two disease-free equilibria,E0(0,0,0,0) and E1(0,M,0,0), which always exist; equilibrium E0 is the state of equilibrium of the uninfected host population only, while for equilibrium E1, it is the equilibria of the uninfected hosts and vectors only and a disease equilibrium E2(I*,M,Xi*,0) which exists if the threshold R01=b12αβKM/(b(μ+d))>1.(ii)
If the equilibrium value of the predator is larger than zero, then there exists a disease-free equilibriumE3(0,X^,0,P^) if the threshold R1=γhM/(e(1+aM))>1; equilibrium E3 is the vector and predator equilibria in the absence of the pathogen and a disease equilibrium E4(I^,X^,X^i,P^) if R1>1 and R02=b12αβKX^/(b(μ+d))>1, where
(7)I*=b12αβMK-b(μ+d)b12αβM+b1α(μ+d),Xi*=(μ+d)I*b1β(K-I*),X^=eγh-ae,P^=rh(1+aX^)(1-X^M),I^=b12αβKX^-b(μ+d)b12αβX^+b1α(μ+d),X^i=(μ+d)I^b1β(K-I^).
InequalityR1>1 aims to ensure P^>0, and by the next generation matrix, we find that
(8)R01=b12αβKMb(μ+d)=Kb1αb·Mb1β(μ+d),R02=b12αβKX^b(μ+d)=Kb1αb·X^b1β(μ+d)
are the basic reproduction number of systems (5) and (2), respectively.
The biological meaning ofR01 and R02 can be interpreted as follows. For R01, since the predator of vectors is not introduced (i.e., P=0), then the total number of vector populations achieves the stable state M and every vector is the susceptible and the number of the new infected vectors produced by each infected host over his/her expected infectious period is M·(b1β/(μ+d)), and when the total number of host population reaches stable state K and everyone is the susceptible, the number of new infected hosts produced by each infected vector (though the bite of the infected vector) over its expected infectious period is K·(b1α/b). The square root arises from the two “generations” required for an infected host or vector to “reproduce” itself.
While for the biological meaning ofR02, since the predator of vectors is introduced (that is P>0), then the total number of vector population achieves the stable state X^=e/(γh-ae); therefore, the number of the new infected vectors produced by each infected host over his/her expected infectious period is X^·(b1β/(μ+d)), and when the total number of host populations reaches stable state K and everyone is the susceptible, the number of new infected hosts produced by each infected vector (though the bite of the infected vector) over its expected infectious period is K·(b1α/b). The square root arises from the two “generations” required for an infected host or vector to “reproduce” itself.
In fact, by [40, 41], the basic reproduction number of (2) is expressed as
(9)R0=b12αβKX(μ+d)(w+(r/M)X+hP/(1+aX)),
where X and P are the disease-free equilibrium values. From the disease-free equilibrium E3(0,X^,0,P^), substitute X=X^, P=P^ into (9), and denote R0 as R02 yields R02=b12αβKX^/(b(μ+d)), which is the basic reproduction number of system (2).
In particular, ifP=0, then from the disease-free equilibrium E1(0,M,0,0), substitute X=M, P=0 into (9), and denote R0 as R01 yields R01=b12αβKM/(b(μ+d)), which is the basic reproduction number of system (5).Remark 2.
TakeR0 as a function of X (0<X≤M); then from (9) R01>R02 if R1>1 (i.e., M>X^). That is, predators result into a reduction in R0.
In particular, we obtain the following three possible equilibria of (5).Lemma 3.
There exist three possible equilibria for system (5), they are E0(0,0,0), E1=(0,M,0), which always exist, and E2=(I*,M,Xi*) which exists when R01>1, where I* and Xi* are given in Lemma 1.From (9), by adding the predator’s attack rate h, then R0 decreases and an increase in predation strength causes a decrease in the equilibrium proportion of infected hosts and vectors.Now, the aim of biological control is to reduce the risk of disease outbreaks in the host population by introducing predators of the vector population. In other words, we need to find the critical condition or the critical values such as predator’s attack rateh to make I(t)→0, Xi(t)→0, as t→+∞. For this purpose, we need to study the dynamics of models (2) and (5).


## 4. Local Stability of the Equilibria for System (2)

For analyzing the effect of predation on the prevalence of vector-transmitted disease, we first analyze the local stability of model (2).Theorem 4.
The local stability of equilibria for system (2) is as follows. (i)
The disease-free equilibriumE0 (uninfected host population only) is unstable.(ii)
The disease-free equilibriumE1 (only uninfected hosts and vectors) is locally asymptotically stable if R01<1 and R1<1.(iii)
The disease equilibriumE2 (without predators) is locally asymptotically stable if R01>1 and R1<1.(iv)
The disease-free equilibriumE3 (the vector and predator equilibria in absence of the pathogen) is locally asymptotically stable if R02<1 and 1<R1<1+γh/(ae(1+aM)).(v)
The disease equilibriumE4 (with predators) is locally asymptotically stable if R02>1 and 1<R1<1+γh/(ae(1+aM)).Proof.
(i) Linearizing around the disease-free equilibriumE0 (uninfected host population only), there exist four eigenvalues, they are λ1=-(μ+d)<0, λ2=r>0, λ3=-w<0, λ4=-e<0. Thus, E0 is unstable.
(ii) Linearizing around the disease-free equilibriumE1 (only uninfected hosts and vectors), two eigenvalues are obtained, they are λ1=-r<0, λ2=(γhM/(1+aM))-e<0 if R1<1. The other two roots are determined by the quadratic equation
(10)λ2+(b+μ+d)λ+b(μ+d)-b12αβMK=0.
The roots of (10) have negative real parts if and only if its coefficients are positive. Coefficient (b+μ+d) is always positive, while for coefficient b(μ+d)-b12αβMK, it is positive if and only if R01<1. Therefore, disease-free equilibrium E1 is locally asymptotically stable if R01<1 and R1<1.
(iii) For the disease equilibriumE2 (without predators) by linearizing around it, we obtain two eigenvalues, they are λ1=-r<0 and λ2=γhM/(1+aM)-e<0 if R1<1. The other two roots are determined by the quadratic equation
(11)λ2+p1λ+q1=0,
where p1=μ+d+b+b1αI*+b1βXi*>0,
(12)q1=(μ+d+b1βXi*)(b+b1αI*)-b12αβ(M-Xi*)(K-I*).
Equation (12) has two characteristic roots with negative real parts if and only if q1>0. From the equilibrium expression I*=(b12αβMK-b(μ+d))/(b12αβM+b1α(μ+d)), we have
(13)b12αβMK=[b12αβM+b1α(μ+d)]I*+b(μ+d).
Substitute (13) into (12) and reduce it; yield  q1=bb1βXi*+b12αβKXi*>0 if R01>1, Therefore, the disease equilibrium E2 is locally asymptotically stable for R01>1 and R1<1.
(iv) Linearizing system (2) about the disease-free equilibrium E3 (the vector and predator equilibria in absence of the pathogen) gives the following characteristic equation corresponding to the Jacobi matrix:
(14)|λ+(μ+d)0-b1βK00λ-r+2rX^M+r(1-X^/M)1+aX^0hX^1+aX^-b1αX^0λ+b00-γr(1-X^/M)1+aX^0λ|=0.
Expanding by the last column and by further simplifying, the above characteristic equation can be reduced to the following equation:
(15)[λ2+(b+μ+d)λ+b(μ+d)-b12αβKX^]×[γrhX^(1-X^/M)(1+aX^)2λ2+(2rX^M+r(1-X^/M)1+aX^-r)λ+γrhX^(1-X^/M)(1+aX^)2]=0.
Thus, the eigenvalues can be obtained by solving
(16)λ2+(b+μ+d)λ+b(μ+d)-b12αβKX^=0,(17)λ2+(2rX^M+r(1-X^/M)1+aX^-r)λ+γrhX^(1-X^/M)(1+aX^)2=0.
The roots of (16) have negative real parts if and only if R02<1. On the other hand, (17) has two roots with negative real parts if and only if its coefficients are positive. Obviously, coefficient γrhX^(1-X^/M)/(1+aX^)2 is always positive if R1>1, while for coefficient (2rX^/M)+(r(1-(X^/M))/(1+aX^))-r, we derive that 2rX^/M+(r(1-(X^/M))/(1+aX^))-r>0 is equivalent to R1<1+γh/(ae(1+aM)) by some mathematical deductions. Therefore, the disease-free equilibrium E3 (with predators) is locally asymptotically stable if R02<1 and 1<R1<1+γh/(ae(1+aM)).
(v) Linearizing system (2) about the disease equilibrium E4 (with predators) gives the following characteristic equation corresponding to the Jacobi matrix:(18)|λ+(μ+d)+b1βX^i0-b1β(K-I^)00λ-r+2rX^M+r(1-X^/M)1+aX^0hX^1+aX^-b1α(X^-X^i)-b1αI^+rX^M-ar(1-X^/M)X^i1+aX^λ+b+b1αI^hX^i1+aX^0-γr(1-X^/M)1+aX^0λ|=0.Expanding by the first column and by further simplifying, the above characteristic equation can be reduced to the following equation:
(19)[(λ+μ+d+b1βX^i)(λ+b+b1αI^)-b12αβ(K-I^)(X^-X^i)]×[(2rX^M+r(1-X^/M)1+aX^-r)λ2+(2rX^M+r(1-X^/M)1+aX^-r)λ+γrhX^(1-X^/M)(1+aX^)2(2rX^M+r(1-X^/M)1+aX^-r)]=0.
From the above analysis, the equation λ2+(2rX^/M+r(1-X^/M)/(1+aX^)-r)λ+γrhX^(1-X^/M)/(1+aX^)2=0 have two negative real parts if and only if 1<R1<1+γh/(ae(1+aM)). So, in order to determine the asymptotically stable E4, we only need to analyze the following equation:
(20)(λ+μ+d+b1βX^i)(λ+b+b1αI^)-b12αβ(K-I^)(X^-X^i)=0.
For simplicity, set
(21)p2=(b+μ+d+b1αI^+b1βX^i),q2=(μ+d+b1βX^i)(b+b1αI^)-b12αβ(K-I^)(X^-X^i).
Then, (20) is equivalent to the following equation:
(22)λ2+p2λ+q2=0,
where p2=(b+μ+d+b1αI^+b1βX^i)>0 if R02>1 and R1>1. For obtaining two roots of quadratic equation (22) with negative real parts, the coefficient q2 should satisfy q2>0:
(23)q2=(μ+d+b1βX^i)(b+b1αI^)-b12αβ(K-I^)(X^-X^i)=(μ+d)b+(μ+d)b1αI^+bb1βX^i+b12αβKX^i+b12αβX^I^-b12αβKX^.
By I^=(b12αβKX^-b(μ+d))/(b12αβX^+b1α(μ+d)), we have
(24)b12αβMK=[b12αβX^+b1α(μ+d)]I^+b(μ+d).
Substitute (24) into (23) and yield  q2=bb1βX^i+b12αβKX^i>0. Therefore, the disease equilibrium E4 (with predators) is locally asymptotically stable if R02>1 and 1<R1<1+γh/(ae(1+aM)). This completes the proof of Theorem 4.


## 5. Global Stability and Hopf Bifurcation of System (2)

The purpose of this section is to discuss the global stability and other dynamics for system (2) and obtain the control condition under which diseases can be controlled by predators preying on vectors. First, we give the following theorem.Theorem 5.
If the equilibrium value of the predator is zero, then for the disease-free equilibriaE1(0,M,0,0) and disease equilibrium E2(I*,M,Xi*,0) (without predators) of system (2), the following properties hold.(i)
IfR01≤1 and R1<1, then lim⁡t→+∞(I,X,Xi,P)=E1.(ii)
IfR01>1 and R1<1, then lim⁡t→+∞(I,X,Xi,P)=E2.Proof.
(i) From the second equation of system (2), dX/dt≤rX(1-X/M) and then lim⁡t→+∞sup⁡⁡X(t)≤M. Since R1<1, we consider ε, such that γh(M+ε)/(e(1+a(M+ε)))<1 and γh(M+ε)/(1+a(M+ε))-e<0. There exists T>0, such that X(t)≤M+ε for t>T. Therefore, P˙≤(γh(M+ε)/(1+a(M+ε))-e)P, and lim⁡t→+∞P(t)=0, if R1<1. So lim⁡t→+∞X(t)=M. Thus, the limit system of (2) can be reduced to
(25)dIdt=b1βXi(K-I)-(μ+d)I,dXidt=b1αI(M-Xi)-bXi.
Consider the Lyapunov function L=b1βKXi+bI. Its derivative along the solution of (25) is
(26)L′=b1βK[b1αI(M-Xi)-bXi]+b[b1βXi(K-I)-(μ+d)I]=I[b12αβK(M-Xi)-bb1βXi-b(μ+d)]≤Ib(μ+d)[b12αβKMb(μ+d)-1]=Ib(μ+d)[R012-1].
Obviously, L′≤0 if R01≤1. Moreover, the equality L′=0 holds if and only if I=Xi=0. That is, the maximal compact invariant set in {I(t),X(t)∈T:L′=0} is {E(0,0)}. By Lyapunov-LaSalle theorem [42], the equilibrium E(0,0) is globally stable if R01≤1; therefore lim⁡t→+∞(I,X,Xi,P)=E1 when R01≤1 and R1<1.
(ii) From the Jacobin of (25) we can verify that system (25) is a cooperative irreducible system in R2+ [43]. By the Jacobian of (25), it follows that if R01<1 the origin of system (25) is locally asymptotically stable and if R01>1 the origin of system (25) is unstable. Following Smith [44], system (25) is strongly concave. Then it follows that if R01>1 system (25) has an equilibrium E(I*,Xi*), which is globally asymptotically stable. Since system (25) is the limiting system of (2), it follows from Theorem  2.3 in [45] that the unique positive equilibrium E2(I*,M,Xi*,0) is a globally asymptotically stable equilibrium of system (2).
This completes the proof of Theorem5.Based on Theorems4 and 5, we have the following corollary for system (5), which is a special case of system (2) (without predators).Corollary 6.
For system (5), the following results hold. (i)
The disease-free equilibriumE0 is a saddle-node with x-axis as its unstable manifold and I-Xi plane as its stable manifold.(ii)
IfR01≤1, then lim⁡t→+∞(I,X,Xi)=E1, in which case, the vector-transmitted disease will eventually die out.(iii)
IfR01>1, then lim⁡t→+∞(I,X,Xi)=E2, in which case, the vector-transmitted disease will persist.Remark 7.
By Corollary6, we see the disease will persist (stability of E2) or die out (stability of E1) in absent of vector predators. Such results fully depend on the basic reproduction number R01of system (5) which is bigger or smaller than one.
For system (2), there exist two important equilibria: the disease-free boundary equilibrium E3(0,X^,0,P^) and the disease equilibrium E4(I^,X^,X^i,P^) when the equilibrium value of the predator (P) is larger than zero. With the help of the dynamics of these two equilibria, we can judge whether the disease can be eradicated in presence of the vector predators. For this purpose, we give the following theorem.Theorem 8.
Assume that the interior equilibriumE2(I*,X*,Xi*) of the reduced system (5) exists and is globally stable (R01>1); then we have(i)
ifR02≤1 and 1<R1≤1+γh/(ae(1+aM)), then lim⁡t→+∞(I,X,Xi,P)=E3, in which case, the infected hosts and the infected vectors will die out. Therefore, the vector-transmitted disease can be eradicated in presence of vector predators.(ii)
ifR02>1 and 1<R1≤1+γh/(ae(1+aM)), then lim⁡t→+∞(I,X,Xi,P)=E4, in which case, the infected hosts and the infected vectors will persist. Therefore, the vector-transmitted disease will persist though in presence of vector predators.(iii)
ifR02<1 and R1>1+γh/(ae(1+aM)), then there exists a stable limit cycle in (X,P) plane for system (2), in which case, despite these oscillations, predation can still eliminate the virus from the system, while vector densities are greater than zero.Before giving the main proof, we need the following lemma.LetI=Xi=0 in system (5) (without considering disease among hosts); then we have the following predator-prey model:
(27)dXdt=r(1-XM)X-hX1+aXP,dPdt=γhX1+aXP-eP.Lemma 9 (see [46]).
For system (27), the following results hold.(i)
There exists the unique positive equilibrium(X^,P^)=(e/(γh-ae),(r/h)(1+aX^)(1-X^/M)) if R1>1.(ii)
The positive equilibrium(X^,P^) is a globally asymptotically focus if 1<R1≤1+γh/(ae(1+aM)) and when R1>1+γh/(ae(1+aM)), (X^,P^) is an unstable focus; moreover, there exists a stable limit cycle for (27). This phenomenon corresponds to a supercritical Hopf bifurcation.System (27) is a classical predator-prey model and some authors have studies on the dynamics of (27) or models similar to (27). Here, we take the attack rate h as bifurcation parameter and give the proof for the occurrence of the bifurcation when h=h~=e(1+aM)/(γ(M-1/a)), where h~ satisfies equation R1=1+γh/(ae(1+aM)).Proof.
For equation (27), there exists the unique positive equilibrium (X^,P^) if R1>1. It is easy to get that characteristic equation of (27) at (X^,P^) is
(28)λ2+pλ+q=0,
where p=(-r+2rX^/M+hP^/(1+aX^)2), q=(γhP^/(1+aX^)2)(hX^/1+aX^).
Obviously, the above characteristic equation has only two roots and they can be expressed as(29)λ1,2=-p±p2-4q2.
Note that q>0. Consequently, when p>0, that is, R1<1+γh/(ae(1+aM)) (which can be obtained by some mathematical deduction) or h<h~, the above two characteristic roots have negative real parts and the interior equilibrium (X^,P^) is asymptotically stable. Therefore, h=h~ is a bifurcation point of (27) about the interior equilibrium (X^,P^).
For characteristic equationλ2+pλ+q=0, let λ=α(h)+iw(h); then there exists a pair of purely imaginary roots ±wi=±qi when p=0 (i.e., h=h~). Therefore, α(h~)=0, w(h~)>0. Differentiating λ2+pλ+q=0 with respect to h, we have
(30)dλ(h)dh=-12λ(h)+p(h)(λ(h)·dp(h)dh+dq(h)dh),
and noticing the fact that p(h~)=-2α(h~)=0 and λ=±wi=±qi when h=h~, so
(31)dλ(h)dh|h=h~=-12wi(wi·dp(h)dh|h=h~+dq(h)dh|h=h~),
then
(32)dα(h)dh|h=h~=-12dp(h)dh|h=h~=-12d(-r+2rX^/M+hP^/(1+aX^)2)dh∣h=h~=12eγ(γh-ae)2×[((1-X^M)((1-X^M)2(1+aX^)2-r(1+aX^)+raM(1-X^M))×(M(1+aX^)2)-1(1-X^M))]∣h=h~.
When h=h~, we have p(h~)=0. On the other hand, by mathematical deducing, we obtain that p(h~)=0 is equivalent to X^=(aM-1)/(2a). Substitute X^=(aM-1)/(2a) into the above equation; it follows that
(33)dα(h)dh|h=h~=14eγ(γh-ae)2(aM+1)2M(1+aX^)2>0.
Thus, (27) will undergo a Hopf bifurcation about the interior equilibrium (X^,P^) as h passes through the value h~ (i.e., R1=1+γh/(ae(1+aM))).
This completes the proof for the occurrence of the bifurcation.Based on Lemma9, we give the proof of Theorem 8.Proof.
(i-ii) By Lemma9, the solutions X(t) and P(t) of system (2) satisfy lim⁡t→+∞(X(t),P(t))=(X^,P^) if 1<R1≤1+γh/(ae(1+aM)). Then the limit system of (2) is reduced to
(34)dIdt=b1β(K-I)Xi-(μ+d)I,dXidt=b1αI(X^-Xi)-bXi.
Similar to the proof of Theorem 5, we can prove that lim⁡t→+∞(I,Xu,Xi,P)=E3 if 1<R1≤1+γh/(ae(1+aM)) and R02<1, and lim⁡t→+∞(I,X,Xi,P)=E4 if R02>1 and 1<R1≤1+γh/(ae(1+aM)).
(iii) From the characteristic equation (15) of equilibrium E3 of system (2), we have lim⁡t→+∞(I(t),Xi(t))=(0,0) if R02<1. Then by Lemma 9, then there exists a stable limit cycle of (X,P) plane for system (2) if R02<1 and R1>1+γh/(ae(1+aM)).
This completes the proof of Theorem8.Moreover, by simulation we obtain that ifR02>1 and R1>1+γh/(ae(1+aM)), then there exists a stable limit cycle for system (2), in which case predation cannot eliminate the virus from the system and eventually leads to stable oscillations (See Figure 5).From Theorem8, if R02>1 and R1>1, then I(t)→I^, Xi(t)→X^i, as t→+∞; thus vector-borne diseases would persist though predators are introduced, while if R02<1 and R1>1, then I(t)→0, Xi(t)→0, as t→+∞, then vector-borne diseases will be extinct by introducing vector predators. Thus, we have the following corollary.Corollary 10.
Assume that the vector-borne disease persists without predators; in other words, the interior equilibriumE3(I*,Xu*,Xi*) of the system (5) exists and is globally asymptotically stable (R01>1); then by introducing vector predators (R1>1 to ensure P>0), we have (i)
the vector-borne disease can beeliminated ifR02≤1;(ii)
the existing vector-borne disease will persist ifR02>1.Remark 11.
From Corollary10, the persistent disease in absence of predators (R01>1) will tend to extinct if R02<1 and will persist if R02>1 by introducing vector predators (R1>1 to ensure P>0). Therefore, the reproduction number R02 is the threshold value determining the persistence or extinction of the existing vector-borne disease.
If we choose predator’s attack rateh as parameter, then from R02=1, we have h=(e/γ)[a+b12αβK/(b(μ+d))]. Denote h*=(e/γ)[a+b12αβK/(b(μ+d))] and then yield the following corollary.Corollary 12.
Assume that the vector-borne disease persists in absence of vector predator (R01>1); then by introducing vector predators, (i)
the vector-transmitted disease can be eliminated if predator’s attack rateh satisfies h>h*;(ii)
the existing vector-borne disease will persist if predator’s attack rateh satisfies
(35)eγ[a+1M]<h<h*.Remark 13.
Corollaries10 and 12 give the conditions to judge when will the existing vector-borne disease be eradicated by introducing predators to reduce vector densities, which provides us with qualitative scheme for disease control.Remark 14.
Based on Corollary10, other similar results can be derived if taking other predator parameters such as the conversation rate γ as parameter. In this paper, we mainly investigate the persistence or extinction of disease under different attack rate h of the predator population, especially in the following numerical simulations.


## 6. Numerical Simulations

In this section, we present some numerical simulations to support the analytical results for Cases1, 2, and 3.Case 1 (the effects of predators on the persistence or eradication of virus transmitted by arthropod vectors).
Keep values ofμ,d,b1,α,β,M,K,r,w,b unchanged in Table 1. By simple calculations, the basic reproduction number R01=5.7735>1. By Corollary 6, without vector predators, the vector-transmitted disease will persist. In such a case, we introduce the predator. Take a=0.01, h=0.5 and keep values γ and e fixed in Table 1 and then by direct computation, we have R02=0.9492<1, R1=25.00>1, R1-1-γh/(ae(1+aM))=-26.00≤0; then by Theorem 8, the disease-free equilibrium E3(0,X^,0,P^) is globally asymptotically stable, which indicates that the existing vector-transmitted disease will die out in presence of vector predators (see Figure 2).
Keep all the values unchanged in Figure2 except that h=0.2, then by simple calculations, R01=5.7735>1, R02=1.5162>1, R1=10.00>1, R1-1-γh/(ae(1+aM))=-11.00≤0. Therefore, by Theorem 8, the disease equilibrium E4(I^,X^,X^i,P^) is globally asymptotically stable, which indicates that the vector-transmitted disease will be persistent though in presence of vector predators (see Figure 3).Table 1
Definition of parameters and values used in simulations (time unit is selected as day).
Parameter
Description
Estimated value/range
Resource
μ
Natural death rate of plants
0.003
[38]
d
Disease-induced mortality rate of plants
0.003
[38]
b
1
Per capita rate of contacts on plant hosts by vectors
1
Assumed
α
The disease transmission probability from infected plant hosts to uninfected vectors (=the acquisition rate of virus)
0.008
[38]
β
The disease transmission probability from infected vectors to uninfected plants (=the inoculation rate of virus)
0.008
[38]
M
Vector carrying capacity
50
Assumed
K
Total plant population
20
Assumed
r
Intrinsic birth rate of vectors
0.2
[38]
w
Natural death rate of vectors
0.12
[38]
b
Natural birth rate of vectors
0.32
[38]
h
Prey searching rate
0.1–0.65
Assumed
a
Predator’s satiety rate
0.01–0.03
Assumed
γ
Predator’s conversation factor
0.15
[46]
e
Predator’s mortality rate
0.1
[46]Numerical simulation for Case1, with a=0.01, h=0.5, and other parameters fixed in Table 1. The vector-borne disease tends to extinct in presence of vector predators.
(a)
(b)
(c)
(d)Numerical simulation for Case1, with a=0.01, h=0.2, and other parameters fixed in Table 1. The vector-borne disease will persist though in presence of vector predators.
(a)
(b)
(c)
(d)Remark 15.
By Figures2 and 3, the disease will be persistent or tend to be extinct in presence of predators, which greatly depends on the predator strength. It can be easily verified that if the attack rate h satisfies h=0.2<h*, then by Corollary 12 the disease will persist. While if the attack rate h is added up to 0.5 and satisfies h=0.5>0.4511=h*, then by Corollary 12, the disease dies out. Moreover, from Figure 2, by enhancing the attack rate h, the reduced vector densities rather than outright eradication can suffice to eradicate disease, which is different to predators used in pest control where predations aim to eradicate pests.Case 2 (the existence of stable limit cycles).
Keep values ofμ,d,b1,α,β,M,K,r,w,b unchanged in Case 1; then R01=5.7735>1; that is, the disease persists without predator. Take a=0.03, h=0.65, and keep values γ and e unchanged in Case 1. By simple calculations, R02=0.8399<1, R1=19.5>1, R1-1-γh/(ae(1+aM))=5.5>0; then by Theorem 8, there exists a stable limit cycle in (X-P) plane for system (2), but the infected host and vector population level tend to zero (see Figure 4).
Keep all values of system (2) unchanged in Figure 4 except that h=0.12; then by simple calculations, R01=5.7735>1, R02=2.1082>1, R1=3.6>1, R1-1-γh/(ae(1+aM))=0.2>0. By numerical simulation, there exists a stable limit cycle for system (2) (see Figure 5).Numerical simulation for Case2, with a=0.03, h=0.65, and other parameters fixed in Table 1. A stable limit cycle occurs in XP plane of system (2). (a) Temporal plots for system variables. (b) Phase portrait in XP plane. (c) Phase portrait in tXP space.
(a)
(b)
(c)Numerical simulation for Case2, with a=0.03, h=0.12, and other parameters fixed in Table 1. A stable limit cycle occurs for system (2).
(a)
(b)
(c)
(d)Remark 16.
From Figures4 and 5, we can see that periodic solutions exist in presence of vector predators, suggesting the occurrence of supercritical Hopf bifurcation. From Figure 5, the infected hosts, the vectors including infected and susceptible ones, and the predators oscillate, leading to the persistence of the vector-transmitted disease though predators are introduced. By enhancing predators’ attack rate h such that h=0.65>0.4644=h*, then by Corollary 12, we find that predation can lead to extinction of the disease, despite the oscillations of the predator and the vector (see Figure 4.).Case 3 (the comparing outcomes between models with and without the predator population).
Keep values ofμ,d,b1,α,β,M,K,r,w,b unchanged in Case 1; then R01=5.7735>1; that is, the disease persists without predator. Take predator’s attack rate h as parameter and keep γ, e, and c unchanged in Case 1; we study how different values of h affect the disease prevalence. In absence of the predator, that is, h=0, then the disease persists (see the above curve of Figure 6(a)). By introducing predators of the vector population, we find that the equilibrium infection levels have been reduced and also the onset of an epidemic has been delayed if the predator attack rate h=0.2. However, the disease has not been eradicated (see the below curve of Figure 6(a)). By increasing the attack rate of predators h, such that h=0.5, then from the below curve of Figure 6(b), we can see that the vector-borne disease can be eradicated by introducing vector predators.The population densities of the infected hosts with and without vector predators.
(a)
(b)Remark 17.
From Figure6, we conclude that an increase in predator’s attack rate can not only reduce equilibrium infection levels, but also delay the onset of an epidemic (see Figure 6(a)). Moreover, predation can eradicate virus if the attack rate h satisfies h>h* (see Figure 6(b)).


## 7. Concluding Remarks

In this paper, we mainly propose a host, vector, and predator model to investigate the impact of the predator on the outbreak and extinction of vector-transmitted disease. Some sufficient conditions ensuring the globally asymptotically stability for the disease-free and disease equilibria are derived in absence and presence of predators. Moreover, complex dynamical phenomena such as Hopf bifurcation appear due to the stable focus losing stability. Taking the attack rateh as the bifurcation parameter, we derive the threshold condition h* based on the stability results such that the vector-transmitted disease can be eradicated by introducing vector predators when h>h*. Some numerical simulations are illustrated to support the theoretical analysis of research, also used for better comparison of the disease prevalence with or without predators. Moreover, since the viewpoint of pest control, multipredators have been testified more effectively on a single prey than one predator and another possible extension would be coupling our host-vector model with the prey and multipredator model studied in [22, 23]. Therefore, we can further continue our work about the effect of multipredator on the prevalence of vector-borne disease.



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