Applied Mathematical Modelling 33 (2009) 2864–2873

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Mathematical modelling to control a pest population by infected pests q

Shulin Sun a,b,*, Lansun Chen b

a School of Mathematics and Computer Science, Shanxi Normal University, Shanxi, Linfen 041004, PR Chinab Department of Applied Mathematics, Dalian University of Technology, Liaoning, Dalian 116024, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 December 2006Received in revised form 26 August 2008Accepted 28 August 2008Available online 6 September 2008

Keywords:Integrated Pest ManagementGlobally asymptotic stabilityPermanence

0307-904X/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.apm.2008.08.018

q This work is supported by National Natural Scie* Corresponding author. Address: School of Math

E-mail address: sunsl_2004@yahoo.com.cn (S. Su

In this paper, we formulate and investigate the pest control models in accordance with themathematical theory of epidemiology. We assume that the release of infected pests is con-tinuous and impulsive, respectively. Therefore, our models are the ordinary differentialequations and the impulsive differential equations. We study the global stability of theequilibria of the ordinary differential equations. The permanence of the impulsive differen-tial equations is proved. By means of numerical simulation, we obtain the critical values ofthe control variable under different methods of release of infected pests. Thus, we provide amathematical evidence in the management of an epidemic controlling a pest.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

There are many ways to control agricultural pests. In [1–3], the authors indicated that biological control is the purposefulintroduction and establishment of one or more natural enemies from region of origin of an exotic pest, specifically for thepurpose of suppressing the abundance of the pest in a new target region to a level at which it no longer causes economicdamage. Another important method for pest control is chemical control. Pesticides are powerful tools used to control pestsbecause they can quickly kill a significant portion of a pest population and they sometimes can provide the only feasiblemethod for preventing economic loss. However, pesticides need to be used judiciously since pesticide pollution is also con-sidered as a major health hazard to human being and to natural enemies. By the long-term practice, it is recognized that theproblem of pest control is that of biology, economics and ecology. Therefore, many scholars put forward Integrated Pest Man-agement (IPM see [4]). Recently, the authors in [5–8] studied models for pest control and obtained some interesting results.

Studies of epidemic models have become the important areas in the mathematical theory of epidemiology and they havelargely been inspired by the work of Anderson and May [9]. Most of research papers assume that the disease incubation isnegligible. A model based on this assumption is often called an SIR or SIRS model. Liu et al. [10] considered a general SEIRSmodel with nonlinear incidence rate bIpSq and showed that this model exhibited a much wider range of dynamical behaviorthan those models with bilinear incidence rate bIS. But, in [10], the authors assumed that the total population was constant.In [11], the SIR epidemiological model was studied, the authors assumed that the susceptible pest satisfied the Logistic Equa-tion, the incidence rate was kISq and the total population was not constant.

In this paper, we investigate the problem to control pest population by infectious disease. In this problem, the controlvariable is the rate of release of infected pests. The purpose is to suppress the pest population below a certain level witha minimum value of the control variable. We assume that the release of infected pests is continuous and impulsive, respec-tively. Therefore, our models are the ordinary differential equations and the impulsive differential equations. The theory and

. All rights reserved.

nce Foundation of China (10471117).ematics and Computer Science, Shanxi Normal University, Shanxi, Linfen 041004, PR China.n).

S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873 2865

application of impulsive differential equations were introduced systematically in [12–15]. Papers on combining pest controlwith infectious disease are not many (see [16,17] and references therein).

In our models, the following assumptions are given.

(H1) The pest population is divided into two classes which are the susceptible class and the infective class.(H2) The susceptible pest satisfies the Logistic growth and the incidence rate is of the form bS2I, where bS2 is the incidencerate per infective individual [11].(H3) The total pest population is not constant.

Hence, our models are the following forms:

S0 ¼ rSð1� SÞ � bS2I;

I0 ¼ bS2I � ðdþxÞI þ u;

(ð1:1Þ

and

S0 ¼ rSð1� SÞ � bS2I; t–ns;I0 ¼ bS2I � ðdþxÞI; t–ns;DI ¼ IðtþÞ � IðtÞ ¼ su; t ¼ ns; n ¼ 0;1;2; . . . ;

8><>: ð1:2Þ

where SðtÞ and IðtÞ denote the amount of susceptible pests and infective pests at time t, respectively; r > 0 is the growth rateof the susceptible pest; b > 0 is the contact rate; d > 0 and x > 0 are the natural death rate and the disease-caused deathrate of the infective pest, respectively; u > 0 is the release rate of infected pests. In (1.2), s is the period of the pulsing, suis the amount of infected pests pulsed each s.

2. Preliminaries

In order to obtain our results, we make the following assumptions again:

(H4) The infected pest cannot recover for all time and cannot cause excessive damage to the crops. The susceptible pestcan destroy the crops.(H5) There exists a critical value SM which is an economically significant level for pest damage.

Therefore, our aim is to control the amount SðtÞ of susceptible pests below the critical value SM by releasing the infectedpest. First, we consider the epidemic model with the pests without outer interference. Thus, the SI epidemiological modelcan be written as

S0 ¼ rSð1� SÞ � bS2I;

I0 ¼ bS2I � ðdþxÞI:

(ð2:1Þ

Clearly, system (2.1) has trivial equilibria E0ð0;0Þ; E1ð1; 0Þ and a positive equilibrium E2ðbS; bIÞ, where bS ¼ ffiffiffiffiffiffiffidþx

b

q; bI ¼ rð1�bSÞ

bbS .Let r0 ¼ b

dþx. Then, r0 is called the threshold of system (2.1).The following lemma is obvious.

Lemma 2.1. R2þ ¼ fðS; IÞ 2 R2 : S P 0; I P 0g is positive invariant for system (2.1).

From [11], we have the following propositions.

Proposition 2.1. The equilibrium E0 is a saddle.

Proposition 2.2. If r0 > 1, then the equilibrium E1 is a saddle; if r0 < 1, then the equilibrium E1 is a stable node; if r0 ¼ 1, thenthe equilibrium E1 ¼ E2 is a saddle-node.

Lemma 2.2. If r0 < 1, the equilibrium E2 does not exist; if r0 > 1, the equilibrium E2 is locally asymptotically stable.

Proof. The nonexistence of the equilibrium E2 is obvious.The characteristic equation of the Jacobian matrix of the positive equilibrium E2ðbS;bIÞ is

k2 þ A1kþ A2 ¼ 0; ð2:2Þ

where

A1 ¼ �½ðr � 2rbS � 2bbSbIÞ þ ðbbS2 � ðdþxÞÞ� ¼ r > 0;

A2 ¼ ðr � 2rbS � 2bbSbIÞðbbS2 � ðdþxÞÞ þ 2b2bS3bI ¼ 2rðdþxÞ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffidþx

b

s !:

2866 S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873

Hence, if r0 > 1, then the real parts of two roots of Eq. (2.2) are both negative, E2 is a locally asymptotically stable. The proofis complete. h

Lemma 2.3. There exists a constant M > 0 such that SðtÞ 6 M; IðtÞ 6 M for each solution ðSðtÞ; IðtÞÞ of (2.1) with all t largeenough.

Proof. Suppose ðSðtÞ; IðtÞÞ is any solution of (2.1). Let VðtÞ ¼ SðtÞ þ IðtÞ. Then

DþVðtÞ þ ðdþxÞVðtÞ ¼ �rS2 þ ðr þ dþxÞS:

Clearly, the right hand of the above formula is bounded. Let K > 0 be the bound. We have

VðtÞ 6 Vð0Þ � Kdþx

� �e�ðdþxÞt þ K

dþx:

Therefore, VðtÞ is ultimately bounded by a constant and there exists a constant M > 0 such that SðtÞ 6 M; IðtÞ 6 M for eachsolution ðSðtÞ; IðtÞÞ of (2.1) with all t large enough. The proof is complete. h

Let D ¼ fðS; IÞ 2 R2 : 0 < S 6 M;0 < I 6 Mg;R2þ

�¼ fðS; IÞ 2 R2 : S > 0; I > 0g. Then D � R2

þ

�� R2

þ is positive invariant for sys-tem (2.1).

Theorem 2.1. If r0 > 1, then the positive equilibrium E2 of system (2.1) is globally asymptotically stable in R2þ

�.

Proof. Let F1ðS; IÞ ¼ rSð1� SÞ � bS2I; F2ðS; IÞ ¼ bS2I � ðdþxÞI. Choose a Dulac function BðS; IÞ ¼ S�1I�1. Then F1ðS; IÞ;F2ðS; IÞ; BðS; IÞ are continuously differentiable functions on the region D, and

oðBF1ÞoS

þ oðBF2ÞoI

¼ �b� rI< 0:

According to the Bendixson–Dulac Theorem [18], there is no closed orbit in the region D. Therefore, together with Lemmas2.2 and 2.3, the equilibrium E2 is globally asymptotically stable. �

Remark 2.1. As the above analysis, we see that the susceptible pest cannot become extinct. Therefore, we only control theamount of susceptible pests below SM . In terms of IPM, if bS 6 SM , then it is not necessary to control a pest in crops. Therefore,we assume bS > SM in the latter Sections.

3. Continuous release of infected pests

In this section, we study system (1.1) with continuous release of infected pests. Obviously, system (1.1) has a boundaryequilibrium E0ð0; u

dþxÞ.Now, we study the existence and uniqueness of the positive equilibrium E� ¼ ðS�; I�Þ in R2

þ.In R2

þ, as Fig. 1, the vertical isoclinal lines Sðr � rS� bSIÞ ¼ 0 are L1 : S ¼ 0; ðI > 0Þ and L2 : I ¼ rð1�SÞbS ; S 2 ð0;1Þ; L1 is a

asymptotic line of L2; the horizontal isoclinal line bS2I � ðdþxÞI þ u ¼ 0 is L3 : I ¼ uðdþxÞ�bS2 ; S 2 ð0;

ffiffiffiffiffiffiffidþx

b

qÞ, and

L : S ¼ffiffiffiffiffiffiffidþx

b

qis a asymptotic line of L3.

Let f1ðSÞ,I ¼ rð1�SÞbS ; S 2 ð0;1Þ; f2ðSÞ,I ¼ u

ðdþxÞ�bS2 ; S 2 0;ffiffiffiffiffiffiffidþx

b

q� �; S0 ¼min 1;

ffiffiffiffiffiffiffidþx

b

qn o. Then, over the interval ð0; S0Þ, we

have df1dS ¼ � r

bS2 < 0 and df2dS ¼

2ubS½ðdþxÞ�bS2 �2

> 0. This implies that f1ðSÞ is strictly monotone decreasing in S and f2ðSÞ is strictly

monotone increasing in S. It follows that there exists a unique point of intersection of L3 and L2. Obviously, this point of inter-section is the positive equilibrium E�ðS�; I�Þ and S� 2 ð0; S0Þ is the function of the parameter u. We can denote S� by f ðuÞ, i.e.S�, f ðuÞ.

Our aim is to suppress the amount of susceptible pests below SM by releasing infected pests. If the positive equilibriumE�ðS�; I�Þ is globally asymptotically stable, then we have S� < SM as u > f�1ðSMÞ. In the following, we analyze the stability ofthe equilibria E0 and E�.

Lemma 3.1. R2þ ¼ fðS; IÞ 2 R2 : S P 0; I P 0g is positive invariant for system (1.1).

Obviously, the equilibrium E0 0; ud1þx

� �is a saddle of system (1.1) in R2

þ.The characteristic equation of the Jacobian matrix of the positive equilibrium E�ðS�; I�Þ is

k2 þ B1kþ B2 ¼ 0; ð3:1Þ

where

B1 ¼ �½ðr � 2rS� � 2bS�I�Þ þ ðbS�2 � ðdþxÞÞ� ¼ r þ u=I� > 0;

B2 ¼ ðr � 2rS� � 2bS�I�ÞðbS�2 � ðdþxÞÞ þ 2b2S�3I� ¼ ruI�þ 2b2S�3I� > 0:

Fig. 1. (a) r0 < 1 and (b) r0 > 1.

S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873 2867

Therefore, if only the positive equilibrium E� exists, then the real parts of two roots of Eq. (3.1) are negative. Hence, E� islocally asymptotically stable in R2

þ.Similar to Lemma 2.3, we have

Lemma 3.2. There exists a constant M1 > 0 such that SðtÞ 6 M1; IðtÞ 6 M1 for each solution ðSðtÞ; IðtÞÞ of (1.1) with all t largeenough.

Let D1 ¼ fðS; IÞ 2 R2 : 0 < S 6 M1;0 < I 6 M1g. Then D1 � R2þ

�� R2

þ is positive invariant for system (1.1).

Theorem 3.1. The positive equilibrium E� of system (1.1) is globally asymptotically stable in R2þ

�.

The proof is similar to that of Theorem 2.1.

Remark 3.1. As Theorem 3.1, if the rate of release of infected pests satisfies u > f�1ðSMÞ, then we can suppress the amount ofsusceptible pests below SM after some time. Thus, we obtain the minimum value of the control variable u which can drive thesystem to the target.

4. Impulsive release of infected pests

In this section, we analyze system (1.2) with impulsive release of infected pests. Firstly, we need some notations, defini-tions and lemmas which are useful for our main results.

Let Rþ ¼ ½0;þ1Þ; xðtÞ ¼ ðSðtÞ; IðtÞÞ 2 R2þ;N be the set of nonnegative integers. Denote f ¼ ðf1; f2ÞT the map defined by the

right hand of the first two equations of system (1.2).Let V : Rþ � R2

þ ! Rþ. Then V is said to belong to class V0 if

(i) V is continuous in ðns; ðnþ 1Þs� � R2þ and for each x 2 R2

þ; n 2 N; limðt;yÞ!ðnsþ ;xÞVðt; yÞ ¼ Vðnsþ; xÞ exists;(ii) V is locally Lipschitzian in x.

For ðt; xÞ 2 ðns; ðnþ 1Þs� � R2þ, we define

DþVðt; xÞ ¼ lim suph!0þ

1h½Vðt þ h; xþ hf ðt; xÞÞ � Vðt; xÞ�:

The solution of system (1.2) is a piecewise continuous function xðtÞ : Rþ ! R2þ; xðtÞ is continuous on ðns; ðnþ 1Þs�; n 2 N

and xðnsþÞ ¼ limt!nsþxðtÞ exists. Obviously, the global existence and uniqueness of solutions of the system (1.2) is guaran-teed by the smoothness properties of f (see [14] for details). The following lemma is obvious.

Lemma 4.1. Suppose xðtÞ is a solution of (1.2) with xð0þÞP 0. Then xðtÞP 0 for all t P 0. Moreover, if xð0þÞ > 0, then xðtÞ > 0for all t P 0.

Definition 4.1. System (1.2) is said to be permanent if there exist constants M P m > 0 such thatm 6 SðtÞ 6 M;m 6 IðtÞ 6 M for t large enough, where ðSðtÞ; IðtÞÞ is any solution of (1.2) with Sð0þÞ > 0; Ið0þÞ > 0.

Lemma 4.2 (Comparison Theory, [14, Theorem 3.1.1]). Let V : Rþ � R2þ ! Rþ and V 2 V0. Assume that

DþVðt; xðtÞÞ 6 gðt;Vðt; xðtÞÞÞ; t–ns;Vðt; xðtþÞÞ 6 wnðVðt; xðtÞÞÞ; t ¼ ns;

(ð4:1Þ

2868 S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873

where g : Rþ � Rþ ! R is continuous in ðns; ðnþ 1Þs� � Rþ and for each z 2 Rþ; n 2 N; limðt;yÞ!ðnsþ ;zÞgðt; yÞ ¼ gðnsþ; zÞ exists;wn : Rþ ! Rþ is nondecreasing. Let rðtÞ ¼ rðt;0; v0Þ be the maximal solution of the scalar impulsive differential equation

v0 ¼ gðt; vÞ; t–ns;vðtþÞ ¼ wnðvðtÞÞ; t ¼ ns;vð0þÞ ¼ v0;

8><>: ð4:2Þ

existing on ½0;1Þ. Then Vð0þ; x0Þ 6 v0 implies that Vðt; xðtÞÞ 6 rðtÞ; t P 0, where xðtÞ ¼ xðt;0; x0Þ is any solution of (1.2) existingon ½0;1Þ.

Remark 4.1. In Lemma 4.2, if the directions of the inequalities in (4.1) are reversed, that is,

DþVðt; xðtÞÞP gðt;Vðt; xðtÞÞÞ; t–ns;Vðt; xðtþÞÞP wnðVðt; xðtÞÞÞ; t ¼ ns;

(

then Vðt; xðtÞÞP qðtÞ; t P t0, where qðtÞ is the minimal solution of (4.2) on ½0;1Þ.

Next, we use the function �ðdþxÞvðtÞ. For convenience, we give some basic properties of the following system:

v0ðtÞ ¼ �ðdþxÞvðtÞ; t–ns;DvðtÞ ¼ vðtþÞ � vðtÞ ¼ su; t ¼ ns;vð0þÞ ¼ Ið0þÞP 0:

8><>: ð4:3Þ

Clearly, v�ðtÞ ¼ sue�ðdþxÞðt�nsÞ

1�e�ðdþxÞs ; t 2 ðns; ðnþ 1Þs�; n 2 N; ðv�ð0þÞ ¼ su1�e�ðdþxÞsÞ is a positive periodic solution of (4.3). The solution

of (4.3) is vðtÞ ¼ ½vð0þÞ � v�ð0þÞ�e�ðdþxÞt þ v�ðtÞ; t 2 ðns; ðnþ 1Þs�; n 2 N. Therefore, we have the following results.

Lemma 4.3. System (4.3) has a positive periodic solution v�ðtÞ and jvðtÞ � v�ðtÞj ! 0 as t !1 for any solution vðtÞ of (4.3).Moreover, vðtÞP v�ðtÞ if vð0þÞP v�ð0þÞ and vðtÞ < v�ðtÞ if vð0þÞ < v�ð0þÞ.

Next, we show that all solutions of (1.2) are uniformly upper bounded.

Lemma 4.4. There exists a constant M > 0 such that SðtÞ 6 M; IðtÞ 6 M for each solution ðSðtÞ; IðtÞÞ of (1.2) with all t largeenough.

Proof. Suppose ðSðtÞ; IðtÞÞ is any solution of (1.2). Let VðtÞ ¼ SðtÞ þ IðtÞ. Then V 2 V0 and

DþVðtÞ þ ðdþxÞVðtÞ ¼ �rS2 þ ðr þ dþxÞSþ u; t–ns:VðnsþÞ ¼ VðnsÞ þ su; n ¼ 1;2; . . .

(ð4:4Þ

Obviously, the right hand of the first equation in (4.4) is bounded, thus, there exists K > 0 such that

DþVðtÞ 6 �ðdþxÞVðtÞ þ K; t–ns;VðnsþÞ ¼ VðnsÞ þ su; n ¼ 1;2; . . .

(

By Comparison Theory, we have

VðtÞ 6 Vð0þÞ � KðdþxÞ

� �e�ðdþxÞt þ suð1� e�nðdþxÞsÞ

1� e�ðdþxÞs e�ðdþxÞðt�nsÞ þ Kdþx

for t 2 ðns; ðnþ 1Þs�. Therefore, VðtÞ is ultimately bounded by a constant and there exists a constant M > 0 such thatSðtÞ 6 M; IðtÞ 6 M for any solution ðSðtÞ; IðtÞÞ of system (1.2) with all t large enough. The proof is complete. �

Now, we study the stability of the boundary periodic solution and the permanence of system (1.2).Obviously, system (1.2) has a s-periodic solution ð0; I�ðtÞÞ, where ð0; I�ðtÞÞ ¼ 0; sue�ðdþxÞðt�nsÞ

1�e�ðdþxÞs

� �; t 2 ðns; ðnþ 1Þs�.

Theorem 4.1. The boundary periodic solution ð0; I�ðtÞÞ of system (1.2) is unstable.

Proof. The stability of periodic solution ð0; I�ðtÞÞmay be determined by considering the behavior of small amplitude pertur-bation of the solution. Let ðSðtÞ; IðtÞÞ be any solution of (1.2). We define SðtÞ ¼ sðtÞ; IðtÞ ¼ iðtÞ þ I�ðtÞ.

The corresponding linear system of (1.2) at ð0; I�ðtÞÞ is

s0ðtÞ ¼ rs; t–ns;i0ðtÞ ¼ �ðdþxÞi; t–ns;sðtþÞ ¼ sðtÞ; t ¼ ns;iðtþÞ ¼ iðtÞ; t ¼ ns;

8>>><>>>: ð4:5Þ

Let UðtÞ be a fundamental matrix of (4.5). Then UðtÞ satisfies

S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873 2869

dUðtÞdt

¼r 00 �ðdþxÞ

� �UðtÞ,AUðtÞ: ð4:6Þ

and Uð0Þ ¼ I, the identity matrix. The resetting impulsive conditions of (4.5) becomes

sðnsþÞiðnsþÞ

� �¼

1 00 1

� �sðnsÞiðnsÞ

� �:

The stability of the periodic solution ð0; I�ðtÞÞ is determined by the eigenvalues of the monodromy matrix

M ¼1 00 1

� �UðsÞ ¼ UðsÞ:

From (4.6), we can obtain UðsÞ ¼ Uð0Þ expðR s

0 AdtÞ,Uð0Þ expðAÞ. Therefore, the Floquet multipliers of system (4.5) are

k1 ¼ ers > 1; k2 ¼ e�ðdþxÞs < 1:

According to Floquet theory [12, Theorem 3.5], ð0; I�ðtÞÞ is unstable since jk1j > 1.

Theorem 4.2. System (1.2) is permanent.

Proof. Suppose ðSðtÞ; IðtÞÞ is any solution of (1.2) with ðSð0þÞ; Ið0þÞÞ > 0. By Lemma 4.4, we can assume SðtÞ 6 M; IðtÞ 6 Mfor t P 0. Choose �2 > 0 small enough such that m2 ¼ sue�ðdþxÞs

1�e�ðdþxÞs � �2 > 0. It follows from Lemmas 4.2 and 4.3 that IðtÞ > m2 forall t large enough.

Next, we prove that there exists an m1 > 0 such that SðtÞ > m1 for all t large enough in two steps.

Step 1. We can choose m3 > 0; �1 > 0 small enough and the upper bound M > 0 large enough such thatd ¼ bm2

3 � ðdþxÞ < 0; r ¼ rs�m3rs�m3b�1sþ m3busd > 0 and r1 ¼ rð1�m3Þ �m3bM < 0. We claim that

SðtÞ < m3 cannot hold for all t P 0. Otherwise,

I0ðtÞ 6 IðtÞ½bm23 � ðdþxÞ� ¼ dIðtÞ:

By Lemmas 4.2 and 4.3, we have IðtÞ 6 yðtÞ and yðtÞ ! y�ðtÞ as t !1, where yðtÞ is the solution of

y0ðtÞ ¼ dyðtÞ; t–ns;Dy ¼ yðtþÞ � yðtÞ ¼ su; t ¼ ns;yð0þÞ ¼ Ið0þÞ > 0:

8><>: ð4:7Þ

and y�ðtÞ ¼ suedðt�nsÞ

1�eds ; t 2 ðns; ðnþ 1Þs�. Therefore, there exists T1 > 0 such that

IðtÞ 6 yðtÞ 6 y�ðtÞ þ �1

and

S0ðtÞP SðtÞ½rð1�m3Þ �m3bðy�ðtÞ þ �1Þ� ð4:8Þ

for t P T1.Let N1 2 N and N1s P T1. Integrating (4.8) on ðns; ðnþ 1Þs�;n P N1, we have

Sððnþ 1ÞsÞP SðnsþÞ expZ ðnþ1Þs

ns½rð1�m3Þ �m3bðy�ðtÞ þ �1Þ�dt

� �¼ SðnsÞer:

Then, SððN1 þ kÞsÞP SðN1sÞekr as k!1, which is a contradiction. Hence, there exists a t1 > 0 such that Sðt1ÞP m3.Step 2. If SðtÞP m3 for all t P t1, then the result is obtained. Hence, we need only to consider those solutions which leave

the region X1 ¼ fðSðtÞ; IðtÞÞ 2 R2þ : SðtÞ < m3g and enter it again. Let t� ¼ infft P t1 : SðtÞ < m3g. Since SðtÞ is contin-

uous we have SðtÞP m3 for t 2 ½t1; t�Þ and Sðt�Þ ¼ m3. Suppose t� 2 ½n1s; ðn1 þ 1ÞsÞ;n1 2 N. Choose n2;n3 2 N suchthat n2s > T2 ¼ 1

d ln �1Mþsu ; e

ðn2þ1Þr1sen3r > 1.

Let T ¼ n2sþ n3s. We claim that there exists t2 2 ½ðn1 þ 1Þs; ðn1 þ 1Þsþ T� such that Sðt2ÞP m3. Otherwise,SðtÞ < m3; t 2 ½ðn1 þ 1Þs; ðn1 þ 1Þsþ T�. Considering (4.7) with yððn1 þ 1ÞsþÞ ¼ Iððn1 þ 1ÞsþÞ, we have

yðtÞ ¼ yððn1 þ 1ÞsþÞ � su1� eds

� �edðt�ðn1þ1ÞsÞ þ y�ðtÞ;

t 2 ðns; ðnþ 1Þs�; n1 þ 1 6 n 6 n1 þ 1þ n2 þ n3.Thus,

jyðtÞ � y�ðtÞj < ðM þ suÞen2ds < �1

and

IðtÞ 6 yðtÞ 6 y�ðtÞ þ �1

2870 S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873

for t 2 ½ðn1 þ 1þ n2Þs; ðn1 þ 1Þsþ T�, which implies (4.8) holds for ðn1 þ 1þ n2Þs 6 t 6 ðn1 þ 1Þsþ T. As in Step 1, we have

Sððn1 þ 1þ n2 þ n3ÞsÞP Sððn1 þ 1þ n2ÞsÞen3r:

It follows from the first equation of (1.2) that

S0ðtÞP SðtÞ½rð1�m3Þ �m3bM� ¼ r1SðtÞ:

Integrating it on ½t�; ðn1 þ 1þ n2Þs�, we have

Sððn1 þ 1þ n2ÞsÞP Sðt�Þeðn2þ1Þr1s ¼ m3eðn2þ1Þr1s:

Thus, Sððn1 þ 1þ n2 þ n3ÞsÞP m3eðn2þ1Þr1sen3r > m3, which is a contradiction. Let �t ¼ inf tPt� fSðtÞP m3g. Then Sð�tÞP m3. Fort 2 ½t�;�tÞ, we have SðtÞP Sðt�Þer1ðt�t�Þ P m3er1ð1þn2þn3Þs,m1.

For t > �t, the same arguments can be continued since Sð�tÞP m3. Hence, SðtÞ > m1 for all t P t1. The proof is complete. �

Remark 4.2. By Theorems 4.1 and 4.2, we can see that the susceptible pest could not be eradicated by releasing infectedpests, they present periodic oscillation and coexist. The problem to control the susceptible pest by releasing infected pestswill be demonstrated using numerical simulation.

5. Numerical simulation

In this section, we illustrate how to control the susceptible pest using the release of infected pests by means of numericalsimulation.

We have known that if bS ¼ ffiffiffiffiffiffiffidþx

b

q6 SM , then it is not necessary to control a pest. Therefore, in Sections 3 and 4, we study

the problem to control the pest population under the condition bS > SM . If the pest is not affected by outer perturbations, thenwe have two cases from Section 2.

(1) Let r ¼ 4; b ¼ 1; d ¼ 0:3; x ¼ 0:8. Then r0 ¼ 0:91 < 1, the positive equilibrium of system (2.1) does not exist and allsolutions of system (2.1) tend to E1ð1;0Þ. See Fig. 2a.

(2) Let r ¼ 4; b ¼ 1; d ¼ 0:3; x ¼ 0:6. Then r0 ¼ 1:11 > 1, the positive equilibrium E2 of system (2.1) exists and all solu-tions of system (2.1) tend to E2ð0:95;0:22Þ. See Fig. 2b. It follows from Fig. 2 that the susceptible pest can not becomeextinct. Therefore, we suppress the amount of susceptible pests below SM by releasing infected pests. We assumeSM ¼ 0:6 < bS ¼ 0:95.

Let r ¼ 4; b ¼ 1; d ¼ 0:3; x ¼ 0:6 in system (1.1). Then u ¼ 1:44,u�1 which is the critical value of the control variable u.Under the condition r0 > 1, we control the susceptible pest by u as follows:

(1) If u ¼ 1 < u�1, then S� ¼ 0:65 > SM and all solutions of system (1.1) tend to E�ð0:65;2:12Þ. See Fig. 3a.(2) If u ¼ 2 > u�1, then S� ¼ 0:55 < SM and all solutions of system (1.1) tend to E�ð0:55;3:32Þ. See Fig. 3b.

Thus, we can see that if r0 > 1 in (1.1), then, in order to suppress the amount of susceptible pests below SM , we mustchoose u > u�1.

Let r ¼ 4; b ¼ 1; d ¼ 0:3; x ¼ 0:8 in system (1.1). Then u ¼ 1:97,u�2 which is the critical value of the control variable u.Under the condition r0 < 1, we control the susceptible pest by u as follows:

0

0.05

0.1

0.15

0.2

0.25

0.3

I

0.2 0.4 0.6 0.8 1 1.2S

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

I

0.2 0.4 0.6 0.8 1 1.2S

a b

Fig. 2. Phase portraits of system (2.1). The initial values are (0.3, 0.2), (1.2, 0.3), (0.1, 0.1). (a) r0 ¼ 0:91 < 1 and (b) r0 ¼ 1:11 > 1.

0.5

1

1.5

2I

0.2 0.4 0.6 0.8 1 1.2S

0.5

1

1.5

2

2.5

3

I

0.2 0.4 0.6 0.8 1 1.2S

a b

Fig. 3. Phase portraits of system (1.1). The initial values are (0.3, 0.2), (1.2, 0.3), (0.1, 0.1). (a) r0 > 1; u ¼ 1 < u�1 and (b) r0 > 1; u ¼ 2 > u�1.

S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873 2871

(1) If u ¼ 1 < u�2, then S� ¼ 0:71 > SM and all solutions of system (1.1) approach E�ð0:71;1:66Þ. See Fig. 4a.(2) If u ¼ 3 > u�2, then S� ¼ 0:52 < SM and all solutions of system (1.1) approach E�ð0:52;3:63Þ. See Fig. 4b.

Therefore, if r0 < 1 in (1.1), we must choose u > u�2 to control the susceptible pest below SM .Next, we study system (1.2) by numerical simulation. Here, we also assume SM ¼ 0:6.Let r ¼ 4; b ¼ 1; d ¼ 0:3; x ¼ 0:8; s ¼ 1 and the initial state ð0:3;0:2Þ in (1.2). Then u � 2:7,u�3 which is the critical va-

lue of the control variable u. Under the condition r0 < 1, we control the susceptible pest by u as follows:

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

I

0.2 0.4 0.6 0.8 1S

0.5

1

1.5

2

2.5

3

3.5

I

0.2 0.4 0.6 0.8 1 1.2S

a b

Fig. 4. Phase portraits of system (1.1). The initial values are (0.3, 0.2), (1.2, 0.3), (0.1, 0.1). (a) r0 < 1; u ¼ 1 < u�2 and (b) r0 < 1; u ¼ 3 > u�2.

0.52

0.54

0.56

0.58

S(t)

30 40 50 60 70t

0.52

0.54

0.56

0.58

0.6

S(t)

30 40 50 60 70t

a b

Fig. 5. Time series of SðtÞ in (1.2) with initial value (0.3, 0.2). (a) r0 < 1; u ¼ 2:8 > u�3 and (b) r0 < 1, u ¼ 2:6 < u�3.

2872 S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873

(1) If u ¼ 2:8 > u�3, then there exists T1 > 0 such that SðtÞ 6 0:596 < SM for t > T1 (see Fig. 5a), and the susceptible pest andthe infected pest coexist. See Fig. 6a.

(2) If u ¼ 2:6 < u�3, then there exists T2 > 0 such that SðtÞ 6 0:608; SM < 0:608 for t > T2 (see Fig. 5b), and the susceptiblepest and the infected pest coexist. See Fig. 6b.

Let r ¼ 4; b ¼ 1; d ¼ 0:3; x ¼ 0:6; s ¼ 1 and the initial state (0.3, 0.2) in (1.2). Then u � 1:85,u�4 which is the criticalvalue of the control variable u. Under the condition r0 > 1, we control the susceptible pest by u as follows:

2.5

3

3.5

4

4.5

5

I

0.52 0.54 0.56 0.58S

2.5

3

3.5

4

4.5

I0.52 0.54 0.56 0.58 0.6

S

a b

Fig. 6. Phase portraits in (1.2) with initial value (0.3, 0.2). (a) r0 < 1; u ¼ 2:8 > u�3 and (b) r0 < 1, u ¼ 2:6 < u�3.

0.53

0.54

0.55

0.56

0.57

0.58

S(t)

30 40 50 60 70t

0.56

0.57

0.58

0.59

0.6

0.61

S(t)

30 40 50 60 70t

a b

Fig. 7. Time series of SðtÞ in (1.2) with initial value (0.3, 0.2). (a) r0 > 1; u ¼ 2 > u�4 and (b) r0 > 1, u ¼ 1:7 < u�4.

2.5

3

3.5

4

I

0.53 0.54 0.55 0.56 0.57 0.58S

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

I

0.56 0.57 0.58 0.59 0.6 0.61S

a b

Fig. 8. Phase portraits in (1.2) with initial value (0.3, 0.2). (a) r0 > 1; u ¼ 2 > u�4 and (b) r0 > 1, u ¼ 1:7 < u�4.

S. Sun, L. Chen / Applied Mathematical Modelling 33 (2009) 2864–2873 2873

(1) If u ¼ 2 > u�4, then there exists T3 > 0 such that SðtÞ 6 0:588 < SM for t > T3 (see Fig. 7a), and the susceptible pest andthe infected pest coexist. See Fig. 8a.

(2) If u ¼ 1:7 < u�4, then there exists T4 > 0 such that SðtÞ 6 0:614; SM < 0:614 for t > T4 (see Fig. 7b), and the susceptiblepest and the infected pest coexist. See Fig. 8b.

6. Conclusion

In this paper, we formulate and investigate the pest control models with discrete and continuous processes in accordancewith the mathematical theory of epidemiology. The important results of the systems are given. By means of numerical sim-ulation, we obtain the thresholds of the control variable u for different release methods of infected pests. Thus, we provide amathematical evidence in the management of an epidemic controlling a pest.

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