Research ArticleThreshold Dynamics of a Huanglongbing Model withLogistic Growth in Periodic Environments

Jianping Wang Shujing Gao Yueli Luo and Dehui Xie

Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques Gannan Normal UniversityGanzhou 341000 China

Correspondence should be addressed to Shujing Gao gaosjmathtomcom

Received 10 January 2014 Accepted 10 February 2014 Published 20 March 2014

Academic Editor Kaifa Wang

Copyright copy 2014 Jianping Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection A newmodel about HLBtransmissionwith Logistic growth in psyllid insect vectors and periodic coefficients has been investigated It is shown that the globaldynamics are determined by the basic reproduction number 119877

0which is defined through the spectral radius of a linear integral

operator If 1198770lt 1 then the disease-free periodic solution is globally asymptotically stable and if 119877

0gt 1 then the disease persists

Numerical values of parameters of the model are evaluated taken from the literatures Furthermore numerical simulations supportour analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitudevalues of the recruitment function of citrus are shown Finally some useful comments on controlling the transmission of HLB aregiven

1 Introduction

Plant disease is an important constraint to crop productionDue to plant diseases more than 10 of global food pro-duction is lost and 800 million people do not have adequatefood in the world [1ndash3] Plant pathologists cannot ignorethe juxtaposition of these figures for food shortage and thereduction of crops caused by plant disease

Nowadays Huanglongbing (HLB) which is a century olddisease caused by the bacteria Candidatus Liberibacter sppis one of the most serious problems of citrus worldwide[4] HLB has been responsible for the near destruction ofcitrus industries in Asia and Africa [4] The main symptomson HLB-infected citrus trees are yellow shoots leaves withblotchy mottle and small lopsided fruits [4 5] The HLB isa phloem-restricted noncultured Gram-negative bacteriumcausing crippling diseases denoting ldquogreeningrdquo in SouthAfrica ldquomottle leaf rdquo in the Philippines ldquodiebackrdquo in Indiaand ldquovein phloem degenerationrdquo in Indonesia The infectedcitrus orchards are usually destroyed or becomeunproductivein 5 to 8 years [4]

Most of the known plant viruses are transmitted byinsect vectors and entirely dependent on the behaviour and

dispersal capacity of their vectors to spread from plant toplant HLB a destructive disease of citrus can be transmittedby grafting from citrus to citrus and by dodder to periwinkleThe citrus psyllid (Diaphorina Citri Kuwayama) is naturaland mainly vector [4] In this paper we mainly consider thatHLB transmitted from tree to tree by Asian citrus psyllidinsect vectors

Mathematical models play an important role in under-standing the epidemiology of vector-transmitted plant dis-eases Since the introduction of HLB a lot of researcheshave been conducted on the epidemiology of the disease andon the vector but the result of these two lines of inquiryintegrated is very fewAnalyticalmodels have also beendevel-oped for the spread of citrus canker [6] butmodels for vector-transmitted bacterial pathogens are still preliminary [7] In[8] the authors proposed a deterministic compartmentalmathematic model to analyze HLB spread between citrusplants They assumed that all coefficients of the model areconstant (autonomous systems) However in the real worldactual data and evidence show that dynamics of diseasetransmission are not as simple as shown in the model In[9] Hall and Hentz have studied seasonal activity of psyllidinsect vectors which is correlated with humidity Seasonal

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 841367 10 pageshttpdxdoiorg1011552014841367

2 Abstract and Applied Analysis

fluctuations in the transmission of infectious diseases implythat the corresponding mathematical models may admitperiodic solutions It is interesting and important to studythe globally dynamics which are determined by thresholdparameter 119877

0in periodic epidemiological models

Based on above introduction we propose a model withperiodic transmission rates to investigate the seasonal HLBepidemics [10 11] In this model we consider Logistic growthterm for dynamics of susceptible psyllid vector Furthermorewe assumed that the infective citrus population is generatedthrough susceptible citrus which was bit by infective psyllidand the susceptible psyllid bit the infective citrus which willbecome infective psyllid Then the periodic system is asfollows

119889119878ℎ (119905)

119889119905= Λ (119905) minus 1205731 (119905) 119878ℎ (119905) 119868V (119905) minus 1205831 (119905) 119878ℎ (119905)

119889119868ℎ (119905)

119889119905= 1205731 (119905) 119878ℎ (119905) 119868V (119905) minus 1205831 (119905) 119868ℎ (119905) minus 119889 (119905) 119868ℎ (119905)

119889119878V (119905)

119889119905= 119887 (119905) (119878V (119905) + 119868V (119905)) [1 minus

119878V (119905) + 119868V (119905)

119898 (119878ℎ (119905) + 119868ℎ (119905))

]

minus 1205732(119905) 119878V (119905) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) 119878V (119905) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(1)

with initial condition

119878ℎ(0) gt 0 119868

ℎ(0) gt 0 119878V (0) gt 0 119868V (0) gt 0

(2)

Here 119878ℎ(119905) 119868ℎ(119905) 119878V(119905) and 119868V(119905) represent susceptible citrus

host infected citrus host susceptible psyllid and infectedpsyllid respectively We can easily see that 119873

ℎ(119905) = 119878

ℎ(119905) +

119868ℎ(119905) and 119873V(119905) = 119878V(119905) + 119868V(119905) are the number of citrus

population and psyllid population respectively Λ(119905) is therecruitment rate of citrus at time 119905 120573

1(119905) is the infected rate of

citrus host at time 119905 1205831(119905) and 119889(119905) are the nature death and

disease induced death rate of citrus host at time 119905 respectively119887(119905) is the intrinsic growth rate of psyllid at time 119905 120573

2(119905) and

1205832(119905) are the infected rate and the nature death rate of psyllid

at time 119905 respectively and119898(gt 0) is themaximumabundanceof psyllid per citrus Λ(119905) 120573

1(119905) 1205831(119905) 119889(119905) 119887(119905) 120573

2(119905) and

1205832(119905) are continuous positive 120596-periodic functionsThe paper is organized as follows In the next section

we give the basic reproduction number of (1) In Sections 3and 4 the results show that the dynamical properties of themodel are completely determined by 119877

0 That is if 119877

0lt 1

the disease-free periodic solution is globally asymptoticallystable and if 119877

0gt 1 the model is permanence In Section 5

we present numerical simulations which demonstrate thetheoretical analysis and a brief discussion of ourmain results

2 Basic Reproduction Number

In the following we introduce some notations and lemmaswhich will be used for our further argument

Let (119877119896 119877119896+) be the standard ordered 119896-dimensional

Euclidean space with a norm sdot For 119906 V isin 119877119896 we denote

119906 ge V if 119906 minus V isin 119877119896+ 119906 gt V if 119906 minus V isin 119877119896

+ 0 and 119906 ≫ V if

119906 minus V isin Int(119877119896+) respectively

Define 119892119871 = max119905isin[0120596)

119892(119905) and 119892119872 = min119905isin[0120596)

119892(119905)where 119892(119905) is a continuous positive 120596-periodic function

Consider the following linear ordinary differential sys-tem

119889119909 (119905)

119889119905= 119860 (119905) 119909 (119905) (3)

where 119860(119905) is a continuous cooperative irreducible and120596-periodic 119896 times 119896 matrix function Denote Φ

119860(119905) be the

fundamental solution matrix of (3) and 119903(Φ119860(120596)) be the

spectral radius ofΦ119860(120596) By the Perron-Frobenius Theorem

we know that 119903(Φ119860(120596)) is the principle eigenvalue of Φ

119860(120596)

that is it is simple and admits an eigenvector Vlowast ≫ 0

Lemma 1 (see [12]) Let 119901 = (1120596) ln 119903(Φ119860(sdot)(120596)) Then there

exists a positive 120596-periodic function V(119905) such that exp(119901119905)V(119905)is a solution of (3)

Consider the following nonautonomous linear equation119889119878ℎ (119905)

119889119905= Λ (119905) minus 1205831 (119905) 119878ℎ (119905) (4)

whereΛ(119905) and 1205831(119905) are the same as in System (1) FromZhang

and Teng ([13 Lemma 21]) and simple calculation we have thefollowing lemma

Lemma 2 System (4) has a unique positive 120596-periodic solu-tion 119878lowast

ℎ(119905) which is globally asymptotically stable

Consider the following nonautonomous Logistic equation119889119878V (119905)

119889119905= 119887 (119905) 119878V (119905) (1 minus

119878V (119905)

119898119878ℎ(119905)) (5)

where 119887(119905) and119898 are the same as in system (1) From Teng andLi ([14 Lemma 2]) and simple calculation we can obtain thefollowing lemma

Lemma3 System (5) has a unique positive120596-periodic solution119878lowast

V (119905) which is globally asymptotically stable where 119878lowastV (119905) =

119898119878lowast

ℎ(119905)

According to Lemmas 2 and 3 it is easy to see that (1) hasa unique disease-free periodic solution (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0)Now we use the generation operator approach (see [15])

to derive the basic reproduction number Applying the sym-bol of the theory in Wang and Zhao [15] for system (1) wehave

F (119905 119909) = (

1205731(119905) 119878ℎ(119905) 119868V (119905)

1205732(119905) 119878V (119905) 119868ℎ (119905)

0

0

)

Abstract and Applied Analysis 3

V+(119905 119909) = (

0

0

Λ (119905)

119887 (119905) (119878V (119905) + 119868V (119905))

)

Vminus(119905 119909) = (

(1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

1205832 (119905) 119868V (119905)

1205831 (119905) 119878ℎ + 1205731 (119905) 119878ℎ (119905) 119868V (119905)

119887 (119905)(119878V (119905) + 119868V (119905))

2

119898(119878ℎ (119905) + 119868ℎ (119905))

+1205732(119905) 119878V (119905) 119868ℎ (119905)

)

(6)

where 119909 = (119868ℎ(119905) 119868V(119905) 119878ℎ(119905) 119878V(119905))

119879 Then System (1) can bewritten as the following form

119889119909 (119905)

119889119905= F (119905 119909 (119905)) minusV (119905 119909 (119905)) (7)

whereV(119905 119909(119905)) =Vminus(119905 119909(119905)) minusV+(119905 119909(119905))It is easy to obtain that the conditions (A1)ndash(A5) in [15]

hold In the following we will check the conditions (A6) and(A7) in [15]

We know that 119909lowast(119905) = (0 0 119878lowast

ℎ(119905) 119878lowast

V (119905)) is the disease-free periodic solution of system (7) Denote

119891 (119905 119909 (119905)) = F (119905 119909 (119905)) minusV (119905 119909 (119905))

119872 (119905) = (120597119891119894(119905 119909lowast(119905))

120597119909119895

)

3le119894119895le4

(8)

where 119891119894(119905 119909(119905)) and 119909

119894are the 119894th components of 119891(119905 119909(119905))

and 119909 respectively According to (6) we have

119872(119905) = (

minus1205831(119905) 0

119887 (119905) 119878lowast2

V (119905)

119898119878lowast2

ℎ(119905)

minus119887 (119905)) (9)

It is easy to see that 119903(Φ119872(120596)) lt 1 where 119903(Φ

119872(120596))

is the spectral radius of Φ119872(120596) This implies that 119909lowast(119905) is

linearly asymptotically stable in the disease-free subspace119883119878= (0 0 119878

ℎ 119878V) isin 119877

4

+ Thus condition (A6) in [15] holds

We further define

119865 (119905) = (120597F119894(119905 119909lowast(119905))

120597119909119895

)

1le119894119895le2

119881 (119905) = (120597V119894(119905 119909lowast(119905))

120597119909119895

)

1le119894119895le2

(10)

whereF119894(119905 119909) andV

119894(119905 119909) are the 119894th components ofF(119905 119909)

andV(119905 119909) respectively Then from (6) we obtain that

119865 (119905) = (0 120573

1(119905) 119878lowast

ℎ(119905)

1205732(119905) 119878lowast

V (119905) 0)

119881 (119905) = (1205831(119905) + 119889 (119905) 0

0 1205832(119905))

(11)

Let 119884(119905 119904) be a 2 times 2matrix solution of the system

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904

119884 (119904 119904) = 119868

(12)

where 119868 is 2 times 2 identity matrix From (11) and (12) we have119903(Φminus119881(120596)) lt 1 Therefore the condition (A7) in [15] also

holdsLet 119862

120596be the ordered Banach space of all 120596-periodic

function from 119877 rarr 1198772 which is equipped with maximum

norm sdot infin

and the positive cone 119862+120596= 120601 isin 119862

120596 120601(119905) ge 0

for all 119905 isin 119877 Define the following linear operator 119871 119862120596rarr

119862120596by

(119871120601) (119905) = int

+infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886

forall119905 isin 119877 120601 isin 119862120596

(13)

Based on the assumptions above and the results of Wang andZhang [15] we can derive the basic reproduction number 119877

0

of system (1) as follows

1198770= 119903 (119871) (14)

and obtain the following conclusion

Theorem 4 For system (1) the following statements are valid

(i) 1198770= 1 if and only if 119903(Φ

119865minus119881(120596)) = 1

(ii) 1198770gt 1 if and only if 119903(Φ

119865minus119881(120596)) gt 1

(iii) 1198770lt 1 if and only if 119903(Φ

119865minus119881(120596)) lt 1

where 119865(119905) and 119881(119905) are defined in (11)

It follows from Theorem 4 that the disease-free periodicsolution (119878

lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) of system (1) is asymptoticallystable if 119877

0lt 1 and it is unstable if 119877

0gt 1

In order to calculate 1198770 we consider the following linear

120596-periodic system

119889119908

119889119905= (minus119881 (119905) +

1

120582119865 (119905))119908 120582 isin (0infin) (15)

Let 119882(119905 119904 120582) 119905 ⩾ 119904 119904 isin 119877 be the evolution operator ofthe System (15) on 1198772 Since 119865(119905) is nonnegative and minus119881(119905)is cooperative then 119903(119882(120596 0 120582)) is continuous and nonin-creasing for 120582 isin (0infin) and lim

120582rarrinfin119903(119882(120596 0 120582)) lt 1

Thus we have the following result which will be used in ournumerical calculation of the basic reproduction ratio 119877

0in

Section 5

Lemma 5 (see [15]) The following statements are valid

(i) If 119903(119882(120596 0 120582)) = 1 has a positive solution 1205820is an

eigenvalue of 119871 and hence 1198770gt 0

(ii) If 1198770gt 0 then 120582 = 119877

0is the unique solution of 119903(119882(120596

0 120582)) = 1(iii) 119877

0= 0 if and only if 119903(119882(120596 0 120582)) lt 1 for all 120582 gt 0

4 Abstract and Applied Analysis

3 Global Stability of Disease-FreePeriodic Solution

In this section we will prove the global asymptotical stabilityof the disease-free periodic solution (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0)Let119873

ℎ(119905) = 119878

ℎ(119905) + 119868

ℎ(119905)119873V(119905) = 119878V(119905) + 119868V(119905) Denote

Ω = (119878ℎ 119868ℎ 119878V 119868V) isin 119877

4

+| 0 le 119878

ℎ+ 119868ℎle 1198731lt +infin

0 le 119878V + 119868V le 1198732 lt +infin

(16)

where 1198731= Λ119871120583119872

1and 119873

2= 119898119873

1 Similar to [16 17] we

firstly prove the following lemmas

Lemma 6 Ω is a positively invariant set for (1)

Proof From the equations in (1) we have

119889119873ℎ(119905)

119889119905= Λ (119905) minus 120583

1(119905)119873ℎ(119905)

le Λ119871minus 120583119872

1119873ℎ(119905)

le 0 if 119873ℎ (119905) ge 1198731

119889119873V (119905)

119889119905= 119887 (119905)119873V (1 minus

119873V

119898119873ℎ

) minus 1205832(119905) 119868V (119905)

le 119887 (119905)119873V (119905) (1 minus119873V (119905)

119898119873ℎ

)

le 0 if 119873V (119905) ge 1198732

(17)

which implies that Ω is a positive invariant compact set for(1) The proof is completed

Lemma 7 Let (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) be any solution of

system (1) It holds that

lim119905rarr+infin

(119873ℎ (119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119873V (119905) minus 119878lowast

V (119905)) = 0(18)

where 119878lowastℎ(119905) 119878lowastV (119905) are defined in Lemmas 2 and 3 respectively

Proof We denote that 1199101(119905) = 119873

ℎ(119905) minus 119878

lowast

ℎ(119905) It follows from

the first equation of (17) that 1198891199101(119905)119889119905 le minus120583

1(119905)1199101(119905) which

implies that lim119905rarr+infin

1199101(119905) = lim

119905rarr+infin(119873ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

Further from Lemma 6 we obtain that for any 120576 gt 0 thereexists a 119879 gt 0 such that

119878lowast

ℎ(119905) minus 120576 le 119873ℎ (119905) le 119878

lowast

ℎ(119905) + 120576 119873V (119905) lt 1198732 forall119905 ge 119879

(19)

Let 1199102(119905) = 119873V(119905) minus 119878

lowast

V (119905) From the second equation of(17) and (19) we get

1198891199102(119905)

119889119905= 119887 (119905)119873V (119905) [1 minus

119873V (119905)

119898119873ℎ (119905)

] minus 1205832(119905) 119868V (119905)

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

le 119887 (119905)119873V (119905) [1 minus119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

]

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

= 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ(119905) + 120576

]

minus 119887 (119905) (119873lowast

V (119905) minus 119878lowast

V (119905))119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ(119905) + 120576)

= minus 119887 (119905)119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

1199102(119905) + Δ (120576)

(20)

for all 119905 gt 119879 where

Δ (120576) = 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ+ 120576

]

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ+ 120576)

(21)

Obviously lim120576rarr0

Δ(120576) = 0 Because 120576 is arbitrarily smallthen lim

119905rarr+infin1199102(119905) = lim

119905rarr+infin(119873V(119905) minus 119878

lowast

V (119905)) = 0 Hencethe proof is completed

Theorem 8 The disease-free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905)

0) is globally asymptotically stable if 1198770lt 1 whereas it is

unstable if 1198770gt 1

Proof From Theorem 4 we have that (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isunstable if 119877

0gt 1 and (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) is locally stable if1198770lt 1Therefore we only need to show the global attractivity

of (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) for 1198770 lt 1Since 119877

0lt 1 by Theorem 4 we can choose 120598

1gt 0

sufficiently small such that

119903 (Φ119865minus119881+119872

1205761

(120596)) lt 1 (22)

where

1198721205981

(119905) = (0 1205981

12059810) (23)

From Lemma 6 and (18) we have that for above men-tioned 120598

1gt 0 there exists a119879

1gt 0 such that 119878

ℎ(119905) le 119878

lowast

ℎ(119905)+1205981

119878V(119905) le 119878lowast

V (119905) + 1205981 for 119905 gt 1198791 It follows from the second and

fourth equations that for 119905 gt 1198791

119889119868ℎ(119905)

119889119905le 1205731(119905) (119878lowast

ℎ(119905) + 120598

1) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905le 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(24)

Abstract and Applied Analysis 5

Consider the following comparison system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) + 1205981) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(25)

In view of Lemma 1 we know that there exists a positive 120596-periodic function V

1(119905) such that 119869(119905) le V

1(119905) exp(119901

1119905) where

119869(119905) = (119868ℎ(119905) 119868V(119905))

119879 and 1199011= (1120596) ln 119903(Φ

119865minus119881+119872120598

(120596)) lt

0 It follows from (22) that lim119905rarr+infin

119868ℎ(119905) = 0 and

lim119905rarr+infin

119868V(119905) = 0 By the comparison of theorem [18] wehave lim

119905rarr+infin119868ℎ(119905) = 0 and lim

119905rarr+infin119868V(119905) = 0 From (18)

we have

lim119905rarr+infin

(119878ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119878V (119905) minus 119878lowast

V (119905)) = 0(26)

Hence the disease free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isglobally attractive This completes the proof

4 Permanence

In this section we show that if 1198770gt 1 then the disease

persistsFirstly we define 119883 = (119878

ℎ 119868ℎ 119878V 119868V) isin 119877

4

+1198830= (119878ℎ 119868ℎ

119878V 119868V) isin 119883 119878ℎge 0 119868

ℎgt 0 119878V ge 0 119868V gt 0 and 120597119883

0=

1198831198830 andwedenote119906(119905 119909

0) as the unique solution of System

(1) with the initial value 1199090= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )Define Poincare map 119875 119883 rarr 119883 associated with System

(1) as follows

119875 (1199090) = 119906 (120596 119909

0) forall119909

0isin 119883 (27)

By Lemma 6 it is easy to see that both119883 and1198830are positively

invariant and 119875 is point dissipative Set

119872120597= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830 | 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830

119898 isin 119885+

(28)

where 119885+ = 0 1 2 We claim that

119872120597= (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 (29)

Obviously 119872120597supe (119878

ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 Next we

want to show 119872120597 (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 = 0 If it

does not hold then there exists a point (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 119872120597

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0

Case 1 1198680ℎ= 0 and 1198680V gt 0 It is obvious that 119868V(119905) gt 0 and

119878ℎ(119905) gt 0 for any 119905 gt 0 Then from the second equation

of System (1) 119889119868ℎ(119905)119889119905|

119905=0= 1205731(0)119878ℎ(0)119868V(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

Case 2 1198680ℎgt 0 and 1198680V = 0 It is obvious that 119868

ℎ(119905) gt 0 and

119878V(119905) gt 0 for any 119905 gt 0 Then from the fourth equationof System (1) 119889119868V(119905)119889119905|119905=0 = 120573

2(0)119878V(0)119868ℎ(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

That is to say for any (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) notin (119878ℎ 0 119878V 0) 119878ℎ ge

0 119878V ge 0 then (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) notin 119872120597Therefore we have119872120597=

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0Next we present the following result of the uniform

persistence of the disease

Theorem 9 Suppose 1198770gt 1 Then there is a positive constant

120598 gt 0 such that each positive solution (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905))

of System (1) satisfies

lim inf119905rarr+infin

119868ℎ(119905) ge 120598 lim inf

119905rarr+infin

119868V (119905) ge 120598 (30)

Proof By Theorem 4 we obtain 119903(Φ119865minus119881

(120596)) gt 1 So we canchoose 120578 gt 0 small enough such that 119903(Φ

119865minus119881minus119872120578

) gt 1 where

119872120578= (

0 120578

120578 0) (31)

Put 1198750= 119878lowast

ℎ(0) 0 119878

lowast

V (0) 0 Now we proceed by contra-diction to prove that

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) ge 120575 (32)

If it does not hold then

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 (33)

for some (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1198830 Without loss of generality

suppose that

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 forall119898 isin 119885+ (34)

By the continuity of the solutions with respect to the initialvalues we obtain

10038171003817100381710038171003817119906 (119905 119875

119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817le 120578

forall119905 isin [0 120596] forall119898 isin 119885+

(35)

For any 119905 ge 0 there exists a 119898 isin 119885+such that 119905 = 119898120596 + 119905

1

where 1199051isin [0 120596] Then we have

10038171003817100381710038171003817119906 (119905 (119878

0

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817

=10038171003817100381710038171003817119906 (1199051 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (1199051 1198750)10038171003817100381710038171003817le 120578

(36)

for all 119905 ge 0 which implies that 119878lowastℎ(119905) minus 120578 le 119878

ℎ(119905) le 119878

lowast

ℎ(119905) + 120578

119878lowast

V (119905) minus 120578 le 119878V(119905) le 119878lowast

V (119905) + 120578 Then from (1) we have

119889119868ℎ(119905)

119889119905ge 1205731(119905) (119878lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905ge 1205732(119905) (119878lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(37)

6 Abstract and Applied Analysis

Table 1 Parameter definitions and values used for numerical simulations of the Huanglongbing model

Parameter Definition Average value Unit ReferenceΛ The recruitment rate of citrus mdash monthminus1 Estimate1205731

Infected rate of citrus mdash monthminus1 Estimate1205831

Nature death rate of citrus 000275ndash0004167 monthminus1 [20]119889 Disease induced death rate of citrus 0016667ndash0027775 monthminus1 [21]119863 Birth rate of psyllid 378327ndash33526137 monthminus1 [20 22]1205732

Infected rate of psyllid mdash monthminus1 Estimate1205832

Nature death rate of psyllid 01169825ndash095052 monthminus1 [23]119898 Max abundance of psyllid per citrus 120ndash1000 mdash [24]

Table 2 Parameter functions for model (1) according to the values of Table 1

Parameter functions Value Reference

1205731(119905) 00042925 + 0003543 cos (212058711990512) Estimate

1205831(119905) 00034585 + 00007085 cos (212058711990512) [20]

119889 (119905) 0022221 + 0005554 cos (212058711990512) [21]

119863 (119905) 186547035 + 148714335 cos (212058711990512) [20 22]

1205732(119905) 0008779171 + 0004838437 cos (212058711990512) Estimate

1205832(119905) 053375125 + 041676875 cos (212058711990512) [23]

119887 (119905) = 119863 (119905) minus 1205832(119905) 18120952 + 1445466475 cos (212058711990512) [20 22 23]

119898 560 [24]

Consider the linear system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732 (119905) (119878

lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(38)

By Lemma 1 and the standard comparison principle we havethat there exists a positive120596-periodic function V

2(119905) such that

119869(119905) = exp(1199012119905)V2(119905) is a solution of System (38) where 119869(119905) =

(119868ℎ(119905) 119868V(119905))

119879 and

1199012=1

120596ln 119903 (Φ

119865minus119881minus119872120578

(120596)) (39)

It follows from 119903(Φ119865minus119881minus119872

120578

(120596)) gt 1 that 1199012gt 0 and 119869(119905) rarr

+infin as 119905 rarr +infin Applying the comparison principle [18] weknow that 119868

ℎ(119905) rarr +infin and 119868V(119905) rarr +infin as 119905 rarr +infin This

is a contradictionThus we have proved that (32) holds and119875is weakly uniformly persistent with respect to (119883

0 1205971198830)

According to the results of Lemma 7 we can easily obtainthat119875 has a global attractor119875

0 It is easy to obtain that119875

0is an

isolated invariant set in119883 and119882119904(1198750)cap1198830= 0We know that

1198750is acyclic in119872

120597and every solution in119872

120597converges to 119875

0

According to Zhao [19] we have that119875 is uniformly persistentwith respect to (119883

0 1205971198830) This implies that the solution of

(1) is uniformly persistent with respect to (1198830 1205971198830) Thus we

have that there exists a 120598 gt 0 such that lim inf119905rarr+infin

119868ℎ(119905) ge 120598

lim inf119905rarr+infin

119868V(119905) ge 120598

5 Numerical Simulationsand Sensitivity Analysis

In this section we will make numerical simulations bymeansof Matlab 71 to support our theoretical results to predictthe trend of the disease and to explore some control andprevention measures Numerical values of parameters ofsystem (1) are given in Table 1 (most of the data are taken fromthe literatures ([20ndash24]))

According to the periodicity of System (1) and Table 1 weset 1205831(119905) = 120572

0

1+ 1205720

2cos(212058711990512) where 1205720

2= (0004167 minus

000275)2 = 00007085 and 12057201= 000275 + 120572

0

2= 00034585

By the similar method we can obtain the other parameterfunctions of model (1) (see Table 2) For the simulations thatfollows we apply the parameters in Table 2 unless otherwisestated

ChooseΛ(119905) = 000265+000235 cos(212058711990512)Then fromLemma 5 we can compute 119877

0= 09844 lt 1 by means of

Matlab 71 FromTheorem 8 we obtain that the infected citruspopulation 119868

ℎ(119905) and the infected psyllid population 119868V(119905) of

system (1) are extinct (see Figures 1 and 2)Choose Λ(119905) = 0005 + 00035 cos(212058711990512) Then from

Lemma 5 we obtain that 1198770= 18342 gt 1 From Theorem 9

we have that the infected citrus population 119868ℎ(119905) and the

infected psyllid population 119868V(119905) of System (1) are permanence(see Figures 3 and 4)

From the formulae for 1198770 we can predict the general

tendency of the epidemic in a long term according to thecurrent situation which is presented in Figures 1 2 3 and 4From the first two figures we know that the epidemic of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

fluctuations in the transmission of infectious diseases implythat the corresponding mathematical models may admitperiodic solutions It is interesting and important to studythe globally dynamics which are determined by thresholdparameter 119877

0in periodic epidemiological models

Based on above introduction we propose a model withperiodic transmission rates to investigate the seasonal HLBepidemics [10 11] In this model we consider Logistic growthterm for dynamics of susceptible psyllid vector Furthermorewe assumed that the infective citrus population is generatedthrough susceptible citrus which was bit by infective psyllidand the susceptible psyllid bit the infective citrus which willbecome infective psyllid Then the periodic system is asfollows

119889119878ℎ (119905)

119889119905= Λ (119905) minus 1205731 (119905) 119878ℎ (119905) 119868V (119905) minus 1205831 (119905) 119878ℎ (119905)

119889119868ℎ (119905)

119889119905= 1205731 (119905) 119878ℎ (119905) 119868V (119905) minus 1205831 (119905) 119868ℎ (119905) minus 119889 (119905) 119868ℎ (119905)

119889119878V (119905)

119889119905= 119887 (119905) (119878V (119905) + 119868V (119905)) [1 minus

119878V (119905) + 119868V (119905)

119898 (119878ℎ (119905) + 119868ℎ (119905))

]

minus 1205732(119905) 119878V (119905) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) 119878V (119905) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(1)

with initial condition

119878ℎ(0) gt 0 119868

ℎ(0) gt 0 119878V (0) gt 0 119868V (0) gt 0

(2)

Here 119878ℎ(119905) 119868ℎ(119905) 119878V(119905) and 119868V(119905) represent susceptible citrus

host infected citrus host susceptible psyllid and infectedpsyllid respectively We can easily see that 119873

ℎ(119905) = 119878

ℎ(119905) +

119868ℎ(119905) and 119873V(119905) = 119878V(119905) + 119868V(119905) are the number of citrus

population and psyllid population respectively Λ(119905) is therecruitment rate of citrus at time 119905 120573

1(119905) is the infected rate of

citrus host at time 119905 1205831(119905) and 119889(119905) are the nature death and

disease induced death rate of citrus host at time 119905 respectively119887(119905) is the intrinsic growth rate of psyllid at time 119905 120573

2(119905) and

1205832(119905) are the infected rate and the nature death rate of psyllid

at time 119905 respectively and119898(gt 0) is themaximumabundanceof psyllid per citrus Λ(119905) 120573

1(119905) 1205831(119905) 119889(119905) 119887(119905) 120573

2(119905) and

1205832(119905) are continuous positive 120596-periodic functionsThe paper is organized as follows In the next section

we give the basic reproduction number of (1) In Sections 3and 4 the results show that the dynamical properties of themodel are completely determined by 119877

0 That is if 119877

0lt 1

the disease-free periodic solution is globally asymptoticallystable and if 119877

0gt 1 the model is permanence In Section 5

we present numerical simulations which demonstrate thetheoretical analysis and a brief discussion of ourmain results

2 Basic Reproduction Number

In the following we introduce some notations and lemmaswhich will be used for our further argument

Let (119877119896 119877119896+) be the standard ordered 119896-dimensional

Euclidean space with a norm sdot For 119906 V isin 119877119896 we denote

119906 ge V if 119906 minus V isin 119877119896+ 119906 gt V if 119906 minus V isin 119877119896

+ 0 and 119906 ≫ V if

119906 minus V isin Int(119877119896+) respectively

Define 119892119871 = max119905isin[0120596)

119892(119905) and 119892119872 = min119905isin[0120596)

119892(119905)where 119892(119905) is a continuous positive 120596-periodic function

Consider the following linear ordinary differential sys-tem

119889119909 (119905)

119889119905= 119860 (119905) 119909 (119905) (3)

where 119860(119905) is a continuous cooperative irreducible and120596-periodic 119896 times 119896 matrix function Denote Φ

119860(119905) be the

fundamental solution matrix of (3) and 119903(Φ119860(120596)) be the

spectral radius ofΦ119860(120596) By the Perron-Frobenius Theorem

we know that 119903(Φ119860(120596)) is the principle eigenvalue of Φ

119860(120596)

that is it is simple and admits an eigenvector Vlowast ≫ 0

Lemma 1 (see [12]) Let 119901 = (1120596) ln 119903(Φ119860(sdot)(120596)) Then there

exists a positive 120596-periodic function V(119905) such that exp(119901119905)V(119905)is a solution of (3)

Consider the following nonautonomous linear equation119889119878ℎ (119905)

119889119905= Λ (119905) minus 1205831 (119905) 119878ℎ (119905) (4)

whereΛ(119905) and 1205831(119905) are the same as in System (1) FromZhang

and Teng ([13 Lemma 21]) and simple calculation we have thefollowing lemma

Lemma 2 System (4) has a unique positive 120596-periodic solu-tion 119878lowast

ℎ(119905) which is globally asymptotically stable

Consider the following nonautonomous Logistic equation119889119878V (119905)

119889119905= 119887 (119905) 119878V (119905) (1 minus

119878V (119905)

119898119878ℎ(119905)) (5)

where 119887(119905) and119898 are the same as in system (1) From Teng andLi ([14 Lemma 2]) and simple calculation we can obtain thefollowing lemma

Lemma3 System (5) has a unique positive120596-periodic solution119878lowast

V (119905) which is globally asymptotically stable where 119878lowastV (119905) =

119898119878lowast

ℎ(119905)

According to Lemmas 2 and 3 it is easy to see that (1) hasa unique disease-free periodic solution (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0)Now we use the generation operator approach (see [15])

to derive the basic reproduction number Applying the sym-bol of the theory in Wang and Zhao [15] for system (1) wehave

F (119905 119909) = (

1205731(119905) 119878ℎ(119905) 119868V (119905)

1205732(119905) 119878V (119905) 119868ℎ (119905)

0

0

)

Abstract and Applied Analysis 3

V+(119905 119909) = (

0

0

Λ (119905)

119887 (119905) (119878V (119905) + 119868V (119905))

)

Vminus(119905 119909) = (

(1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

1205832 (119905) 119868V (119905)

1205831 (119905) 119878ℎ + 1205731 (119905) 119878ℎ (119905) 119868V (119905)

119887 (119905)(119878V (119905) + 119868V (119905))

2

119898(119878ℎ (119905) + 119868ℎ (119905))

+1205732(119905) 119878V (119905) 119868ℎ (119905)

)

(6)

where 119909 = (119868ℎ(119905) 119868V(119905) 119878ℎ(119905) 119878V(119905))

119879 Then System (1) can bewritten as the following form

119889119909 (119905)

119889119905= F (119905 119909 (119905)) minusV (119905 119909 (119905)) (7)

whereV(119905 119909(119905)) =Vminus(119905 119909(119905)) minusV+(119905 119909(119905))It is easy to obtain that the conditions (A1)ndash(A5) in [15]

hold In the following we will check the conditions (A6) and(A7) in [15]

We know that 119909lowast(119905) = (0 0 119878lowast

ℎ(119905) 119878lowast

V (119905)) is the disease-free periodic solution of system (7) Denote

119891 (119905 119909 (119905)) = F (119905 119909 (119905)) minusV (119905 119909 (119905))

119872 (119905) = (120597119891119894(119905 119909lowast(119905))

120597119909119895

)

3le119894119895le4

(8)

where 119891119894(119905 119909(119905)) and 119909

119894are the 119894th components of 119891(119905 119909(119905))

and 119909 respectively According to (6) we have

119872(119905) = (

minus1205831(119905) 0

119887 (119905) 119878lowast2

V (119905)

119898119878lowast2

ℎ(119905)

minus119887 (119905)) (9)

It is easy to see that 119903(Φ119872(120596)) lt 1 where 119903(Φ

119872(120596))

is the spectral radius of Φ119872(120596) This implies that 119909lowast(119905) is

linearly asymptotically stable in the disease-free subspace119883119878= (0 0 119878

ℎ 119878V) isin 119877

4

+ Thus condition (A6) in [15] holds

We further define

119865 (119905) = (120597F119894(119905 119909lowast(119905))

120597119909119895

)

1le119894119895le2

119881 (119905) = (120597V119894(119905 119909lowast(119905))

120597119909119895

)

1le119894119895le2

(10)

whereF119894(119905 119909) andV

119894(119905 119909) are the 119894th components ofF(119905 119909)

andV(119905 119909) respectively Then from (6) we obtain that

119865 (119905) = (0 120573

1(119905) 119878lowast

ℎ(119905)

1205732(119905) 119878lowast

V (119905) 0)

119881 (119905) = (1205831(119905) + 119889 (119905) 0

0 1205832(119905))

(11)

Let 119884(119905 119904) be a 2 times 2matrix solution of the system

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904

119884 (119904 119904) = 119868

(12)

where 119868 is 2 times 2 identity matrix From (11) and (12) we have119903(Φminus119881(120596)) lt 1 Therefore the condition (A7) in [15] also

holdsLet 119862

120596be the ordered Banach space of all 120596-periodic

function from 119877 rarr 1198772 which is equipped with maximum

norm sdot infin

and the positive cone 119862+120596= 120601 isin 119862

120596 120601(119905) ge 0

for all 119905 isin 119877 Define the following linear operator 119871 119862120596rarr

119862120596by

(119871120601) (119905) = int

+infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886

forall119905 isin 119877 120601 isin 119862120596

(13)

Based on the assumptions above and the results of Wang andZhang [15] we can derive the basic reproduction number 119877

0

of system (1) as follows

1198770= 119903 (119871) (14)

and obtain the following conclusion

Theorem 4 For system (1) the following statements are valid

(i) 1198770= 1 if and only if 119903(Φ

119865minus119881(120596)) = 1

(ii) 1198770gt 1 if and only if 119903(Φ

119865minus119881(120596)) gt 1

(iii) 1198770lt 1 if and only if 119903(Φ

119865minus119881(120596)) lt 1

where 119865(119905) and 119881(119905) are defined in (11)

It follows from Theorem 4 that the disease-free periodicsolution (119878

lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) of system (1) is asymptoticallystable if 119877

0lt 1 and it is unstable if 119877

0gt 1

In order to calculate 1198770 we consider the following linear

120596-periodic system

119889119908

119889119905= (minus119881 (119905) +

1

120582119865 (119905))119908 120582 isin (0infin) (15)

Let 119882(119905 119904 120582) 119905 ⩾ 119904 119904 isin 119877 be the evolution operator ofthe System (15) on 1198772 Since 119865(119905) is nonnegative and minus119881(119905)is cooperative then 119903(119882(120596 0 120582)) is continuous and nonin-creasing for 120582 isin (0infin) and lim

120582rarrinfin119903(119882(120596 0 120582)) lt 1

Thus we have the following result which will be used in ournumerical calculation of the basic reproduction ratio 119877

0in

Section 5

Lemma 5 (see [15]) The following statements are valid

(i) If 119903(119882(120596 0 120582)) = 1 has a positive solution 1205820is an

eigenvalue of 119871 and hence 1198770gt 0

(ii) If 1198770gt 0 then 120582 = 119877

0is the unique solution of 119903(119882(120596

0 120582)) = 1(iii) 119877

0= 0 if and only if 119903(119882(120596 0 120582)) lt 1 for all 120582 gt 0

4 Abstract and Applied Analysis

3 Global Stability of Disease-FreePeriodic Solution

In this section we will prove the global asymptotical stabilityof the disease-free periodic solution (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0)Let119873

ℎ(119905) = 119878

ℎ(119905) + 119868

ℎ(119905)119873V(119905) = 119878V(119905) + 119868V(119905) Denote

Ω = (119878ℎ 119868ℎ 119878V 119868V) isin 119877

4

+| 0 le 119878

ℎ+ 119868ℎle 1198731lt +infin

0 le 119878V + 119868V le 1198732 lt +infin

(16)

where 1198731= Λ119871120583119872

1and 119873

2= 119898119873

1 Similar to [16 17] we

firstly prove the following lemmas

Lemma 6 Ω is a positively invariant set for (1)

Proof From the equations in (1) we have

119889119873ℎ(119905)

119889119905= Λ (119905) minus 120583

1(119905)119873ℎ(119905)

le Λ119871minus 120583119872

1119873ℎ(119905)

le 0 if 119873ℎ (119905) ge 1198731

119889119873V (119905)

119889119905= 119887 (119905)119873V (1 minus

119873V

119898119873ℎ

) minus 1205832(119905) 119868V (119905)

le 119887 (119905)119873V (119905) (1 minus119873V (119905)

119898119873ℎ

)

le 0 if 119873V (119905) ge 1198732

(17)

which implies that Ω is a positive invariant compact set for(1) The proof is completed

Lemma 7 Let (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) be any solution of

system (1) It holds that

lim119905rarr+infin

(119873ℎ (119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119873V (119905) minus 119878lowast

V (119905)) = 0(18)

where 119878lowastℎ(119905) 119878lowastV (119905) are defined in Lemmas 2 and 3 respectively

Proof We denote that 1199101(119905) = 119873

ℎ(119905) minus 119878

lowast

ℎ(119905) It follows from

the first equation of (17) that 1198891199101(119905)119889119905 le minus120583

1(119905)1199101(119905) which

implies that lim119905rarr+infin

1199101(119905) = lim

119905rarr+infin(119873ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

Further from Lemma 6 we obtain that for any 120576 gt 0 thereexists a 119879 gt 0 such that

119878lowast

ℎ(119905) minus 120576 le 119873ℎ (119905) le 119878

lowast

ℎ(119905) + 120576 119873V (119905) lt 1198732 forall119905 ge 119879

(19)

Let 1199102(119905) = 119873V(119905) minus 119878

lowast

V (119905) From the second equation of(17) and (19) we get

1198891199102(119905)

119889119905= 119887 (119905)119873V (119905) [1 minus

119873V (119905)

119898119873ℎ (119905)

] minus 1205832(119905) 119868V (119905)

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

le 119887 (119905)119873V (119905) [1 minus119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

]

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

= 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ(119905) + 120576

]

minus 119887 (119905) (119873lowast

V (119905) minus 119878lowast

V (119905))119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ(119905) + 120576)

= minus 119887 (119905)119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

1199102(119905) + Δ (120576)

(20)

for all 119905 gt 119879 where

Δ (120576) = 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ+ 120576

]

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ+ 120576)

(21)

Obviously lim120576rarr0

Δ(120576) = 0 Because 120576 is arbitrarily smallthen lim

119905rarr+infin1199102(119905) = lim

119905rarr+infin(119873V(119905) minus 119878

lowast

V (119905)) = 0 Hencethe proof is completed

Theorem 8 The disease-free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905)

0) is globally asymptotically stable if 1198770lt 1 whereas it is

unstable if 1198770gt 1

Proof From Theorem 4 we have that (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isunstable if 119877

0gt 1 and (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) is locally stable if1198770lt 1Therefore we only need to show the global attractivity

of (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) for 1198770 lt 1Since 119877

0lt 1 by Theorem 4 we can choose 120598

1gt 0

sufficiently small such that

119903 (Φ119865minus119881+119872

1205761

(120596)) lt 1 (22)

where

1198721205981

(119905) = (0 1205981

12059810) (23)

From Lemma 6 and (18) we have that for above men-tioned 120598

1gt 0 there exists a119879

1gt 0 such that 119878

ℎ(119905) le 119878

lowast

ℎ(119905)+1205981

119878V(119905) le 119878lowast

V (119905) + 1205981 for 119905 gt 1198791 It follows from the second and

fourth equations that for 119905 gt 1198791

119889119868ℎ(119905)

119889119905le 1205731(119905) (119878lowast

ℎ(119905) + 120598

1) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905le 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(24)

Abstract and Applied Analysis 5

Consider the following comparison system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) + 1205981) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(25)

In view of Lemma 1 we know that there exists a positive 120596-periodic function V

1(119905) such that 119869(119905) le V

1(119905) exp(119901

1119905) where

119869(119905) = (119868ℎ(119905) 119868V(119905))

119879 and 1199011= (1120596) ln 119903(Φ

119865minus119881+119872120598

(120596)) lt

0 It follows from (22) that lim119905rarr+infin

119868ℎ(119905) = 0 and

lim119905rarr+infin

119868V(119905) = 0 By the comparison of theorem [18] wehave lim

119905rarr+infin119868ℎ(119905) = 0 and lim

119905rarr+infin119868V(119905) = 0 From (18)

we have

lim119905rarr+infin

(119878ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119878V (119905) minus 119878lowast

V (119905)) = 0(26)

Hence the disease free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isglobally attractive This completes the proof

4 Permanence

In this section we show that if 1198770gt 1 then the disease

persistsFirstly we define 119883 = (119878

ℎ 119868ℎ 119878V 119868V) isin 119877

4

+1198830= (119878ℎ 119868ℎ

119878V 119868V) isin 119883 119878ℎge 0 119868

ℎgt 0 119878V ge 0 119868V gt 0 and 120597119883

0=

1198831198830 andwedenote119906(119905 119909

0) as the unique solution of System

(1) with the initial value 1199090= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )Define Poincare map 119875 119883 rarr 119883 associated with System

(1) as follows

119875 (1199090) = 119906 (120596 119909

0) forall119909

0isin 119883 (27)

By Lemma 6 it is easy to see that both119883 and1198830are positively

invariant and 119875 is point dissipative Set

119872120597= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830 | 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830

119898 isin 119885+

(28)

where 119885+ = 0 1 2 We claim that

119872120597= (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 (29)

Obviously 119872120597supe (119878

ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 Next we

want to show 119872120597 (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 = 0 If it

does not hold then there exists a point (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 119872120597

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0

Case 1 1198680ℎ= 0 and 1198680V gt 0 It is obvious that 119868V(119905) gt 0 and

119878ℎ(119905) gt 0 for any 119905 gt 0 Then from the second equation

of System (1) 119889119868ℎ(119905)119889119905|

119905=0= 1205731(0)119878ℎ(0)119868V(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

Case 2 1198680ℎgt 0 and 1198680V = 0 It is obvious that 119868

ℎ(119905) gt 0 and

119878V(119905) gt 0 for any 119905 gt 0 Then from the fourth equationof System (1) 119889119868V(119905)119889119905|119905=0 = 120573

2(0)119878V(0)119868ℎ(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

That is to say for any (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) notin (119878ℎ 0 119878V 0) 119878ℎ ge

0 119878V ge 0 then (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) notin 119872120597Therefore we have119872120597=

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0Next we present the following result of the uniform

persistence of the disease

Theorem 9 Suppose 1198770gt 1 Then there is a positive constant

120598 gt 0 such that each positive solution (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905))

of System (1) satisfies

lim inf119905rarr+infin

119868ℎ(119905) ge 120598 lim inf

119905rarr+infin

119868V (119905) ge 120598 (30)

Proof By Theorem 4 we obtain 119903(Φ119865minus119881

(120596)) gt 1 So we canchoose 120578 gt 0 small enough such that 119903(Φ

119865minus119881minus119872120578

) gt 1 where

119872120578= (

0 120578

120578 0) (31)

Put 1198750= 119878lowast

ℎ(0) 0 119878

lowast

V (0) 0 Now we proceed by contra-diction to prove that

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) ge 120575 (32)

If it does not hold then

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 (33)

for some (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1198830 Without loss of generality

suppose that

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 forall119898 isin 119885+ (34)

By the continuity of the solutions with respect to the initialvalues we obtain

10038171003817100381710038171003817119906 (119905 119875

119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817le 120578

forall119905 isin [0 120596] forall119898 isin 119885+

(35)

For any 119905 ge 0 there exists a 119898 isin 119885+such that 119905 = 119898120596 + 119905

1

where 1199051isin [0 120596] Then we have

10038171003817100381710038171003817119906 (119905 (119878

0

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817

=10038171003817100381710038171003817119906 (1199051 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (1199051 1198750)10038171003817100381710038171003817le 120578

(36)

for all 119905 ge 0 which implies that 119878lowastℎ(119905) minus 120578 le 119878

ℎ(119905) le 119878

lowast

ℎ(119905) + 120578

119878lowast

V (119905) minus 120578 le 119878V(119905) le 119878lowast

V (119905) + 120578 Then from (1) we have

119889119868ℎ(119905)

119889119905ge 1205731(119905) (119878lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905ge 1205732(119905) (119878lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(37)

6 Abstract and Applied Analysis

Table 1 Parameter definitions and values used for numerical simulations of the Huanglongbing model

Parameter Definition Average value Unit ReferenceΛ The recruitment rate of citrus mdash monthminus1 Estimate1205731

Infected rate of citrus mdash monthminus1 Estimate1205831

Nature death rate of citrus 000275ndash0004167 monthminus1 [20]119889 Disease induced death rate of citrus 0016667ndash0027775 monthminus1 [21]119863 Birth rate of psyllid 378327ndash33526137 monthminus1 [20 22]1205732

Infected rate of psyllid mdash monthminus1 Estimate1205832

Nature death rate of psyllid 01169825ndash095052 monthminus1 [23]119898 Max abundance of psyllid per citrus 120ndash1000 mdash [24]

Table 2 Parameter functions for model (1) according to the values of Table 1

Parameter functions Value Reference

1205731(119905) 00042925 + 0003543 cos (212058711990512) Estimate

1205831(119905) 00034585 + 00007085 cos (212058711990512) [20]

119889 (119905) 0022221 + 0005554 cos (212058711990512) [21]

119863 (119905) 186547035 + 148714335 cos (212058711990512) [20 22]

1205732(119905) 0008779171 + 0004838437 cos (212058711990512) Estimate

1205832(119905) 053375125 + 041676875 cos (212058711990512) [23]

119887 (119905) = 119863 (119905) minus 1205832(119905) 18120952 + 1445466475 cos (212058711990512) [20 22 23]

119898 560 [24]

Consider the linear system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732 (119905) (119878

lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(38)

By Lemma 1 and the standard comparison principle we havethat there exists a positive120596-periodic function V

2(119905) such that

119869(119905) = exp(1199012119905)V2(119905) is a solution of System (38) where 119869(119905) =

(119868ℎ(119905) 119868V(119905))

119879 and

1199012=1

120596ln 119903 (Φ

119865minus119881minus119872120578

(120596)) (39)

It follows from 119903(Φ119865minus119881minus119872

120578

(120596)) gt 1 that 1199012gt 0 and 119869(119905) rarr

+infin as 119905 rarr +infin Applying the comparison principle [18] weknow that 119868

ℎ(119905) rarr +infin and 119868V(119905) rarr +infin as 119905 rarr +infin This

is a contradictionThus we have proved that (32) holds and119875is weakly uniformly persistent with respect to (119883

0 1205971198830)

According to the results of Lemma 7 we can easily obtainthat119875 has a global attractor119875

0 It is easy to obtain that119875

0is an

isolated invariant set in119883 and119882119904(1198750)cap1198830= 0We know that

1198750is acyclic in119872

120597and every solution in119872

120597converges to 119875

0

According to Zhao [19] we have that119875 is uniformly persistentwith respect to (119883

0 1205971198830) This implies that the solution of

(1) is uniformly persistent with respect to (1198830 1205971198830) Thus we

have that there exists a 120598 gt 0 such that lim inf119905rarr+infin

119868ℎ(119905) ge 120598

lim inf119905rarr+infin

119868V(119905) ge 120598

5 Numerical Simulationsand Sensitivity Analysis

In this section we will make numerical simulations bymeansof Matlab 71 to support our theoretical results to predictthe trend of the disease and to explore some control andprevention measures Numerical values of parameters ofsystem (1) are given in Table 1 (most of the data are taken fromthe literatures ([20ndash24]))

According to the periodicity of System (1) and Table 1 weset 1205831(119905) = 120572

0

1+ 1205720

2cos(212058711990512) where 1205720

2= (0004167 minus

000275)2 = 00007085 and 12057201= 000275 + 120572

0

2= 00034585

By the similar method we can obtain the other parameterfunctions of model (1) (see Table 2) For the simulations thatfollows we apply the parameters in Table 2 unless otherwisestated

ChooseΛ(119905) = 000265+000235 cos(212058711990512)Then fromLemma 5 we can compute 119877

0= 09844 lt 1 by means of

Matlab 71 FromTheorem 8 we obtain that the infected citruspopulation 119868

ℎ(119905) and the infected psyllid population 119868V(119905) of

system (1) are extinct (see Figures 1 and 2)Choose Λ(119905) = 0005 + 00035 cos(212058711990512) Then from

Lemma 5 we obtain that 1198770= 18342 gt 1 From Theorem 9

we have that the infected citrus population 119868ℎ(119905) and the

infected psyllid population 119868V(119905) of System (1) are permanence(see Figures 3 and 4)

From the formulae for 1198770 we can predict the general

tendency of the epidemic in a long term according to thecurrent situation which is presented in Figures 1 2 3 and 4From the first two figures we know that the epidemic of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

V+(119905 119909) = (

0

0

Λ (119905)

119887 (119905) (119878V (119905) + 119868V (119905))

)

Vminus(119905 119909) = (

(1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

1205832 (119905) 119868V (119905)

1205831 (119905) 119878ℎ + 1205731 (119905) 119878ℎ (119905) 119868V (119905)

119887 (119905)(119878V (119905) + 119868V (119905))

2

119898(119878ℎ (119905) + 119868ℎ (119905))

+1205732(119905) 119878V (119905) 119868ℎ (119905)

)

(6)

where 119909 = (119868ℎ(119905) 119868V(119905) 119878ℎ(119905) 119878V(119905))

119879 Then System (1) can bewritten as the following form

119889119909 (119905)

119889119905= F (119905 119909 (119905)) minusV (119905 119909 (119905)) (7)

whereV(119905 119909(119905)) =Vminus(119905 119909(119905)) minusV+(119905 119909(119905))It is easy to obtain that the conditions (A1)ndash(A5) in [15]

hold In the following we will check the conditions (A6) and(A7) in [15]

We know that 119909lowast(119905) = (0 0 119878lowast

ℎ(119905) 119878lowast

V (119905)) is the disease-free periodic solution of system (7) Denote

119891 (119905 119909 (119905)) = F (119905 119909 (119905)) minusV (119905 119909 (119905))

119872 (119905) = (120597119891119894(119905 119909lowast(119905))

120597119909119895

)

3le119894119895le4

(8)

where 119891119894(119905 119909(119905)) and 119909

119894are the 119894th components of 119891(119905 119909(119905))

and 119909 respectively According to (6) we have

119872(119905) = (

minus1205831(119905) 0

119887 (119905) 119878lowast2

V (119905)

119898119878lowast2

ℎ(119905)

minus119887 (119905)) (9)

It is easy to see that 119903(Φ119872(120596)) lt 1 where 119903(Φ

119872(120596))

is the spectral radius of Φ119872(120596) This implies that 119909lowast(119905) is

linearly asymptotically stable in the disease-free subspace119883119878= (0 0 119878

ℎ 119878V) isin 119877

4

+ Thus condition (A6) in [15] holds

We further define

119865 (119905) = (120597F119894(119905 119909lowast(119905))

120597119909119895

)

1le119894119895le2

119881 (119905) = (120597V119894(119905 119909lowast(119905))

120597119909119895

)

1le119894119895le2

(10)

whereF119894(119905 119909) andV

119894(119905 119909) are the 119894th components ofF(119905 119909)

andV(119905 119909) respectively Then from (6) we obtain that

119865 (119905) = (0 120573

1(119905) 119878lowast

ℎ(119905)

1205732(119905) 119878lowast

V (119905) 0)

119881 (119905) = (1205831(119905) + 119889 (119905) 0

0 1205832(119905))

(11)

Let 119884(119905 119904) be a 2 times 2matrix solution of the system

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904

119884 (119904 119904) = 119868

(12)

where 119868 is 2 times 2 identity matrix From (11) and (12) we have119903(Φminus119881(120596)) lt 1 Therefore the condition (A7) in [15] also

holdsLet 119862

120596be the ordered Banach space of all 120596-periodic

function from 119877 rarr 1198772 which is equipped with maximum

norm sdot infin

and the positive cone 119862+120596= 120601 isin 119862

120596 120601(119905) ge 0

for all 119905 isin 119877 Define the following linear operator 119871 119862120596rarr

119862120596by

(119871120601) (119905) = int

+infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886

forall119905 isin 119877 120601 isin 119862120596

(13)

Based on the assumptions above and the results of Wang andZhang [15] we can derive the basic reproduction number 119877

0

of system (1) as follows

1198770= 119903 (119871) (14)

and obtain the following conclusion

Theorem 4 For system (1) the following statements are valid

(i) 1198770= 1 if and only if 119903(Φ

119865minus119881(120596)) = 1

(ii) 1198770gt 1 if and only if 119903(Φ

119865minus119881(120596)) gt 1

(iii) 1198770lt 1 if and only if 119903(Φ

119865minus119881(120596)) lt 1

where 119865(119905) and 119881(119905) are defined in (11)

It follows from Theorem 4 that the disease-free periodicsolution (119878

lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) of system (1) is asymptoticallystable if 119877

0lt 1 and it is unstable if 119877

0gt 1

In order to calculate 1198770 we consider the following linear

120596-periodic system

119889119908

119889119905= (minus119881 (119905) +

1

120582119865 (119905))119908 120582 isin (0infin) (15)

Let 119882(119905 119904 120582) 119905 ⩾ 119904 119904 isin 119877 be the evolution operator ofthe System (15) on 1198772 Since 119865(119905) is nonnegative and minus119881(119905)is cooperative then 119903(119882(120596 0 120582)) is continuous and nonin-creasing for 120582 isin (0infin) and lim

120582rarrinfin119903(119882(120596 0 120582)) lt 1

Thus we have the following result which will be used in ournumerical calculation of the basic reproduction ratio 119877

0in

Section 5

Lemma 5 (see [15]) The following statements are valid

(i) If 119903(119882(120596 0 120582)) = 1 has a positive solution 1205820is an

eigenvalue of 119871 and hence 1198770gt 0

(ii) If 1198770gt 0 then 120582 = 119877

0is the unique solution of 119903(119882(120596

0 120582)) = 1(iii) 119877

0= 0 if and only if 119903(119882(120596 0 120582)) lt 1 for all 120582 gt 0

4 Abstract and Applied Analysis

3 Global Stability of Disease-FreePeriodic Solution

In this section we will prove the global asymptotical stabilityof the disease-free periodic solution (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0)Let119873

ℎ(119905) = 119878

ℎ(119905) + 119868

ℎ(119905)119873V(119905) = 119878V(119905) + 119868V(119905) Denote

Ω = (119878ℎ 119868ℎ 119878V 119868V) isin 119877

4

+| 0 le 119878

ℎ+ 119868ℎle 1198731lt +infin

0 le 119878V + 119868V le 1198732 lt +infin

(16)

where 1198731= Λ119871120583119872

1and 119873

2= 119898119873

1 Similar to [16 17] we

firstly prove the following lemmas

Lemma 6 Ω is a positively invariant set for (1)

Proof From the equations in (1) we have

119889119873ℎ(119905)

119889119905= Λ (119905) minus 120583

1(119905)119873ℎ(119905)

le Λ119871minus 120583119872

1119873ℎ(119905)

le 0 if 119873ℎ (119905) ge 1198731

119889119873V (119905)

119889119905= 119887 (119905)119873V (1 minus

119873V

119898119873ℎ

) minus 1205832(119905) 119868V (119905)

le 119887 (119905)119873V (119905) (1 minus119873V (119905)

119898119873ℎ

)

le 0 if 119873V (119905) ge 1198732

(17)

which implies that Ω is a positive invariant compact set for(1) The proof is completed

Lemma 7 Let (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) be any solution of

system (1) It holds that

lim119905rarr+infin

(119873ℎ (119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119873V (119905) minus 119878lowast

V (119905)) = 0(18)

where 119878lowastℎ(119905) 119878lowastV (119905) are defined in Lemmas 2 and 3 respectively

Proof We denote that 1199101(119905) = 119873

ℎ(119905) minus 119878

lowast

ℎ(119905) It follows from

the first equation of (17) that 1198891199101(119905)119889119905 le minus120583

1(119905)1199101(119905) which

implies that lim119905rarr+infin

1199101(119905) = lim

119905rarr+infin(119873ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

Further from Lemma 6 we obtain that for any 120576 gt 0 thereexists a 119879 gt 0 such that

119878lowast

ℎ(119905) minus 120576 le 119873ℎ (119905) le 119878

lowast

ℎ(119905) + 120576 119873V (119905) lt 1198732 forall119905 ge 119879

(19)

Let 1199102(119905) = 119873V(119905) minus 119878

lowast

V (119905) From the second equation of(17) and (19) we get

1198891199102(119905)

119889119905= 119887 (119905)119873V (119905) [1 minus

119873V (119905)

119898119873ℎ (119905)

] minus 1205832(119905) 119868V (119905)

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

le 119887 (119905)119873V (119905) [1 minus119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

]

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

= 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ(119905) + 120576

]

minus 119887 (119905) (119873lowast

V (119905) minus 119878lowast

V (119905))119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ(119905) + 120576)

= minus 119887 (119905)119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

1199102(119905) + Δ (120576)

(20)

for all 119905 gt 119879 where

Δ (120576) = 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ+ 120576

]

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ+ 120576)

(21)

Obviously lim120576rarr0

Δ(120576) = 0 Because 120576 is arbitrarily smallthen lim

119905rarr+infin1199102(119905) = lim

119905rarr+infin(119873V(119905) minus 119878

lowast

V (119905)) = 0 Hencethe proof is completed

Theorem 8 The disease-free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905)

0) is globally asymptotically stable if 1198770lt 1 whereas it is

unstable if 1198770gt 1

Proof From Theorem 4 we have that (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isunstable if 119877

0gt 1 and (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) is locally stable if1198770lt 1Therefore we only need to show the global attractivity

of (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) for 1198770 lt 1Since 119877

0lt 1 by Theorem 4 we can choose 120598

1gt 0

sufficiently small such that

119903 (Φ119865minus119881+119872

1205761

(120596)) lt 1 (22)

where

1198721205981

(119905) = (0 1205981

12059810) (23)

From Lemma 6 and (18) we have that for above men-tioned 120598

1gt 0 there exists a119879

1gt 0 such that 119878

ℎ(119905) le 119878

lowast

ℎ(119905)+1205981

119878V(119905) le 119878lowast

V (119905) + 1205981 for 119905 gt 1198791 It follows from the second and

fourth equations that for 119905 gt 1198791

119889119868ℎ(119905)

119889119905le 1205731(119905) (119878lowast

ℎ(119905) + 120598

1) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905le 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(24)

Abstract and Applied Analysis 5

Consider the following comparison system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) + 1205981) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(25)

In view of Lemma 1 we know that there exists a positive 120596-periodic function V

1(119905) such that 119869(119905) le V

1(119905) exp(119901

1119905) where

119869(119905) = (119868ℎ(119905) 119868V(119905))

119879 and 1199011= (1120596) ln 119903(Φ

119865minus119881+119872120598

(120596)) lt

0 It follows from (22) that lim119905rarr+infin

119868ℎ(119905) = 0 and

lim119905rarr+infin

119868V(119905) = 0 By the comparison of theorem [18] wehave lim

119905rarr+infin119868ℎ(119905) = 0 and lim

119905rarr+infin119868V(119905) = 0 From (18)

we have

lim119905rarr+infin

(119878ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119878V (119905) minus 119878lowast

V (119905)) = 0(26)

Hence the disease free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isglobally attractive This completes the proof

4 Permanence

In this section we show that if 1198770gt 1 then the disease

persistsFirstly we define 119883 = (119878

ℎ 119868ℎ 119878V 119868V) isin 119877

4

+1198830= (119878ℎ 119868ℎ

119878V 119868V) isin 119883 119878ℎge 0 119868

ℎgt 0 119878V ge 0 119868V gt 0 and 120597119883

0=

1198831198830 andwedenote119906(119905 119909

0) as the unique solution of System

(1) with the initial value 1199090= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )Define Poincare map 119875 119883 rarr 119883 associated with System

(1) as follows

119875 (1199090) = 119906 (120596 119909

0) forall119909

0isin 119883 (27)

By Lemma 6 it is easy to see that both119883 and1198830are positively

invariant and 119875 is point dissipative Set

119872120597= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830 | 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830

119898 isin 119885+

(28)

where 119885+ = 0 1 2 We claim that

119872120597= (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 (29)

Obviously 119872120597supe (119878

ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 Next we

want to show 119872120597 (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 = 0 If it

does not hold then there exists a point (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 119872120597

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0

Case 1 1198680ℎ= 0 and 1198680V gt 0 It is obvious that 119868V(119905) gt 0 and

119878ℎ(119905) gt 0 for any 119905 gt 0 Then from the second equation

of System (1) 119889119868ℎ(119905)119889119905|

119905=0= 1205731(0)119878ℎ(0)119868V(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

Case 2 1198680ℎgt 0 and 1198680V = 0 It is obvious that 119868

ℎ(119905) gt 0 and

119878V(119905) gt 0 for any 119905 gt 0 Then from the fourth equationof System (1) 119889119868V(119905)119889119905|119905=0 = 120573

2(0)119878V(0)119868ℎ(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

That is to say for any (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) notin (119878ℎ 0 119878V 0) 119878ℎ ge

0 119878V ge 0 then (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) notin 119872120597Therefore we have119872120597=

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0Next we present the following result of the uniform

persistence of the disease

Theorem 9 Suppose 1198770gt 1 Then there is a positive constant

120598 gt 0 such that each positive solution (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905))

of System (1) satisfies

lim inf119905rarr+infin

119868ℎ(119905) ge 120598 lim inf

119905rarr+infin

119868V (119905) ge 120598 (30)

Proof By Theorem 4 we obtain 119903(Φ119865minus119881

(120596)) gt 1 So we canchoose 120578 gt 0 small enough such that 119903(Φ

119865minus119881minus119872120578

) gt 1 where

119872120578= (

0 120578

120578 0) (31)

Put 1198750= 119878lowast

ℎ(0) 0 119878

lowast

V (0) 0 Now we proceed by contra-diction to prove that

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) ge 120575 (32)

If it does not hold then

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 (33)

for some (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1198830 Without loss of generality

suppose that

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 forall119898 isin 119885+ (34)

By the continuity of the solutions with respect to the initialvalues we obtain

10038171003817100381710038171003817119906 (119905 119875

119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817le 120578

forall119905 isin [0 120596] forall119898 isin 119885+

(35)

For any 119905 ge 0 there exists a 119898 isin 119885+such that 119905 = 119898120596 + 119905

1

where 1199051isin [0 120596] Then we have

10038171003817100381710038171003817119906 (119905 (119878

0

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817

=10038171003817100381710038171003817119906 (1199051 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (1199051 1198750)10038171003817100381710038171003817le 120578

(36)

for all 119905 ge 0 which implies that 119878lowastℎ(119905) minus 120578 le 119878

ℎ(119905) le 119878

lowast

ℎ(119905) + 120578

119878lowast

V (119905) minus 120578 le 119878V(119905) le 119878lowast

V (119905) + 120578 Then from (1) we have

119889119868ℎ(119905)

119889119905ge 1205731(119905) (119878lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905ge 1205732(119905) (119878lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(37)

6 Abstract and Applied Analysis

Table 1 Parameter definitions and values used for numerical simulations of the Huanglongbing model

Parameter Definition Average value Unit ReferenceΛ The recruitment rate of citrus mdash monthminus1 Estimate1205731

Infected rate of citrus mdash monthminus1 Estimate1205831

Nature death rate of citrus 000275ndash0004167 monthminus1 [20]119889 Disease induced death rate of citrus 0016667ndash0027775 monthminus1 [21]119863 Birth rate of psyllid 378327ndash33526137 monthminus1 [20 22]1205732

Infected rate of psyllid mdash monthminus1 Estimate1205832

Nature death rate of psyllid 01169825ndash095052 monthminus1 [23]119898 Max abundance of psyllid per citrus 120ndash1000 mdash [24]

Table 2 Parameter functions for model (1) according to the values of Table 1

Parameter functions Value Reference

1205731(119905) 00042925 + 0003543 cos (212058711990512) Estimate

1205831(119905) 00034585 + 00007085 cos (212058711990512) [20]

119889 (119905) 0022221 + 0005554 cos (212058711990512) [21]

119863 (119905) 186547035 + 148714335 cos (212058711990512) [20 22]

1205732(119905) 0008779171 + 0004838437 cos (212058711990512) Estimate

1205832(119905) 053375125 + 041676875 cos (212058711990512) [23]

119887 (119905) = 119863 (119905) minus 1205832(119905) 18120952 + 1445466475 cos (212058711990512) [20 22 23]

119898 560 [24]

Consider the linear system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732 (119905) (119878

lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(38)

By Lemma 1 and the standard comparison principle we havethat there exists a positive120596-periodic function V

2(119905) such that

119869(119905) = exp(1199012119905)V2(119905) is a solution of System (38) where 119869(119905) =

(119868ℎ(119905) 119868V(119905))

119879 and

1199012=1

120596ln 119903 (Φ

119865minus119881minus119872120578

(120596)) (39)

It follows from 119903(Φ119865minus119881minus119872

120578

(120596)) gt 1 that 1199012gt 0 and 119869(119905) rarr

+infin as 119905 rarr +infin Applying the comparison principle [18] weknow that 119868

ℎ(119905) rarr +infin and 119868V(119905) rarr +infin as 119905 rarr +infin This

is a contradictionThus we have proved that (32) holds and119875is weakly uniformly persistent with respect to (119883

0 1205971198830)

According to the results of Lemma 7 we can easily obtainthat119875 has a global attractor119875

0 It is easy to obtain that119875

0is an

isolated invariant set in119883 and119882119904(1198750)cap1198830= 0We know that

1198750is acyclic in119872

120597and every solution in119872

120597converges to 119875

0

According to Zhao [19] we have that119875 is uniformly persistentwith respect to (119883

0 1205971198830) This implies that the solution of

(1) is uniformly persistent with respect to (1198830 1205971198830) Thus we

have that there exists a 120598 gt 0 such that lim inf119905rarr+infin

119868ℎ(119905) ge 120598

lim inf119905rarr+infin

119868V(119905) ge 120598

5 Numerical Simulationsand Sensitivity Analysis

In this section we will make numerical simulations bymeansof Matlab 71 to support our theoretical results to predictthe trend of the disease and to explore some control andprevention measures Numerical values of parameters ofsystem (1) are given in Table 1 (most of the data are taken fromthe literatures ([20ndash24]))

According to the periodicity of System (1) and Table 1 weset 1205831(119905) = 120572

0

1+ 1205720

2cos(212058711990512) where 1205720

2= (0004167 minus

000275)2 = 00007085 and 12057201= 000275 + 120572

0

2= 00034585

By the similar method we can obtain the other parameterfunctions of model (1) (see Table 2) For the simulations thatfollows we apply the parameters in Table 2 unless otherwisestated

ChooseΛ(119905) = 000265+000235 cos(212058711990512)Then fromLemma 5 we can compute 119877

0= 09844 lt 1 by means of

Matlab 71 FromTheorem 8 we obtain that the infected citruspopulation 119868

ℎ(119905) and the infected psyllid population 119868V(119905) of

system (1) are extinct (see Figures 1 and 2)Choose Λ(119905) = 0005 + 00035 cos(212058711990512) Then from

Lemma 5 we obtain that 1198770= 18342 gt 1 From Theorem 9

we have that the infected citrus population 119868ℎ(119905) and the

infected psyllid population 119868V(119905) of System (1) are permanence(see Figures 3 and 4)

From the formulae for 1198770 we can predict the general

tendency of the epidemic in a long term according to thecurrent situation which is presented in Figures 1 2 3 and 4From the first two figures we know that the epidemic of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

3 Global Stability of Disease-FreePeriodic Solution

In this section we will prove the global asymptotical stabilityof the disease-free periodic solution (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0)Let119873

ℎ(119905) = 119878

ℎ(119905) + 119868

ℎ(119905)119873V(119905) = 119878V(119905) + 119868V(119905) Denote

Ω = (119878ℎ 119868ℎ 119878V 119868V) isin 119877

4

+| 0 le 119878

ℎ+ 119868ℎle 1198731lt +infin

0 le 119878V + 119868V le 1198732 lt +infin

(16)

where 1198731= Λ119871120583119872

1and 119873

2= 119898119873

1 Similar to [16 17] we

firstly prove the following lemmas

Lemma 6 Ω is a positively invariant set for (1)

Proof From the equations in (1) we have

119889119873ℎ(119905)

119889119905= Λ (119905) minus 120583

1(119905)119873ℎ(119905)

le Λ119871minus 120583119872

1119873ℎ(119905)

le 0 if 119873ℎ (119905) ge 1198731

119889119873V (119905)

119889119905= 119887 (119905)119873V (1 minus

119873V

119898119873ℎ

) minus 1205832(119905) 119868V (119905)

le 119887 (119905)119873V (119905) (1 minus119873V (119905)

119898119873ℎ

)

le 0 if 119873V (119905) ge 1198732

(17)

which implies that Ω is a positive invariant compact set for(1) The proof is completed

Lemma 7 Let (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) be any solution of

system (1) It holds that

lim119905rarr+infin

(119873ℎ (119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119873V (119905) minus 119878lowast

V (119905)) = 0(18)

where 119878lowastℎ(119905) 119878lowastV (119905) are defined in Lemmas 2 and 3 respectively

Proof We denote that 1199101(119905) = 119873

ℎ(119905) minus 119878

lowast

ℎ(119905) It follows from

the first equation of (17) that 1198891199101(119905)119889119905 le minus120583

1(119905)1199101(119905) which

implies that lim119905rarr+infin

1199101(119905) = lim

119905rarr+infin(119873ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

Further from Lemma 6 we obtain that for any 120576 gt 0 thereexists a 119879 gt 0 such that

119878lowast

ℎ(119905) minus 120576 le 119873ℎ (119905) le 119878

lowast

ℎ(119905) + 120576 119873V (119905) lt 1198732 forall119905 ge 119879

(19)

Let 1199102(119905) = 119873V(119905) minus 119878

lowast

V (119905) From the second equation of(17) and (19) we get

1198891199102(119905)

119889119905= 119887 (119905)119873V (119905) [1 minus

119873V (119905)

119898119873ℎ (119905)

] minus 1205832(119905) 119868V (119905)

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

le 119887 (119905)119873V (119905) [1 minus119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

]

minus 119887 (119905) 119878lowast

V (119905) [1 minus119878lowast

V (119905)

119898119878lowast

ℎ(119905)]

= 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ(119905) + 120576

]

minus 119887 (119905) (119873lowast

V (119905) minus 119878lowast

V (119905))119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ(119905) + 120576)

= minus 119887 (119905)119873V (119905)

119898 (119878lowast

ℎ(119905) + 120576)

1199102(119905) + Δ (120576)

(20)

for all 119905 gt 119879 where

Δ (120576) = 119887 (119905) (119873V (119905) minus 119878lowast

V (119905)) [1 minus119878lowast

ℎ(119905)

119878lowast

ℎ+ 120576

]

+ 119887 (119905)119878lowast2

V (119905) 120576

119898119878lowast

ℎ(119905) (119878lowast

ℎ+ 120576)

(21)

Obviously lim120576rarr0

Δ(120576) = 0 Because 120576 is arbitrarily smallthen lim

119905rarr+infin1199102(119905) = lim

119905rarr+infin(119873V(119905) minus 119878

lowast

V (119905)) = 0 Hencethe proof is completed

Theorem 8 The disease-free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905)

0) is globally asymptotically stable if 1198770lt 1 whereas it is

unstable if 1198770gt 1

Proof From Theorem 4 we have that (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isunstable if 119877

0gt 1 and (119878lowast

ℎ(119905) 0 119878

lowast

V (119905) 0) is locally stable if1198770lt 1Therefore we only need to show the global attractivity

of (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) for 1198770 lt 1Since 119877

0lt 1 by Theorem 4 we can choose 120598

1gt 0

sufficiently small such that

119903 (Φ119865minus119881+119872

1205761

(120596)) lt 1 (22)

where

1198721205981

(119905) = (0 1205981

12059810) (23)

From Lemma 6 and (18) we have that for above men-tioned 120598

1gt 0 there exists a119879

1gt 0 such that 119878

ℎ(119905) le 119878

lowast

ℎ(119905)+1205981

119878V(119905) le 119878lowast

V (119905) + 1205981 for 119905 gt 1198791 It follows from the second and

fourth equations that for 119905 gt 1198791

119889119868ℎ(119905)

119889119905le 1205731(119905) (119878lowast

ℎ(119905) + 120598

1) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905le 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(24)

Abstract and Applied Analysis 5

Consider the following comparison system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) + 1205981) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(25)

In view of Lemma 1 we know that there exists a positive 120596-periodic function V

1(119905) such that 119869(119905) le V

1(119905) exp(119901

1119905) where

119869(119905) = (119868ℎ(119905) 119868V(119905))

119879 and 1199011= (1120596) ln 119903(Φ

119865minus119881+119872120598

(120596)) lt

0 It follows from (22) that lim119905rarr+infin

119868ℎ(119905) = 0 and

lim119905rarr+infin

119868V(119905) = 0 By the comparison of theorem [18] wehave lim

119905rarr+infin119868ℎ(119905) = 0 and lim

119905rarr+infin119868V(119905) = 0 From (18)

we have

lim119905rarr+infin

(119878ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119878V (119905) minus 119878lowast

V (119905)) = 0(26)

Hence the disease free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isglobally attractive This completes the proof

4 Permanence

In this section we show that if 1198770gt 1 then the disease

persistsFirstly we define 119883 = (119878

ℎ 119868ℎ 119878V 119868V) isin 119877

4

+1198830= (119878ℎ 119868ℎ

119878V 119868V) isin 119883 119878ℎge 0 119868

ℎgt 0 119878V ge 0 119868V gt 0 and 120597119883

0=

1198831198830 andwedenote119906(119905 119909

0) as the unique solution of System

(1) with the initial value 1199090= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )Define Poincare map 119875 119883 rarr 119883 associated with System

(1) as follows

119875 (1199090) = 119906 (120596 119909

0) forall119909

0isin 119883 (27)

By Lemma 6 it is easy to see that both119883 and1198830are positively

invariant and 119875 is point dissipative Set

119872120597= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830 | 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830

119898 isin 119885+

(28)

where 119885+ = 0 1 2 We claim that

119872120597= (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 (29)

Obviously 119872120597supe (119878

ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 Next we

want to show 119872120597 (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 = 0 If it

does not hold then there exists a point (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 119872120597

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0

Case 1 1198680ℎ= 0 and 1198680V gt 0 It is obvious that 119868V(119905) gt 0 and

119878ℎ(119905) gt 0 for any 119905 gt 0 Then from the second equation

of System (1) 119889119868ℎ(119905)119889119905|

119905=0= 1205731(0)119878ℎ(0)119868V(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

Case 2 1198680ℎgt 0 and 1198680V = 0 It is obvious that 119868

ℎ(119905) gt 0 and

119878V(119905) gt 0 for any 119905 gt 0 Then from the fourth equationof System (1) 119889119868V(119905)119889119905|119905=0 = 120573

2(0)119878V(0)119868ℎ(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

That is to say for any (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) notin (119878ℎ 0 119878V 0) 119878ℎ ge

0 119878V ge 0 then (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) notin 119872120597Therefore we have119872120597=

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0Next we present the following result of the uniform

persistence of the disease

Theorem 9 Suppose 1198770gt 1 Then there is a positive constant

120598 gt 0 such that each positive solution (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905))

of System (1) satisfies

lim inf119905rarr+infin

119868ℎ(119905) ge 120598 lim inf

119905rarr+infin

119868V (119905) ge 120598 (30)

Proof By Theorem 4 we obtain 119903(Φ119865minus119881

(120596)) gt 1 So we canchoose 120578 gt 0 small enough such that 119903(Φ

119865minus119881minus119872120578

) gt 1 where

119872120578= (

0 120578

120578 0) (31)

Put 1198750= 119878lowast

ℎ(0) 0 119878

lowast

V (0) 0 Now we proceed by contra-diction to prove that

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) ge 120575 (32)

If it does not hold then

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 (33)

for some (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1198830 Without loss of generality

suppose that

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 forall119898 isin 119885+ (34)

By the continuity of the solutions with respect to the initialvalues we obtain

10038171003817100381710038171003817119906 (119905 119875

119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817le 120578

forall119905 isin [0 120596] forall119898 isin 119885+

(35)

For any 119905 ge 0 there exists a 119898 isin 119885+such that 119905 = 119898120596 + 119905

1

where 1199051isin [0 120596] Then we have

10038171003817100381710038171003817119906 (119905 (119878

0

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817

=10038171003817100381710038171003817119906 (1199051 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (1199051 1198750)10038171003817100381710038171003817le 120578

(36)

for all 119905 ge 0 which implies that 119878lowastℎ(119905) minus 120578 le 119878

ℎ(119905) le 119878

lowast

ℎ(119905) + 120578

119878lowast

V (119905) minus 120578 le 119878V(119905) le 119878lowast

V (119905) + 120578 Then from (1) we have

119889119868ℎ(119905)

119889119905ge 1205731(119905) (119878lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905ge 1205732(119905) (119878lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(37)

6 Abstract and Applied Analysis

Table 1 Parameter definitions and values used for numerical simulations of the Huanglongbing model

Parameter Definition Average value Unit ReferenceΛ The recruitment rate of citrus mdash monthminus1 Estimate1205731

Infected rate of citrus mdash monthminus1 Estimate1205831

Nature death rate of citrus 000275ndash0004167 monthminus1 [20]119889 Disease induced death rate of citrus 0016667ndash0027775 monthminus1 [21]119863 Birth rate of psyllid 378327ndash33526137 monthminus1 [20 22]1205732

Infected rate of psyllid mdash monthminus1 Estimate1205832

Nature death rate of psyllid 01169825ndash095052 monthminus1 [23]119898 Max abundance of psyllid per citrus 120ndash1000 mdash [24]

Table 2 Parameter functions for model (1) according to the values of Table 1

Parameter functions Value Reference

1205731(119905) 00042925 + 0003543 cos (212058711990512) Estimate

1205831(119905) 00034585 + 00007085 cos (212058711990512) [20]

119889 (119905) 0022221 + 0005554 cos (212058711990512) [21]

119863 (119905) 186547035 + 148714335 cos (212058711990512) [20 22]

1205732(119905) 0008779171 + 0004838437 cos (212058711990512) Estimate

1205832(119905) 053375125 + 041676875 cos (212058711990512) [23]

119887 (119905) = 119863 (119905) minus 1205832(119905) 18120952 + 1445466475 cos (212058711990512) [20 22 23]

119898 560 [24]

Consider the linear system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732 (119905) (119878

lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(38)

By Lemma 1 and the standard comparison principle we havethat there exists a positive120596-periodic function V

2(119905) such that

119869(119905) = exp(1199012119905)V2(119905) is a solution of System (38) where 119869(119905) =

(119868ℎ(119905) 119868V(119905))

119879 and

1199012=1

120596ln 119903 (Φ

119865minus119881minus119872120578

(120596)) (39)

It follows from 119903(Φ119865minus119881minus119872

120578

(120596)) gt 1 that 1199012gt 0 and 119869(119905) rarr

+infin as 119905 rarr +infin Applying the comparison principle [18] weknow that 119868

ℎ(119905) rarr +infin and 119868V(119905) rarr +infin as 119905 rarr +infin This

is a contradictionThus we have proved that (32) holds and119875is weakly uniformly persistent with respect to (119883

0 1205971198830)

According to the results of Lemma 7 we can easily obtainthat119875 has a global attractor119875

0 It is easy to obtain that119875

0is an

isolated invariant set in119883 and119882119904(1198750)cap1198830= 0We know that

1198750is acyclic in119872

120597and every solution in119872

120597converges to 119875

0

According to Zhao [19] we have that119875 is uniformly persistentwith respect to (119883

0 1205971198830) This implies that the solution of

(1) is uniformly persistent with respect to (1198830 1205971198830) Thus we

have that there exists a 120598 gt 0 such that lim inf119905rarr+infin

119868ℎ(119905) ge 120598

lim inf119905rarr+infin

119868V(119905) ge 120598

5 Numerical Simulationsand Sensitivity Analysis

In this section we will make numerical simulations bymeansof Matlab 71 to support our theoretical results to predictthe trend of the disease and to explore some control andprevention measures Numerical values of parameters ofsystem (1) are given in Table 1 (most of the data are taken fromthe literatures ([20ndash24]))

According to the periodicity of System (1) and Table 1 weset 1205831(119905) = 120572

0

1+ 1205720

2cos(212058711990512) where 1205720

2= (0004167 minus

000275)2 = 00007085 and 12057201= 000275 + 120572

0

2= 00034585

By the similar method we can obtain the other parameterfunctions of model (1) (see Table 2) For the simulations thatfollows we apply the parameters in Table 2 unless otherwisestated

ChooseΛ(119905) = 000265+000235 cos(212058711990512)Then fromLemma 5 we can compute 119877

0= 09844 lt 1 by means of

Matlab 71 FromTheorem 8 we obtain that the infected citruspopulation 119868

ℎ(119905) and the infected psyllid population 119868V(119905) of

system (1) are extinct (see Figures 1 and 2)Choose Λ(119905) = 0005 + 00035 cos(212058711990512) Then from

Lemma 5 we obtain that 1198770= 18342 gt 1 From Theorem 9

we have that the infected citrus population 119868ℎ(119905) and the

infected psyllid population 119868V(119905) of System (1) are permanence(see Figures 3 and 4)

From the formulae for 1198770 we can predict the general

tendency of the epidemic in a long term according to thecurrent situation which is presented in Figures 1 2 3 and 4From the first two figures we know that the epidemic of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Consider the following comparison system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) + 1205981) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732(119905) (119878lowast

V (119905) + 1205981) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(25)

In view of Lemma 1 we know that there exists a positive 120596-periodic function V

1(119905) such that 119869(119905) le V

1(119905) exp(119901

1119905) where

119869(119905) = (119868ℎ(119905) 119868V(119905))

119879 and 1199011= (1120596) ln 119903(Φ

119865minus119881+119872120598

(120596)) lt

0 It follows from (22) that lim119905rarr+infin

119868ℎ(119905) = 0 and

lim119905rarr+infin

119868V(119905) = 0 By the comparison of theorem [18] wehave lim

119905rarr+infin119868ℎ(119905) = 0 and lim

119905rarr+infin119868V(119905) = 0 From (18)

we have

lim119905rarr+infin

(119878ℎ(119905) minus 119878

lowast

ℎ(119905)) = 0

lim119905rarr+infin

(119878V (119905) minus 119878lowast

V (119905)) = 0(26)

Hence the disease free periodic solution (119878lowastℎ(119905) 0 119878

lowast

V (119905) 0) isglobally attractive This completes the proof

4 Permanence

In this section we show that if 1198770gt 1 then the disease

persistsFirstly we define 119883 = (119878

ℎ 119868ℎ 119878V 119868V) isin 119877

4

+1198830= (119878ℎ 119868ℎ

119878V 119868V) isin 119883 119878ℎge 0 119868

ℎgt 0 119878V ge 0 119868V gt 0 and 120597119883

0=

1198831198830 andwedenote119906(119905 119909

0) as the unique solution of System

(1) with the initial value 1199090= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )Define Poincare map 119875 119883 rarr 119883 associated with System

(1) as follows

119875 (1199090) = 119906 (120596 119909

0) forall119909

0isin 119883 (27)

By Lemma 6 it is easy to see that both119883 and1198830are positively

invariant and 119875 is point dissipative Set

119872120597= (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830 | 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1205971198830

119898 isin 119885+

(28)

where 119885+ = 0 1 2 We claim that

119872120597= (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 (29)

Obviously 119872120597supe (119878

ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 Next we

want to show 119872120597 (119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0 = 0 If it

does not hold then there exists a point (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 119872120597

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0

Case 1 1198680ℎ= 0 and 1198680V gt 0 It is obvious that 119868V(119905) gt 0 and

119878ℎ(119905) gt 0 for any 119905 gt 0 Then from the second equation

of System (1) 119889119868ℎ(119905)119889119905|

119905=0= 1205731(0)119878ℎ(0)119868V(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

Case 2 1198680ℎgt 0 and 1198680V = 0 It is obvious that 119868

ℎ(119905) gt 0 and

119878V(119905) gt 0 for any 119905 gt 0 Then from the fourth equationof System (1) 119889119868V(119905)119889119905|119905=0 = 120573

2(0)119878V(0)119868ℎ(0) gt 0 holds It

follows that (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905)) notin 120597119883

0for 0 lt 119905 ≪ 1

This is a contradiction

That is to say for any (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) notin (119878ℎ 0 119878V 0) 119878ℎ ge

0 119878V ge 0 then (1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) notin 119872120597Therefore we have119872120597=

(119878ℎ 0 119878V 0) 119878ℎ ge 0 119878V ge 0Next we present the following result of the uniform

persistence of the disease

Theorem 9 Suppose 1198770gt 1 Then there is a positive constant

120598 gt 0 such that each positive solution (119878ℎ(119905) 119868ℎ(119905) 119878V(119905) 119868V(119905))

of System (1) satisfies

lim inf119905rarr+infin

119868ℎ(119905) ge 120598 lim inf

119905rarr+infin

119868V (119905) ge 120598 (30)

Proof By Theorem 4 we obtain 119903(Φ119865minus119881

(120596)) gt 1 So we canchoose 120578 gt 0 small enough such that 119903(Φ

119865minus119881minus119872120578

) gt 1 where

119872120578= (

0 120578

120578 0) (31)

Put 1198750= 119878lowast

ℎ(0) 0 119878

lowast

V (0) 0 Now we proceed by contra-diction to prove that

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) ge 120575 (32)

If it does not hold then

lim sup119898rarr+infin

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 (33)

for some (1198780ℎ 1198680

ℎ 1198780

V 1198680

V ) isin 1198830 Without loss of generality

suppose that

119889 (119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V ) 1198750) lt 120575 forall119898 isin 119885+ (34)

By the continuity of the solutions with respect to the initialvalues we obtain

10038171003817100381710038171003817119906 (119905 119875

119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817le 120578

forall119905 isin [0 120596] forall119898 isin 119885+

(35)

For any 119905 ge 0 there exists a 119898 isin 119885+such that 119905 = 119898120596 + 119905

1

where 1199051isin [0 120596] Then we have

10038171003817100381710038171003817119906 (119905 (119878

0

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (119905 1198750)10038171003817100381710038171003817

=10038171003817100381710038171003817119906 (1199051 119875119898(1198780

ℎ 1198680

ℎ 1198780

V 1198680

V )) minus 119906 (1199051 1198750)10038171003817100381710038171003817le 120578

(36)

for all 119905 ge 0 which implies that 119878lowastℎ(119905) minus 120578 le 119878

ℎ(119905) le 119878

lowast

ℎ(119905) + 120578

119878lowast

V (119905) minus 120578 le 119878V(119905) le 119878lowast

V (119905) + 120578 Then from (1) we have

119889119868ℎ(119905)

119889119905ge 1205731(119905) (119878lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905ge 1205732(119905) (119878lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(37)

6 Abstract and Applied Analysis

Table 1 Parameter definitions and values used for numerical simulations of the Huanglongbing model

Parameter Definition Average value Unit ReferenceΛ The recruitment rate of citrus mdash monthminus1 Estimate1205731

Infected rate of citrus mdash monthminus1 Estimate1205831

Nature death rate of citrus 000275ndash0004167 monthminus1 [20]119889 Disease induced death rate of citrus 0016667ndash0027775 monthminus1 [21]119863 Birth rate of psyllid 378327ndash33526137 monthminus1 [20 22]1205732

Infected rate of psyllid mdash monthminus1 Estimate1205832

Nature death rate of psyllid 01169825ndash095052 monthminus1 [23]119898 Max abundance of psyllid per citrus 120ndash1000 mdash [24]

Table 2 Parameter functions for model (1) according to the values of Table 1

Parameter functions Value Reference

1205731(119905) 00042925 + 0003543 cos (212058711990512) Estimate

1205831(119905) 00034585 + 00007085 cos (212058711990512) [20]

119889 (119905) 0022221 + 0005554 cos (212058711990512) [21]

119863 (119905) 186547035 + 148714335 cos (212058711990512) [20 22]

1205732(119905) 0008779171 + 0004838437 cos (212058711990512) Estimate

1205832(119905) 053375125 + 041676875 cos (212058711990512) [23]

119887 (119905) = 119863 (119905) minus 1205832(119905) 18120952 + 1445466475 cos (212058711990512) [20 22 23]

119898 560 [24]

Consider the linear system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732 (119905) (119878

lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(38)

By Lemma 1 and the standard comparison principle we havethat there exists a positive120596-periodic function V

2(119905) such that

119869(119905) = exp(1199012119905)V2(119905) is a solution of System (38) where 119869(119905) =

(119868ℎ(119905) 119868V(119905))

119879 and

1199012=1

120596ln 119903 (Φ

119865minus119881minus119872120578

(120596)) (39)

It follows from 119903(Φ119865minus119881minus119872

120578

(120596)) gt 1 that 1199012gt 0 and 119869(119905) rarr

+infin as 119905 rarr +infin Applying the comparison principle [18] weknow that 119868

ℎ(119905) rarr +infin and 119868V(119905) rarr +infin as 119905 rarr +infin This

is a contradictionThus we have proved that (32) holds and119875is weakly uniformly persistent with respect to (119883

0 1205971198830)

According to the results of Lemma 7 we can easily obtainthat119875 has a global attractor119875

0 It is easy to obtain that119875

0is an

isolated invariant set in119883 and119882119904(1198750)cap1198830= 0We know that

1198750is acyclic in119872

120597and every solution in119872

120597converges to 119875

0

According to Zhao [19] we have that119875 is uniformly persistentwith respect to (119883

0 1205971198830) This implies that the solution of

(1) is uniformly persistent with respect to (1198830 1205971198830) Thus we

have that there exists a 120598 gt 0 such that lim inf119905rarr+infin

119868ℎ(119905) ge 120598

lim inf119905rarr+infin

119868V(119905) ge 120598

5 Numerical Simulationsand Sensitivity Analysis

In this section we will make numerical simulations bymeansof Matlab 71 to support our theoretical results to predictthe trend of the disease and to explore some control andprevention measures Numerical values of parameters ofsystem (1) are given in Table 1 (most of the data are taken fromthe literatures ([20ndash24]))

According to the periodicity of System (1) and Table 1 weset 1205831(119905) = 120572

0

1+ 1205720

2cos(212058711990512) where 1205720

2= (0004167 minus

000275)2 = 00007085 and 12057201= 000275 + 120572

0

2= 00034585

By the similar method we can obtain the other parameterfunctions of model (1) (see Table 2) For the simulations thatfollows we apply the parameters in Table 2 unless otherwisestated

ChooseΛ(119905) = 000265+000235 cos(212058711990512)Then fromLemma 5 we can compute 119877

0= 09844 lt 1 by means of

Matlab 71 FromTheorem 8 we obtain that the infected citruspopulation 119868

ℎ(119905) and the infected psyllid population 119868V(119905) of

system (1) are extinct (see Figures 1 and 2)Choose Λ(119905) = 0005 + 00035 cos(212058711990512) Then from

Lemma 5 we obtain that 1198770= 18342 gt 1 From Theorem 9

we have that the infected citrus population 119868ℎ(119905) and the

infected psyllid population 119868V(119905) of System (1) are permanence(see Figures 3 and 4)

From the formulae for 1198770 we can predict the general

tendency of the epidemic in a long term according to thecurrent situation which is presented in Figures 1 2 3 and 4From the first two figures we know that the epidemic of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Table 1 Parameter definitions and values used for numerical simulations of the Huanglongbing model

Parameter Definition Average value Unit ReferenceΛ The recruitment rate of citrus mdash monthminus1 Estimate1205731

Infected rate of citrus mdash monthminus1 Estimate1205831

Nature death rate of citrus 000275ndash0004167 monthminus1 [20]119889 Disease induced death rate of citrus 0016667ndash0027775 monthminus1 [21]119863 Birth rate of psyllid 378327ndash33526137 monthminus1 [20 22]1205732

Infected rate of psyllid mdash monthminus1 Estimate1205832

Nature death rate of psyllid 01169825ndash095052 monthminus1 [23]119898 Max abundance of psyllid per citrus 120ndash1000 mdash [24]

Table 2 Parameter functions for model (1) according to the values of Table 1

Parameter functions Value Reference

1205731(119905) 00042925 + 0003543 cos (212058711990512) Estimate

1205831(119905) 00034585 + 00007085 cos (212058711990512) [20]

119889 (119905) 0022221 + 0005554 cos (212058711990512) [21]

119863 (119905) 186547035 + 148714335 cos (212058711990512) [20 22]

1205732(119905) 0008779171 + 0004838437 cos (212058711990512) Estimate

1205832(119905) 053375125 + 041676875 cos (212058711990512) [23]

119887 (119905) = 119863 (119905) minus 1205832(119905) 18120952 + 1445466475 cos (212058711990512) [20 22 23]

119898 560 [24]

Consider the linear system

119889119868ℎ (119905)

119889119905= 1205731 (119905) (119878

lowast

ℎ(119905) minus 120578) 119868V (119905) minus (1205831 (119905) + 119889 (119905)) 119868ℎ (119905)

119889119868V (119905)

119889119905= 1205732 (119905) (119878

lowast

V (119905) minus 120578) 119868ℎ (119905) minus 1205832 (119905) 119868V (119905)

(38)

By Lemma 1 and the standard comparison principle we havethat there exists a positive120596-periodic function V

2(119905) such that

119869(119905) = exp(1199012119905)V2(119905) is a solution of System (38) where 119869(119905) =

(119868ℎ(119905) 119868V(119905))

119879 and

1199012=1

120596ln 119903 (Φ

119865minus119881minus119872120578

(120596)) (39)

It follows from 119903(Φ119865minus119881minus119872

120578

(120596)) gt 1 that 1199012gt 0 and 119869(119905) rarr

+infin as 119905 rarr +infin Applying the comparison principle [18] weknow that 119868

ℎ(119905) rarr +infin and 119868V(119905) rarr +infin as 119905 rarr +infin This

is a contradictionThus we have proved that (32) holds and119875is weakly uniformly persistent with respect to (119883

0 1205971198830)

According to the results of Lemma 7 we can easily obtainthat119875 has a global attractor119875

0 It is easy to obtain that119875

0is an

isolated invariant set in119883 and119882119904(1198750)cap1198830= 0We know that

1198750is acyclic in119872

120597and every solution in119872

120597converges to 119875

0

According to Zhao [19] we have that119875 is uniformly persistentwith respect to (119883

0 1205971198830) This implies that the solution of

(1) is uniformly persistent with respect to (1198830 1205971198830) Thus we

have that there exists a 120598 gt 0 such that lim inf119905rarr+infin

119868ℎ(119905) ge 120598

lim inf119905rarr+infin

119868V(119905) ge 120598

5 Numerical Simulationsand Sensitivity Analysis

In this section we will make numerical simulations bymeansof Matlab 71 to support our theoretical results to predictthe trend of the disease and to explore some control andprevention measures Numerical values of parameters ofsystem (1) are given in Table 1 (most of the data are taken fromthe literatures ([20ndash24]))

According to the periodicity of System (1) and Table 1 weset 1205831(119905) = 120572

0

1+ 1205720

2cos(212058711990512) where 1205720

2= (0004167 minus

000275)2 = 00007085 and 12057201= 000275 + 120572

0

2= 00034585

By the similar method we can obtain the other parameterfunctions of model (1) (see Table 2) For the simulations thatfollows we apply the parameters in Table 2 unless otherwisestated

ChooseΛ(119905) = 000265+000235 cos(212058711990512)Then fromLemma 5 we can compute 119877

0= 09844 lt 1 by means of

Matlab 71 FromTheorem 8 we obtain that the infected citruspopulation 119868

ℎ(119905) and the infected psyllid population 119868V(119905) of

system (1) are extinct (see Figures 1 and 2)Choose Λ(119905) = 0005 + 00035 cos(212058711990512) Then from

Lemma 5 we obtain that 1198770= 18342 gt 1 From Theorem 9

we have that the infected citrus population 119868ℎ(119905) and the

infected psyllid population 119868V(119905) of System (1) are permanence(see Figures 3 and 4)

From the formulae for 1198770 we can predict the general

tendency of the epidemic in a long term according to thecurrent situation which is presented in Figures 1 2 3 and 4From the first two figures we know that the epidemic of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

002

004

006

008

01

012

0 500 1000 1500 2000 25000

t (month)

R0 = 09844

Ih(t)

(a)

0 100 200 300 400 5000

002

004

006

008

01

012

t (month)

R0 = 09844

Ih(t)

(b)

Figure 1 Graphs of numerical simulations of (1) showings the tendency of the infected citrus population (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

R0 = 09844

0

02

04

06

08

1

12

14

Iv(t)

0 500 1000 1500 2000 25000

t (month)

(a)

02

04

06

08

1

12

14

R0 = 09844

Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 2 It is similar to Figure 1

Huanglongbing can be rising in a short time but cannot beoutbreak with the current prevention and control measuresFrom Figures 3 and 4 we can see that the epidemic ofHuanglongbing dropped heavily after 100 months whilethere is still tendency to a stable periodic solution in a longtime

Next we perform some sensitivity analysis to determinethe influence 119877

0on the parameters Λ(119905) 120573

1(119905) and 120573

2(119905)

We choose function Λ(119905) = Λ01+ Λ0

2cos(212058711990512) where

Λ0

1 Λ02denote the average and amplitude values of Λ(119905)

respectively and Λ0

1= (112) int

12

0Λ(119905)119889119905 From Figure 5

we can observe that the blue line is linear relation between1198770and Λ

0

2 and 119877

0increases as Λ0

2increases The red

curve reflects the influence of the average value of Λ(119905) on1198770 Figure 5 shows that Λ0

1is more sensitive than Λ

0

2on

the basic reproduction number 1198770 Therefore in the real

world decreasing the average recruitment rate of citrus is thevaluable way to control Huanglongbing

Now we consider the combined influence of 1205731(119905) and

1205732(119905) on 119877

0 Set Λ(119905) = 00027 + 000235 cos(212058711990512)

1205731(119905) = 119886

1+ 1198871cos(212058711990512) and 120573

2(119905) = 119886

2+ 1198872cos(212058711990512)

Moreover we know that 1198861

= (112) int12

01205731(119905)119889119905 and

1198862= (112) int

12

01205732(119905)119889119905 Other parameters can be seen in

Table 2

Case (I) We fix 1198871= 0003543 and 119887

2= 0004838437

and let 1198861vary from 000001 to 0015 and 119886

2from 000001

to 002 For this case it is interesting to examine how theaverage values of adequate contact rate 120573

1(119905) and 120573

2(119905) affect

the basic reproduction number 1198770 Numerical results shown

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

R0 = 18342

006

008

01

012

014

016

018

02Ih(t)

0 500 1000 1500 2000 2500t (month)

(a)

006

008

01

012

014

016

018

02

R0 = 18342

Ih(t)

0 100 200 300 400 500t (month)

(b)

Figure 3 The figures show that the infected citrus population is permanence (a) 119905 isin [0 2500] (b) 119905 isin [0 500]

02

04

06

08

1

12

14

16

18

2

R0 = 18342

0 500 1000 1500 2000 25000

t (month)

Iv(t)

(a)

R0 = 18342

02

04

06

08

1

12

14

16

18

2Iv(t)

0 100 200 300 400 5000

t (month)

(b)

Figure 4 It is similar to Figure 3

in Figure 6 imply that the basic reproduction number1198770may

be less than 1 when 1198861or 1198862is small enough And the results

also imply that 1198770increases as 119886

1and 1198862increase Further we

can observe that from Figure 6(i)the smaller the values of 1198861

or 1198862are the more sensitive 119877

0is (ii) increasing 119886

2may be

more sensitive for 1198770when 119886

1is fixed (iii) increasing 119886

1may

be more sensitive for 1198770when 119886

2is fixed

Case (II) We fix 1198861= 00042925 and 119886

2= 000877917 and

let 1198871vary from 0000001 to 0005 and 119887

2from 0000002 to

0006Then we obtain the result of numerical simulation andit is shown in Figure 7 Obviously Figure 7 shows that 119877

0is

linearly related to both 1198871and 119887

2with the pattern that 119877

0

decreases to a relatively small value (less than 1) only when1198871and 1198872are very small

By the above graphs of the basic reproduction number1198770on the average values of recruitment rate of citrus Λ(119905)

and adequate contact rate 1205731(119905) 1205732(119905) we know that the basic

reproduction number 1198770is a monotonic increasing function

by the average values From the sensitivity analysis diagramswe observe that 119877

0falls to less than 1 by decreasing the values

of those parameters

6 Conclusion

In this paper we have analyzed a HLB transmission modelwith Logistic growth in periodic environments It is provedthat 119877

0is the threshold for distinguishing the disease extinc-

tion or permanence The disease-free periodic solution is

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

0 1 2 3 4 502

04

06

08

1

12

14

16

18

2

R0

times10minus3

Λ0

1= 00027

Λ0

2= [0000001 0005]

Λ0

1= [0000001 0005]Λ0

2= 000235

Λ0

1or Λ0

2

Figure 5 The graph shows the sensitivity of the basic reproductionnumber 119877

0to the changes of Λ(119905)

00005

0010015

000050010015

0020

05

1

15

2

25

3

R0

a1a2

Figure 6 The graph of 1198770in terms of 119886

1and 119886

2

globally asymptotically stable and the disease dies out when1198770lt 1 When 119877

0gt 1 the disease persists

The numerical simulations shown in Figure 5 show thatthere are some parameter ranges of Λ

1and Λ

2such that the

threshold parameter 1198770is smaller than 1 It indicates a useful

way to eradicate Huanglongbing by limiting the recruitmentof citrus including the average value and amplitude ofrecruitment function

The results shown in Figure 6 (Figure 7) show that if theamplitudes of infected functions 119887

1 1198872(the average infected

rate 1198861 1198862) are fixed we can control the infection of citrus

and psyllid by limiting the average infected rates 1198861 1198862(the

amplitudes of infected functions 1198871 1198872)

According to the above theoretical analysis andnumericalsimulations we can conclude that the recruitment of citrusand the infection of citrus and psyllid have significanteffects on Huanglongbing transmission In order to preventthe epidemic disease from generating endemic making anappropriate reduction of the recruitment rate of citrus and

01

23

45

02

46

09

1

11

12

13

b1b2

R0

times10minus3

times10minus3

Figure 7 The graph of 1198770in terms of 119887

1and 1198872

decreasing the contact rate between psyllid and the citrus areeffective measures to control Huanglongbing

Disclosure

The paper is approved by all authors for publication Theauthors would like to declare that the work described wasoriginal research that has not been published previously andnot under consideration for publication elsewhere

Conflict of Interests

No conflict of interests exists in the submission of this paper

Acknowledgments

The research has been supported by the Natural ScienceFoundation of China (11261004) the Natural Science Foun-dation of Jiangxi Province (20122BAB211010) the Scienceand Technology Plan Projects of Jiangxi Provincial EducationDepartment (GJJ13646) and the Postgraduate InnovationFund of Jiangxi Province (YC2012-S121)

References

[1] P Christou and R M Twyman ldquoThe potential of geneticallyenhanced plants to address food insecurityrdquo Nutrition ResearchReviews vol 17 no 1 pp 23ndash42 2004

[2] FAO The State of Food Insecurity in the World (SOFI) FAORome Italy 2000 httpwwwfaoorgFOCUSESOFI00sofi001-ehtm

[3] C James ldquoGlobal food securityrdquo in International Congress ofPlant Pathology Pittsburgh Pa USA August 1998

[4] J M Bove ldquoHuanglongbing a destructive newly-emergingcentury-old disease of citrusrdquo Journal of Plant Pathology vol88 no 1 pp 7ndash37 2006

[5] S E Halbert and K L Manjunath ldquoAsian citrus psyllids (Stern-orrhyncha Psyllidae) and greening disease of citrus a literaturereview and assessment of risk in Floridardquo Florida Entomologistvol 87 no 3 pp 330ndash353 2004

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[6] S Parnell T R Gottwald C A Gilligan N J Cunniffe and FVan Den Bosch ldquoThe effect of landscape pattern on the optimaleradication zone of an invading epidemicrdquo Phytopathology vol100 no 7 pp 638ndash644 2010

[7] R FMizell III C Tipping P C Andersen B V BrodbeckW BHunter and T Northfield ldquoBehavioral model for Homalodiscavitripennis (Hemiptera Cicadellidae) optimization of hostplant utilization andmanagement implicationsrdquo EnvironmentalEntomology vol 37 no 5 pp 1049ndash1062 2008

[8] G A Braga S Ternes et al ldquoModelagem Matematica daDinamica TemporaldoHLB emCitrosrdquo in Proceedings of the 8thCongresso Brasileiro de Agroinformatica Bento Goncalves 2011

[9] DGHall andMGHentz ldquoSeasonal flight activity by theAsiancitrus psyllid in east central Floridardquo Entomologia et Applicatavol 139 no 1 pp 75ndash85 2011

[10] X Song and A U Neumann ldquoGlobal stability and periodicsolution of the viral dynamicsrdquo Journal ofMathematical Analysisand Applications vol 329 no 1 pp 281ndash297 2007

[11] K Wang Z Teng and H Jiang ldquoOn the permanence forn-species non-autonomous Lotka-Volterra competitive systemwith infinite delays and feedback controlsrdquo International Journalof Biomathematics vol 1 no 1 pp 29ndash43 2008

[12] F Zhang and X-Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[13] T Zhang and Z Teng ldquoOn a nonautonomous SEIRS model inepidemiologyrdquo Bulletin of Mathematical Biology vol 69 no 8pp 2537ndash2559 2007

[14] Z Teng and Z Li ldquoPermanence and asymptotic behavior of thetv-species nonautonomous lotka-volterra competitive systemsrdquoComputers and Mathematics with Applications vol 39 no 7-8pp 107ndash116 2000

[15] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[16] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[17] Y Nakata Permanence and Global Asymptotic Stability ForPopulation Models in Mathematical Biology Waseda UniversityTokyo Japan 2010

[18] H Smith and P Waltman The Theory of the Chemostat Cam-bridge University Press Cambridge Mass USA 1995

[19] X ZhaoDynamical Systems in Population Biology Spring NewYork NY USA 2003

[20] X M Deng ldquoFormming process and basis and technologicalpoints of the theory emphasis on control citrus psylla for inte-grated control Huanglongbingrdquo Chinese Agricultural ScienceBulletin vol 25 no 23 pp 358ndash363 2009 (Chinese)

[21] T Li C Z Cheng et al ldquoDetection of the bearing rate ofliberobacter asiaticum in citrus psylla and its host plantrdquo ActaAgriculturae Universitatis Jiangxiensis vol 29 no 5 pp 743ndash745 2007 (Chinese)

[22] G F Chen and X M Deng ldquoDynamic observation adult citruspsyllid quantity live through the winter in spring and winterrdquoSouth China Fruits vol 39 no 4 pp 36ndash38 2010 (Chinese)

[23] XMDeng G F Chen et al ldquoThe newly process of Huanglong-bing in citrusrdquo Guangxi Horticulture vol 17 no 3 pp 49ndash512006

[24] R G drsquoA Vilamiu S Ternes B A Guilherme et al ldquoA modelfor Huanglongbing spread between citrus plants includingdelay times and human interventionrdquo in Proceedings of theInternational Conference of Numerical Analysis and AppliedMathematics (ICNAAM rsquo12) vol 1479 pp 2315ndash2319 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of