modelling the effects of snail control and health...

Research ArticleModelling the Effects of Snail Control and Health Education inClonorchiasis Infection in Foshan China

Shujing Gao 1 Ruixia Yuan2 Yujiang Liu1 and Xinzhu Meng 3

1Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques Gannan Normal UniversityGanzhou 341000 China2College of Business Economics Shanghai Business School Shanghai 200336 China3College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Shujing Gao gaosjmath126com

Received 18 January 2019 Revised 11 May 2019 Accepted 16 June 2019 Published 9 July 2019

Academic Editor Hassan Zargarzadeh

Copyright copy 2019 Shujing Gao 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

Clonorchiasis is the most important food-borne parasitic disease in China In this paper a mathematical model of clonorchiasissinensis is proposed to mimic its transmission dynamics to assess the effects of intervention strategies such as snail control healtheducation and chemotherapy A threshold dynamics in terms of the basic reproductive number 1198770 has been established that is if1198770 lt 1 the disease dies out and the disease-free periodic solution is globally asymptotically stable and if 1198770 gt 1 then the diseasebreaks out The effects of different control measures are compared by numerical simulations The numerical results suggest that itis necessary to strengthen health education and improve faeces management and illustrate that snail control is the most effectiveway to be implemented in the clonorchiasis sinensis control in Foshan

1 Introduction

Clonorchiasis sinensis (C sinensis) is a major food-borneparasitosis which is actively infected in China the Demo-cratic Peoplersquos Republic of Korea the Republic of KoreaRussia and Viet Nam [1] Currently it is estimated that morethan 200 million people are at risk of infection [2] andabout 35 million people are infected globally including 15million in China [3] There are two major endemic regionsin Chinamdashnamely provinces in the southeast includingGuangdong and Guangxi and provinces in the northeastsuch asHeilongjiang and Jilin [4] Human beings are infectedthrough ingestion of raw or undercooked fish which containsthe metacercariae of liver flukes [5ndash7] The clonorchiasisinfection (CI) may cause serious liver and biliary systemdamage and affect many sectors of human health [4 8]

Foshan is located in the central of Pearl River Deltain Guangdong It enjoys subtropical monsoon climate withwarm weather abundant sunshine and rainfall Agricultureis based on rice cultivation and freshwater aquaculture indus-try Mulberry fish pond is above 50 of the total farmlandarea Local residents like to eat raw fish Additionally coupled

with misconceptions such as the belief that consumption ofalcohol or spicy food can prevent infection [9] Foshan hasbeen becoming a heavily clonorchiasis-endemic area There-fore in order to reduce theCI rate a number of strategies havebeen proposed including chemotherapy (morbidity controlwith praziquantel) information education communicationand faeces management (sanitation improvement and animalmanagement) In [10] it is reported that after the implementof integrated control strategies the CI rate of population inFoshan decreased dramatically from 5684 in 1989 to 347in 2000 But the rate shows the resurgence since 2005 owingto the general investigation project eased from 2001 to 2004and the tradition of consuming raw freshwater fish or shellfishresuscitated (see Figure 1) This shows that health educationdoes play important role in the control of C sinensis

The amphibious snail is one of intermediate hosts of Csinensis As far as we know snail control with molluscicideis one of important control measures for schistosomiasis buthas not been widely applied in practice for clonorchiasiscontrol However some biologists have tried to study therole of snail-killing in controlling the spread of C sinensisFor example Yang [11] carried out comprehensive measures

HindawiComplexityVolume 2019 Article ID 5878424 14 pageshttpsdoiorg10115520195878424

2 Complexity

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Figure 1 Positive infection rate in Foshan from 1980 to 2010

in Daliang Village Yangshan County Guangdong Provinceincluding health education chemotherapy and snail controlwith molluscicide The CI rate decreased from 482 in 1975to 29 in 1976 to 0 in 1979-1982 Therefore it is of greatpractical significance to assess different tools and strategiesfor large-scale control of clonorchiasis by using mathematicalmodelling

There have been studies of modeling clonorchiasis asso-ciated with the human being in China [12 13] However noneof the studies were to quantitatively consider the relationshipbetween CI rate and the number of molluscicide to spraywhen chemotherapy and health education are implementedin combination Mathematical models can be important fordetermining the optimal number of molluscicide spraying soas to control the transmission of clonorchiasis In this papera clonorchiasis spread model with an integrated strategyis developed and studied in which seasonal variation andimpulsive control are considered

This paper is organized as follows In Section 2 wepropose a C sinensis model with pulse snail-killing healtheducation and chemotherapy control in periodic environ-ment Some results are stated which will be essential toour proof We calculate the basic reproductive number forthe model in Section 3 In Section 4 we illustrate that thebasic reproductive number serves as a threshold parameterthat determines the disease to be extinction or endemicIn Section 5 different control programs are compared andsensitivity analysis is done to evaluate the snails controlstrategy by numerical simulations Section 6 gives a briefdiscussion and future work

2 Model and Preliminaries

Clonorchiasis belongs to vector-borne disease including twointermediate hosts snail and fish C sinensis is transmit-ted indirectly among the human hosts first intermediate

snails and second intermediate fishes in the sense that free-swimming stages (miracidia cercariae) and ingestion stage(metacercariae) are interposed In addition to human beingsspecially many mammals such as dogs cats pigs rodentsfoxes and possibly any fish eating mammal can serve asdefinitive hosts for C sinensis and humans are the maindefinitive hosts [14] In order to simplify the mathematicalmodel we only consider human hosts here Figure 2 showsa schematic description of the transmission of clonorchissinensis in definitive host snail host and fish host

In this section we mainly formulate an impulsive epi-demic model to describe the transmission dynamics of Csinensis

We firstly formulate a model for the spread of C sinensisincorporating health education and chemotherapy strategiesThe total human population at any time 119905 denoted by 119873(119905)is the sum of individual populations in each compartmentwhich includes susceptible 119878ℎ(119905) infected 119868ℎ(119905) and recovered119877ℎ(119905) We assume that the total human population remains aconstant denoting 119873ℎ(119905) = 119873ℎ for all 119905 ge 0

Following the idea of Dai andGao [13] we divide the snailpopulation and fish population into disjoint classes suscepti-ble (119878119904 119878119891) and infected (119868119904 119868119891) respectively We suppose thatthe infection rates of susceptible human susceptible snailand susceptible fish are described by1205731 (119905) 119878ℎ (119905) 119868119891 (119905) 1205732 (119905) 119878119904 (119905) 119868ℎ (119905) 1205733 (119905) 119878119891 (119905) 119868119904 (119905) (1)

respectively where 1205731(119905) is per capita (successful) infectionrate of human hosts by one fish at time 119905 1205732(119905) is per capita(successful) infection rate of snails by one human at time 119905and 1205733(119905) is per capita (successful) infection rate of fish byone snail at time 119905

Complexity 3

Susceptible Infected

Definitive host population

SusceptibleInfected

Snail population


Fish population

Metacercariae

Wormrsquosegg

Miracidia

Cercariae

Figure 2 A transmission diagram of Clonorchis sinensis

Some residents would temporarily change their badeating habit owing to health education Let 120572(119905) denote theproportion of human host at time 119905 from susceptible torecovered human hosts owing to health education 120574(119905) is therecovery rate due to chemotherapy and 120575(119905) is the proportionof human host at time 119905 of transition from recovered tosusceptible human hosts due to the loss of heath awarenessand misconceptions Suppose that sail population and fishpopulation do not result in death and increase at periodicrecruitment rates Λ 2(119905) and Λ 3(119905) and also decrease at thenatural death rates1205832 (119905) and1205833(119905) respectively Incorporatingabove assumptions we have the following model in whichseasonal variation and control measures (health educationand chemotherapy) are taken into consideration

1198781015840ℎ (119905) = 1205831 (119905)119873ℎ (119905) minus 1205731 (119905) 119878ℎ (119905) 119868119891 (119905) minus 120572 (119905) 119878ℎ (119905)minus 1205831 (119905) 119878ℎ (119905) + 120575 (119905) 119877ℎ (119905) 1198681015840ℎ (119905) = 1205731 (119905) 119878ℎ (119905) 119868119891 (119905) minus 120574 (119905) 119868ℎ (119905) minus 1205831 (119905) 119868ℎ (119905) 1198771015840ℎ (119905) = 120572 (119905) 119878ℎ (119905) + 120574 (119905) 119868ℎ (119905) minus 1205831 (119905) 119877ℎ (119905)minus 120575 (119905) 119877ℎ (119905) 1198781015840119904 (119905) = Λ 2 (119905) minus 1205732 (119905) 119878119904 (119905) 119868ℎ (119905) minus 1205832 (119905) 119878119904 (119905) 1198681015840119904 (119905) = 1205732 (119905) 119878119904 (119905) 119868ℎ (119905) minus 1205832 (119905) 119868119904 (119905) 1198781015840119891 (119905) = Λ 3 (119905) minus 1205733 (119905) 119878119891 (119905) 119868119904 (119905) minus 1205833 (119905) 119878119891 (119905) 1198681015840119891 (119905) = 1205733 (119905) 119878119891 (119905) 119868119904 (119905) minus 1205833 (119905) 119868119891 (119905)

(2)

where all parameters are positive periodic andcontinuousfunctions with period 12 (months)

Since 119877ℎ(119905) = 119873ℎ minus 119878ℎ(119905) minus 119868ℎ(119905) then (2) can yield1198781015840ℎ (119905) = (1205831 (119905) + 120575 (119905))119873ℎ minus 1205731 (119905) 119878ℎ (119905) 119868119891 (119905)minus (120572 (119905) + 1205831 (119905) + 120575 (119905)) 119878ℎ (119905)minus 120575 (119905) 119868ℎ (119905) 1198681015840ℎ (119905) = 1205731 (119905) 119878ℎ (119905) 119868119891 (119905) minus 120574 (119905) 119868ℎ (119905) minus 1205831 (119905) 119868ℎ (119905) 1198781015840119904 (119905) = Λ 2 (119905) minus 1205732 (119905) 119878119904 (119905) 119868ℎ (119905) minus 1205832 (119905) 119878119904 (119905) 1198681015840119904 (119905) = 1205732 (119905) 119878119904 (119905) 119868ℎ (119905) minus 1205832 (119905) 119868119904 (119905) 1198781015840119891 (119905) = Λ 3 (119905) minus 1205733 (119905) 119878119891 (119905) 119868119904 (119905) minus 1205833 (119905) 119878119891 (119905) 1198681015840119891 (119905) = 1205733 (119905) 119878119891 (119905) 119868119904 (119905) minus 1205833 (119905) 119868119891 (119905)

(3)

Snail control is conducted withmolluscicide in fish pondsand canals Note that molluscicides are most commonlysprayed to be taken on different dose and different time-interval in a year The phenomenon exhibits impulsive effectson the transmission of C sinensis Suppose the number ofmolluscicide to spray is 119902 in one year 119905119896 (119896 isin N) is the pulsetime and the elimination rate of snails at time 119905119896 is 120579119896 Thenwe have impulsive equations119878119904 (119905+) = (1 minus 120579119896) 119878119904 (119905) 119868119904 (119905+) = (1 minus 120579119896) 119868119904 (119905)

for 119905 = 119905119896 (4)

where 119878119904(119905+) = limℎ997888rarr0+119878119904(119905 + ℎ) 119868119904(119905+) = limℎ997888rarr0+119868119904(119905 + ℎ)and 120579119896+119902 = 120579119896 119905119896+119902 = 119905119896 + 12 (119896 isin N)

The system consisting of (3) and (4) is an impulsivedifferential system For simplicity we will refer to the systemas ldquothe model systemrdquo in the rest of this paper

4 Complexity

Based on the biological background of the model systemwe always assume that all solutions of the model systemsatisfy the following initial conditions119878119894 (0+) ge 0 and 119868119894 (0+) ge 0 for 119894 = ℎ 119904 119891 (5)

It is not difficult to prove that the positive cone of 1198776+ isflow invariant relative to the model system

Let119873119904(119905) and119873119891(119905) denote the density of sail populationand fish population at time 119905 respectively From (3) and (4)we have1198731015840

s (119905) = Λ 2 (119905) minus 1205832 (119905)119873119904 (119905) 119905 = 119905119896 119896 isin N119873119904 (119905+) = (1 minus 120579119896)119873119904 (119905) 119905 = 119905119896 119896 isin N (6)

and 1198731015840119891 (119905) = Λ 3 (119905) minus 1205833 (119905)119873119891 (119905) (7)

For a continuous positive 120596minusperiodic function 119891(119905) weset 119891119872 = sup119905isin[0120596]119891(119905) and 119891119871 = inf 119905isin[0120596]119891(119905)Define a setΩ = (119878ℎ (119905) 119868ℎ (119905) 119878119904 (119905) 119868119904 (119905) 119878119891 (119905) 119868119891 (119905))isin 1198776+ | 0 le 119878ℎ (119905) + 119868ℎ (119905) le 119873ℎ 0 le 119878119904 (119905) + 119868119904 (119905)le 119873lowast

119904 0 le 119878119891 (119905) + 119868119891 (119905) le 119873lowast119891 (8)

where119873lowast119904 = Λ1198722 1205831198712 and119873lowast

119891 = Λ1198723 1205831198713 Lemma 1 Ω is a positively invariant set of the model system

Proof Let (119878ℎ(119905) 119868ℎ(119905) 119878119904(119905) 119868119904(119905) 119878119891(119905) 119868119891(119905)) be any positivesolution of the model system with initial condition (5)

For system (6) we consider the following auxiliarysystem 1198671015840

119904 (119905) = Λ 2 (119905) minus 1205832 (119905)119867119904 (119905) 119867119904 (0+) = 119873119904 (0+) (9)

Thus we have1198671015840119904 (119905) le Λ1198722 minus 1205831198712119867119904 (119905) lt 0 if 119867119904 (119905) gt Λ11987221205831198712 = 119873lowast

119904 (10)

Then it follows from comparison theorem we havelim sup119905997888rarr+infin119873119904(119905) le 119873lowast

119904 Applying similar methodwe can obtain that lim sup119905997888rarr+infin119873119891(119905) le 119873lowast

119891 This completesthe proof

Lemma 2 Consider the following impulsive differential equa-tion 1199111015840 (119905) = 119886 (119905) minus 119887 (119905) 119911 (119905) 119905 = 119905119896119911 (119905+) = (1 minus 120579119896) 119911 (119905) 119905 = 119905119896 (11)

where 119886(119905) and 119887(119905) are continuous and positive 120596-periodicfunctions and there is a positive integer 119902 such that 120579119896+119902 = 120579119896

and 119905119896+119902 = 119905119896 + 120596 for all 119896 isin 119873 Then there exists a uniquepositive periodic solution of system (11)

119911lowast (119905) = 119894prod119897=1(1 minus 120579119897) 119911lowast0 119890int119905119899120596+1199050 minus119887(120591)119889120591

+ 119890int119905119899120596+119905119894 minus119887(120591)119889120591 int119905119899120596+119905119894

119886 (119904) 119890int119904119899120596+119905119894 119887(120591)119889120591119889119904+ 119894sum119897=1

119894prod119895=l(1 minus 120579119895) 119890int119905119899120596+119905119895minus1 minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(12)

for 119899120596+ 119905119894 lt 119905 le 119899120596+ 119905119894+1 119894 = 0 1 119902 minus 1 which is globallyasymptotically stable where

119911lowast0 = [1 minus 119902prod119897=1(1 minus 120579119897) 119890int119899120596+119905119902119899120596+1199050 minus119887(120591)119889120591]minus1

sdot 119902sum119897=1

119902prod119895=119897(1 minus 120579119895) 119890int119899120596+119905119902119899120596+119905119895minus1

minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(13)

The proof of Lemma 2 can be seen in the AppendixBy Lemma 2 it is easy to see that the following conclusion

holds true

Theorem 3 The model system always has a unique disease-free periodic solution (119878lowastℎ(119905) 0 119878lowast119904 (119905) 0 119878lowast119891(119905) 0)

Before proving the main results we introduce the follow-ing notations

Let (119877119899 119877119899+) be the standard ordered n-dimensionalEuclidean space with a norm sdot For 984858 ] isin 119877119899 we denote984858 ge ] if 984858 minus ] isin 119877119899+ 984858 gt ] if 120583 minus ] isin 119877119899+ 0 and 984858 ≫ ] if984858 minus ] isin Int (119877119899+) respectively

LetΨ119860(119905 119904) (119905 ge 119904) be the evolution operator of the linear120596-periodic system (119905) = 119860 (119905) 119909 (119905) 119909 isin R119899 (14)

Consider the linear impulsive system1199091015840 (119905) = 119860 (119905) 119909 119905 = 119905119896 119896 isin N119909 (119905+) = 119875119896119909 (119905119896) 119905 = 119905119896 119896 isin N119909 (0+) = 119909 (0) = 1199090 1199050 = 0 (15)

which satisfies the following three conditions

(a1) 119860(sdot) isin C(RR119899times119899) 119860(119905 + 120596) = 119860(119905) where 120596 is apositive real number

Complexity 5

(a2) 119875119896 isin R119899times119899 det119875119896 = 0 119905119896 lt 119905119896+1 (119896 isin N) andlim119896997888rarr+infin 119905119896 = +infin

(a3) There exists 119902 isin N+ such that 119875119896+119902 = 119875119896 119905119896+119902 = 119905119896 + 120596(119896 isin N)

Denote

Φ119860119875119896 (120596) fl 119902prod119894=1(119875119902minus119894+1Ψ119860 (119905119902minus119894+1 119905119902minus119894)) (16)

By the Perron-Frobenius theorem its spectral radius119903(Φ119860119875119896(120596)) is the principal eigenvalue of Φ119860119875119896(120596) inthe sense that it is simple and admits an eigenvector]lowast ≫ 0

The following lemma is useful for our future discussion

Lemma 4 (see [15]) Let 120578 = (1120596) ln 119903(Φ119860119875119896(120596))Then thereexists a positive 120596minusperiodic function V(119905) such that 119890120578119905V(119905) is asolution of system (15)

3 The Basic Reproductive Number 1198770To use the computation approach of the basic reproductivenumber (see [15]) we set 119909 = (1199091 1199092 1199093 1199094 1199095 1199096)119879 =(119868ℎ 119868119904 119868119891 119878ℎ 119878119904 119878119891)119879 and rewrite system (3) as1199091015840 (119905) = F (119905 119909 (119905)) minusV (119905 119909 (119905)) (17)

whereF is the new infection rate andV is the decay rate ortransfer rate and

F (119905 119909) =((((((((

1205731 (119905) 1199094 (119905) 1199093 (119905)1205732 (119905) 1199095 (119905) 1199091 (119905)1205733 (119905) 1199096 (119905) 1199092 (119905)000))))))))

V (119905 119909) =((((((((

(1205831 (119905) + 120574 (119905)) 1199091 (119905)1205832 (119905) 1199092 (119905)1205833 (119905) 1199093 (119905)minus (1205831 (119905) + 120575 (119905))119873ℎ + 1205731 (119905) 1199094 (119905) 1199093 (119905) + (120572 (119905) + 1205831 (119905) + 120575 (119905)) 1199094 (119905) + 120575 (119905) 1199091 (119905)minusΛ 2 (119905) + 1205732 (119905) 1199095 (119905) 1199091 (119905) + 1205832 (119905) 1199095 (119905)minusΛ 3 (119905) + 1205733 (119905) 1199096 (119905) 1199092 (119905) + 1205833 (119905) 1199096 (119905)))))))))

(18)

We call the rearranged model system as ldquonew modelrdquoAccording to Lemma 2 we easily know that the modelsystem has a unique disease-free periodic solution 119909lowast(119905) =(119878lowastℎ(119905) 0 119878lowast119904 (119905) 0 119878lowast119891(119905) 0)Then 119909lowast(119905) = (0 0 0 119878lowastℎ (119905) 119878lowast119904 (119905)119878lowast119891(119905)) is the unique disease-free periodic solution of the newmodel

Let 119862+120596 = 120601 isin 119862(119877 1198773+) | 120601(119905 + 120596) = 120601(119905) for all 119905 isin 119877which is equipped with maximum norm sdot infin It follows thatthe new infection matrix at 119909lowast(119905) is119865 (119905) = (120597F119894 (119905 119909lowast (119905))120597119909119895 )

1le119894 119895le3

= ( 0 0 1205731 (119905) 119878lowastℎ (119905)1205732 (119905) 119878lowast119904 (119905) 0 00 1205733 (119905) 119878lowast119891 (119905) 0 ) (19)

and the evolution of the initial infective members introducedat 119909lowast(119905) is described by

119910 (119905) = minus119881 (119905) 119910 119905 = 119905119896 119896 isin N119910 (119905+119896 ) = 119875119896119910 (119905119896) 119905 = 119905119896 119896 isin N (20)

Let119884(119905 119904) be the evolution operator of (20) and define a linearoperator 119871 on 119862+12 by(119871120601) (119905) = int+infin

0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886forall119905 isin 119877 (21)

It follows from [15] that the basic reproductive number 1198770 ofthe model system is given by 1198770 = 119903(119871)

6 Complexity

Following [15] by direct computation we have

119875119896 = (1 0 00 1 minus 120579119896 00 0 1) 119876119896 = (1 0 00 1 minus 120579119896 00 0 1 )

(22)

119872(119905) = (120597119891119894 (119905 119909lowast (119905))120597119909119895 )4le119894 119895le6

= (minus1205831 (119905) minus 120572 (119905) minus 120575 (119905) 0 00 minus1205832 (119905) 00 0 minus1205833 (119905)) (23)

and

119881 (119905) = (120597V119894 (119905 119909lowast (119905))120597119909119895 )1le119894 119895le3

= (1205831 (119905) + 120574 (119905) 0 00 1205832 (119905) 00 0) 1205833 (119905)) (24)

Obviously 119903(Φ119872119876119896(12)) lt 1 and 119903(Φminus119881119875119896(12)) lt 1 and thenthe assumptions (1198671) minus (1198678) in [15] hold By Theorem 32 in[15] we can obtain that the following result

Theorem5 For the model system the following statements arevalid

(i) 1198770 = 1 lArrrArr 119903(Φ(119865minus119881)119875119896(12)) = 1(ii) 1198770 gt 1 lArrrArr 119903(Φ(119865minus119881)119875119896(12)) gt 1(iii) 1198770 lt 1 lArrrArr 119903(Φ(119865minus119881)119875119896(12)) lt 1

4 Main Results

In this section we show that the basic reproductive numberis a threshold parameter that determines dynamics of thedisease The first result shows that C sinensis dies out if 1198770 lt1Theorem 6 The disease-free periodic solution 119909lowast(119905) =(119878lowastℎ(119905) 0 119878lowast119904 (119905) 0 119878lowast119891(119905) 0) of the model system is asymptoti-cally stable if 1198770 lt 1 and unstable if 1198770 gt 1

Proof The linearized system of the newmodel at the disease-free periodic solution 119909lowast(119905) is1199091015840 (119905) = (119865 (119905) minus 119881 (119905) 0119860 (119905) 119872 (119905)) 119909 (119905) 119905 = 119905119896 119896 isin N119909 (119905+) = (119875119896 0119861119896 119876119896)119909 (119905) 119905 = 119905119896 119896 isin N

(25)

where 119865(119905) 119881(119905)119872(119905) 119875119896 119876119896 are defined in (19) (22)-(24)and119860(119905) and 119861119896 are zeromatrices Then the Floquet multipli-ers of system (25) are the eigenvalues of 119903(Φ(119865minus119881)119875119896(12)) and119903(Φ119872119876119896(12)) By Theorem 5 we have all Floquet multipliersof system (25) are less than 1 provided that 1198770 lt 1 Thereforethe disease-free periodic solution 119909lowast(119905) of the model systemis asymptotically stable And if 1198770 gt 1 it is unstable Thiscompletes the proof

Theorem 7 If 1198770 lt 1 then the disease-free periodic solution119909lowast(119905) of the model system is globally asymptotically stable

Proof By Theorem 6 we know that if 1198770 lt 1 then 119909lowast(119905)is locally asymptotically stable Thus we only show that itattracts all nonnegative solution of the model system

ByTheorem 5 we know 1198770 lt 1 implies 119903(Φ(119865minus119881)119875119896(12)) lt1Thus there is a sufficiently small 120576 gt 0 such that119903 (Φ(119865minus119881+119872120576)119875119896 (12)) lt 1 (26)

where

119872120576 = ( 0 0 1205731 (119905) 1205761205732 (119905) 120576 0 00 1205733 (119905) 120576 0 ) (27)

Let (119878ℎ(119905) 119868ℎ(119905) 119878119904(119905) 119868119904(119905) 119878119891(119905) 119868119891(119905)) be any solution of themodel system In view of Lemma 1 the first equation ofsystem (3) yields1198781015840ℎ (119905) le (1205831 (119905) + 120575 (119905))119873ℎminus (120572 (119905) + 1205831 (119905) + 120575 (119905)) 119878ℎ (119905) (28)

Consider the following comparison system1198781015840ℎ (119905) = (1205831 (119905) + 120575 (119905))119873ℎminus (120572 (119905) + 1205831 (119905) + 120575 (119905)) 119878ℎ (119905) 119878ℎ (0+) = 119878ℎ (0+) (29)

By [16] we know that the first equation of (29) admits apositive periodic solution 119878lowastℎ (119905) which is globally asymptot-ically stable Thus 119878ℎ(119905) 997888rarr 119878lowastℎ (119905) as 119905 997888rarr +infin By thecomparison theorem we have 119878ℎ(119905) le 119878ℎ(119905) Hence thereexists a sufficiently large 1205850 gt 0 and given above 120576 gt 0 suchthat 119878ℎ(119905) le 119878lowastℎ (119905) + 120576 for 119905 ge 1205850 In the same way we can alsoprove that 119878119891(119905) le 119878lowast119891(119905) + 120576 for 119905 ge 12058510158400

Complexity 7

From the model system we have1198781015840119904 (119905) le Λ 2 (119905) minus 1205832 (119905) 119878119904 (119905) 119905 = 119905119896 119896 isin 119885119878119904 (119905+) = (1 minus 120579119896) 119878119904 (119905) le 119878119904 (119905) 119905 = 119905119896 119896 isin 119885 (30)

By comparison theorem in impulsive equations and abovemethod we can obtain 119878119904(119905) le 119878lowast119904 (119905) + 120576 for 119905 ge 120585101584010158400

Let 120585lowast = max 1205850 12058510158400 120585101584010158400 According to above discussionwe have for 119905 ge 120585lowast 119878ℎ (119905) le 119878lowastℎ (119905) + 120576119878119891 (119905) le 119878lowast119891 (119905) + 120576119878119904 (119905) le 119878lowast119904 (119905) + 120576 (31)

From (3) (4) and (31) we have that for 119905 ge 120585lowast1198681015840ℎ (119905) le 1205731 (119905) (119878lowastℎ + 120576) 119868119891 minus (1205831 (119905) + 120574 (119905)) 119868ℎ1198681015840119904 (119905) le 1205732 (119905) (119878lowast119904 + 120576) 119868ℎ minus 1205832 (119905) 1198681199041198681015840119891 (119905) le 1205733 (119905) (119878lowast119891 + 120576) 119868119904 minus 1205833 (119905) 119868119891 119905 = 119905119896 119896 isin N119868ℎ (119905+119896 ) = 119868ℎ (119905119896) 119868119904 (119905+119896 ) = (1 minus 120579119896) 119868119904 (119905119896) 119868119891 (119905+119896 ) = 119868119891 (119905119896) 119905 = 119905119896 119896 isin N

(32)

Consider the following linear approximation system1199091015840 (119905) = (119865 (119905) minus 119881 (119905) + 119872120576) 119909 (119905) 119905 = 119905119896 119896 isin N119909 (119905+119896 ) = 119875119896119909 (119905119896) 119905 = 119905119896 119896 isin N (33)

where 119865 119881 119875119896 are defined as (19) (23) and (24)By Lemma 4 we have that there exists a positive 12-

period function 119909lowast(119905) such that 119909(119905) = 119890120578119905119909lowast(119905) is a solutionof system (33) where 120578 = (112) ln 119903(Φ(119865minus119881+119872120576)119875119896 (12)) 119909(119905) =(119868ℎ(119905) 119868119904(119905) 119868119891(119905)) It follows from (26) that 120578 lt 0Thereforewe have 119909(119905) 997888rarr 0 as 119905 997888rarr +infinThis implies that the zerosolution of system (33) is globally attractive if 1198770 lt 1

For any nonnegative initial value (119868ℎ(0+) 119868119904(0+) 119868119891(0+))119879of system (32) there exists a sufficiently large 119886 gt 0 suchthat (119868ℎ(0+) 119868119904(0+) 119868119891(0+))119879 le 119886119909(0+) holds Applying thecomparison principle we have (119868ℎ(119905) 119868119904(119905) 119868119891(119905))119879 le 119886119909(119905)for all 119905 gt 1205850 where 119886119909(119905) is also the solution of system (33)Therefore we have 119868ℎ(119905) 997888rarr 0 119868119904(119905) 997888rarr 0 and 119868119891(119905) 997888rarr 0as 119905 997888rarr +infin By the theory of asymptotically semiflows itfollows that

lim119905997888rarr+infin

119878ℎ (119905) = 119878lowastℎ (119905) lim

119905997888rarr+infin119878119904 (119905) = 119878lowast119904 (119905)

and lim119905997888rarr+infin

119878119891 (119905) = 119878lowast119891 (119905) (34)

Therefore the disease-free periodic solution 119909lowast(119905) of themodel system is globally asymptotically stable This com-pletes the proof

The subsequent result shows that the disease is uniformlypersistent if 1198770 gt 1

Define119883 fl (119878ℎ 119868ℎ 119878119904 119868119904 119878119891 119868119891) isin 1198776 | 119878119894 ge 0 119868119894 ge 0 119894= ℎ 119904 119891 1198830 fl (119878ℎ 119868ℎ 119878119904 119868119904 119878119891 119868119891) isin 119883 | 119868ℎ gt 0 119868119904 gt 0 119868119891gt 0 1205971198830 fl 119883 1198830(35)

Let 119875 119883 997888rarr 119883 be the Poincare map associated with themodel system that is119875 (1199090) = 119906 (12+ 1199090) forall1199090 isin 119883 (36)

where 1199090 = (1198780ℎ 1198680ℎ 1198780119904 1198680119904 1198780119891 1198680119891) and 119906(119905 1199090) is the uniquesolution of the model system with 119906(0+ 1199090) = 1199090 It is easy tosee that 119875119898 (1199090) = 119906 (12119898+ 1199090) forall119898 isin 119873 (37)

Letting 119880lowast = (119878lowastℎ (0+) 0 119878lowast119904 (0+) 0 119878lowast119891(0+) 0) we have119875119898 (119880lowast) = 119906 (12119898+ 119880lowast) forall119898 isin 119873 (38)

According to Lemma 1 we can easily see that119883 and1198830 arepositively invariant and Poincare map 119875 is point dissipative

Next we establish the following lemma which will beuseful in subsequent main result

Lemma 8 If the basic reproductive number 1198770 gt 1 then thereexists 120576lowast gt 0 such that for any 1199090 isin 1198830 with 1199090 minus 119880lowast le 120576lowastwe have

lim sup119898997888rarrinfin

119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)

Proof In view ofTheorem 5 we know that 1198770 gt 1 if and onlyif 119903(Φ(119865minus119881)119875119896(12)) gt 1 and then there is a 120576 gt 0 sufficientlysmall such that 119903 (Φ(119865minus119881minus119872120576)119875119896 (12)) gt 1 (40)

where119872120576 is given in (27)By the continuity of the solutions with respect to the

initial values there exists 120576lowast gt 0 such that for any 1199090 isin 1198830with 1199090 minus 119880lowast le 120576lowast and 119905 isin [0 12)10038171003817100381710038171003817119906 (119905 1199090) minus 119906 (119905 119880lowast)10038171003817100381710038171003817 le 120576 (41)

Next we claim that (39) holds Suppose the claim is notvalid Then there exists 1199090 isin 1198830 such that


119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity

Without loss of generality we assume that119889 (119875119898 (1199090) 119880lowast) lt 120576lowast for all 119898 ge 0 (43)

It follows from (41) and (43) that10038171003817100381710038171003817119906 (119905 119875119898 (1199090)) minus 119906 (119905 119880lowast)10038171003817100381710038171003817 lt 120576forall119898 ge 0 forall119905 isin [0 12) (44)

For any 119905 ge 0 there exist nonnegative integer 1198981015840 and 1198791 isin[0 12) such that 119905 = 121198981015840 + 1198791 From (44) we have10038171003817100381710038171003817119906 (119905 1199090) minus 119906 (119905 119880lowast)10038171003817100381710038171003817= 10038171003817100381710038171003817119906 (1198791 1198751198981015840 (1199090)) minus 119906 (1198791 119880lowast)10038171003817100381710038171003817 lt 120576 (45)

Note that (119878ℎ(119905) 119868ℎ(119905) 119878119904(119905) 119868119904(119905) 119878119891(119905) 119868119891(119905)) = 119906(119905 1199090)Thus1003817100381710038171003817119868ℎ (119905)1003817100381710038171003817 lt 1205761003817100381710038171003817119868119904 (119905)1003817100381710038171003817 lt 12057610038171003817100381710038171003817119868119891 (119905)10038171003817100381710038171003817 lt 120576for all 119905 ge 0

(46)

By the first equation of (3) and (46) we have1198781015840ℎ (119905) ge (1205831 (119905) + 120575 (119905))119873ℎ minus 1205731 (119905) 119878ℎ (119905) 120576minus (120572 (119905) + 1205831 (119905) + 120575 (119905)) 119878ℎ (119905) minus 120575 (119905) 120576 (47)

By comparison theorem we have 119878ℎ(119905) ge 119878ℎ(119905) and 119878ℎ(119905) 997888rarr119878lowastℎ(119905) as 120576 997888rarr 0 where 119878ℎ(119905) is the solution of the followingsystem1198781015840ℎ (119905) = (1205831 (119905) + 120575 (119905))119873ℎ minus 1205731 (119905) 119878ℎ (119905) 120576 minus 120575 (119905) 120576minus (120572 (119905) + 1205831 (119905) + 120575 (119905)) 119878ℎ (119905) 119878ℎ (0+) = 119878ℎ (0+) (48)

Therefore for above mentioned 120576 there exists 1198792 gt 0sufficiently large such that119878ℎ (119905) ge 119878ℎ (119905) ge 119878lowastℎ (119905) minus 120576 for 119905 ge 1198792 (49)

Using the same method we can get there exists 1198793 ge 1198792sufficiently large such that119878119891 (119905) ge 119878119891 (119905) ge 119878lowast119891 (119905) minus 120576119878119904 (119905) ge 119878119904 (119905) ge 119878lowast119904 (119905) minus 120576

for 119905 ge 1198793 (50)

From (3) (4) (49) and (50) we obtain for 119905 ge 11987931198681015840ℎ (119905) ge 1205731 (119905) (119878lowastℎ (119905) minus 120576) 119868119891 (119905)minus (1205831 (119905) + 120574 (119905)) 119868ℎ (119905) 1198681015840119904 (119905) ge 1205732 (119905) (119878lowast119904 (119905) minus 120576) 119868ℎ (119905) minus 1205832 (119905) 119868119904 (119905) 1198681015840119891 (119905) ge 1205733 (119905) (119878lowast119891 (119905) minus 120576) 119868119904 (119905) minus 1205833 (119905) 119868119891 (119905) 119905 = 119905119896 119896 isin N119868ℎ (119905+119896 ) = 119868ℎ (119905119896) 119868119904 (119905+119896 ) = (1 minus 120579119896) 119868119904 (119905119896) 119868119891 (119905+119896 ) = 119868119891 (119905119896) 119905 = 119905119896 119896 isin N

(51)

Consider the following impulsive linear approximationsystem

1199091015840 (119905) = (119865 (119905) minus 119881 (119905) minus 119872120576) 119909 (119905) 119905 = 119905119896 119896 isin N119909 (119905+119896 ) = 119875119896119909 (119905119896) 119905 = 119905119896 119896 isin N (52)

By Lemma 4 we know that there exists a positive 12-periodfunction119909lowast(119905) = (119868lowastℎ (119905) 119868lowast119904 (119905) 119868lowast119891(119905)) such that119909(119905) = 1198901205781119905119909lowast(119905)is a solution of (52) where 119909(119905) = (119868ℎ(119905) 119868119904(119905) 119868119891(119905)) and 1205781 =(112) ln 119903(Φ(119865minus119881minus119872120576)119875119896 (12)) It follows from (40) that 1205781 gt 0Obviously we can choose 12 gt 1198793 and a proper 119887 gt 0 suchthat

119868ℎ (12+) ge 119887119868lowastℎ (0+) 119868119904 (12+) ge 119887119868lowast119904 (0+) and 119868119891 ( (12+) ge 119887119868lowast119891 (0+) (53)

By the comparison theorem we have that for all 119905 ge 12119868ℎ (119905) ge 119868ℎ (119905) ge 119887e1205781(119905minus12)119868lowastℎ (119905 minus 12) 119868119904 (119905) ge 119868119904 (119905) ge 1198871198901205781(119905minus12)119868lowast119904 (119905 minus 12) 119868119891 (119905) ge 119868119891 (119905) ge 1198871198901205781(119905minus12)119868lowast119891 (119905 minus 12)

(54)

Then we can obtain that 119868ℎ(119905) 997888rarr +infin 119868119904(119905) 997888rarr +infin and119868119891(119905) 997888rarr +infin as 119905 997888rarr +infin which is a contradiction Theproof of Lemma 8 is completed

Complexity 9

Theorem 9 If the basic reproductive number 1198770 gt 1 thenthere exist constants 120575119894 gt 0 (119894 = 1 2 3) such that for anysolution of the model system with initial value(1198780ℎ 1198680ℎ 1198780119904 1198680119904 1198780119891 1198680119891) isin 1198830 (55)

satisfies

lim inf119905997888rarr+infin

119868ℎ (119905) ge 1205751lim inf119905997888rarr+infin

119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)

Proof Denote119872120597 = (1198780ℎ 1198680ℎ 1198780119904 1198680119904 1198780119891 1198680119891)isin 1205971198830 | 119867119898 (1198780ℎ 1198680ℎ 1198780119904 1198680119904 1198780119891 1198680119891) isin 1205971198830 119898 isin N (57)

We claim that119872120597 = (119878ℎ 0 119878119904 0 119878119891 0) isin 1205971198830 | 119878ℎ ge 0 119878119904 ge 0 119878119891ge 0 (58)

Obviously 119872120597 supe (119878ℎ 0 119878119904 0 119878119891 0) isin 1205971198830 | 119878ℎ ge0 119878119904 ge 0 119878119891 ge 0 Thus we only need to prove 119872120597 (119878ℎ 0 119878119904 0 119878119891 0) isin 120597X0 | 119878ℎ ge 0 119878119904 ge 0 119878119891 ge 0 = 0 If it isnot true then there exists a point 1199090 = (1198780ℎ 1198680ℎ 1198780119904 1198680119904 1198780119891 1198680119891) isin119872120597 (119878ℎ 0 119878119904 0 119878119891 0) isin 1205971198830 | 119878ℎ ge 0 119878119904 ge 0 119878119891 ge 0

There are six cases to consider (i) 119868ℎ gt 0 119868119904 = 0 119868119891 = 0(ii) 119868ℎ = 0 119868119904 gt 0 119868119891 = 0 (iii) 119868ℎ = 0 119868119904 = 0 119868119891 gt 0 (iv)119868ℎ = 0 119868119904 gt 0 119868119891 gt 0 (v) 119868ℎ gt 0 119868119904 = 0 119868119891 gt 0 (vi) 119868ℎ gt 0119868119904 gt 0 119868119891 = 0Case A For case (i) that is 119868ℎ gt 0 119868119904 = 0 119868119891 = 0 it iseasily seen that 119868ℎ(119905) gt 0 and 119878119904(119905) gt 0 119878119891(119905) gt 0 for all119905 gt 0 Then from the fourth equation of (3) (119889119868119904(119905)119889119905)|119905=0 =1205732(0)119878119904(0+)119868ℎ(0+) gt 0 Thus 119868119904(119905) gt 0 for 0 lt 119905 ≪ 1 It followsfrom the sixth equation of (3) that 119868119891(119905) gt 0 for 0 lt 119905 ≪ 1It is easy to obtain that (119878ℎ(119905) 119868ℎ(119905) 119878119904(119905) 119868119904(119905) 119878119891(119905) 119868119891(119905)) notin1205971198830 for 0 lt 119905 ≪ 1 This is a contradiction Usingthe same method we can prove the second and the thirdcases

Case B For case (iv) that is 119868ℎ = 0 119868119904 gt 0 119868119891 gt 0 itis easily seen that 119868119904(119905) gt 0 119868119891(119905) gt 0 and 119878ℎ(119905) gt 0for all 119905 gt 0 Then from the second equation of system(3) (119889119868ℎ(119905)119889119905)|119905=0 = 1205731(0)119878ℎ(0+)119868119891(0+) gt 0 Obviously(119878ℎ(119905) 119868ℎ(119905) 119878119904(119905) 119868119904(119905) 119878119891(119905) 119868119891(119905)) notin 1205971198830 for 0 lt 119905 ≪ 1This is a contradiction We can also discuss the last two casesusing the same method

From the above discussion we have that for any initial1199090 notin (1198780ℎ 0 1198780119904 0 1198780119891 0) | 119878ℎ ge 0 119878119904 ge 0 119878119891 ge 0 then 1199090 notin119872120597 Thus 119872120597 = (119878ℎ 0 119878119904 0 119878119891 0) isin 1205971198830 | 119878ℎ ge 0 119878119904 ge0 119878119891 ge 0Obviously Piocare map 119875 has a global attractor 119880lowast119880lowast is an isolated invariant set in 119883 and 119882119904(119880lowast) cap 1198830 =

Further 119880lowast is acyclic in 119872120597 and every solution in 119872120597converges to 119880lowast By Lemma 8 we know that 119875 is weaklyuniformly persistent with respect to (1198830 1205971198830) According toZhao [17] we have that 119875 is uniformly persistent with respectto (1198830 1205971198830) That is there exist constants 120575119894 gt 0 (119894 = 1 2 3)such that for any solution of the model system with initialvalue (1198780ℎ 1198680ℎ 1198780119904 1198680119904 1198780119891 1198680119891) isin 1198830 (59)

satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)

The proof of Theorem 9 is completed

5 Numerical Simulations

In this section wewillmake numerical simulations inMatlabUsing the theory of impulsive equations and analysis methoddynamic behavior of the model has been studied and athreshold for a disease to be extinct or endemic has beenestablished

Before carrying out the numerical simulations we haveto estimate the model parameters Some data are takenfrom literature [14 18] Some data are estimated from theclonorchiasis surveillance data of Foshan city Guangdongprovince China The values of the model parameters arelisted in Table 1

Set 119902 = 2 1199051 = 4 1199052 = 6 In view of the clonorchiasissurveillance data the initial value of the model system istaken as follows 119878ℎ(0) = 2368 times 106 119868ℎ(0) = 16038 times 106119878119904(0) = 7488 times 1010 119868119904(0) = 1872 times 1010 119878119891(0) = 176 times 108119868119891(0) = 44 times 107

If we fix 1205791 = 0 and 1205792 = 0 that is no snail controlstrategy is implemented With numerical simulation we get1198770 = 18858 gt 1 If we fix 1205791 = 02 and 1205792 = 0 that is20 snail-killing once a year is carried out which results in1198770 = 10719 gt 1 these of course lead to the persistence ofthe disease as clearly indicated by Figures 3 and 4 If we fix1205791 = 02 and 1205792 = 01 then 1198770 = 09140 lt 1 This suggeststhe extinction of the disease (see Figure 5)

Figure 6 shows the sensitivity of the basic reproductivenumber 1198770 to the elimination rate of snails 1205791 and 1205792 It isapparent that the elimination rate of snails are very sensitiveto 1198770 when 120579119894 lt 02 (119894 = 1 2)

Health education snail control and feces treatment arethree effective strategies to control clonorchiasis Parameters120572 (or 120575) 120579119894 and 1205732 reflect the efforts of health educationsnail control and feces treatment respectively In Table 2six projects are studied to determine the impact of controlstrategies on the basic reproductive number The first andsecond projects are only considered the health educationstrategy If 120572 is doubled and 120575 is half then 1198770 decreases from

10 Complexity

Table 1 The values of parameters in the model system

Parameter Value Unit Reference1205831(119905) 00011 119898119900119899119905ℎminus1 [18]1205731(119905) 75times10minus11 +2times10minus11 sin 1205871199056 119898119900119899119905ℎminus1 Estimation120572(119905) 00167 + 0002 sin 1205871199056 119898119900119899119905ℎminus1 Estimation120575(119905) 002778 + 0009 sin 1205871199056 119898119900119899119905ℎminus1 Estimation120574(119905) 0076 + 0006 sin 1205871199056 119898119900119899119905ℎminus1 [14]Λ 2(119905) 13 times 109 + 10 times 108 sin 1205871199056 119898119900119899119905ℎminus1 [10 14]1205832(119905) 001389 + 001 sin 1205871199056 119898119900119899119905ℎminus1 [19]1205732(119905) 38 times 10minus8 + 5 times 10minus10 sin 1205871199056 119898119900119899119905ℎminus1 EstimationΛ 3(119905) 46 times 106 + 2 times 105 sin 1205871199056 119898119900119899119905ℎminus1 Estimation1205733(119905) 178times10minus12+5times10minus14 sin 1205871199056 119898119900119899119905ℎminus1 Estimation1205833(119905) 00208 + 0101 sin 1205871199056 119898119900119899119905ℎminus1 [10]

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105

Figure 3 This figure shows movement paths of 119868ℎ 119868119904 and 119868119891 as functions of time 119905 1198770 = 18858 gt 1 where parameters 1205791 = 0 and 1205792 = 0 Thedisease is permanent

18858 to 14855 And if 120572 is tripled and 120575 is reduced by one-third then 1198770 decreases from 18858 to 12192 Combininghealth education and snail control strategies in the thirdproject 120572 is doubled and 120575 is in half 1205791 = 014 by computersimulation It is shown that 1198770 decreases to 09674This indi-cates that the disease will die out Combining snail controland faeces treatment strategies the last project shows 1198770 alsobecomes less than 1 From Table 2 it is evident that healtheducations snail control and faeces treatment are helpfulfor the disease control They can all decrease the fractionof infected individuals however snail-killing is the mosteffective way to control the transmission of clonorchiasis

6 Conclusions and Discussion

Clonorchiasis has been one of significant public health threatsinChina Facing up to the epidemic situation both the centraland local governments have been exploring an effectiveand sustainable strategy on the control of clonorchiasis inendemic areas Various prevention and controlmeasures havebeen proposed by many researchers Chemotherapy withpraziquantel is one of vital strategies but the use of prazi-quantel for humans and animals has been only temporarilyeffective There is no doubt that the chemotherapy-basedcontrol strategy would no longer be an ideal measure

Complexity 11

Table 2 Comparison of different control strategies

Plan Health educationefforts

Snailcontrol 1205791 Faeces

treatment 1198770 Extinction Permanence

1 120572(119905) times 2 0 1205732(119905) times 1 14855 No Yes120575(119905) divide 22 120572(119905) times 3 0 1205732(119905) times 1 12192 No Yes120575(119905) divide 33 120572(119905) times 2 014 1205732(119905) times 1 09674 Yes No120575(119905) divide 24 120572(119905) times 1 0 1205732(119905) divide 2 14968 No Yes120575(119905) divide 15 120572(119905) times 2 0 1205732(119905) divide 2 11790 No Yes120575(119905) divide 26 120572(119905) times 1 014 1205732(119905) divide 2 09747 Yes No120575(119905) divide 1

0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105

Figure 4This figures showmovement paths of 119868ℎ 119868119904 and 119868119891 as functions of time 119905 1198770 = 10719 gt 1 where parameters 1205791 = 02 and 1205792 = 0Thedisease is permanent

Currently the integrated control strategy mainly focuseson ensuring fish cooked well prior to consumption healtheducation and improvement of sanitation to prevent parasiteeggs reaching fish habitats Controlling snail is widely usedto control of schistosomiasis [20] but it has not yet beenapplied to control clonorchiasis in practice even if it has beentheoretically confirmed In this paper we proposed a dynamicclonorchiasis model with impulsive snail-killing to studythe effects of snail control strategy on the transmission ofclonorchiasis We also derived the basic reproductive numberfor general epidemic model with seasonality and impulsivecontrol which extended previous works on such models inthat the impulsive control actions can be carried out at several

time points within a period Furthermore we performed thenumerical simulation and sensitivity analysis to recognize theimpact of crucial model parameters on 1198770 We found thatcontrolling snail was the best effectivemethod for eliminatingclonorchiasis

As Hepatitis B virus mainly attacks the liver C sinensisinfection also causes liver diseases as well as biliary condi-tions Both Hepatitis B and C sinensis are present in almostthe same regions in China [8] Therefore using HBVCIcoinfection model to assess different tools and strategies forlarge-scale control of Hepatitis B and clonorchiasis shouldbe an interesting issue which will be considered in thefuture

12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105

Figure 5 This figure shows movement paths of 119868ℎ 119868119904 and 119868119891 as functions of time 119905 1198770 = 09140 lt 1 where parameters 1205791 = 02 and 1205792 = 01The disease will be distinct

0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2

Figure 6 The graph is a contour plot showing the regions in the (1205791 1205792) plane in which 1198770 falls in different intervals

Appendix

Proof of Lemma 2 Integrating and solving the first equationof system (11) between pulses for 119899120596 + 119905119894 lt 119905 le 119899120596 + 119905119894+1 (119894 =0 1 119902 minus 1)119911 (119905) = 119911 (119899120596 + 119905+119894 ) 119890minus int119905119899120596+119905119894 119887(120591)119889120591+ 119890minusint119905119899120596+119905119894 119887(120591)119889120591 int119905

119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)

where 119911(119899120596+119905+119894 ) = (1minus120579119894)119911(119899120596+119905119894) It follows from the aboveequation and the second equation of system (11) that

119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13

and119911 (119899120596 + 119905+2 ) = (1 minus 1205792) (119911 (119899120596 + 119905+1 ) 119890minus int119899120596+1199052119899120596+1199051 119887(120591)119889120591+ 119890minus int119899120596+1199052119899120596+1199051

119887(120591)119889120591 int119899120596+1199052119899120596+1199051

119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904) = (1minus 1205791) (1 minus 1205792) 119911 (119899120596 + 1199050) 119890minus int119899120596+1199052119899120596+1199050 119887(120591)119889120591 + (1 minus 1205791)sdot (1 minus 1205792) 119890minus int119899120596+1199052119899120596+1199051 119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904

+ (1 minus 1205792) 119890minus int119899120596+1199052119899120596+1199051119887(120591)119889120591 int119899120596+1199052

119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)

Using the inductive method we know 119911(119899120596 + 119905+119902 ) = 119911((119899 +1)120596 + 119905+0 ) and119911 (119899120596 + 119905+119902) = 119902prod

119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1

119902prod119895=119897(1 minus 120579119895) 119890minusint119899120596+119905119902119899120596+119905119895minus1 119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)

Set 119880119899 = 119911(119899120596 + 1199050) From (A4) and 119905119902 minus 1199050 = 120596 we have119880119899+1 = 119902prod

119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1

119902prod119895=119897(1 minus 120579119895) 119890minus int119899120596+119905119902119899120596+119905119895minus1 119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)

where 119891 is the stroboscopic map It is easy to see that system(A5) has a unique positive equilibrium

119911lowast0 = 119911 (0+) = [1 minus 119902prod119897=1(1 minus 120579119897) 119890int119899120596+119905119902119899120596+1199050 minus119887(120591)119889120591]minus1

times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)

Since 119891(119880119899) is a straight line with slope less than 1 weobtain that 119911lowast0 is globally asymptotically stable It impliesthat the corresponding periodic solution of system (11) 119911lowast(119905)is globally asymptotically stable The proof of Lemma 2 iscompleted

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The work by Shujing Gao was partially supported by TheNatural Science Foundation of China (11561004)

References

[1] R Anderson and R May Infectious Diseases of HumansDynamics and Control Oxford University Press USA 1991

[2] S Hong and Y Fang ldquoClonorchis sinensis and clonorchiasis anupdaterdquoParasitology International vol 61 no 1 pp 17ndash24 2012

[3] N Bacaer and R Ouifki ldquoGrowth rate and basic reproductionnumber for population models with a simple periodic factorrdquoMathematical Biosciences vol 210 no 2 pp 647ndash658 2007

[4] M Qian Y Chen S Liang G Yang and X Zhou ldquoTheglobal epidemiology of clonorchiasis and its relation withcholangiocarcinomardquo Infectious Diseases of Poverty vol 1 no4 pp 1ndash11 2012

[5] Z Lun R B Gasser D Lai et al ldquoClonorchiasis a keyfoodborne zoonosis in Chinardquo The Lancet Infectious Diseasesvol 5 no 1 pp 31ndash41 2005

[6] J Keiser and J Utzinger ldquoFood-borne trematodiasesrdquo ClinicalMicrobiology Reviews vol 22 no 3 pp 466ndash483 2009

[7] B Sripa S Kaewkes P M Intapan W Maleewong andP J Brindley ldquoFood-borne trematodiases in Southeast Asiaepidemiology pathology clinical manifestation and controlrdquoAdvances in Parasitology vol 72 pp 305ndash350 2010

[8] M Qian Y Chen and F Yan ldquoTime to tackle clonorchiasis inChinardquo Infectious Diseases of Poverty vol 2 no 4 pp 1ndash4 2013

[9] PWei Studies on the killing eff ect of Clonorchis sinensis metacer-cariae by non-heated treatment and metacercariae histochemicalproperties analysis MSc thesis Guangxi University NanningChina 2013

[10] Q Gaun Z Huang X Huang et al ldquoThe infection status andepidemiological characteristicsof clonorchiasis in Foshan 1980-2010rdquo South China Journal of Preventive Medicine vol 41 no 3pp 276ndash279 2015

[11] B X Yang Z Z Chen L C Huo et al ldquoInvestigationon epidemiology of clonorchiasis and discussion of controlstrategy andmeasuresrdquoChinese Journal of Public Health vol 10no 3 pp 303ndash306 1994

[12] R Yuan J Huang X Zhang and S Ruan ldquoModeling thetransmission dynamics of clonorchiasis in Foshan ChinardquoScientific Reports vol 8 article 15176 pp 1ndash9 2018

[13] Y Dai and S Gao ldquoThreshold and stability results for clonorchi-asis epidemic modelrdquo Journal of Science Technology and Envi-ronment vol 2 no 2 pp 1ndash13 2013

[14] T Li Z Yang andMWang ldquoCorrelation between clonorchiasisincidences and climatic factors in Guangzhou Chinardquo Parasitesamp Vectors vol 7 no 1 article 29 2014

14 Complexity

[15] S Gao Y Tu and J Wang ldquoBasic reproductive number fora general hybrid epidemic modelrdquo Advances in DifferenceEquations vol 2018 article 310 pp 1ndash9 2018

[16] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005

[17] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[18] E T Chiyaka and W Garira ldquoMathematical analysis of thetransmission dynamics of schistosomiasis in the human-snailhostsrdquo Journal of Biological Systems vol 17 no 3 pp 397ndash4232009

[19] S Gao Y He Y Liu G Yang and X Zhou ldquoField transmissionintensity of Schistosoma japonicum measured by basic repro-duction ratio frommodified Barboursmodelrdquo Parasites Vectorsvol 6 article 141 2013

[20] T Lin Q W Jiang D D Lin et al ldquoClassification study on themarshland in endemic areas of Schistosoma japonicum usingsatellite TM image datardquoChinese Journal of PreventiveMedicinevol 35 no 5 pp 312ndash314 2001

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Submit your manuscripts atwwwhindawicom

2 Complexity

60

50

40

30

20

10

0

Year

Posit

ive i

nfec

tion

rate

()

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2005

-2006

2008

2009

2010

Figure 1 Positive infection rate in Foshan from 1980 to 2010

in Daliang Village Yangshan County Guangdong Provinceincluding health education chemotherapy and snail controlwith molluscicide The CI rate decreased from 482 in 1975to 29 in 1976 to 0 in 1979-1982 Therefore it is of greatpractical significance to assess different tools and strategiesfor large-scale control of clonorchiasis by using mathematicalmodelling

There have been studies of modeling clonorchiasis asso-ciated with the human being in China [12 13] However noneof the studies were to quantitatively consider the relationshipbetween CI rate and the number of molluscicide to spraywhen chemotherapy and health education are implementedin combination Mathematical models can be important fordetermining the optimal number of molluscicide spraying soas to control the transmission of clonorchiasis In this papera clonorchiasis spread model with an integrated strategyis developed and studied in which seasonal variation andimpulsive control are considered

This paper is organized as follows In Section 2 wepropose a C sinensis model with pulse snail-killing healtheducation and chemotherapy control in periodic environ-ment Some results are stated which will be essential toour proof We calculate the basic reproductive number forthe model in Section 3 In Section 4 we illustrate that thebasic reproductive number serves as a threshold parameterthat determines the disease to be extinction or endemicIn Section 5 different control programs are compared andsensitivity analysis is done to evaluate the snails controlstrategy by numerical simulations Section 6 gives a briefdiscussion and future work

2 Model and Preliminaries

Clonorchiasis belongs to vector-borne disease including twointermediate hosts snail and fish C sinensis is transmit-ted indirectly among the human hosts first intermediate

snails and second intermediate fishes in the sense that free-swimming stages (miracidia cercariae) and ingestion stage(metacercariae) are interposed In addition to human beingsspecially many mammals such as dogs cats pigs rodentsfoxes and possibly any fish eating mammal can serve asdefinitive hosts for C sinensis and humans are the maindefinitive hosts [14] In order to simplify the mathematicalmodel we only consider human hosts here Figure 2 showsa schematic description of the transmission of clonorchissinensis in definitive host snail host and fish host

In this section we mainly formulate an impulsive epi-demic model to describe the transmission dynamics of Csinensis

We firstly formulate a model for the spread of C sinensisincorporating health education and chemotherapy strategiesThe total human population at any time 119905 denoted by 119873(119905)is the sum of individual populations in each compartmentwhich includes susceptible 119878ℎ(119905) infected 119868ℎ(119905) and recovered119877ℎ(119905) We assume that the total human population remains aconstant denoting 119873ℎ(119905) = 119873ℎ for all 119905 ge 0

Following the idea of Dai andGao [13] we divide the snailpopulation and fish population into disjoint classes suscepti-ble (119878119904 119878119891) and infected (119868119904 119868119891) respectively We suppose thatthe infection rates of susceptible human susceptible snailand susceptible fish are described by1205731 (119905) 119878ℎ (119905) 119868119891 (119905) 1205732 (119905) 119878119904 (119905) 119868ℎ (119905) 1205733 (119905) 119878119891 (119905) 119868119904 (119905) (1)

respectively where 1205731(119905) is per capita (successful) infectionrate of human hosts by one fish at time 119905 1205732(119905) is per capita(successful) infection rate of snails by one human at time 119905and 1205733(119905) is per capita (successful) infection rate of fish byone snail at time 119905

Complexity 3



SusceptibleInfected

Snail population


Fish population

Metacercariae

Wormrsquosegg

Miracidia

Cercariae




(2)



(3)


for 119905 = 119905119896 (4)



4 Complexity





and 1198731015840119891 (119905) = Λ 3 (119905) minus 1205833 (119905)119873119891 (119905) (7)


119904 0 le 119878119891 (119905) + 119868119891 (119905) le 119873lowast119891 (8)





119904 (119905) = Λ 2 (119905) minus 1205832 (119905)119867119904 (119905) 119867119904 (0+) = 119873119904 (0+) (9)


119904 (10)








+ 119890int119905119899120596+119905119894 minus119887(120591)119889120591 int119905119899120596+119905119894

119886 (119904) 119890int119904119899120596+119905119894 119887(120591)119889120591119889119904+ 119894sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(12)



sdot 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(13)


holds true








Complexity 5



Denote







F (119905 119909) =((((((((

1205731 (119905) 1199094 (119905) 1199093 (119905)1205732 (119905) 1199095 (119905) 1199091 (119905)1205733 (119905) 1199096 (119905) 1199092 (119905)000))))))))

V (119905 119909) =((((((((


(18)



1le119894 119895le3







6 Complexity


119875119896 = (1 0 00 1 minus 120579119896 00 0 1) 119876119896 = (1 0 00 1 minus 120579119896 00 0 1 )

(22)

119872(119905) = (120597119891119894 (119905 119909lowast (119905))120597119909119895 )4le119894 119895le6


and

119881 (119905) = (120597V119894 (119905 119909lowast (119905))120597119909119895 )1le119894 119895le3

= (1205831 (119905) + 120574 (119905) 0 00 1205832 (119905) 00 0) 1205833 (119905)) (24)




4 Main Results



(25)





where

119872120576 = ( 0 0 1205731 (119905) 1205761205732 (119905) 120576 0 00 1205733 (119905) 120576 0 ) (27)




Complexity 7





(32)






119878ℎ (119905) = 119878lowastℎ (119905) lim



119878119891 (119905) = 119878lowast119891 (119905) (34)











119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)






119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3



SusceptibleInfected

Snail population


Fish population

Metacercariae

Wormrsquosegg

Miracidia

Cercariae




(2)



(3)


for 119905 = 119905119896 (4)



4 Complexity





and 1198731015840119891 (119905) = Λ 3 (119905) minus 1205833 (119905)119873119891 (119905) (7)


119904 0 le 119878119891 (119905) + 119868119891 (119905) le 119873lowast119891 (8)





119904 (119905) = Λ 2 (119905) minus 1205832 (119905)119867119904 (119905) 119867119904 (0+) = 119873119904 (0+) (9)


119904 (10)








+ 119890int119905119899120596+119905119894 minus119887(120591)119889120591 int119905119899120596+119905119894

119886 (119904) 119890int119904119899120596+119905119894 119887(120591)119889120591119889119904+ 119894sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(12)



sdot 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(13)


holds true








Complexity 5



Denote







F (119905 119909) =((((((((

1205731 (119905) 1199094 (119905) 1199093 (119905)1205732 (119905) 1199095 (119905) 1199091 (119905)1205733 (119905) 1199096 (119905) 1199092 (119905)000))))))))

V (119905 119909) =((((((((


(18)



1le119894 119895le3







6 Complexity


119875119896 = (1 0 00 1 minus 120579119896 00 0 1) 119876119896 = (1 0 00 1 minus 120579119896 00 0 1 )

(22)

119872(119905) = (120597119891119894 (119905 119909lowast (119905))120597119909119895 )4le119894 119895le6


and

119881 (119905) = (120597V119894 (119905 119909lowast (119905))120597119909119895 )1le119894 119895le3

= (1205831 (119905) + 120574 (119905) 0 00 1205832 (119905) 00 0) 1205833 (119905)) (24)




4 Main Results



(25)





where

119872120576 = ( 0 0 1205731 (119905) 1205761205732 (119905) 120576 0 00 1205733 (119905) 120576 0 ) (27)




Complexity 7





(32)






119878ℎ (119905) = 119878lowastℎ (119905) lim



119878119891 (119905) = 119878lowast119891 (119905) (34)











119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)






119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity





and 1198731015840119891 (119905) = Λ 3 (119905) minus 1205833 (119905)119873119891 (119905) (7)


119904 0 le 119878119891 (119905) + 119868119891 (119905) le 119873lowast119891 (8)





119904 (119905) = Λ 2 (119905) minus 1205832 (119905)119867119904 (119905) 119867119904 (0+) = 119873119904 (0+) (9)


119904 (10)








+ 119890int119905119899120596+119905119894 minus119887(120591)119889120591 int119905119899120596+119905119894

119886 (119904) 119890int119904119899120596+119905119894 119887(120591)119889120591119889119904+ 119894sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(12)



sdot 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(13)


holds true








Complexity 5



Denote







F (119905 119909) =((((((((

1205731 (119905) 1199094 (119905) 1199093 (119905)1205732 (119905) 1199095 (119905) 1199091 (119905)1205733 (119905) 1199096 (119905) 1199092 (119905)000))))))))

V (119905 119909) =((((((((


(18)



1le119894 119895le3







6 Complexity


119875119896 = (1 0 00 1 minus 120579119896 00 0 1) 119876119896 = (1 0 00 1 minus 120579119896 00 0 1 )

(22)

119872(119905) = (120597119891119894 (119905 119909lowast (119905))120597119909119895 )4le119894 119895le6


and

119881 (119905) = (120597V119894 (119905 119909lowast (119905))120597119909119895 )1le119894 119895le3

= (1205831 (119905) + 120574 (119905) 0 00 1205832 (119905) 00 0) 1205833 (119905)) (24)




4 Main Results



(25)





where

119872120576 = ( 0 0 1205731 (119905) 1205761205732 (119905) 120576 0 00 1205733 (119905) 120576 0 ) (27)




Complexity 7





(32)






119878ℎ (119905) = 119878lowastℎ (119905) lim



119878119891 (119905) = 119878lowast119891 (119905) (34)











119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)






119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5



Denote







F (119905 119909) =((((((((

1205731 (119905) 1199094 (119905) 1199093 (119905)1205732 (119905) 1199095 (119905) 1199091 (119905)1205733 (119905) 1199096 (119905) 1199092 (119905)000))))))))

V (119905 119909) =((((((((


(18)



1le119894 119895le3







6 Complexity


119875119896 = (1 0 00 1 minus 120579119896 00 0 1) 119876119896 = (1 0 00 1 minus 120579119896 00 0 1 )

(22)

119872(119905) = (120597119891119894 (119905 119909lowast (119905))120597119909119895 )4le119894 119895le6


and

119881 (119905) = (120597V119894 (119905 119909lowast (119905))120597119909119895 )1le119894 119895le3

= (1205831 (119905) + 120574 (119905) 0 00 1205832 (119905) 00 0) 1205833 (119905)) (24)




4 Main Results



(25)





where

119872120576 = ( 0 0 1205731 (119905) 1205761205732 (119905) 120576 0 00 1205733 (119905) 120576 0 ) (27)




Complexity 7





(32)






119878ℎ (119905) = 119878lowastℎ (119905) lim



119878119891 (119905) = 119878lowast119891 (119905) (34)











119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)






119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity


119875119896 = (1 0 00 1 minus 120579119896 00 0 1) 119876119896 = (1 0 00 1 minus 120579119896 00 0 1 )

(22)

119872(119905) = (120597119891119894 (119905 119909lowast (119905))120597119909119895 )4le119894 119895le6


and

119881 (119905) = (120597V119894 (119905 119909lowast (119905))120597119909119895 )1le119894 119895le3

= (1205831 (119905) + 120574 (119905) 0 00 1205832 (119905) 00 0) 1205833 (119905)) (24)




4 Main Results



(25)





where

119872120576 = ( 0 0 1205731 (119905) 1205761205732 (119905) 120576 0 00 1205733 (119905) 120576 0 ) (27)




Complexity 7





(32)






119878ℎ (119905) = 119878lowastℎ (119905) lim



119878119891 (119905) = 119878lowast119891 (119905) (34)











119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)






119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7





(32)






119878ℎ (119905) = 119878lowastℎ (119905) lim



119878119891 (119905) = 119878lowast119891 (119905) (34)











119889 (119875119898 (1199090) 119880lowast) ge 120576lowast (39)






119889 (119875119898 (1199090) 119880lowast) lt 120576lowast (42)

8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity





(46)





for 119905 ge 1198793 (50)


(51)






(54)


Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9


satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (56)








satisfies



119868119904 (119905) ge 1205752and lim inf

119905997888rarr+infin119868119891 (119905) ge 1205753 (60)









10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity



0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11






0 100 200 300 400 500 600 7000

200

400

0 100 200 300 400 500 600 7000

5

10

0 100 200 300 400 500 600 7000

5

10

t (month)

t (month)

t (month)

）Ｂ）

）Ｍ

Ｒ 106

Ｒ 105





12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity

0 100 200 300 400 500 600 7000

100200300400

t ( month )

0 100 200 300 400 500 600 70001234

t ( month )

0 100 200 300 400 500 600 7000

5

10

t ( month )

）Ｂ）

）ＭＲ 10

6

Ｒ 105


0708

1

0 01 02 03 04 05 06 07 08 090

01 1211

09

08

07

06

06

0614

02

03

04

04

04

04

04

05

06

07

08

09

1

2


Appendix


119899120596+119905119894119886 (119904) 119890int119904119899120596+119905119894 119887(119906)119889119906119889119904 (A1)


119911 (119899120596 + 119905+1 ) = (1 minus 1205791) (119911 (119899120596 + 1199050) 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591

+ 119890minus int119899120596+1199051119899120596+1199050119887(120591)119889120591 int119899120596+1199051

119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904) (A2)

Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13


119887(120591)119889120591 int119899120596+1199052119899120596+1199051


119899120596+1199050119886 (119904) 119890int119904119899120596+1199050 119887(119906)119889119906119889119904


119899120596+1199051119886 (119904) 119890int119904119899120596+1199051 119887(119906)119889119906119889119904

(A3)


119897=1(1 minus 120579119897) 119911 (119899120596 + 1199050) 119890minus int119899120596+119905119902119899120596+1199050

119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904(A4)


119897=1(1 minus 120579119897) 119880119899119890minus int119899120596+119905119902119899120596+1199050 119887(120591)119889120591

+ 119902sum119897=1


sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(119906)119889119906119889119904 ≜ 119891 (119880119899) (A5)



times 119902sum119897=1


minus119887(120591)119889120591

sdot int119899120596+119905119895119899120596+119905119895minus1

119886 (119904) 119890int119904119899120596+119905119895minus1 119887(120591)119889120591119889119904(A6)


Data Availability




Acknowledgments


References















14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity














Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








modelling the effects of snail control and health...

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