Research ArticleA Dynamic Model of Human and Livestock Tuberculosis Spreadand Control in Urumqi Xinjiang China

Shan Liu1 Aiqiao Li2 Xiaomei Feng3 Xueliang Zhang4 and Kai Wang4

1Department of Public Health Xinjiang Medical University Urumqi 830011 China2Urumqi Animal Disease Control and Diagnosis Center Urumqi 830063 China3Department of Mathematics Yuncheng University Yuncheng 044000 China4Department of Medical Engineering and Technology Xinjiang Medical University Urumqi 830011 China

Correspondence should be addressed to Kai Wang wangkaimathsinacom

Received 18 February 2016 Accepted 12 June 2016

Academic Editor Konstantin Blyuss

Copyright copy 2016 Shan Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We establish a dynamical model for tuberculosis of humans and cows For the model we firstly give the basic reproduction number1198770 Furthermore we discuss the dynamical behaviors of themodel By epidemiological investigation of tuberculosis among humans

and livestock from 2007 to 2014 in Urumqi Xinjiang China we estimate the parameters of the model and study the transmissiontrend of the disease in Urumqi Xinjiang China The reproduction number in Urumqi for the model is estimated to be 01811 (95confidence interval 0123ndash0281) Finally we perform some sensitivity analysis of several model parameters and give some usefulcomments on controlling the transmission of tuberculosis

1 Introduction

Tuberculosis (TB) is a worldwide public health problemthat is chronic infectious disease of respiratory tract asthe main route of transmission In 1993 WHO declared astate of the global TB in emergency Even if we alreadyknow how to effectively prevent and cure TB through thehalf a century of development and progress there are stillmore than 16 million people who died of TB In 2014TB killed 15 million people (11 million HIV-negative and04 million HIV-positive) The toll comprised 890 000 men480 000 women and 140 000 children India Indonesia andChina had the largest number of cases 23 10 and10 of the global total respectively [1] TB is caused byMycobacterium tuberculosis and spread via air-borne dropletsfrom a cough or sneeze The majority of infected individualsnever develop TB and only few people would induce activeTB

Bovine tuberculosis (BTB) is zoonotic infectious diseasethat is by the OIE (Office International des Epizooties)classified as class B animal epidemics Infected cattle canact as the primary source of infection in other animals and

humans The main route of transmission is the respiratoryand digestive tract Healthy people and animals will beinfected by contacting infected animals or drinking their rawmilk [2 3]

BTB is caused by Mycobacterium bovis and Mycobac-terium tuberculosis BTB is a major infection of work cattleand cows Most of high-yielding dairy cows and young cattleare infected by BTB [4] After being infected with TB cowswill decrease milk production and working cattle becomeemaciated finally most infected cattle died of heart failureBTB not only restricts the development of the livestockindustry but also threatens peoplersquos health It has become aworldwide public health problem [5 6]

BTB has a very long history in Xinjiang and it has a widepopularity and has a serious impact on the animal husbandryin Xinjiang According to Xinjiang related archives andrecords livestock infection of TB existed in Xinjiang beforethe founding of China [7] After the founding of ChinaXinjiang quarantined bovine tuberculosis in the early 1950sQuarantine was via conjunctival sac or use the intradermalallergic reaction method All positive cows detected in thecountry documents or policy specific requirements are to be

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 3410320 10 pageshttpdxdoiorg10115520163410320

2 Computational and Mathematical Methods in Medicine

slaughtered From 1990 to 2007 a total of 1098651 head ofcattle were quarantined in Xinjiang positive rate was 088But because of the shortage of the subsidy funds only a partof positive cattle was slaughtered [6]

Mathematical model is the important tool to measurecontrol strategies against various infectious diseases [8]Mathematical models have played a significant role in under-standing the complexity of TB transmission dynamics Theoriginal mathematical models for TB were developed byBlower et al in 1995 [8] They established a simple modeland a complex model to explain the spread of TB in thepopulation They demonstrated that it takes one to severalhundred years for a TB epidemic to rise fall and reach a sta-ble endemic level Since then a large number ofmathematicalmodels have been created for tuberculosis [9ndash18] Blower etal introduced chemoprophylaxis and treatment in previousmodels due to drug sensitivity and drug resistance expansion[9] They concluded that in order to control TB treatmentfailure rates must be lower in developing countries than indeveloped countries

Although many studies of dynamical TB models spreadbetween humans have been reported little work has beenperformed on such models spread between humans andanimals up to now The purpose of this paper is to propose aTB model between humans and cows to investigate the BTBepidemic situation and analyze the effect of current controlstrategies in Urumqi In this paper based on the referenceof the literature exploring TB transmission mechanismbetween humans and cows the dynamic model is estab-lished

The paper is organized as follows In Section 2 we intro-duce the data sources The model establishing and analysiswere shown in Section 3 including the calculation of thebasic reproductive number and the discussion of positiveequilibrium points The parameter estimation and sensitivityanalysis of the model were carried out in Section 4 Adiscussion is given in Section 5

2 Data Sources

This paper used data from the human and livestock TBepidemiological investigation in Urumqi [19] The epidemio-logical investigation was to find the rules and characteristicsbetween human and livestock TB in Urumqi and control thespread of TB better

21 Object and Method

211 Object We targeted 14 large-scale dairy farms and 8counties of grazed cows in total of 82271 cows in Urumqi

212 Method

(i) Bovine tuberculin intradermal allergy it is accordingto (The Animal Tuberculosis Diagnosis Technology(GBT 18645-2002))

(ii) The EU intradermal allergic reaction it is the same asthe bovine PPD (purified protein derivative) allergic

reaction test On the other side of the cow neckinjected imported bovine type PPD and avian typePPD at the same time located between 12 cm and15 cm doses were 3000 IUhead and 2500 IUheadThe results of the experiments were measured by thesame person before and after injection

(iii) Interferon-120574

(1) antigen stimulation for each cow 5mL heparinanticoagulant blood was collected and trans-ported to the laboratory within 30 h at roomtemperature Each sample of heparin anticoagu-lant blood 15mLwas taken and injected to threedifferent holes The 100 gL bovine type PPDavian type PPD and negative control phosphatebuffer solution (PBS) were taken and joinedto heparin anticoagulant blood respectivelythoroughlymixing after being incorporated intocontaining 5 CO

2incubator in 37∘C for 16 h

With Transferpettor absorbing the supernatantthe supernatant was transferred to centrifugetube (15mL) Namely it is to stimulate theinterferon-120574 in the supernatant

(2) cattle interferon-120574 enzyme-linked immune sor-bent assay (ELISA) For cetuximab coated plateon each hole by adding 50 120583L sample dilutionliquid then adding 50120583L measured samplesor control mixing at room temperature 1 hwashing each hole to join 100 120583L enzyme labeledantibodies at room temperature for 1 h andwashing after each hole joining 100120583L sub-strate at room temperature avoiding light for30min after adding stop solution the OD valueof bovine type PPD stimulation supernatantsminus OD value of PBS supernatant is morethan or equal 01 and OD value of bovine typePPD stimulation supernatants minus OD valueof avian PPD stimulation supernatants is morethan or equal 01 is positive otherwise nega-tive

22 Result

221 The Cow TB Quarantine Results A total of 82271cows were quarantined using tuberculin (PPD) intradermalallergic reaction for 8 years in Urumqi of 14 large-scale dairyfarms and 8 counties (see Table 1) Results the result showsthat there are 333 positive cows in quarantined cows sopositive rate is 040 For 14 large-scale dairy farms 35634cows were quarantined and the positive rate was 051Large-scale dairy farm from 2007 to 2014 TB positive rateswere 037 064 055 052 155 009 015 and018 (see Table 2) For eight counties in Urumqi in thecows of scattered households 46637 cows were quarantinedand the positive rate was 032 The positive rates were006 063 017 032 060 023 014 and 0(seeTable 3)Thepositive rate of cows of scattered householdswas lower than large-scale dairy farm

Computational and Mathematical Methods in Medicine 3

Table 1 The cow TB quarantine statistics in 2007ndash2014

Time Cow herds(head)

TBquarantine(head)

Positive(head)

Positive rate()

2007 21232 10084 28 0282008 20789 15359 97 0632009 24527 12099 39 0322010 20789 9543 35 0372011 15066 10304 99 0962012 13211 9071 15 0172013 9429 9830 14 0142014 14638 5981 6 010Total 139681 82271 333 040

Table 2The cow TB quarantine statistics of scale cow field in 2007ndash2014

TimeTB

quarantine(head)

Positive(head)

Positive rate()

2007 6955 26 0372008 6732 43 0642009 4865 27 0552010 2155 11 0522011 3926 61 1552012 4263 4 0092013 3418 5 0152014 3360 6 018Total 35634 183 051

Table 3 The cow TB quarantine statistics of grazed cows in 2007ndash2014

TimeTB

quarantine(head)

Positive(head)

Positive rate()

2007 3129 2 0062008 8627 54 0632009 7234 12 0172010 7428 24 0322011 6378 38 0602012 4808 11 0232013 6412 9 0142014 2621 0 0Total 46637 150 032

222 Comparison Results of Different BTB QuarantineMethod Use the comparison of the allergy test and 120574-interferon test of developed countries to test 124 cowsof which 199 TB positive samples use the domestic neckallergy quarantineThe results were as follows domestic PPDintradermal allergy and abroad allergy coincidence rate was6859 and interferon-120574 detection of coincidence rate was

Table 4 The point estimate and interval estimation of TB positivecows in Urumqi city in 2007ndash2014

Time Cow herds(head)

Positive rate()

Pointestimate 95 CI

2007 21232 028 59 [37 81]

2008 20789 063 131 [105 157]

2009 24527 032 79 [54 104]

2010 20789 037 76 [51 101]

2011 15066 096 145 [116 173]

2012 13211 017 22 [11 33]

2013 9429 014 13 [6 20]

2014 14638 010 15 [3 26]

79 Abroad allergy and interferon-120574 detection coincidencerate was 864

223 Different Methods of TB Quarantine Compared withPathological Autopsy Results Domestic pure neck allergyquarantine abroad allergy test interferon-120574 test and patho-logic autopsy results compared with positive coincidence rateare as follows it was observed that the coincidence ratebetween the lesion and 120574-interferon detection was 962 andthe coincidence rate with the foreign comparative allergy was923 Test proved that interferon-120574 compared with abroadallergy was of strong specificity

224 Urumqi Cow TB Bacteria Isolation and Identifica-tion Results Harbin Veterinary Research Institute in Chinamakes for the submission of material disease isolated 42strains of acid-fast bacteria Among them are already iden-tified 12 strains of Mycobacterium tuberculosis complexincluding 6 strains of M tuberculosis and Mycobacteriumbovis Mycobacterium tuberculosis complex separation ratewas 39 Spoligotyping and VNTR-MIRU classificationmethod of 12 strains of Mycobacterium tuberculosis complexisolates genotyping results showed that 12 strains of isolatesof tuberculosis bacterium present eight genotypes and threeof them have unique genotype strains China Animal HealthCenter was isolated to 20 strains of bacteria from 26 autopsypositive cows The classification identification of the bacteriaof 20 isolated strains showed that there were three epidemicstrains of bovine type accounting for 65 bovine type BCGaccounting for 5 and other mycobacteria accounting for30 respectively

In 2011 Sanlu milk powder caused damage to a lot ofpeople because of toxic ingredientsmelamine Dairy industryhad a great adverse impact after this point As a consequenceof the not acquired raw milk farmers sold and slaughtereda large number of cows so that the large number of cowsdeclined sharply

Therefore we can get the point estimate and intervalestimation of TB positive cows in Urumqi city in 2007ndash2014(see Table 4)

4 Computational and Mathematical Methods in Medicine

Cow

Human

Ac

Ah

dhShdhEh

Eh

IcSc

Sh Ih Rh

Qc

dcSc

1205731ScIc + 120573

2ScIh

1205733ShIc + 120573

4ShIh

120575cIc

(dc + 120572c)Qc(dc + 120572c + 120583

c)Ic

120590Rh

120588Eh120574Ih

(dh + 120572h)IhdhRh

Figure 1 Transmission diagram of TB among humans and cows

3 The Transmission Model

31 Model Formulation We use a TB model to study thetransmission of TB in Urumqi Xinjiang China [6 13 19]Model consists of two parts cow TBmodel captures the tem-poral dynamics of three groups susceptible cows (119878

119888) cows

infected withMycobacterium tuberculosis (119868119888) and cows that

are removed after infection with Mycobacterium tuberculosis(119876119888) (including quarantined and slaughtered cows) human

TB model captures the temporal dynamics of four groupssusceptible individuals (119878

ℎ) latently infected individuals (119864

ℎ)

active infectious TB cases (119868ℎ) and recovered (119877

ℎ) The

transmission flow among humans and cows is illustrated inFigure 1

The model is described by the following system of sevenordinary differential equations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119876119888

119889119905= 120575119888119868119888minus (119889119888+ 120572119888) 119876119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(1)

The parameters of the model are explained below 119860119888

is recruiting of susceptible cows 119889119888is natural death rate of

cows 1205731is the rate of cows infected TB via cows 120573

2is the

rate of cows infected TB via humans 120572119888is mortality rate

due to TB of cows 120583119888is the slaughter rate to infected cows

120575119888is the quarantine rate to infected cows 119860

ℎis recruiting

of susceptible humans 119889ℎis the removal rate of livestock

workers in dairy farm 1205733is the rate of humans infected TB

via cows 1205734is the rate of humans infected TB via humans 120588

is the progression rate to TB 120572ℎis mortality rate due to TB

of humans 120574 is the cure rate to TB 120590 is the rate of relapse toactive TB

32 Model Analysis Notice that 119876119888is independent of the

first six equations and we start by considering the first sixequations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(2)

Simple algebraic calculation shows that model (2)always has a unique disease-free equilibrium 119864

0(119860119888119889119888

0 119860ℎ119889ℎ 0 0 0) According to the concepts of next genera-

tion matrix and reproduction number presented in [24 25]we define

119865 = (

1205731119878119888119868119888+ 1205732119878119888119868ℎ

1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎ

0

)

119881 = (

(119889119888+ 120572119888+ 120583119888) 119868119888+ 120575119888119868119888

120588119864ℎ+ 119889ℎ119864ℎ

120574119868ℎminus 120590119877ℎ+ (119889ℎ+ 120572ℎ) 119868ℎminus 120588119864ℎ

)

(3)

Noting that the disease-free equilibrium of model (2) is 1198640

then

119865 = (

12057311198781198880 1205732119878119888

1205733119878ℎ0 1205734119878ℎ

0 0 0

)

119881 = (

119889119888+ 120572119888+ 120583119888+ 120575119888

0 0

0 120588 + 119889ℎ

0

0 minus120588 120574 + 119889ℎ+ 120572ℎ

)

(4)

Hence the next generation matrix is

Computational and Mathematical Methods in Medicine 5

119865119881minus1=(

1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205732119860119888

119889119888(120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205734119860ℎ

119889ℎ(120574 + 119889

ℎ+ 120572ℎ)

0 0 0

) (5)

The basic reproduction number is given by 120588(119865119881minus1) and

1198770=minus119886 + radic1198862 minus 4119887

2

119886

= minus1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

minus1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

119887

=1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

minus1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

(6)

According to the conclusions of the literature [24 25] thefollowing results are obtained

Theorem 1 When 1198770lt 1 119864

0is local stable when 119877

0gt 1 119864

0

is unstableUsing a similar argument as in the proof of proposition 33

in [26] we can show that when 1198770gt 1 model (2) has at least

one endemic equilibrium 119864lowast On the stability of the endemic

equilibrium one has the following theorem

Theorem 2 Assume that 1198770gt 1 the endemic equilibrium 119864

lowast

is globally asymptotically stable

Proof Let

1198811= 119878119888minus 119878lowast

119888minus 119878lowast

119888ln119878119888

119878lowast119888

+ 119868119888minus 119868lowast

119888minus 119868lowast

119888ln119868119888

119868lowast119888

1198812= 119878ℎminus 119878lowast

ℎminus 119878lowast

ℎln119878ℎ

119878lowast

ℎ

+ 119864ℎminus 119864lowast

ℎminus 119864lowast

ℎln119864ℎ

119864lowast

ℎ

1198813= 119868ℎminus 119868lowast

ℎminus 119868lowast

ℎln119868ℎ

119868lowast

ℎ

1198814= 119877ℎminus 119877lowast

ℎminus 119877lowast

ℎln119877ℎ

119877lowast

ℎ

(7)

Differentiating 119881119894(119894 = 1 2 3 4) along the solutions of model

(2) then

1198811015840

1= (1 minus

119878lowast

119888

119878119888

) (119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888)

+ (1 minus119868lowast

119888

119868119888

)

sdot (1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888+ 120575119888) 119868119888)

(8)

Further using the equilibrium satisfying equations we have

1198811015840

1= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868119888

119878lowast119888119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868ℎ

119878lowast119888119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119868lowast

119888

119868119888

)(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119868lowast

119888

119868119888

)(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878119888119868119888

119878lowast119888119868lowast119888

minus119878lowast

119888

119878119888

+119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

minus119878119888

119878lowast119888

+ 1)

+ 1205732119878lowast

119888119868lowast

ℎ(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

+ 1)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(2 minus

119878lowast

119888

119878119888

minus119878119888

119878lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(2 minus

119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

)

le 1205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

(9)

6 Computational and Mathematical Methods in Medicine

Through the same calculation we obtain

1198811015840

2= (1 minus

119878lowast

ℎ

119878ℎ

) (119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ)

+ (1 minus119864lowast

ℎ

119864ℎ

) (1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus (120588 + 119889

ℎ) 119864ℎ)

= minus119889ℎ

(119878ℎminus 119878lowast

ℎ)2

119878ℎ

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868119888

119878lowast

ℎ119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(Sℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

+ 1)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

+ 1)

= 1205733119878lowast

ℎ119868lowast

119888(2 minus

119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(2 minus

119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

)

1198811015840

3= (1 minus

119868lowast

ℎ

119868ℎ

) (120588119864ℎ+ 120590119877ℎminus (119889ℎ+ 120572ℎ+ 120574) 119868ℎ)

= 120588119864lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

le 120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

(10)

Similarly it is easy that

1198811015840

4= (1 minus

119877lowast

ℎ

119877ℎ

) (120574119868ℎminus 119889ℎ119877ℎ)

= 120574119868lowast

ℎ(1 minus

119877lowast

ℎ

119877ℎ

)(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

)

le 120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

)

(11)

Now construct the following Lyapunov function

119871 = 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881198811+ 120588120574120573

2119864lowast

ℎ119868lowast

ℎ119878lowast

1198881198812

+ (12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888)1198813

+ (12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888+ 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888) 1198814

(12)

Then

le 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

+ 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

) + 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) + 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) le 0

(13)

It can be verified that the largest invariant set where 1198711015840 = 0 issingleton 119864lowast Therefore by LaSallersquos invariance principle 119864lowastis globally asymptotically stable

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

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Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

2 Computational and Mathematical Methods in Medicine

slaughtered From 1990 to 2007 a total of 1098651 head ofcattle were quarantined in Xinjiang positive rate was 088But because of the shortage of the subsidy funds only a partof positive cattle was slaughtered [6]

Mathematical model is the important tool to measurecontrol strategies against various infectious diseases [8]Mathematical models have played a significant role in under-standing the complexity of TB transmission dynamics Theoriginal mathematical models for TB were developed byBlower et al in 1995 [8] They established a simple modeland a complex model to explain the spread of TB in thepopulation They demonstrated that it takes one to severalhundred years for a TB epidemic to rise fall and reach a sta-ble endemic level Since then a large number ofmathematicalmodels have been created for tuberculosis [9ndash18] Blower etal introduced chemoprophylaxis and treatment in previousmodels due to drug sensitivity and drug resistance expansion[9] They concluded that in order to control TB treatmentfailure rates must be lower in developing countries than indeveloped countries

Although many studies of dynamical TB models spreadbetween humans have been reported little work has beenperformed on such models spread between humans andanimals up to now The purpose of this paper is to propose aTB model between humans and cows to investigate the BTBepidemic situation and analyze the effect of current controlstrategies in Urumqi In this paper based on the referenceof the literature exploring TB transmission mechanismbetween humans and cows the dynamic model is estab-lished

The paper is organized as follows In Section 2 we intro-duce the data sources The model establishing and analysiswere shown in Section 3 including the calculation of thebasic reproductive number and the discussion of positiveequilibrium points The parameter estimation and sensitivityanalysis of the model were carried out in Section 4 Adiscussion is given in Section 5

2 Data Sources

This paper used data from the human and livestock TBepidemiological investigation in Urumqi [19] The epidemio-logical investigation was to find the rules and characteristicsbetween human and livestock TB in Urumqi and control thespread of TB better

21 Object and Method

211 Object We targeted 14 large-scale dairy farms and 8counties of grazed cows in total of 82271 cows in Urumqi

212 Method

(i) Bovine tuberculin intradermal allergy it is accordingto (The Animal Tuberculosis Diagnosis Technology(GBT 18645-2002))

(ii) The EU intradermal allergic reaction it is the same asthe bovine PPD (purified protein derivative) allergic

reaction test On the other side of the cow neckinjected imported bovine type PPD and avian typePPD at the same time located between 12 cm and15 cm doses were 3000 IUhead and 2500 IUheadThe results of the experiments were measured by thesame person before and after injection

(iii) Interferon-120574

(1) antigen stimulation for each cow 5mL heparinanticoagulant blood was collected and trans-ported to the laboratory within 30 h at roomtemperature Each sample of heparin anticoagu-lant blood 15mLwas taken and injected to threedifferent holes The 100 gL bovine type PPDavian type PPD and negative control phosphatebuffer solution (PBS) were taken and joinedto heparin anticoagulant blood respectivelythoroughlymixing after being incorporated intocontaining 5 CO

2incubator in 37∘C for 16 h

With Transferpettor absorbing the supernatantthe supernatant was transferred to centrifugetube (15mL) Namely it is to stimulate theinterferon-120574 in the supernatant

(2) cattle interferon-120574 enzyme-linked immune sor-bent assay (ELISA) For cetuximab coated plateon each hole by adding 50 120583L sample dilutionliquid then adding 50120583L measured samplesor control mixing at room temperature 1 hwashing each hole to join 100 120583L enzyme labeledantibodies at room temperature for 1 h andwashing after each hole joining 100120583L sub-strate at room temperature avoiding light for30min after adding stop solution the OD valueof bovine type PPD stimulation supernatantsminus OD value of PBS supernatant is morethan or equal 01 and OD value of bovine typePPD stimulation supernatants minus OD valueof avian PPD stimulation supernatants is morethan or equal 01 is positive otherwise nega-tive

22 Result

221 The Cow TB Quarantine Results A total of 82271cows were quarantined using tuberculin (PPD) intradermalallergic reaction for 8 years in Urumqi of 14 large-scale dairyfarms and 8 counties (see Table 1) Results the result showsthat there are 333 positive cows in quarantined cows sopositive rate is 040 For 14 large-scale dairy farms 35634cows were quarantined and the positive rate was 051Large-scale dairy farm from 2007 to 2014 TB positive rateswere 037 064 055 052 155 009 015 and018 (see Table 2) For eight counties in Urumqi in thecows of scattered households 46637 cows were quarantinedand the positive rate was 032 The positive rates were006 063 017 032 060 023 014 and 0(seeTable 3)Thepositive rate of cows of scattered householdswas lower than large-scale dairy farm

Computational and Mathematical Methods in Medicine 3

Table 1 The cow TB quarantine statistics in 2007ndash2014

Time Cow herds(head)

TBquarantine(head)

Positive(head)

Positive rate()

2007 21232 10084 28 0282008 20789 15359 97 0632009 24527 12099 39 0322010 20789 9543 35 0372011 15066 10304 99 0962012 13211 9071 15 0172013 9429 9830 14 0142014 14638 5981 6 010Total 139681 82271 333 040

Table 2The cow TB quarantine statistics of scale cow field in 2007ndash2014

TimeTB

quarantine(head)

Positive(head)

Positive rate()

2007 6955 26 0372008 6732 43 0642009 4865 27 0552010 2155 11 0522011 3926 61 1552012 4263 4 0092013 3418 5 0152014 3360 6 018Total 35634 183 051

Table 3 The cow TB quarantine statistics of grazed cows in 2007ndash2014

TimeTB

quarantine(head)

Positive(head)

Positive rate()

2007 3129 2 0062008 8627 54 0632009 7234 12 0172010 7428 24 0322011 6378 38 0602012 4808 11 0232013 6412 9 0142014 2621 0 0Total 46637 150 032

222 Comparison Results of Different BTB QuarantineMethod Use the comparison of the allergy test and 120574-interferon test of developed countries to test 124 cowsof which 199 TB positive samples use the domestic neckallergy quarantineThe results were as follows domestic PPDintradermal allergy and abroad allergy coincidence rate was6859 and interferon-120574 detection of coincidence rate was

Table 4 The point estimate and interval estimation of TB positivecows in Urumqi city in 2007ndash2014

Time Cow herds(head)

Positive rate()

Pointestimate 95 CI

2007 21232 028 59 [37 81]

2008 20789 063 131 [105 157]

2009 24527 032 79 [54 104]

2010 20789 037 76 [51 101]

2011 15066 096 145 [116 173]

2012 13211 017 22 [11 33]

2013 9429 014 13 [6 20]

2014 14638 010 15 [3 26]

79 Abroad allergy and interferon-120574 detection coincidencerate was 864

223 Different Methods of TB Quarantine Compared withPathological Autopsy Results Domestic pure neck allergyquarantine abroad allergy test interferon-120574 test and patho-logic autopsy results compared with positive coincidence rateare as follows it was observed that the coincidence ratebetween the lesion and 120574-interferon detection was 962 andthe coincidence rate with the foreign comparative allergy was923 Test proved that interferon-120574 compared with abroadallergy was of strong specificity

224 Urumqi Cow TB Bacteria Isolation and Identifica-tion Results Harbin Veterinary Research Institute in Chinamakes for the submission of material disease isolated 42strains of acid-fast bacteria Among them are already iden-tified 12 strains of Mycobacterium tuberculosis complexincluding 6 strains of M tuberculosis and Mycobacteriumbovis Mycobacterium tuberculosis complex separation ratewas 39 Spoligotyping and VNTR-MIRU classificationmethod of 12 strains of Mycobacterium tuberculosis complexisolates genotyping results showed that 12 strains of isolatesof tuberculosis bacterium present eight genotypes and threeof them have unique genotype strains China Animal HealthCenter was isolated to 20 strains of bacteria from 26 autopsypositive cows The classification identification of the bacteriaof 20 isolated strains showed that there were three epidemicstrains of bovine type accounting for 65 bovine type BCGaccounting for 5 and other mycobacteria accounting for30 respectively

In 2011 Sanlu milk powder caused damage to a lot ofpeople because of toxic ingredientsmelamine Dairy industryhad a great adverse impact after this point As a consequenceof the not acquired raw milk farmers sold and slaughtereda large number of cows so that the large number of cowsdeclined sharply

Therefore we can get the point estimate and intervalestimation of TB positive cows in Urumqi city in 2007ndash2014(see Table 4)

4 Computational and Mathematical Methods in Medicine

Cow

Human

Ac

Ah

dhShdhEh

Eh

IcSc

Sh Ih Rh

Qc

dcSc

1205731ScIc + 120573

2ScIh

1205733ShIc + 120573

4ShIh

120575cIc

(dc + 120572c)Qc(dc + 120572c + 120583

c)Ic

120590Rh

120588Eh120574Ih

(dh + 120572h)IhdhRh

Figure 1 Transmission diagram of TB among humans and cows

3 The Transmission Model

31 Model Formulation We use a TB model to study thetransmission of TB in Urumqi Xinjiang China [6 13 19]Model consists of two parts cow TBmodel captures the tem-poral dynamics of three groups susceptible cows (119878

119888) cows

infected withMycobacterium tuberculosis (119868119888) and cows that

are removed after infection with Mycobacterium tuberculosis(119876119888) (including quarantined and slaughtered cows) human

TB model captures the temporal dynamics of four groupssusceptible individuals (119878

ℎ) latently infected individuals (119864

ℎ)

active infectious TB cases (119868ℎ) and recovered (119877

ℎ) The

transmission flow among humans and cows is illustrated inFigure 1

The model is described by the following system of sevenordinary differential equations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119876119888

119889119905= 120575119888119868119888minus (119889119888+ 120572119888) 119876119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(1)

The parameters of the model are explained below 119860119888

is recruiting of susceptible cows 119889119888is natural death rate of

cows 1205731is the rate of cows infected TB via cows 120573

2is the

rate of cows infected TB via humans 120572119888is mortality rate

due to TB of cows 120583119888is the slaughter rate to infected cows

120575119888is the quarantine rate to infected cows 119860

ℎis recruiting

of susceptible humans 119889ℎis the removal rate of livestock

workers in dairy farm 1205733is the rate of humans infected TB

via cows 1205734is the rate of humans infected TB via humans 120588

is the progression rate to TB 120572ℎis mortality rate due to TB

of humans 120574 is the cure rate to TB 120590 is the rate of relapse toactive TB

32 Model Analysis Notice that 119876119888is independent of the

first six equations and we start by considering the first sixequations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(2)

Simple algebraic calculation shows that model (2)always has a unique disease-free equilibrium 119864

0(119860119888119889119888

0 119860ℎ119889ℎ 0 0 0) According to the concepts of next genera-

tion matrix and reproduction number presented in [24 25]we define

119865 = (

1205731119878119888119868119888+ 1205732119878119888119868ℎ

1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎ

0

)

119881 = (

(119889119888+ 120572119888+ 120583119888) 119868119888+ 120575119888119868119888

120588119864ℎ+ 119889ℎ119864ℎ

120574119868ℎminus 120590119877ℎ+ (119889ℎ+ 120572ℎ) 119868ℎminus 120588119864ℎ

)

(3)

Noting that the disease-free equilibrium of model (2) is 1198640

then

119865 = (

12057311198781198880 1205732119878119888

1205733119878ℎ0 1205734119878ℎ

0 0 0

)

119881 = (

119889119888+ 120572119888+ 120583119888+ 120575119888

0 0

0 120588 + 119889ℎ

0

0 minus120588 120574 + 119889ℎ+ 120572ℎ

)

(4)

Hence the next generation matrix is

Computational and Mathematical Methods in Medicine 5

119865119881minus1=(

1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205732119860119888

119889119888(120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205734119860ℎ

119889ℎ(120574 + 119889

ℎ+ 120572ℎ)

0 0 0

) (5)

The basic reproduction number is given by 120588(119865119881minus1) and

1198770=minus119886 + radic1198862 minus 4119887

2

119886

= minus1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

minus1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

119887

=1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

minus1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

(6)

According to the conclusions of the literature [24 25] thefollowing results are obtained

Theorem 1 When 1198770lt 1 119864

0is local stable when 119877

0gt 1 119864

0

is unstableUsing a similar argument as in the proof of proposition 33

in [26] we can show that when 1198770gt 1 model (2) has at least

one endemic equilibrium 119864lowast On the stability of the endemic

equilibrium one has the following theorem

Theorem 2 Assume that 1198770gt 1 the endemic equilibrium 119864

lowast

is globally asymptotically stable

Proof Let

1198811= 119878119888minus 119878lowast

119888minus 119878lowast

119888ln119878119888

119878lowast119888

+ 119868119888minus 119868lowast

119888minus 119868lowast

119888ln119868119888

119868lowast119888

1198812= 119878ℎminus 119878lowast

ℎminus 119878lowast

ℎln119878ℎ

119878lowast

ℎ

+ 119864ℎminus 119864lowast

ℎminus 119864lowast

ℎln119864ℎ

119864lowast

ℎ

1198813= 119868ℎminus 119868lowast

ℎminus 119868lowast

ℎln119868ℎ

119868lowast

ℎ

1198814= 119877ℎminus 119877lowast

ℎminus 119877lowast

ℎln119877ℎ

119877lowast

ℎ

(7)

Differentiating 119881119894(119894 = 1 2 3 4) along the solutions of model

(2) then

1198811015840

1= (1 minus

119878lowast

119888

119878119888

) (119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888)

+ (1 minus119868lowast

119888

119868119888

)

sdot (1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888+ 120575119888) 119868119888)

(8)

Further using the equilibrium satisfying equations we have

1198811015840

1= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868119888

119878lowast119888119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868ℎ

119878lowast119888119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119868lowast

119888

119868119888

)(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119868lowast

119888

119868119888

)(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878119888119868119888

119878lowast119888119868lowast119888

minus119878lowast

119888

119878119888

+119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

minus119878119888

119878lowast119888

+ 1)

+ 1205732119878lowast

119888119868lowast

ℎ(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

+ 1)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(2 minus

119878lowast

119888

119878119888

minus119878119888

119878lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(2 minus

119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

)

le 1205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

(9)

6 Computational and Mathematical Methods in Medicine

Through the same calculation we obtain

1198811015840

2= (1 minus

119878lowast

ℎ

119878ℎ

) (119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ)

+ (1 minus119864lowast

ℎ

119864ℎ

) (1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus (120588 + 119889

ℎ) 119864ℎ)

= minus119889ℎ

(119878ℎminus 119878lowast

ℎ)2

119878ℎ

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868119888

119878lowast

ℎ119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(Sℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

+ 1)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

+ 1)

= 1205733119878lowast

ℎ119868lowast

119888(2 minus

119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(2 minus

119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

)

1198811015840

3= (1 minus

119868lowast

ℎ

119868ℎ

) (120588119864ℎ+ 120590119877ℎminus (119889ℎ+ 120572ℎ+ 120574) 119868ℎ)

= 120588119864lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

le 120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

(10)

Similarly it is easy that

1198811015840

4= (1 minus

119877lowast

ℎ

119877ℎ

) (120574119868ℎminus 119889ℎ119877ℎ)

= 120574119868lowast

ℎ(1 minus

119877lowast

ℎ

119877ℎ

)(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

)

le 120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

)

(11)

Now construct the following Lyapunov function

119871 = 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881198811+ 120588120574120573

2119864lowast

ℎ119868lowast

ℎ119878lowast

1198881198812

+ (12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888)1198813

+ (12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888+ 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888) 1198814

(12)

Then

le 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

+ 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

) + 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) + 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) le 0

(13)

It can be verified that the largest invariant set where 1198711015840 = 0 issingleton 119864lowast Therefore by LaSallersquos invariance principle 119864lowastis globally asymptotically stable

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

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of

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Disease Markers

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Oxidative Medicine and Cellular Longevity

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Research and TreatmentAIDS

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 3

Table 1 The cow TB quarantine statistics in 2007ndash2014

Time Cow herds(head)

TBquarantine(head)

Positive(head)

Positive rate()

2007 21232 10084 28 0282008 20789 15359 97 0632009 24527 12099 39 0322010 20789 9543 35 0372011 15066 10304 99 0962012 13211 9071 15 0172013 9429 9830 14 0142014 14638 5981 6 010Total 139681 82271 333 040

Table 2The cow TB quarantine statistics of scale cow field in 2007ndash2014

TimeTB

quarantine(head)

Positive(head)

Positive rate()

2007 6955 26 0372008 6732 43 0642009 4865 27 0552010 2155 11 0522011 3926 61 1552012 4263 4 0092013 3418 5 0152014 3360 6 018Total 35634 183 051

Table 3 The cow TB quarantine statistics of grazed cows in 2007ndash2014

TimeTB

quarantine(head)

Positive(head)

Positive rate()

2007 3129 2 0062008 8627 54 0632009 7234 12 0172010 7428 24 0322011 6378 38 0602012 4808 11 0232013 6412 9 0142014 2621 0 0Total 46637 150 032

222 Comparison Results of Different BTB QuarantineMethod Use the comparison of the allergy test and 120574-interferon test of developed countries to test 124 cowsof which 199 TB positive samples use the domestic neckallergy quarantineThe results were as follows domestic PPDintradermal allergy and abroad allergy coincidence rate was6859 and interferon-120574 detection of coincidence rate was

Table 4 The point estimate and interval estimation of TB positivecows in Urumqi city in 2007ndash2014

Time Cow herds(head)

Positive rate()

Pointestimate 95 CI

2007 21232 028 59 [37 81]

2008 20789 063 131 [105 157]

2009 24527 032 79 [54 104]

2010 20789 037 76 [51 101]

2011 15066 096 145 [116 173]

2012 13211 017 22 [11 33]

2013 9429 014 13 [6 20]

2014 14638 010 15 [3 26]

79 Abroad allergy and interferon-120574 detection coincidencerate was 864

223 Different Methods of TB Quarantine Compared withPathological Autopsy Results Domestic pure neck allergyquarantine abroad allergy test interferon-120574 test and patho-logic autopsy results compared with positive coincidence rateare as follows it was observed that the coincidence ratebetween the lesion and 120574-interferon detection was 962 andthe coincidence rate with the foreign comparative allergy was923 Test proved that interferon-120574 compared with abroadallergy was of strong specificity

224 Urumqi Cow TB Bacteria Isolation and Identifica-tion Results Harbin Veterinary Research Institute in Chinamakes for the submission of material disease isolated 42strains of acid-fast bacteria Among them are already iden-tified 12 strains of Mycobacterium tuberculosis complexincluding 6 strains of M tuberculosis and Mycobacteriumbovis Mycobacterium tuberculosis complex separation ratewas 39 Spoligotyping and VNTR-MIRU classificationmethod of 12 strains of Mycobacterium tuberculosis complexisolates genotyping results showed that 12 strains of isolatesof tuberculosis bacterium present eight genotypes and threeof them have unique genotype strains China Animal HealthCenter was isolated to 20 strains of bacteria from 26 autopsypositive cows The classification identification of the bacteriaof 20 isolated strains showed that there were three epidemicstrains of bovine type accounting for 65 bovine type BCGaccounting for 5 and other mycobacteria accounting for30 respectively

In 2011 Sanlu milk powder caused damage to a lot ofpeople because of toxic ingredientsmelamine Dairy industryhad a great adverse impact after this point As a consequenceof the not acquired raw milk farmers sold and slaughtereda large number of cows so that the large number of cowsdeclined sharply

Therefore we can get the point estimate and intervalestimation of TB positive cows in Urumqi city in 2007ndash2014(see Table 4)

4 Computational and Mathematical Methods in Medicine

Cow

Human

Ac

Ah

dhShdhEh

Eh

IcSc

Sh Ih Rh

Qc

dcSc

1205731ScIc + 120573

2ScIh

1205733ShIc + 120573

4ShIh

120575cIc

(dc + 120572c)Qc(dc + 120572c + 120583

c)Ic

120590Rh

120588Eh120574Ih

(dh + 120572h)IhdhRh

Figure 1 Transmission diagram of TB among humans and cows

3 The Transmission Model

31 Model Formulation We use a TB model to study thetransmission of TB in Urumqi Xinjiang China [6 13 19]Model consists of two parts cow TBmodel captures the tem-poral dynamics of three groups susceptible cows (119878

119888) cows

infected withMycobacterium tuberculosis (119868119888) and cows that

are removed after infection with Mycobacterium tuberculosis(119876119888) (including quarantined and slaughtered cows) human

TB model captures the temporal dynamics of four groupssusceptible individuals (119878

ℎ) latently infected individuals (119864

ℎ)

active infectious TB cases (119868ℎ) and recovered (119877

ℎ) The

transmission flow among humans and cows is illustrated inFigure 1

The model is described by the following system of sevenordinary differential equations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119876119888

119889119905= 120575119888119868119888minus (119889119888+ 120572119888) 119876119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(1)

The parameters of the model are explained below 119860119888

is recruiting of susceptible cows 119889119888is natural death rate of

cows 1205731is the rate of cows infected TB via cows 120573

2is the

rate of cows infected TB via humans 120572119888is mortality rate

due to TB of cows 120583119888is the slaughter rate to infected cows

120575119888is the quarantine rate to infected cows 119860

ℎis recruiting

of susceptible humans 119889ℎis the removal rate of livestock

workers in dairy farm 1205733is the rate of humans infected TB

via cows 1205734is the rate of humans infected TB via humans 120588

is the progression rate to TB 120572ℎis mortality rate due to TB

of humans 120574 is the cure rate to TB 120590 is the rate of relapse toactive TB

32 Model Analysis Notice that 119876119888is independent of the

first six equations and we start by considering the first sixequations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(2)

Simple algebraic calculation shows that model (2)always has a unique disease-free equilibrium 119864

0(119860119888119889119888

0 119860ℎ119889ℎ 0 0 0) According to the concepts of next genera-

tion matrix and reproduction number presented in [24 25]we define

119865 = (

1205731119878119888119868119888+ 1205732119878119888119868ℎ

1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎ

0

)

119881 = (

(119889119888+ 120572119888+ 120583119888) 119868119888+ 120575119888119868119888

120588119864ℎ+ 119889ℎ119864ℎ

120574119868ℎminus 120590119877ℎ+ (119889ℎ+ 120572ℎ) 119868ℎminus 120588119864ℎ

)

(3)

Noting that the disease-free equilibrium of model (2) is 1198640

then

119865 = (

12057311198781198880 1205732119878119888

1205733119878ℎ0 1205734119878ℎ

0 0 0

)

119881 = (

119889119888+ 120572119888+ 120583119888+ 120575119888

0 0

0 120588 + 119889ℎ

0

0 minus120588 120574 + 119889ℎ+ 120572ℎ

)

(4)

Hence the next generation matrix is

Computational and Mathematical Methods in Medicine 5

119865119881minus1=(

1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205732119860119888

119889119888(120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205734119860ℎ

119889ℎ(120574 + 119889

ℎ+ 120572ℎ)

0 0 0

) (5)

The basic reproduction number is given by 120588(119865119881minus1) and

1198770=minus119886 + radic1198862 minus 4119887

2

119886

= minus1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

minus1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

119887

=1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

minus1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

(6)

According to the conclusions of the literature [24 25] thefollowing results are obtained

Theorem 1 When 1198770lt 1 119864

0is local stable when 119877

0gt 1 119864

0

is unstableUsing a similar argument as in the proof of proposition 33

in [26] we can show that when 1198770gt 1 model (2) has at least

one endemic equilibrium 119864lowast On the stability of the endemic

equilibrium one has the following theorem

Theorem 2 Assume that 1198770gt 1 the endemic equilibrium 119864

lowast

is globally asymptotically stable

Proof Let

1198811= 119878119888minus 119878lowast

119888minus 119878lowast

119888ln119878119888

119878lowast119888

+ 119868119888minus 119868lowast

119888minus 119868lowast

119888ln119868119888

119868lowast119888

1198812= 119878ℎminus 119878lowast

ℎminus 119878lowast

ℎln119878ℎ

119878lowast

ℎ

+ 119864ℎminus 119864lowast

ℎminus 119864lowast

ℎln119864ℎ

119864lowast

ℎ

1198813= 119868ℎminus 119868lowast

ℎminus 119868lowast

ℎln119868ℎ

119868lowast

ℎ

1198814= 119877ℎminus 119877lowast

ℎminus 119877lowast

ℎln119877ℎ

119877lowast

ℎ

(7)

Differentiating 119881119894(119894 = 1 2 3 4) along the solutions of model

(2) then

1198811015840

1= (1 minus

119878lowast

119888

119878119888

) (119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888)

+ (1 minus119868lowast

119888

119868119888

)

sdot (1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888+ 120575119888) 119868119888)

(8)

Further using the equilibrium satisfying equations we have

1198811015840

1= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868119888

119878lowast119888119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868ℎ

119878lowast119888119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119868lowast

119888

119868119888

)(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119868lowast

119888

119868119888

)(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878119888119868119888

119878lowast119888119868lowast119888

minus119878lowast

119888

119878119888

+119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

minus119878119888

119878lowast119888

+ 1)

+ 1205732119878lowast

119888119868lowast

ℎ(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

+ 1)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(2 minus

119878lowast

119888

119878119888

minus119878119888

119878lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(2 minus

119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

)

le 1205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

(9)

6 Computational and Mathematical Methods in Medicine

Through the same calculation we obtain

1198811015840

2= (1 minus

119878lowast

ℎ

119878ℎ

) (119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ)

+ (1 minus119864lowast

ℎ

119864ℎ

) (1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus (120588 + 119889

ℎ) 119864ℎ)

= minus119889ℎ

(119878ℎminus 119878lowast

ℎ)2

119878ℎ

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868119888

119878lowast

ℎ119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(Sℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

+ 1)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

+ 1)

= 1205733119878lowast

ℎ119868lowast

119888(2 minus

119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(2 minus

119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

)

1198811015840

3= (1 minus

119868lowast

ℎ

119868ℎ

) (120588119864ℎ+ 120590119877ℎminus (119889ℎ+ 120572ℎ+ 120574) 119868ℎ)

= 120588119864lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

le 120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

(10)

Similarly it is easy that

1198811015840

4= (1 minus

119877lowast

ℎ

119877ℎ

) (120574119868ℎminus 119889ℎ119877ℎ)

= 120574119868lowast

ℎ(1 minus

119877lowast

ℎ

119877ℎ

)(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

)

le 120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

)

(11)

Now construct the following Lyapunov function

119871 = 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881198811+ 120588120574120573

2119864lowast

ℎ119868lowast

ℎ119878lowast

1198881198812

+ (12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888)1198813

+ (12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888+ 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888) 1198814

(12)

Then

le 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

+ 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

) + 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) + 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) le 0

(13)

It can be verified that the largest invariant set where 1198711015840 = 0 issingleton 119864lowast Therefore by LaSallersquos invariance principle 119864lowastis globally asymptotically stable

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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EndocrinologyInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

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Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

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Research and TreatmentAIDS

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

4 Computational and Mathematical Methods in Medicine

Cow

Human

Ac

Ah

dhShdhEh

Eh

IcSc

Sh Ih Rh

Qc

dcSc

1205731ScIc + 120573

2ScIh

1205733ShIc + 120573

4ShIh

120575cIc

(dc + 120572c)Qc(dc + 120572c + 120583

c)Ic

120590Rh

120588Eh120574Ih

(dh + 120572h)IhdhRh

Figure 1 Transmission diagram of TB among humans and cows

3 The Transmission Model

31 Model Formulation We use a TB model to study thetransmission of TB in Urumqi Xinjiang China [6 13 19]Model consists of two parts cow TBmodel captures the tem-poral dynamics of three groups susceptible cows (119878

119888) cows

infected withMycobacterium tuberculosis (119868119888) and cows that

are removed after infection with Mycobacterium tuberculosis(119876119888) (including quarantined and slaughtered cows) human

TB model captures the temporal dynamics of four groupssusceptible individuals (119878

ℎ) latently infected individuals (119864

ℎ)

active infectious TB cases (119868ℎ) and recovered (119877

ℎ) The

transmission flow among humans and cows is illustrated inFigure 1

The model is described by the following system of sevenordinary differential equations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119876119888

119889119905= 120575119888119868119888minus (119889119888+ 120572119888) 119876119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(1)

The parameters of the model are explained below 119860119888

is recruiting of susceptible cows 119889119888is natural death rate of

cows 1205731is the rate of cows infected TB via cows 120573

2is the

rate of cows infected TB via humans 120572119888is mortality rate

due to TB of cows 120583119888is the slaughter rate to infected cows

120575119888is the quarantine rate to infected cows 119860

ℎis recruiting

of susceptible humans 119889ℎis the removal rate of livestock

workers in dairy farm 1205733is the rate of humans infected TB

via cows 1205734is the rate of humans infected TB via humans 120588

is the progression rate to TB 120572ℎis mortality rate due to TB

of humans 120574 is the cure rate to TB 120590 is the rate of relapse toactive TB

32 Model Analysis Notice that 119876119888is independent of the

first six equations and we start by considering the first sixequations

119889119878119888

119889119905= 119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888

119889119868119888

119889119905= 1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888) 119868119888minus 120575119888119868119888

119889119878ℎ

119889119905= 119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ

119889119864ℎ

119889119905= 1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus 120588119864ℎminus 119889ℎ119864ℎ

119889119868ℎ

119889119905= 120588119864ℎminus 120574119868ℎminus (119889ℎ+ 120572ℎ) 119868ℎ+ 120590119877ℎ

119889119877ℎ

119889119905= 120574119868ℎminus 120590119877ℎminus 119889ℎ119877ℎ

(2)

Simple algebraic calculation shows that model (2)always has a unique disease-free equilibrium 119864

0(119860119888119889119888

0 119860ℎ119889ℎ 0 0 0) According to the concepts of next genera-

tion matrix and reproduction number presented in [24 25]we define

119865 = (

1205731119878119888119868119888+ 1205732119878119888119868ℎ

1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎ

0

)

119881 = (

(119889119888+ 120572119888+ 120583119888) 119868119888+ 120575119888119868119888

120588119864ℎ+ 119889ℎ119864ℎ

120574119868ℎminus 120590119877ℎ+ (119889ℎ+ 120572ℎ) 119868ℎminus 120588119864ℎ

)

(3)

Noting that the disease-free equilibrium of model (2) is 1198640

then

119865 = (

12057311198781198880 1205732119878119888

1205733119878ℎ0 1205734119878ℎ

0 0 0

)

119881 = (

119889119888+ 120572119888+ 120583119888+ 120575119888

0 0

0 120588 + 119889ℎ

0

0 minus120588 120574 + 119889ℎ+ 120572ℎ

)

(4)

Hence the next generation matrix is

Computational and Mathematical Methods in Medicine 5

119865119881minus1=(

1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205732119860119888

119889119888(120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205734119860ℎ

119889ℎ(120574 + 119889

ℎ+ 120572ℎ)

0 0 0

) (5)

The basic reproduction number is given by 120588(119865119881minus1) and

1198770=minus119886 + radic1198862 minus 4119887

2

119886

= minus1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

minus1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

119887

=1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

minus1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

(6)

According to the conclusions of the literature [24 25] thefollowing results are obtained

Theorem 1 When 1198770lt 1 119864

0is local stable when 119877

0gt 1 119864

0

is unstableUsing a similar argument as in the proof of proposition 33

in [26] we can show that when 1198770gt 1 model (2) has at least

one endemic equilibrium 119864lowast On the stability of the endemic

equilibrium one has the following theorem

Theorem 2 Assume that 1198770gt 1 the endemic equilibrium 119864

lowast

is globally asymptotically stable

Proof Let

1198811= 119878119888minus 119878lowast

119888minus 119878lowast

119888ln119878119888

119878lowast119888

+ 119868119888minus 119868lowast

119888minus 119868lowast

119888ln119868119888

119868lowast119888

1198812= 119878ℎminus 119878lowast

ℎminus 119878lowast

ℎln119878ℎ

119878lowast

ℎ

+ 119864ℎminus 119864lowast

ℎminus 119864lowast

ℎln119864ℎ

119864lowast

ℎ

1198813= 119868ℎminus 119868lowast

ℎminus 119868lowast

ℎln119868ℎ

119868lowast

ℎ

1198814= 119877ℎminus 119877lowast

ℎminus 119877lowast

ℎln119877ℎ

119877lowast

ℎ

(7)

Differentiating 119881119894(119894 = 1 2 3 4) along the solutions of model

(2) then

1198811015840

1= (1 minus

119878lowast

119888

119878119888

) (119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888)

+ (1 minus119868lowast

119888

119868119888

)

sdot (1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888+ 120575119888) 119868119888)

(8)

Further using the equilibrium satisfying equations we have

1198811015840

1= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868119888

119878lowast119888119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868ℎ

119878lowast119888119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119868lowast

119888

119868119888

)(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119868lowast

119888

119868119888

)(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878119888119868119888

119878lowast119888119868lowast119888

minus119878lowast

119888

119878119888

+119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

minus119878119888

119878lowast119888

+ 1)

+ 1205732119878lowast

119888119868lowast

ℎ(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

+ 1)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(2 minus

119878lowast

119888

119878119888

minus119878119888

119878lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(2 minus

119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

)

le 1205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

(9)

6 Computational and Mathematical Methods in Medicine

Through the same calculation we obtain

1198811015840

2= (1 minus

119878lowast

ℎ

119878ℎ

) (119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ)

+ (1 minus119864lowast

ℎ

119864ℎ

) (1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus (120588 + 119889

ℎ) 119864ℎ)

= minus119889ℎ

(119878ℎminus 119878lowast

ℎ)2

119878ℎ

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868119888

119878lowast

ℎ119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(Sℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

+ 1)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

+ 1)

= 1205733119878lowast

ℎ119868lowast

119888(2 minus

119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(2 minus

119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

)

1198811015840

3= (1 minus

119868lowast

ℎ

119868ℎ

) (120588119864ℎ+ 120590119877ℎminus (119889ℎ+ 120572ℎ+ 120574) 119868ℎ)

= 120588119864lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

le 120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

(10)

Similarly it is easy that

1198811015840

4= (1 minus

119877lowast

ℎ

119877ℎ

) (120574119868ℎminus 119889ℎ119877ℎ)

= 120574119868lowast

ℎ(1 minus

119877lowast

ℎ

119877ℎ

)(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

)

le 120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

)

(11)

Now construct the following Lyapunov function

119871 = 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881198811+ 120588120574120573

2119864lowast

ℎ119868lowast

ℎ119878lowast

1198881198812

+ (12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888)1198813

+ (12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888+ 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888) 1198814

(12)

Then

le 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

+ 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

) + 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) + 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) le 0

(13)

It can be verified that the largest invariant set where 1198711015840 = 0 issingleton 119864lowast Therefore by LaSallersquos invariance principle 119864lowastis globally asymptotically stable

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OncologyJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

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PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

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Diabetes ResearchJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 5

119865119881minus1=(

1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205732119860119888

119889119888(120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205734119860ℎ

119889ℎ(120574 + 119889

ℎ+ 120572ℎ)

0 0 0

) (5)

The basic reproduction number is given by 120588(119865119881minus1) and

1198770=minus119886 + radic1198862 minus 4119887

2

119886

= minus1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

minus1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

119887

=1205731119860119888

119889119888(119889119888+ 120572119888+ 120583119888+ 120575119888)

1205734119860ℎ120588

119889ℎ(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

minus1205732119860119888120588

119889119888(120588 + 119889

ℎ) (120574 + 119889

ℎ+ 120572ℎ)

1205733119860ℎ

119889ℎ(119889119888+ 120572119888+ 120583119888+ 120575119888)

(6)

According to the conclusions of the literature [24 25] thefollowing results are obtained

Theorem 1 When 1198770lt 1 119864

0is local stable when 119877

0gt 1 119864

0

is unstableUsing a similar argument as in the proof of proposition 33

in [26] we can show that when 1198770gt 1 model (2) has at least

one endemic equilibrium 119864lowast On the stability of the endemic

equilibrium one has the following theorem

Theorem 2 Assume that 1198770gt 1 the endemic equilibrium 119864

lowast

is globally asymptotically stable

Proof Let

1198811= 119878119888minus 119878lowast

119888minus 119878lowast

119888ln119878119888

119878lowast119888

+ 119868119888minus 119868lowast

119888minus 119868lowast

119888ln119868119888

119868lowast119888

1198812= 119878ℎminus 119878lowast

ℎminus 119878lowast

ℎln119878ℎ

119878lowast

ℎ

+ 119864ℎminus 119864lowast

ℎminus 119864lowast

ℎln119864ℎ

119864lowast

ℎ

1198813= 119868ℎminus 119868lowast

ℎminus 119868lowast

ℎln119868ℎ

119868lowast

ℎ

1198814= 119877ℎminus 119877lowast

ℎminus 119877lowast

ℎln119877ℎ

119877lowast

ℎ

(7)

Differentiating 119881119894(119894 = 1 2 3 4) along the solutions of model

(2) then

1198811015840

1= (1 minus

119878lowast

119888

119878119888

) (119860119888minus 1205731119878119888119868119888minus 1205732119878119888119868ℎminus 119889119888119878119888)

+ (1 minus119868lowast

119888

119868119888

)

sdot (1205731119878119888119868119888+ 1205732119878119888119868ℎminus (119889119888+ 120572119888+ 120583119888+ 120575119888) 119868119888)

(8)

Further using the equilibrium satisfying equations we have

1198811015840

1= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868119888

119878lowast119888119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878lowast

119888

119878119888

)(1 minus119878119888119868ℎ

119878lowast119888119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119868lowast

119888

119868119888

)(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119868lowast

119888

119868119888

)(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(1 minus

119878119888119868119888

119878lowast119888119868lowast119888

minus119878lowast

119888

119878119888

+119868119888

119868lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(1 minus

119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

)

+ 1205731119878lowast

119888119868lowast

119888(119878119888119868119888

119878lowast119888119868lowast119888

minus119868119888

119868lowast119888

minus119878119888

119878lowast119888

+ 1)

+ 1205732119878lowast

119888119868lowast

ℎ(119878119888119868ℎ

119878lowast119888119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

+ 1)

= minus119889119888

(119878119888minus 119878lowast

119888)2

119878119888

+ 1205731119878lowast

119888119868lowast

119888(2 minus

119878lowast

119888

119878119888

minus119878119888

119878lowast119888

)

+ 1205732119878lowast

119888119868lowast

ℎ(2 minus

119878lowast

119888

119878119888

+119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus119868ℎ119878119888119868lowast

119888

119868lowast

ℎ119878lowast119888119868119888

)

le 1205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

(9)

6 Computational and Mathematical Methods in Medicine

Through the same calculation we obtain

1198811015840

2= (1 minus

119878lowast

ℎ

119878ℎ

) (119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ)

+ (1 minus119864lowast

ℎ

119864ℎ

) (1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus (120588 + 119889

ℎ) 119864ℎ)

= minus119889ℎ

(119878ℎminus 119878lowast

ℎ)2

119878ℎ

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868119888

119878lowast

ℎ119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(Sℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

+ 1)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

+ 1)

= 1205733119878lowast

ℎ119868lowast

119888(2 minus

119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(2 minus

119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

)

1198811015840

3= (1 minus

119868lowast

ℎ

119868ℎ

) (120588119864ℎ+ 120590119877ℎminus (119889ℎ+ 120572ℎ+ 120574) 119868ℎ)

= 120588119864lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

le 120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

(10)

Similarly it is easy that

1198811015840

4= (1 minus

119877lowast

ℎ

119877ℎ

) (120574119868ℎminus 119889ℎ119877ℎ)

= 120574119868lowast

ℎ(1 minus

119877lowast

ℎ

119877ℎ

)(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

)

le 120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

)

(11)

Now construct the following Lyapunov function

119871 = 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881198811+ 120588120574120573

2119864lowast

ℎ119868lowast

ℎ119878lowast

1198881198812

+ (12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888)1198813

+ (12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888+ 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888) 1198814

(12)

Then

le 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

+ 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

) + 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) + 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) le 0

(13)

It can be verified that the largest invariant set where 1198711015840 = 0 issingleton 119864lowast Therefore by LaSallersquos invariance principle 119864lowastis globally asymptotically stable

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

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MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

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BioMed Research International

OncologyJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

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PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ObesityJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

6 Computational and Mathematical Methods in Medicine

Through the same calculation we obtain

1198811015840

2= (1 minus

119878lowast

ℎ

119878ℎ

) (119860ℎminus 1205733119878ℎ119868119888minus 1205734119878ℎ119868ℎminus 119889ℎ119878ℎ)

+ (1 minus119864lowast

ℎ

119864ℎ

) (1205733119878ℎ119868119888+ 1205734119878ℎ119868ℎminus (120588 + 119889

ℎ) 119864ℎ)

= minus119889ℎ

(119878ℎminus 119878lowast

ℎ)2

119878ℎ

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868119888

119878lowast

ℎ119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878lowast

ℎ

119878ℎ

)(1 minus119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119864lowast

ℎ

119864ℎ

)(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(1 minus

119878ℎ119868119888

119878lowast

ℎ119868lowast119888

minus119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(1 minus

119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

)

+ 1205733119878lowast

ℎ119868lowast

119888(Sℎ119868119888

119878lowast

ℎ119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

+ 1)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119878ℎ119868ℎ

119878lowast

ℎ119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

+ 1)

= 1205733119878lowast

ℎ119868lowast

119888(2 minus

119878lowast

ℎ

119878ℎ

+119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus119868119888119878ℎ119864lowast

ℎ

119868lowast119888119878lowast

ℎ119864ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(2 minus

119878lowast

ℎ

119878ℎ

+119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus119868ℎ119878ℎ119864lowast

ℎ

119868lowast

ℎ119878lowast

ℎ119864ℎ

)

le 1205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

)

+ 1205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

)

1198811015840

3= (1 minus

119868lowast

ℎ

119868ℎ

) (120588119864ℎ+ 120590119877ℎminus (119889ℎ+ 120572ℎ+ 120574) 119868ℎ)

= 120588119864lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(1 minus

119868lowast

ℎ

119868ℎ

)(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

)

le 120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

(10)

Similarly it is easy that

1198811015840

4= (1 minus

119877lowast

ℎ

119877ℎ

) (120574119868ℎminus 119889ℎ119877ℎ)

= 120574119868lowast

ℎ(1 minus

119877lowast

ℎ

119877ℎ

)(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

)

le 120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

)

(11)

Now construct the following Lyapunov function

119871 = 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881198811+ 120588120574120573

2119864lowast

ℎ119868lowast

ℎ119878lowast

1198881198812

+ (12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888)1198813

+ (12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888+ 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888) 1198814

(12)

Then

le 1205881205741205733119878lowast

ℎ119864lowast

ℎ119868lowast

1198881205732119878lowast

119888119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119868119888

119868lowast119888

minus ln119868ℎ

119868lowast

ℎ

+ ln119868119888

119868lowast119888

)

+ 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205733119878lowast

ℎ119868lowast

119888(119868119888

119868lowast119888

minus119864ℎ

119864lowast

ℎ

minus ln119868119888

119868lowast119888

+ ln119864ℎ

119864lowast

ℎ

) + 1205881205741205732119864lowast

ℎ119868lowast

ℎ119878lowast

1198881205734119878lowast

ℎ119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119864ℎ

119864lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119864ℎ

119864lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205734119878lowast

ℎ119868lowast

ℎ

2119878lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120588119864lowast

ℎ(119864ℎ

119864lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119864ℎ

119864lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

)

+ 12057412057321205733119878lowast

ℎ119868lowast

ℎ119878lowast

119888119868lowast

119888120590119877lowast

ℎ(119877ℎ

119877lowast

ℎ

minus119868ℎ

119868lowast

ℎ

minus ln119877ℎ

119877lowast

ℎ

+ ln119868ℎ

119868lowast

ℎ

) + 12059012057321205734119878lowast

ℎ119868lowast

ℎ119877lowast

ℎ119878lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) + 12059012057321205733119878lowast

ℎ119877lowast

ℎ119878lowast

119888119868lowast

119888120574119868lowast

ℎ(119868ℎ

119868lowast

ℎ

minus119877ℎ

119877lowast

ℎ

minus ln119868ℎ

119868lowast

ℎ

+ ln119877ℎ

119877lowast

ℎ

) le 0

(13)

It can be verified that the largest invariant set where 1198711015840 = 0 issingleton 119864lowast Therefore by LaSallersquos invariance principle 119864lowastis globally asymptotically stable

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

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Disease Markers

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Oxidative Medicine and Cellular Longevity

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PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ObesityJournal of

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Research and TreatmentAIDS

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 7

Table 5 Descriptions and values of parameters in model

Parameter Value Interpretation Source119860119888

38538 Recruiting of susceptible cows [19]119889119888

15 Natural death rate of cows Estimation1205731

10995 times 10minus5 The rate of cows infected TB via cows Fitting

1205732

57803 times 10minus5 The rate of cows infected TB via humans Fitting

120572119888

0 Mortality rate due to TB of cows Estimation120583119888

085 The slaughter rate to infected cows Estimation120575119888

012 The isolation rate to infected cows Estimation119860ℎ

36 Recruiting of susceptible humans [20ndash23]119889ℎ

004 The removal rate of livestock worker in dairy farm [20ndash23]1205733

16252 times 10minus5 The rate of humans infected TB via cows Fitting

1205734

0 The rate of humans infected TB via humans Estimation120588 13 Progression rate to TB [8]120572ℎ

0139 Mortality rate due to TB of humans [8]120574 0058 Cure rate to TB [8]120590 001 Rate of relapse to active TB [8]

Table 6 The point estimation and 95 Bootstrap confidenceinterval for the parameters and 119877

0

Parameter Point estimate 95 Bootstrap CI1205731

10995 times 10minus5

[749 times 10minus6 171 times 10

minus5]

1205732

57803 times 10minus5

[272 times 10minus5 953 times 10

minus5]

1205733

16252 times 10minus5

[176 times 10minus19 445 times 10

minus18]

1198770

01811 [0123 0281]

4 Model Application

41 Parameter Estimation The values of parameters formodel (1) are listed in Table 5 According to the nationalpolicy the positive livestock infected TB should be slaugh-tered however due to the lack of funds and the nontimelypayment of the slaughter of livestock resulting in the fact thatTB positive livestock are not completely slaughtered So wechoose 120583

119888= 085 and 120575

119888= 012

We use 2007ndash2014 in Urumqi dairy herds number andpositive rate data to estimate the parameters of the model weestimate that the initial condition of infected cows is 119868

119888(0) =

59 The other initial conditions are assumed to be 119878119888(0) =

21000 119876119888(0) = 70 119878

ℎ(0) = 800 119864

ℎ(0) = 100 119868

ℎ(0) = 30

and 119877ℎ(0) = 20 respectively

The parameters 1205731 1205732 and 120573

3are obtained by fitting

the model to data We ignored humans infected TB viahumans hence we make 120573

4= 0 By least-square fitting

and Bootstrap method we can obtain the point estimationand confidence interval for transmission coefficientwhich arelisted in Table 6 respectively

Based on Table 6 we obtained the basic reproductionnumber 119877

0asymp 01811 The result shows that disease will not

break out under current situation by Theorem 1 We give ahistogram of119877

0obtained by using the Bootstrapmethod (see

Figure 2) In 2011 some of the large-scale dairy farm ownerschanged and the new buy cows from other places so therate of TB positive cows is very high We regard this point

005 01 015 02 025 03 035 04 0450

01

02

03

04

05

06

07Fr

eque

ncy

R0

Figure 2 The frequency histogram for 1198770

as outlier We discard this point estimate the number of TBpositive cows in 2007ndash2014 and draw the 95 confidenceinterval (see Figure 3) The result shows that the fitting effectis good in fact we estimate the number of TB positive cowsin 2007ndash2014 and provide the confidence belt by all of thedata (see Figure 4) We can predict the general tendency ofthe epidemic according to the current situation which ispresented in Figure 5 The prediction shows that disease willvanish around 2020 (see Figure 5)

42 Sensitivity Analysis For the sensitivity analysis Latinhypercube sampling was used to sample parameters thatappear in the derived expression for basic reproductionnumber 119877

0 Uncertainty and sensitivity analysis based on

Latin hypercube sampling has been previously applied todisease transmission models Thus in order to examine thesensitivity of our results to parameter variations we use Latin

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

8 Computational and Mathematical Methods in Medicine

2007 2008 2009 2010 2011 2012 2013 20140

20

40

60

80

100

120

140

160

180

200

220

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 3 The cow TB positive fitting model in 2007ndash2014

2007 2008 2009 2010 2011 2012 2013 2014

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 4 The cow TB positive fitting model for Bootstrap in 2007ndash2014

2008 2010 2012 2014 2016 2018 2020 2022 2024

0

50

100

150

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year)

Figure 5 The tendency of the number of cow positive TB cases

Table 7 Partial rank correlation coefficients (PRCCs) for aggregate1198770and each input parameter

Input parameter PRCC 119901 value119889119888

minus05549 0120572119888

minus07175 0120583119888

minus09008 0120575119888

minus07349 0120588 00223 03210120574 minus00187 04046120590 00270 022931205731

05341 01205732

minus00177 042921205733

05001 0

06

PRCC

04

02

0

minus02

minus04

minus06

minus08

minus1

lowast

lowast

lowast

lowast

lowastlowast

1205733

1205732

1205731

120590120574120588120575c120583c

120572cdc

Figure 6 Partial rank correlation coefficients (PRCCs) result for thedependence of 119877

0on each parameter

hypercube sampling to examine the dependence of basicreproduction number 119877

0

We choose sample size 119899 = 2000 parameters of interestas the input variables and the value of 119877

0as the output

variable The PRCC values of ten parameters are listed inTable 7 and shown in Figure 6 The ordering of these PRCCscorresponds to the level of statistical influence the parameterhas on the variability for the basic reproduction number 119877

0

The larger the PRCCs in absolute value the more importantthe parameter in responding to the change in 119877

0 Plus sign

or minus sign means the influence is positive or negativerespectively Figure 6 shows that 120573

1and 120573

3have positive

impact upon1198770 whilst119889

119888120572119888120583119888 and 120575

119888have negative impact

We also know that 1198770is not sensitive to parameters 120588 120574 120590

and 1205732

Table 7 shows that the slaughter rate to infected cows120583119888(|PRCC| = 09008) has the greatest impact on 119877

0 Then

the quarantine rate 120575119888(|PRCC| = 07349) to infected cows

has the greater impact on 1198770 Hence from sensitivity and

mathematical analysis we conclude that the most effectiveapproach to reduce the TB infection is to increase parameters120583119888and 120575119888

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 9

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200Th

e num

ber o

f cow

s TB

posit

ive c

ases

2014 2016 2018 2020 2022 20240

20

40

60

80

100

120

140

160

180

200

The n

umbe

r of c

ows T

B po

sitiv

e cas

es

T (year) T (year)

120583c= 085

120583c= 075

120583c= 065

120583c= 055

120575c = 012

120575c = 022

120575c = 032

120575c = 042

Figure 7 The influence of parameters 120583119888and 120575

119888on the number of cow positive TB cases

In the following we focus on parameters 120583119888and 120575

119888 The

influence of parameters 120583119888and 120575

119888on the number of cows

TB positive cases is shown in Figure 7 We can see fromFigure 7 that with the increase in slaughter rate the positiverate of TB in dairy cows will be greatly reduced Similarly thisphenomenon is also reflected in the effect of quarantine rateon the number of TB positive cows appropriate increase ofthe quarantine rate of TB positive cows can also be a goodcontrol of the spread of TB

It is very significant to investigate the effect of slaughterrate and quarantine rate on basic reproduction number 119877

0

Due to the lack of funds and the nontimely payment ofthe slaughter of cattle resulting in the fact that TB positivecattle are not completely slaughtered when the slaughterrate can not reach a high proportion of cases appropriateimprovement to the quarantine of sick cattle can also controlthe epidemic of BTB

5 Discussion

TB infection exists widely in the world In Xinjiang TB isone of the major infectious diseases that seriously endangerthe health of people Xinjiang is one of the large pastoralareas in China The prevalence of BTB not only restricts thedevelopment of the livestock industry in Xinjiang but alsothreatens peoplersquos health To investigate the prevalence of BTBin Urumqi a total of 82271 cows in Urumqi areas from 14large-scale dairy farms and 8 counties of grazed cows werequarantined [19] We establish a dynamical model for TBof humans and cows We get the disease-free equilibriumpoint discuss the positive equilibrium point estimate theparameters and conduct the sensitivity analysis The sensi-tivity coefficients (PRCCs) of the parameters with respect tothe basic reproduction number are shown in Figure 6 Theresults indicate that the slaughter rate and quarantine rate arethe main factors affecting the spread of BTB so the standard

slaughter and quarantine management of the TB positivecows will inhibit the spread of BTB effectivelyThe simulationresults reveal the main trend of BTB epidemic in Urumqi andalso a prediction for the trend of the BTB infection In 2011some of the large-scale dairy farm owners changed and thenewbrought cows fromother places so the rate of TBpositivecowswas very highThis point has a little impact on our fittingeffect Finally we predict the number of TB positive cows inUrumqi from 2014 to 2024 Figure 5 shows that the number ofTBpositive cowswill be close to zero in ourmodel Accordingto recent epidemiological investigation BTB effective controlhad been obtained in Urumqi The result shows that thecurrent control measures are effective

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11301451 11461073 and 11501498)Urumqi City Technology Bureau Project (P07121001) andSubproject of National 973 Programme (2006CB504401)

References

[1] WHO ldquoGlobal tuberculosis report 2015rdquo Global TuberculosisReport 2015

[2] A Z Guo and H C Chen ldquoThe epidemiological characteriza-tion and control strategy of bovine tuberculosisrdquo China DairyCattle no 11 pp 38ndash45 2010

[3] C H Lei D L Ran J L Yu L Jiang Y Liu and Y S ZhangldquoMonitoring and analysis of bovine tuberculosisrdquo XinjiangAgricultural Sciences vol 49 no 1 pp 150ndash154 2012

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

10 Computational and Mathematical Methods in Medicine

[4] H Jia T Xin X Y Guo W F Yuan S H Hou and H FZhu ldquoBovine tuberculosis impacts on human health and itsdiagnostic methodsrdquo Journal of Microbes and Infections vol 9no 1 pp 6ndash13 2014

[5] Y X Shi Q Y Yang L I Ai-Qiao andC S Zhang ldquoPreliminaryapproach to tactics of cattle TB prevention and control inxinjiangrdquo Grass-Feeding Livestock no 1 pp 76ndash77 2010

[6] W X Wang G L Sun and A Q Li ldquoCharacteristic ofepidemiology about tuberculosis in the UrumqirdquoGrass-FeedingLivestock no 2 pp 74ndash76 2011

[7] X Jin ldquoThe epidemic state of tuberculosis and its controlstrategies in xinjiang from 1979 to 2000rdquo Endemic DiseasesBulletin no 1 pp 50ndash52 2003

[8] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[9] SM Blower PM Small andPCHopewell ldquoControl strategiesfor tuberculosis epidemics new models for old problemsrdquoScience vol 273 no 5274 pp 497ndash500 1996

[10] T C Porco and S M Blower ldquoQuantifying the intrinsictransmission dynamics of tuberculosisrdquo Theoretical PopulationBiology vol 54 no 2 pp 117ndash132 1998

[11] M Mehra N Cossrow C Kambili R Underwood R Makkarand R Potluri ldquoAssessment of tuberculosis burden in Chinausing a dynamic disease simulation modelrdquo International Jour-nal of Tuberculosis and Lung Disease vol 17 no 9 pp 1186ndash11942013

[12] S Whang S Choi and E Jung ldquoA dynamic model for tuber-culosis transmission and optimal treatment strategies in SouthKoreardquo Journal ofTheoretical Biology vol 279 no 1 pp 120ndash1312011

[13] E Brooks-Pollock G O Roberts andM J Keeling ldquoA dynamicmodel of bovine tuberculosis spread and control in GreatBritainrdquo Nature vol 511 no 7508 pp 228ndash231 2014

[14] J G Yang and L W Zhang ldquoStability of an age-structured epi-demic model with latent periodrdquo Journal of Xuchang Universityvol 29 no 5 pp 4ndash8 2010

[15] L Liu X-Q Zhao and Y Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[16] L Liu and Y Wang ldquoA mathematical study of a TB model withtreatment interruptions and two latent periodsrdquo Computationaland Mathematical Methods in Medicine vol 2014 Article ID932186 15 pages 2014

[17] Y Yang J Li Z Ma and L Liu ldquoGlobal stability of two modelswith incomplete treatment for tuberculosisrdquo Chaos Solitons ampFractals vol 43 no 1ndash12 pp 79ndash85 2010

[18] L Liu Y Wu and G You ldquoGlobal dynamics for a tb modelincorporating case detection and noninfectious tb casesrdquo FarEast Journal of Mathematical Sciences vol 2 no 2 pp 157ndash1802012

[19] A Q Li J G Zhao andD J Hu ldquoEpidemiological investigationand control of dairy cow tuberculosis in urumqirdquoChina AnimalQuarantine no 10 pp 52ndash53 2012

[20] S H Lin ldquoInvestigation on the production management ofdairy farms in the Xinjiang in 2011rdquo China Dairy no 9 pp 18ndash21 2012

[21] H P Chen ldquoInvestigation report on the status of the dairy farmworkers in 2011rdquo China Dairy no 8 pp 6ndash11 2012

[22] Y Q Feng and H P Chen ldquoInvestigation report on theproductionmanagement of dairy farms in 21 provinces of Chinain 2011rdquo China Dairy no 6 pp 10ndash18 2012

[23] Y Q Feng and H P Chen ldquoInvestigation report on the pro-duction management of dairy farms in 21 provinces of China in2011rdquo China Dairy no 9 pp 22ndash25 2012

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P V D Driessche and J Watmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[26] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom