Research ArticleSeasonality Impact on the TransmissionDynamics of Tuberculosis

Yali Yang123 Chenping Guo1 Luju Liu4 Tianhua Zhang5 and Weiping Liu5

1Science College Air Force Engineering University Xirsquoan Shaanxi 710051 China2College of Mathematics and Information Science Shaanxi Normal University Xirsquoan Shaanxi 710062 China3Centre for Disease Modelling York University Toronto ON Canada M3J 1P34School of Mathematics and Statistics Henan University of Science and Technology Luoyang 471023 China5Institute of Tuberculosis Prevention and Treatment in Shaanxi Xirsquoan Shaanxi 710048 China

Correspondence should be addressed to Yali Yang yylhgrmathstatyorkuca

Received 10 November 2015 Revised 14 January 2016 Accepted 1 February 2016

Academic Editor Ruy M Ribeiro

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

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show theseasonality fluctuations in Shaanxi of China A seasonality TB epidemic model with periodic varying contact rate reactivation rateand disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB Simulationsshow that the basic reproduction number of time-averaged autonomous systemsmay underestimate or overestimate infection risksin some cases which may be up to the value of period The basic reproduction number of the seasonality model is appropriatelygiven which determines the extinction and uniformpersistence of TB disease If it is less than one then the disease-free equilibriumis globally asymptotically stable if it is greater than one the system at least has a positive periodic solution and the disease willpersist Moreover numerical simulations demonstrate these theorem results

1 Introduction

Tuberculosis (TB) remains one of the worldrsquos deadliestcommunicable diseases In 2013 it was estimated that 90million people developed TB and 15 million died from thedisease and TB is slowly declining each year [1] Accordingto the online global TB data collection system China aloneaccounted for 11 of the total cases which is the secondcountry of 22 TB high-burden countries only after IndiaShaanxi is one of themore serious TB provinces inChina andits reported new cases each year reach about 25000 which isthe second in the number of cases of infectious diseases inShaanxi only next to the hepatitis B virus (HBV)

Some researchers have investigated the influence of sea-sonal variations on the transmission dynamics of infectiousdiseases [2ndash4] And seasonal variation in TB incidence hasbeen described in many countries and cities such as IndiaUnited States Russia New York city and Hong Kong [5ndash9] TB is a seasonal disease in China [10 11] but it remainsunknown in Shaanxi

From 2004 to 2012 there are 273305 reported notifiableactive TB cases in Shaanxi Monthly reports of notifiableactive TB cases from January 2004 to December 2012 inShaanxi (Table 1) are available on the data-center of Chinapublic health science [12] We apply seasonal filters inMATLAB program to deseasonalize the time series then theoriginal time series is decomposed into three componentstrend curve and season and irregular noise The trend curveis the long-term and medium-to-long term movement ofthe series it also contains consequential turning pointsthe seasonal component is within one-year (12 months)fluctuations about the trend that recur in a similar way inthe same month or quarter every year and the irregularcomponent is the residual component that still remains aftertrend curve and seasonal component are removed from theoriginal series

Figure 1 shows the original time series of active TBcases from January 2004 to December 2012 in ShaanxiFigures 2(a) and 2(b) show the isolated trend curve and theseasonal component respectively To show the correctness

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 8713924 12 pageshttpdxdoiorg10115520168713924

2 Computational and Mathematical Methods in Medicine

Table 1 Shaanxi TB cases month report from January 2004 to December 2012 [12]

Monthyear 2004 2005 2006 2007 2008 2009 2010 2011 2012January 2761 3781 3727 3751 3250 2696 2586 2290 2309February 3057 2613 3118 2914 3033 2486 2038 2187 2440March 3519 4865 3774 3178 3531 2956 2382 2489 2531April 3284 4739 3254 2996 3005 2606 2435 2260 2276May 3064 3602 3064 2909 3182 2556 2314 2325 2258June 3008 3514 2660 2812 2872 2343 2148 2173 1975July 3076 3142 2644 2508 2674 2317 2091 1957 1923August 2822 3293 2569 2480 2540 2234 2076 2061 1878September 2465 3121 2248 2317 2478 2229 1998 1969 1651October 2517 2554 2328 2204 1602 2133 2036 1966 1662November 2306 2653 1721 1989 1954 2021 1849 1837 1595December 1580 1946 1300 1352 1581 1953 1662 1799 1548Sum 33459 39823 32407 31410 32702 28530 25615 25313 24046

1 12 24 36 48 60 72 84 96 108

1500

2000

2500

3000

3500

4000

4500

5000

Original series

Figure 1The original time series of pulmonary TB cases in Shaanxiof China January 2004 to December 2012

of this multiplicative decomposition the original series arecompared to a series reconstructed using the componentestimates in Figure 3(a) and giving irregular noise componentin Figure 3(b) It has shown that the multiplicative decompo-sition is fit for the TB data of Shaanxi

In Figure 2(a) trend component shows that there is a fastupward trend after 2004 and in 2005 (the data in 2004 is lowjust because this is the first year for reported data and somestaff may just begin to the report system so some patientsmay be omitted) then a downward trend from 2005 to 2007and then a slowly upward in 2008 a steadily decreasing trendfrom 2009 to 2012 In Figure 2(b) firstly seasonal componentshows that seasonal amplitude decreases each year from 2004to 2012 secondly it illustrates there exists a seasonal period119879 12 months and along with peak and trough months thefirst- and second-peakmonth of TBnotification inMarch andJanuary respectively and between them there exists a troughFebruary between these two months (February may be apeak month but for Chinese lunar new year Spring Festivalthere exists a special reason that patients may not choose

diagnosis for regarding illness as an unlucky thing in thistraditional festival) the troughmonth is in December TB is aseasonal disease in Shaanxi In addition the peak and troughof TB transmission actually are in winter and in autumnrespectively due to the delay which tends to last 4ndash8 weeksUnderstanding TB seasonality may help TB programs betterto plan and allocate resources for TB control activities [8]Motivated by this we formulate a seasonality 119878119871119868119877 epidemicmodel with periodic coefficients for TB and study its globaldynamics in this paper

This paper is organized as follows In Section 2 a season-ality TB model is formulated In Section 3 a unique disease-free equilibrium is obtained and the basic reproductionnumber 119877

119879for the periodic model is given in detail Further-

more some numerical simulations are used to compare theaverage basic reproduction number [119877

119879] to 119877

119879in different

cases Threshold dynamics of the TB model is analyzedin Section 4 Meanwhile some numerical simulations areprovided to validate analytical results Finally conclusions aregiven in Section 5

2 Model Formulation

The total population is divided into four compartmentssusceptible (119878) latent (119871) infectious (119868) and recovered (119877)individuals

From Figure 2(b) new TB cases have shown the periodicmonthly trend and the possible causes of the seasonal patternShaanxi has a continental monsoon climate and has fourdistinct seasons Seasons in Shaanxi are defined as spring(FebruaryndashApril) summer (MayndashJuly) autumn (AugustndashOctober) and winter (NovemberndashJanuary) TB is usuallyacquired through airborne infection from active TB casesits transmission and progress tend to be effected by theclimate within one year (12 months) In particular the indooractivities are much more in winter than in a warm climatewhich improves the probability of susceptible individualsexposed toMycobacterium tuberculosis (Mtb) from the infec-tious individuals in a room with windows closed for a longerperiod of time [13] thus infection rate may have the periodic

Computational and Mathematical Methods in Medicine 3

1 12 24 36 48 60 72 84 96 108

2000

2500

3000

3500

(a) Trend1 12 24 36 48 60 72 84 96 108

0608

11214

(b) Seasonal component

Figure 2 (a) The trend component of time series for pulmonary TB cases in Shaanxi of China January 2004 to December 2012 (b) Theseasonal component of time series for pulmonary TB cases in Shaanxi of China January 2004 to December 2012

1 12 24 36 48 60 72 84 96 108

2000

3000

4000

5000

Original seriesTrend and seasonal components

(a) TB cases

1 12 24 36 48 60 72 84 96 108070809

11112

(b) Irregular component

Figure 3 (a) Compare the original series to a series reconstructed using the component estimates of time series for pulmonary TB cases inShaanxi of China January 2004 to December 2012 (b) The irregular noise component of time series for pulmonary TB cases in Shaanxi ofChina January 2004 to December 2012

influence In addition during these months near or in thehighest peak month of TB cases cold weather and lackof sunshine which may reduce human immunity cause ahigher disease-induced rate Individuals with lower VitaminD level may be more reactivated for TB [14] Thus disease-induced rate and reactivation rate may also have the periodicinfluence To describe and study the TB transmission inShaanxi three periodic coefficients are selected (i) infectionrate coefficient 120573(119905) and the bilinear incidence 120573(119905)119878119868 whichare applied in this model (ii) reactivation rate coefficient120574(119905) at which an individual leaves the latent compartment forbecoming infectious and (iii) disease-induced rate coefficient120572(119905) which is the disease-induced death rate coefficientsfor individuals in compartment 119868 In view of the periodictrend of monthly 120573(119905) 120574(119905) and 120572(119905) are assumed to bepositive periodic continuous function of 119905 with period 119879In some TB models the fast and slow progression has beenconsidered [15ndash17] Based on those a seasonality TB modelwith fast and slow progression and periodic coefficients isformulated in this sectionThe transfer among compartmentsis schematically depicted in Figure 4 It leads to the followingmodel of ordinary differential equations

1198781015840= Λ minus 120573 (119905) 119878119868 minus 120583119878

1198711015840= (1 minus 119901) 120573 (119905) 119878119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840= 119901120573 (119905) 119878119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

1198771015840= 120590119868 minus 120583119877

(1)

p120573(t)SI

ΛS

120583S 120583L 120583I 120572(t)I 120583R

120590IRIL

(1 minus p)120573(t)SI 120574(t)L

Figure 4 The transfer diagram for model (1)

with initial condition (119878(0) 119871(0) 119868(0) 119877(0)) = (1198780 1198710 1198680

1198770) isin R4

+ and all parameters are positive Here Λ is the

recruitment rate and parameter 120583 is the natural death ratecoefficient 120590 is the rate coefficient at which an infective indi-vidual leaves the infectious compartment to the recovered119901 (0 le 119901 lt 1) is the fraction of infected individuals who arefast developing into infected cases and enter the infectiouscompartment directly while 1 minus 119901 is the fraction of infectedindividuals who are slowly developing into infected cases andtransferred to the latent compartment

Since 119877 does not appear in the other equations of system(1) system (1) is equivalent to the following system

1198781015840= Λ minus 120573 (119905) 119878119868 minus 120583119878

1198711015840= (1 minus 119901) 120573 (119905) 119878119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840= 119901120573 (119905) 119878119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(2)

4 Computational and Mathematical Methods in Medicine

Theorem 1 Every forward solution (119878(119905) 119871(119905) 119868(119905)) of system(2) eventually enters Ω = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583

and Ω is a positively invariant set for system (2)

Proof From system (1) it follows that

(119878 + 119871 + 119868 + 119877)1015840= Λ minus 120583 (119878 + 119871 + 119868 + 119877) minus 120572 (119905) 119868

le Λ minus 120583 (119878 + 119871 + 119868 + 119877)

(3)

and then

lim sup119905rarrinfin

(119878 + 119871 + 119868 + 119877) leΛ

120583 (4)

It implies that region 119883 = (119878 119871 119868 119877) isin R4+ 119878 + 119871 + 119868 +

119877 le Λ120583 is a positively invariant set for system (1) ThenΩ = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583 is a positive invariant

with respect to system (2)Therefore system (2) is dissipativeand its global attractor is contained inΩ

In the rest of this paper system (2)will be studied in regionΩ

3 Disease-Free Equilibrium andthe Basic Reproduction Number

To study system (2) some notations are introducedLet (RR119899

+) be the standard ordered 119899-dimensional

Euclidean space with a norm sdot For 119906 V isin R119899 denote 119906 ge Vif 119906 minus V isin R119899

+ 119906 gt V if 119906 minus V isin R119899

+ 0 and 119906 ≫ V if

119906 minus V isin Int(R119899+)

Let 119860(119905) be a continuous cooperative irreducible and119899 times 119899 matrix function with period 119879 gt 0 and let Φ

119860(119905)

be the fundamental solution matrix of the linear ordinarydifferential equation

119889119909

119889119905= 119860 (119905) 119909 (5)

And let 120588(Φ119860(119879)) be the spectral radius ofΦ

119860(119879) By Perron-

Frobenius theorem 120588(Φ119860(119879)) is the principle eigenvalue of

Φ119860(119879) in the sense that it is simple and admits an eigenvector

Vlowast ≫ 0There is a unique disease-free steady state 119864

0 that is

(Λ120583 0 0) for system (2)In the following the basic reproduction number 119877

119879

will be introduced for system (2) according to the generalprocedure presented in [2]

With 120594 fl (119871 119868 119878) system (2) becomes

1205941015840(119905) = F (120594) minusV (120594) (6)

where

F (120594) = (

(1 minus 119901) 120573 (119905) 119878119868

119901120573 (119905) 119878119868

0

)

V (120594) = (

120574 (119905) 119871 + 120583119871

minus120574 (119905) 119871 + 120590119868 + 120572 (119905) 119868 + 120583119868

minusΛ + 120583119878

)

(7)

Furthermore here comes

119865 (119905) = (

0(1 minus 119901) 120573 (119905) Λ

120583

0119901120573 (119905) Λ

120583

)

119881 (119905) = (

120574 (119905) + 120583 0

minus120574 (119905) 120590 + 120572 (119905) + 120583)

(8)

Then 119865(119905) is nonnegative and minus119881(119905) is cooperative in thesense that the off-diagonal elements ofminus119881(119905) are nonnegativeThus it is easy to verify that system (2) satisfies the assump-tions (A1)ndash(A7) in [2]

Define 119884(119905 119904) 119905 ge 119904 which is a 2 times 2 matrix and is theevolution operator of the linear 119879-periodic system

119889119910

119889119905= minus119881 (119905) 119910 (9)

That is for each 119904 isin R 119884(119905 119904) satisfies

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119864 (10)

where 119864 is the 2 times 2 identity matrix Thus the monodromymatrix Φ

minus119881(119905)of (9) equals 119884(119905 0) 119905 ge 0 Assume that the

population is near the disease-free periodic state 1198640 And

suppose that 120601(119904) 119879-periodic in 119904 is the initial distributionof infectious individuals Then 119865(119904)120601(119904) is the rate of newinfections produced by the infected individuals who wereintroduced at time 119904 Given 119905 ge 119904 119884(119905 119904)119865(119904)120601(119904) gives thedistribution of those infected individuals who were newlyinfected at time 119904 and remain in infected compartments at 119905It follows that

120595 (119905) fl int

119905

minusinfin

119884 (119905 119904) 119865 (119904) 120601 (119904) 119889119904

= int

infin

0

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

(11)

is the distribution of accumulative new infections at time 119905produced by all those infected individuals 120601(119904) introduced attime 119904 (119904 le 119905) Let 119862

119879be the ordered Banach space of all 119879-

periodic functions fromR toR119899 which is equipped with themaximum norm sdot

infinand the positive cone 119862+

119879= 120601 isin 119862

119879

120601(119905) ge 0 119905 isin R Define a linear operator119867 119862119879rarr 119862119879by

(119867120601) (119905) = int

infin

0

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

forall119905 isin R 120601 isin 119862119879

(12)

Then according to Wang and Zhao [2] the basic reproduc-tion number 119877

119879is defined as

119877119879fl 120588 (119867) (13)

for the periodic epidemic system (2) where 120588(119867) denotes thespectral radius of the matrix119867

Computational and Mathematical Methods in Medicine 5

In the constant case that is 120573(119905) equiv 120573 120574(119905) equiv 120574 120575(119905) equiv 120575forall119905 gt 0 then 119865(119905) equiv 119865 119881(119905) equiv 119881 forall119905 gt 0 in which

119865 = (

0(1 minus 119901) 120573Λ

120583

0119901120573Λ

120583

)

119881 = (

120574 + 120583 0

minus120574 120590 + 120572 + 120583)

(14)

By van den Driessche and Watmough [18] here comes

119877119879= 120588 (119865119881

minus1) =

120573

120583 + 120572 + 120590(119901 +

120574

120583 + 120574(1 minus 119901)) (15)

In the periodic case in order to characterize 119877119879 consider

the following linear 119879-periodic equation

119889119908

119889119905= (minus119881 (119905) +

119865 (119905)

120582)119908 forall119905 isin R (16)

with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) be the evolutionoperator of system (16) on R2 and 119877

119879can be calculated in

numerically according to Lemma 2

Lemma2 (Wang and Zhao [2]Theorem 21) For system (2)the following statements are valid

(i) If 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 120582

0

is an eigenvalue of119867 and hence 119877119879gt 0

(ii) If 119877119879gt 0 then 120582 = 119877

119879is the unique solution of

120588(119882(119879 0 120582)) = 1(iii) 119877

119879= 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0

And the threshold behaviors occur

Lemma 3 (Wang and Zhao [2]Theorem 22) For system (2)the following statements are valid

(i) 119877119879= 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) = 1

(ii) 119877119879gt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) gt 1

(iii) 119877119879lt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) lt 1

Thus the disease-free periodic solution 1198640for system (2) is

locally asymptotically stable if 119877119879lt 1 and unstable if 119877

119879gt 1

Define

[119891] fl1

119879int

119879

0

119891 (119905) 119889119905 (17)

as the average for a continuous periodic function 119891(119905) withthe period 119879 Let [119877

119879] be the basic reproduction number of

the autonomous systems obtained from the average of system(2) that is

1198781015840= Λ minus [120573] 119878119868 minus 120583119878

1198711015840= (1 minus 119901) [120573] 119878119868 minus [120574] 119871 minus 120583119871

1198681015840= 119901 [120573] 119878119868 + [120574] 119871 minus 120590119868 minus [120572] 119868 minus 120583119868

(18)

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 5 For system (2) the graph of the average basic reproductionnumber [119877

119879] and the basic reproduction number 119877

119879with respect

to 1198870which varies from 0003 to 0045 and Λ = 08 120583 = 0008

119901 = 008 1198920= 0003 119896

1= 1198962= 1198963= 1 120590 = 05 119879 = 12 and

1198860= 008

An example is given to show that the basic reproductionnumber of the time-averaged autonomous systems mayunderestimate estimate or overestimate the infection risk

Example 4 Consider the following

120573 (119905) = 1198870 (1 + 1198961 cos(120587 (119905 + 1)

(1198792)))

120574 (119905) = 1198920 (1 + 1198962 cos(120587 (119905 minus 1)

(1198792)))

120572 (119905) = 1198860 (1 + 1198963 cos(120587 (119905 minus 1)

(1198792)))

(19)

Now Lemma 2 is applied to calculate the basic reproduc-tion number 119877

119879of system (2)

Firstly Λ = 08 120583 = 0008 119901 = 008 1198920= 0003

1198961= 1198962= 1198963= 1 120590 = 05 119879 = 12 and 119886

0= 008

in system (2) by numerical computation it can acquire thecurves of the average basic reproduction number [119877

119879] and the

basic reproduction number 119877119879when 119887

0varies respectively

in Figure 5 It can be seen that [119877119879] is always greater than

119877119879as 1198870is ranging from 0003 to 0045 Secondly when

1198870= 0015 in system (2) 119892

0varies from 0003 to 0007

and other parameters are the same as those of Figure 5 thenthe numerical calculations indicate [119877

119879] is greater than 119877

119879

in Figure 6 as 1198920is varying Thirdly 119887

0= 0015 in system

(2) 1198860varies from 001 to 005 and other parameters are the

same as those of Figure 5 then the numerical calculationsindicate [119877

119879] is greater than 119877

119879in Figure 7 as 119886

0is varying

Summing up the above Figures 5 6 and 7 imply that therisk of infection may be overestimated if the average basicreproduction number [119877

119879] is used

6 Computational and Mathematical Methods in Medicine

3 35 4 45 5 55 6 65 705

06

07

08

09

1

11

12

13

14

15

[RT]RT

The trends of RT and [RT] as g0 is changing

g0 times10minus3

Figure 6 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198920which varies from 0003 to 0007 when 119887

0= 0015 and other

parameter values are the same as those of Figure 5

001 0015 002 0025 003 0035 004 0045 00508

085

09

095

1

105

11

115

The trends of RT and [RT] as a0 is changing

[RT]RT

a0

Figure 7 For system (2) the graph of [119877119879] and 119877

119879with respect

to 1198860which varies from 001 to 005 when 119887

0= 00165 and other

parameter values are the same as those of Figure 5

But on the other hand if 119879 = 1 from Figures 8 9 and10 the numerical calculations indicate [119877

119879] is less than 119877

119879

in Figures 8 9 and 10 respectively These imply that therisk of infection may be underestimated if the average basicreproduction number [119877

119879] is used

Especially if 119879 = 5 from Figures 11 12 and 13 thenumerical calculations indicate [119877

119879] is almost equal to 119877

119879in

Figures 11 12 and 13 respectivelyThese imply that the risk ofinfectionmay be estimated by the average basic reproductionnumber [119877

119879] and [119877

119879] can be used in some conditions

0005 001 0015 002 0025 003 0035 004 0045

1

2

3

4

5

6

7

8The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 8 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 1 and other parametervalues are the same as those of Figure 5

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

g0 times10minus3

The trends of RT and [RT] as g0 is changing

Figure 9 For system (2) the graph of [119877119879] and119877

119879with respect to 119892

0

which varies from 0003 to 0007 when 119879 = 1 and other parametervalues are the same as those of Figure 6

If 1198870= 00165 and 119896

1 1198962 1198963vary in [0 1] for system (2)

respectively with other parameters unchanged as Figure 5numerical computation indicates (see Figures 14 15 and 16)the average basic reproduction number overestimates thedisease transmission risk

Finally if 1198870= 0015 and 119879 varies in [0 22] for system (2)

with other parameters unchanged as those shown in Figure 5numerical computation gives the relation between the basicreproduction number 119877

119879and period 119879 in Figure 17 which

indicates the average basic reproduction number [119877119879]may be

Computational and Mathematical Methods in Medicine 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

2 Computational and Mathematical Methods in Medicine

Table 1 Shaanxi TB cases month report from January 2004 to December 2012 [12]

Monthyear 2004 2005 2006 2007 2008 2009 2010 2011 2012January 2761 3781 3727 3751 3250 2696 2586 2290 2309February 3057 2613 3118 2914 3033 2486 2038 2187 2440March 3519 4865 3774 3178 3531 2956 2382 2489 2531April 3284 4739 3254 2996 3005 2606 2435 2260 2276May 3064 3602 3064 2909 3182 2556 2314 2325 2258June 3008 3514 2660 2812 2872 2343 2148 2173 1975July 3076 3142 2644 2508 2674 2317 2091 1957 1923August 2822 3293 2569 2480 2540 2234 2076 2061 1878September 2465 3121 2248 2317 2478 2229 1998 1969 1651October 2517 2554 2328 2204 1602 2133 2036 1966 1662November 2306 2653 1721 1989 1954 2021 1849 1837 1595December 1580 1946 1300 1352 1581 1953 1662 1799 1548Sum 33459 39823 32407 31410 32702 28530 25615 25313 24046

1 12 24 36 48 60 72 84 96 108

1500

2000

2500

3000

3500

4000

4500

5000

Original series

Figure 1The original time series of pulmonary TB cases in Shaanxiof China January 2004 to December 2012

of this multiplicative decomposition the original series arecompared to a series reconstructed using the componentestimates in Figure 3(a) and giving irregular noise componentin Figure 3(b) It has shown that the multiplicative decompo-sition is fit for the TB data of Shaanxi

In Figure 2(a) trend component shows that there is a fastupward trend after 2004 and in 2005 (the data in 2004 is lowjust because this is the first year for reported data and somestaff may just begin to the report system so some patientsmay be omitted) then a downward trend from 2005 to 2007and then a slowly upward in 2008 a steadily decreasing trendfrom 2009 to 2012 In Figure 2(b) firstly seasonal componentshows that seasonal amplitude decreases each year from 2004to 2012 secondly it illustrates there exists a seasonal period119879 12 months and along with peak and trough months thefirst- and second-peakmonth of TBnotification inMarch andJanuary respectively and between them there exists a troughFebruary between these two months (February may be apeak month but for Chinese lunar new year Spring Festivalthere exists a special reason that patients may not choose

diagnosis for regarding illness as an unlucky thing in thistraditional festival) the troughmonth is in December TB is aseasonal disease in Shaanxi In addition the peak and troughof TB transmission actually are in winter and in autumnrespectively due to the delay which tends to last 4ndash8 weeksUnderstanding TB seasonality may help TB programs betterto plan and allocate resources for TB control activities [8]Motivated by this we formulate a seasonality 119878119871119868119877 epidemicmodel with periodic coefficients for TB and study its globaldynamics in this paper

This paper is organized as follows In Section 2 a season-ality TB model is formulated In Section 3 a unique disease-free equilibrium is obtained and the basic reproductionnumber 119877

119879for the periodic model is given in detail Further-

more some numerical simulations are used to compare theaverage basic reproduction number [119877

119879] to 119877

119879in different

cases Threshold dynamics of the TB model is analyzedin Section 4 Meanwhile some numerical simulations areprovided to validate analytical results Finally conclusions aregiven in Section 5

2 Model Formulation

The total population is divided into four compartmentssusceptible (119878) latent (119871) infectious (119868) and recovered (119877)individuals

From Figure 2(b) new TB cases have shown the periodicmonthly trend and the possible causes of the seasonal patternShaanxi has a continental monsoon climate and has fourdistinct seasons Seasons in Shaanxi are defined as spring(FebruaryndashApril) summer (MayndashJuly) autumn (AugustndashOctober) and winter (NovemberndashJanuary) TB is usuallyacquired through airborne infection from active TB casesits transmission and progress tend to be effected by theclimate within one year (12 months) In particular the indooractivities are much more in winter than in a warm climatewhich improves the probability of susceptible individualsexposed toMycobacterium tuberculosis (Mtb) from the infec-tious individuals in a room with windows closed for a longerperiod of time [13] thus infection rate may have the periodic

Computational and Mathematical Methods in Medicine 3

1 12 24 36 48 60 72 84 96 108

2000

2500

3000

3500

(a) Trend1 12 24 36 48 60 72 84 96 108

0608

11214

(b) Seasonal component

Figure 2 (a) The trend component of time series for pulmonary TB cases in Shaanxi of China January 2004 to December 2012 (b) Theseasonal component of time series for pulmonary TB cases in Shaanxi of China January 2004 to December 2012

1 12 24 36 48 60 72 84 96 108

2000

3000

4000

5000

Original seriesTrend and seasonal components

(a) TB cases

1 12 24 36 48 60 72 84 96 108070809

11112

(b) Irregular component

Figure 3 (a) Compare the original series to a series reconstructed using the component estimates of time series for pulmonary TB cases inShaanxi of China January 2004 to December 2012 (b) The irregular noise component of time series for pulmonary TB cases in Shaanxi ofChina January 2004 to December 2012

influence In addition during these months near or in thehighest peak month of TB cases cold weather and lackof sunshine which may reduce human immunity cause ahigher disease-induced rate Individuals with lower VitaminD level may be more reactivated for TB [14] Thus disease-induced rate and reactivation rate may also have the periodicinfluence To describe and study the TB transmission inShaanxi three periodic coefficients are selected (i) infectionrate coefficient 120573(119905) and the bilinear incidence 120573(119905)119878119868 whichare applied in this model (ii) reactivation rate coefficient120574(119905) at which an individual leaves the latent compartment forbecoming infectious and (iii) disease-induced rate coefficient120572(119905) which is the disease-induced death rate coefficientsfor individuals in compartment 119868 In view of the periodictrend of monthly 120573(119905) 120574(119905) and 120572(119905) are assumed to bepositive periodic continuous function of 119905 with period 119879In some TB models the fast and slow progression has beenconsidered [15ndash17] Based on those a seasonality TB modelwith fast and slow progression and periodic coefficients isformulated in this sectionThe transfer among compartmentsis schematically depicted in Figure 4 It leads to the followingmodel of ordinary differential equations

1198781015840= Λ minus 120573 (119905) 119878119868 minus 120583119878

1198711015840= (1 minus 119901) 120573 (119905) 119878119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840= 119901120573 (119905) 119878119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

1198771015840= 120590119868 minus 120583119877

(1)

p120573(t)SI

ΛS

120583S 120583L 120583I 120572(t)I 120583R

120590IRIL

(1 minus p)120573(t)SI 120574(t)L

Figure 4 The transfer diagram for model (1)

with initial condition (119878(0) 119871(0) 119868(0) 119877(0)) = (1198780 1198710 1198680

1198770) isin R4

+ and all parameters are positive Here Λ is the

recruitment rate and parameter 120583 is the natural death ratecoefficient 120590 is the rate coefficient at which an infective indi-vidual leaves the infectious compartment to the recovered119901 (0 le 119901 lt 1) is the fraction of infected individuals who arefast developing into infected cases and enter the infectiouscompartment directly while 1 minus 119901 is the fraction of infectedindividuals who are slowly developing into infected cases andtransferred to the latent compartment

Since 119877 does not appear in the other equations of system(1) system (1) is equivalent to the following system

1198781015840= Λ minus 120573 (119905) 119878119868 minus 120583119878

1198711015840= (1 minus 119901) 120573 (119905) 119878119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840= 119901120573 (119905) 119878119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(2)

4 Computational and Mathematical Methods in Medicine

Theorem 1 Every forward solution (119878(119905) 119871(119905) 119868(119905)) of system(2) eventually enters Ω = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583

and Ω is a positively invariant set for system (2)

Proof From system (1) it follows that

(119878 + 119871 + 119868 + 119877)1015840= Λ minus 120583 (119878 + 119871 + 119868 + 119877) minus 120572 (119905) 119868

le Λ minus 120583 (119878 + 119871 + 119868 + 119877)

(3)

and then

lim sup119905rarrinfin

(119878 + 119871 + 119868 + 119877) leΛ

120583 (4)

It implies that region 119883 = (119878 119871 119868 119877) isin R4+ 119878 + 119871 + 119868 +

119877 le Λ120583 is a positively invariant set for system (1) ThenΩ = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583 is a positive invariant

with respect to system (2)Therefore system (2) is dissipativeand its global attractor is contained inΩ

In the rest of this paper system (2)will be studied in regionΩ

3 Disease-Free Equilibrium andthe Basic Reproduction Number

To study system (2) some notations are introducedLet (RR119899

+) be the standard ordered 119899-dimensional

Euclidean space with a norm sdot For 119906 V isin R119899 denote 119906 ge Vif 119906 minus V isin R119899

+ 119906 gt V if 119906 minus V isin R119899

+ 0 and 119906 ≫ V if

119906 minus V isin Int(R119899+)

Let 119860(119905) be a continuous cooperative irreducible and119899 times 119899 matrix function with period 119879 gt 0 and let Φ

119860(119905)

be the fundamental solution matrix of the linear ordinarydifferential equation

119889119909

119889119905= 119860 (119905) 119909 (5)

And let 120588(Φ119860(119879)) be the spectral radius ofΦ

119860(119879) By Perron-

Frobenius theorem 120588(Φ119860(119879)) is the principle eigenvalue of

Φ119860(119879) in the sense that it is simple and admits an eigenvector

Vlowast ≫ 0There is a unique disease-free steady state 119864

0 that is

(Λ120583 0 0) for system (2)In the following the basic reproduction number 119877

119879

will be introduced for system (2) according to the generalprocedure presented in [2]

With 120594 fl (119871 119868 119878) system (2) becomes

1205941015840(119905) = F (120594) minusV (120594) (6)

where

F (120594) = (

(1 minus 119901) 120573 (119905) 119878119868

119901120573 (119905) 119878119868

0

)

V (120594) = (

120574 (119905) 119871 + 120583119871

minus120574 (119905) 119871 + 120590119868 + 120572 (119905) 119868 + 120583119868

minusΛ + 120583119878

)

(7)

Furthermore here comes

119865 (119905) = (

0(1 minus 119901) 120573 (119905) Λ

120583

0119901120573 (119905) Λ

120583

)

119881 (119905) = (

120574 (119905) + 120583 0

minus120574 (119905) 120590 + 120572 (119905) + 120583)

(8)

Then 119865(119905) is nonnegative and minus119881(119905) is cooperative in thesense that the off-diagonal elements ofminus119881(119905) are nonnegativeThus it is easy to verify that system (2) satisfies the assump-tions (A1)ndash(A7) in [2]

Define 119884(119905 119904) 119905 ge 119904 which is a 2 times 2 matrix and is theevolution operator of the linear 119879-periodic system

119889119910

119889119905= minus119881 (119905) 119910 (9)

That is for each 119904 isin R 119884(119905 119904) satisfies

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119864 (10)

where 119864 is the 2 times 2 identity matrix Thus the monodromymatrix Φ

minus119881(119905)of (9) equals 119884(119905 0) 119905 ge 0 Assume that the

population is near the disease-free periodic state 1198640 And

suppose that 120601(119904) 119879-periodic in 119904 is the initial distributionof infectious individuals Then 119865(119904)120601(119904) is the rate of newinfections produced by the infected individuals who wereintroduced at time 119904 Given 119905 ge 119904 119884(119905 119904)119865(119904)120601(119904) gives thedistribution of those infected individuals who were newlyinfected at time 119904 and remain in infected compartments at 119905It follows that

120595 (119905) fl int

119905

minusinfin

119884 (119905 119904) 119865 (119904) 120601 (119904) 119889119904

= int

infin

0

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

(11)

is the distribution of accumulative new infections at time 119905produced by all those infected individuals 120601(119904) introduced attime 119904 (119904 le 119905) Let 119862

119879be the ordered Banach space of all 119879-

periodic functions fromR toR119899 which is equipped with themaximum norm sdot

infinand the positive cone 119862+

119879= 120601 isin 119862

119879

120601(119905) ge 0 119905 isin R Define a linear operator119867 119862119879rarr 119862119879by

(119867120601) (119905) = int

infin

0

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

forall119905 isin R 120601 isin 119862119879

(12)

Then according to Wang and Zhao [2] the basic reproduc-tion number 119877

119879is defined as

119877119879fl 120588 (119867) (13)

for the periodic epidemic system (2) where 120588(119867) denotes thespectral radius of the matrix119867

Computational and Mathematical Methods in Medicine 5

In the constant case that is 120573(119905) equiv 120573 120574(119905) equiv 120574 120575(119905) equiv 120575forall119905 gt 0 then 119865(119905) equiv 119865 119881(119905) equiv 119881 forall119905 gt 0 in which

119865 = (

0(1 minus 119901) 120573Λ

120583

0119901120573Λ

120583

)

119881 = (

120574 + 120583 0

minus120574 120590 + 120572 + 120583)

(14)

By van den Driessche and Watmough [18] here comes

119877119879= 120588 (119865119881

minus1) =

120573

120583 + 120572 + 120590(119901 +

120574

120583 + 120574(1 minus 119901)) (15)

In the periodic case in order to characterize 119877119879 consider

the following linear 119879-periodic equation

119889119908

119889119905= (minus119881 (119905) +

119865 (119905)

120582)119908 forall119905 isin R (16)

with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) be the evolutionoperator of system (16) on R2 and 119877

119879can be calculated in

numerically according to Lemma 2

Lemma2 (Wang and Zhao [2]Theorem 21) For system (2)the following statements are valid

(i) If 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 120582

0

is an eigenvalue of119867 and hence 119877119879gt 0

(ii) If 119877119879gt 0 then 120582 = 119877

119879is the unique solution of

120588(119882(119879 0 120582)) = 1(iii) 119877

119879= 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0

And the threshold behaviors occur

Lemma 3 (Wang and Zhao [2]Theorem 22) For system (2)the following statements are valid

(i) 119877119879= 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) = 1

(ii) 119877119879gt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) gt 1

(iii) 119877119879lt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) lt 1

Thus the disease-free periodic solution 1198640for system (2) is

locally asymptotically stable if 119877119879lt 1 and unstable if 119877

119879gt 1

Define

[119891] fl1

119879int

119879

0

119891 (119905) 119889119905 (17)

as the average for a continuous periodic function 119891(119905) withthe period 119879 Let [119877

119879] be the basic reproduction number of

the autonomous systems obtained from the average of system(2) that is

1198781015840= Λ minus [120573] 119878119868 minus 120583119878

1198711015840= (1 minus 119901) [120573] 119878119868 minus [120574] 119871 minus 120583119871

1198681015840= 119901 [120573] 119878119868 + [120574] 119871 minus 120590119868 minus [120572] 119868 minus 120583119868

(18)

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 5 For system (2) the graph of the average basic reproductionnumber [119877

119879] and the basic reproduction number 119877

119879with respect

to 1198870which varies from 0003 to 0045 and Λ = 08 120583 = 0008

119901 = 008 1198920= 0003 119896

1= 1198962= 1198963= 1 120590 = 05 119879 = 12 and

1198860= 008

An example is given to show that the basic reproductionnumber of the time-averaged autonomous systems mayunderestimate estimate or overestimate the infection risk

Example 4 Consider the following

120573 (119905) = 1198870 (1 + 1198961 cos(120587 (119905 + 1)

(1198792)))

120574 (119905) = 1198920 (1 + 1198962 cos(120587 (119905 minus 1)

(1198792)))

120572 (119905) = 1198860 (1 + 1198963 cos(120587 (119905 minus 1)

(1198792)))

(19)

Now Lemma 2 is applied to calculate the basic reproduc-tion number 119877

119879of system (2)

Firstly Λ = 08 120583 = 0008 119901 = 008 1198920= 0003

1198961= 1198962= 1198963= 1 120590 = 05 119879 = 12 and 119886

0= 008

in system (2) by numerical computation it can acquire thecurves of the average basic reproduction number [119877

119879] and the

basic reproduction number 119877119879when 119887

0varies respectively

in Figure 5 It can be seen that [119877119879] is always greater than

119877119879as 1198870is ranging from 0003 to 0045 Secondly when

1198870= 0015 in system (2) 119892

0varies from 0003 to 0007

and other parameters are the same as those of Figure 5 thenthe numerical calculations indicate [119877

119879] is greater than 119877

119879

in Figure 6 as 1198920is varying Thirdly 119887

0= 0015 in system

(2) 1198860varies from 001 to 005 and other parameters are the

same as those of Figure 5 then the numerical calculationsindicate [119877

119879] is greater than 119877

119879in Figure 7 as 119886

0is varying

Summing up the above Figures 5 6 and 7 imply that therisk of infection may be overestimated if the average basicreproduction number [119877

119879] is used

6 Computational and Mathematical Methods in Medicine

3 35 4 45 5 55 6 65 705

06

07

08

09

1

11

12

13

14

15

[RT]RT

The trends of RT and [RT] as g0 is changing

g0 times10minus3

Figure 6 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198920which varies from 0003 to 0007 when 119887

0= 0015 and other

parameter values are the same as those of Figure 5

001 0015 002 0025 003 0035 004 0045 00508

085

09

095

1

105

11

115

The trends of RT and [RT] as a0 is changing

[RT]RT

a0

Figure 7 For system (2) the graph of [119877119879] and 119877

119879with respect

to 1198860which varies from 001 to 005 when 119887

0= 00165 and other

parameter values are the same as those of Figure 5

But on the other hand if 119879 = 1 from Figures 8 9 and10 the numerical calculations indicate [119877

119879] is less than 119877

119879

in Figures 8 9 and 10 respectively These imply that therisk of infection may be underestimated if the average basicreproduction number [119877

119879] is used

Especially if 119879 = 5 from Figures 11 12 and 13 thenumerical calculations indicate [119877

119879] is almost equal to 119877

119879in

Figures 11 12 and 13 respectivelyThese imply that the risk ofinfectionmay be estimated by the average basic reproductionnumber [119877

119879] and [119877

119879] can be used in some conditions

0005 001 0015 002 0025 003 0035 004 0045

1

2

3

4

5

6

7

8The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 8 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 1 and other parametervalues are the same as those of Figure 5

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

g0 times10minus3

The trends of RT and [RT] as g0 is changing

Figure 9 For system (2) the graph of [119877119879] and119877

119879with respect to 119892

0

which varies from 0003 to 0007 when 119879 = 1 and other parametervalues are the same as those of Figure 6

If 1198870= 00165 and 119896

1 1198962 1198963vary in [0 1] for system (2)

respectively with other parameters unchanged as Figure 5numerical computation indicates (see Figures 14 15 and 16)the average basic reproduction number overestimates thedisease transmission risk

Finally if 1198870= 0015 and 119879 varies in [0 22] for system (2)

with other parameters unchanged as those shown in Figure 5numerical computation gives the relation between the basicreproduction number 119877

119879and period 119879 in Figure 17 which

indicates the average basic reproduction number [119877119879]may be

Computational and Mathematical Methods in Medicine 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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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Computational and Mathematical Methods in Medicine

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Research and TreatmentAIDS

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 3

1 12 24 36 48 60 72 84 96 108

2000

2500

3000

3500

(a) Trend1 12 24 36 48 60 72 84 96 108

0608

11214

(b) Seasonal component

Figure 2 (a) The trend component of time series for pulmonary TB cases in Shaanxi of China January 2004 to December 2012 (b) Theseasonal component of time series for pulmonary TB cases in Shaanxi of China January 2004 to December 2012

1 12 24 36 48 60 72 84 96 108

2000

3000

4000

5000

Original seriesTrend and seasonal components

(a) TB cases

1 12 24 36 48 60 72 84 96 108070809

11112

(b) Irregular component

Figure 3 (a) Compare the original series to a series reconstructed using the component estimates of time series for pulmonary TB cases inShaanxi of China January 2004 to December 2012 (b) The irregular noise component of time series for pulmonary TB cases in Shaanxi ofChina January 2004 to December 2012

influence In addition during these months near or in thehighest peak month of TB cases cold weather and lackof sunshine which may reduce human immunity cause ahigher disease-induced rate Individuals with lower VitaminD level may be more reactivated for TB [14] Thus disease-induced rate and reactivation rate may also have the periodicinfluence To describe and study the TB transmission inShaanxi three periodic coefficients are selected (i) infectionrate coefficient 120573(119905) and the bilinear incidence 120573(119905)119878119868 whichare applied in this model (ii) reactivation rate coefficient120574(119905) at which an individual leaves the latent compartment forbecoming infectious and (iii) disease-induced rate coefficient120572(119905) which is the disease-induced death rate coefficientsfor individuals in compartment 119868 In view of the periodictrend of monthly 120573(119905) 120574(119905) and 120572(119905) are assumed to bepositive periodic continuous function of 119905 with period 119879In some TB models the fast and slow progression has beenconsidered [15ndash17] Based on those a seasonality TB modelwith fast and slow progression and periodic coefficients isformulated in this sectionThe transfer among compartmentsis schematically depicted in Figure 4 It leads to the followingmodel of ordinary differential equations

1198781015840= Λ minus 120573 (119905) 119878119868 minus 120583119878

1198711015840= (1 minus 119901) 120573 (119905) 119878119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840= 119901120573 (119905) 119878119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

1198771015840= 120590119868 minus 120583119877

(1)

p120573(t)SI

ΛS

120583S 120583L 120583I 120572(t)I 120583R

120590IRIL

(1 minus p)120573(t)SI 120574(t)L

Figure 4 The transfer diagram for model (1)

with initial condition (119878(0) 119871(0) 119868(0) 119877(0)) = (1198780 1198710 1198680

1198770) isin R4

+ and all parameters are positive Here Λ is the

recruitment rate and parameter 120583 is the natural death ratecoefficient 120590 is the rate coefficient at which an infective indi-vidual leaves the infectious compartment to the recovered119901 (0 le 119901 lt 1) is the fraction of infected individuals who arefast developing into infected cases and enter the infectiouscompartment directly while 1 minus 119901 is the fraction of infectedindividuals who are slowly developing into infected cases andtransferred to the latent compartment

Since 119877 does not appear in the other equations of system(1) system (1) is equivalent to the following system

1198781015840= Λ minus 120573 (119905) 119878119868 minus 120583119878

1198711015840= (1 minus 119901) 120573 (119905) 119878119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840= 119901120573 (119905) 119878119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(2)

4 Computational and Mathematical Methods in Medicine

Theorem 1 Every forward solution (119878(119905) 119871(119905) 119868(119905)) of system(2) eventually enters Ω = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583

and Ω is a positively invariant set for system (2)

Proof From system (1) it follows that

(119878 + 119871 + 119868 + 119877)1015840= Λ minus 120583 (119878 + 119871 + 119868 + 119877) minus 120572 (119905) 119868

le Λ minus 120583 (119878 + 119871 + 119868 + 119877)

(3)

and then

lim sup119905rarrinfin

(119878 + 119871 + 119868 + 119877) leΛ

120583 (4)

It implies that region 119883 = (119878 119871 119868 119877) isin R4+ 119878 + 119871 + 119868 +

119877 le Λ120583 is a positively invariant set for system (1) ThenΩ = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583 is a positive invariant

with respect to system (2)Therefore system (2) is dissipativeand its global attractor is contained inΩ

In the rest of this paper system (2)will be studied in regionΩ

3 Disease-Free Equilibrium andthe Basic Reproduction Number

To study system (2) some notations are introducedLet (RR119899

+) be the standard ordered 119899-dimensional

Euclidean space with a norm sdot For 119906 V isin R119899 denote 119906 ge Vif 119906 minus V isin R119899

+ 119906 gt V if 119906 minus V isin R119899

+ 0 and 119906 ≫ V if

119906 minus V isin Int(R119899+)

Let 119860(119905) be a continuous cooperative irreducible and119899 times 119899 matrix function with period 119879 gt 0 and let Φ

119860(119905)

be the fundamental solution matrix of the linear ordinarydifferential equation

119889119909

119889119905= 119860 (119905) 119909 (5)

And let 120588(Φ119860(119879)) be the spectral radius ofΦ

119860(119879) By Perron-

Frobenius theorem 120588(Φ119860(119879)) is the principle eigenvalue of

Φ119860(119879) in the sense that it is simple and admits an eigenvector

Vlowast ≫ 0There is a unique disease-free steady state 119864

0 that is

(Λ120583 0 0) for system (2)In the following the basic reproduction number 119877

119879

will be introduced for system (2) according to the generalprocedure presented in [2]

With 120594 fl (119871 119868 119878) system (2) becomes

1205941015840(119905) = F (120594) minusV (120594) (6)

where

F (120594) = (

(1 minus 119901) 120573 (119905) 119878119868

119901120573 (119905) 119878119868

0

)

V (120594) = (

120574 (119905) 119871 + 120583119871

minus120574 (119905) 119871 + 120590119868 + 120572 (119905) 119868 + 120583119868

minusΛ + 120583119878

)

(7)

Furthermore here comes

119865 (119905) = (

0(1 minus 119901) 120573 (119905) Λ

120583

0119901120573 (119905) Λ

120583

)

119881 (119905) = (

120574 (119905) + 120583 0

minus120574 (119905) 120590 + 120572 (119905) + 120583)

(8)

Then 119865(119905) is nonnegative and minus119881(119905) is cooperative in thesense that the off-diagonal elements ofminus119881(119905) are nonnegativeThus it is easy to verify that system (2) satisfies the assump-tions (A1)ndash(A7) in [2]

Define 119884(119905 119904) 119905 ge 119904 which is a 2 times 2 matrix and is theevolution operator of the linear 119879-periodic system

119889119910

119889119905= minus119881 (119905) 119910 (9)

That is for each 119904 isin R 119884(119905 119904) satisfies

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119864 (10)

where 119864 is the 2 times 2 identity matrix Thus the monodromymatrix Φ

minus119881(119905)of (9) equals 119884(119905 0) 119905 ge 0 Assume that the

population is near the disease-free periodic state 1198640 And

suppose that 120601(119904) 119879-periodic in 119904 is the initial distributionof infectious individuals Then 119865(119904)120601(119904) is the rate of newinfections produced by the infected individuals who wereintroduced at time 119904 Given 119905 ge 119904 119884(119905 119904)119865(119904)120601(119904) gives thedistribution of those infected individuals who were newlyinfected at time 119904 and remain in infected compartments at 119905It follows that

120595 (119905) fl int

119905

minusinfin

119884 (119905 119904) 119865 (119904) 120601 (119904) 119889119904

= int

infin

0

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

(11)

is the distribution of accumulative new infections at time 119905produced by all those infected individuals 120601(119904) introduced attime 119904 (119904 le 119905) Let 119862

119879be the ordered Banach space of all 119879-

periodic functions fromR toR119899 which is equipped with themaximum norm sdot

infinand the positive cone 119862+

119879= 120601 isin 119862

119879

120601(119905) ge 0 119905 isin R Define a linear operator119867 119862119879rarr 119862119879by

(119867120601) (119905) = int

infin

0

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

forall119905 isin R 120601 isin 119862119879

(12)

Then according to Wang and Zhao [2] the basic reproduc-tion number 119877

119879is defined as

119877119879fl 120588 (119867) (13)

for the periodic epidemic system (2) where 120588(119867) denotes thespectral radius of the matrix119867

Computational and Mathematical Methods in Medicine 5

In the constant case that is 120573(119905) equiv 120573 120574(119905) equiv 120574 120575(119905) equiv 120575forall119905 gt 0 then 119865(119905) equiv 119865 119881(119905) equiv 119881 forall119905 gt 0 in which

119865 = (

0(1 minus 119901) 120573Λ

120583

0119901120573Λ

120583

)

119881 = (

120574 + 120583 0

minus120574 120590 + 120572 + 120583)

(14)

By van den Driessche and Watmough [18] here comes

119877119879= 120588 (119865119881

minus1) =

120573

120583 + 120572 + 120590(119901 +

120574

120583 + 120574(1 minus 119901)) (15)

In the periodic case in order to characterize 119877119879 consider

the following linear 119879-periodic equation

119889119908

119889119905= (minus119881 (119905) +

119865 (119905)

120582)119908 forall119905 isin R (16)

with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) be the evolutionoperator of system (16) on R2 and 119877

119879can be calculated in

numerically according to Lemma 2

Lemma2 (Wang and Zhao [2]Theorem 21) For system (2)the following statements are valid

(i) If 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 120582

0

is an eigenvalue of119867 and hence 119877119879gt 0

(ii) If 119877119879gt 0 then 120582 = 119877

119879is the unique solution of

120588(119882(119879 0 120582)) = 1(iii) 119877

119879= 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0

And the threshold behaviors occur

Lemma 3 (Wang and Zhao [2]Theorem 22) For system (2)the following statements are valid

(i) 119877119879= 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) = 1

(ii) 119877119879gt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) gt 1

(iii) 119877119879lt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) lt 1

Thus the disease-free periodic solution 1198640for system (2) is

locally asymptotically stable if 119877119879lt 1 and unstable if 119877

119879gt 1

Define

[119891] fl1

119879int

119879

0

119891 (119905) 119889119905 (17)

as the average for a continuous periodic function 119891(119905) withthe period 119879 Let [119877

119879] be the basic reproduction number of

the autonomous systems obtained from the average of system(2) that is

1198781015840= Λ minus [120573] 119878119868 minus 120583119878

1198711015840= (1 minus 119901) [120573] 119878119868 minus [120574] 119871 minus 120583119871

1198681015840= 119901 [120573] 119878119868 + [120574] 119871 minus 120590119868 minus [120572] 119868 minus 120583119868

(18)

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 5 For system (2) the graph of the average basic reproductionnumber [119877

119879] and the basic reproduction number 119877

119879with respect

to 1198870which varies from 0003 to 0045 and Λ = 08 120583 = 0008

119901 = 008 1198920= 0003 119896

1= 1198962= 1198963= 1 120590 = 05 119879 = 12 and

1198860= 008

An example is given to show that the basic reproductionnumber of the time-averaged autonomous systems mayunderestimate estimate or overestimate the infection risk

Example 4 Consider the following

120573 (119905) = 1198870 (1 + 1198961 cos(120587 (119905 + 1)

(1198792)))

120574 (119905) = 1198920 (1 + 1198962 cos(120587 (119905 minus 1)

(1198792)))

120572 (119905) = 1198860 (1 + 1198963 cos(120587 (119905 minus 1)

(1198792)))

(19)

Now Lemma 2 is applied to calculate the basic reproduc-tion number 119877

119879of system (2)

Firstly Λ = 08 120583 = 0008 119901 = 008 1198920= 0003

1198961= 1198962= 1198963= 1 120590 = 05 119879 = 12 and 119886

0= 008

in system (2) by numerical computation it can acquire thecurves of the average basic reproduction number [119877

119879] and the

basic reproduction number 119877119879when 119887

0varies respectively

in Figure 5 It can be seen that [119877119879] is always greater than

119877119879as 1198870is ranging from 0003 to 0045 Secondly when

1198870= 0015 in system (2) 119892

0varies from 0003 to 0007

and other parameters are the same as those of Figure 5 thenthe numerical calculations indicate [119877

119879] is greater than 119877

119879

in Figure 6 as 1198920is varying Thirdly 119887

0= 0015 in system

(2) 1198860varies from 001 to 005 and other parameters are the

same as those of Figure 5 then the numerical calculationsindicate [119877

119879] is greater than 119877

119879in Figure 7 as 119886

0is varying

Summing up the above Figures 5 6 and 7 imply that therisk of infection may be overestimated if the average basicreproduction number [119877

119879] is used

6 Computational and Mathematical Methods in Medicine

3 35 4 45 5 55 6 65 705

06

07

08

09

1

11

12

13

14

15

[RT]RT

The trends of RT and [RT] as g0 is changing

g0 times10minus3

Figure 6 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198920which varies from 0003 to 0007 when 119887

0= 0015 and other

parameter values are the same as those of Figure 5

001 0015 002 0025 003 0035 004 0045 00508

085

09

095

1

105

11

115

The trends of RT and [RT] as a0 is changing

[RT]RT

a0

Figure 7 For system (2) the graph of [119877119879] and 119877

119879with respect

to 1198860which varies from 001 to 005 when 119887

0= 00165 and other

parameter values are the same as those of Figure 5

But on the other hand if 119879 = 1 from Figures 8 9 and10 the numerical calculations indicate [119877

119879] is less than 119877

119879

in Figures 8 9 and 10 respectively These imply that therisk of infection may be underestimated if the average basicreproduction number [119877

119879] is used

Especially if 119879 = 5 from Figures 11 12 and 13 thenumerical calculations indicate [119877

119879] is almost equal to 119877

119879in

Figures 11 12 and 13 respectivelyThese imply that the risk ofinfectionmay be estimated by the average basic reproductionnumber [119877

119879] and [119877

119879] can be used in some conditions

0005 001 0015 002 0025 003 0035 004 0045

1

2

3

4

5

6

7

8The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 8 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 1 and other parametervalues are the same as those of Figure 5

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

g0 times10minus3

The trends of RT and [RT] as g0 is changing

Figure 9 For system (2) the graph of [119877119879] and119877

119879with respect to 119892

0

which varies from 0003 to 0007 when 119879 = 1 and other parametervalues are the same as those of Figure 6

If 1198870= 00165 and 119896

1 1198962 1198963vary in [0 1] for system (2)

respectively with other parameters unchanged as Figure 5numerical computation indicates (see Figures 14 15 and 16)the average basic reproduction number overestimates thedisease transmission risk

Finally if 1198870= 0015 and 119879 varies in [0 22] for system (2)

with other parameters unchanged as those shown in Figure 5numerical computation gives the relation between the basicreproduction number 119877

119879and period 119879 in Figure 17 which

indicates the average basic reproduction number [119877119879]may be

Computational and Mathematical Methods in Medicine 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

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Diabetes ResearchJournal of

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Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Theorem 1 Every forward solution (119878(119905) 119871(119905) 119868(119905)) of system(2) eventually enters Ω = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583

and Ω is a positively invariant set for system (2)

Proof From system (1) it follows that

(119878 + 119871 + 119868 + 119877)1015840= Λ minus 120583 (119878 + 119871 + 119868 + 119877) minus 120572 (119905) 119868

le Λ minus 120583 (119878 + 119871 + 119868 + 119877)

(3)

and then

lim sup119905rarrinfin

(119878 + 119871 + 119868 + 119877) leΛ

120583 (4)

It implies that region 119883 = (119878 119871 119868 119877) isin R4+ 119878 + 119871 + 119868 +

119877 le Λ120583 is a positively invariant set for system (1) ThenΩ = (119878 119871 119868) isin R3

+ 119878 + 119871 + 119868 le Λ120583 is a positive invariant

with respect to system (2)Therefore system (2) is dissipativeand its global attractor is contained inΩ

In the rest of this paper system (2)will be studied in regionΩ

3 Disease-Free Equilibrium andthe Basic Reproduction Number

To study system (2) some notations are introducedLet (RR119899

+) be the standard ordered 119899-dimensional

Euclidean space with a norm sdot For 119906 V isin R119899 denote 119906 ge Vif 119906 minus V isin R119899

+ 119906 gt V if 119906 minus V isin R119899

+ 0 and 119906 ≫ V if

119906 minus V isin Int(R119899+)

Let 119860(119905) be a continuous cooperative irreducible and119899 times 119899 matrix function with period 119879 gt 0 and let Φ

119860(119905)

be the fundamental solution matrix of the linear ordinarydifferential equation

119889119909

119889119905= 119860 (119905) 119909 (5)

And let 120588(Φ119860(119879)) be the spectral radius ofΦ

119860(119879) By Perron-

Frobenius theorem 120588(Φ119860(119879)) is the principle eigenvalue of

Φ119860(119879) in the sense that it is simple and admits an eigenvector

Vlowast ≫ 0There is a unique disease-free steady state 119864

0 that is

(Λ120583 0 0) for system (2)In the following the basic reproduction number 119877

119879

will be introduced for system (2) according to the generalprocedure presented in [2]

With 120594 fl (119871 119868 119878) system (2) becomes

1205941015840(119905) = F (120594) minusV (120594) (6)

where

F (120594) = (

(1 minus 119901) 120573 (119905) 119878119868

119901120573 (119905) 119878119868

0

)

V (120594) = (

120574 (119905) 119871 + 120583119871

minus120574 (119905) 119871 + 120590119868 + 120572 (119905) 119868 + 120583119868

minusΛ + 120583119878

)

(7)

Furthermore here comes

119865 (119905) = (

0(1 minus 119901) 120573 (119905) Λ

120583

0119901120573 (119905) Λ

120583

)

119881 (119905) = (

120574 (119905) + 120583 0

minus120574 (119905) 120590 + 120572 (119905) + 120583)

(8)

Then 119865(119905) is nonnegative and minus119881(119905) is cooperative in thesense that the off-diagonal elements ofminus119881(119905) are nonnegativeThus it is easy to verify that system (2) satisfies the assump-tions (A1)ndash(A7) in [2]

Define 119884(119905 119904) 119905 ge 119904 which is a 2 times 2 matrix and is theevolution operator of the linear 119879-periodic system

119889119910

119889119905= minus119881 (119905) 119910 (9)

That is for each 119904 isin R 119884(119905 119904) satisfies

119889119884 (119905 119904)

119889119905= minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119864 (10)

where 119864 is the 2 times 2 identity matrix Thus the monodromymatrix Φ

minus119881(119905)of (9) equals 119884(119905 0) 119905 ge 0 Assume that the

population is near the disease-free periodic state 1198640 And

suppose that 120601(119904) 119879-periodic in 119904 is the initial distributionof infectious individuals Then 119865(119904)120601(119904) is the rate of newinfections produced by the infected individuals who wereintroduced at time 119904 Given 119905 ge 119904 119884(119905 119904)119865(119904)120601(119904) gives thedistribution of those infected individuals who were newlyinfected at time 119904 and remain in infected compartments at 119905It follows that

120595 (119905) fl int

119905

minusinfin

119884 (119905 119904) 119865 (119904) 120601 (119904) 119889119904

= int

infin

0

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

(11)

is the distribution of accumulative new infections at time 119905produced by all those infected individuals 120601(119904) introduced attime 119904 (119904 le 119905) Let 119862

119879be the ordered Banach space of all 119879-

periodic functions fromR toR119899 which is equipped with themaximum norm sdot

infinand the positive cone 119862+

119879= 120601 isin 119862

119879

120601(119905) ge 0 119905 isin R Define a linear operator119867 119862119879rarr 119862119879by

(119867120601) (119905) = int

infin

0

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

forall119905 isin R 120601 isin 119862119879

(12)

Then according to Wang and Zhao [2] the basic reproduc-tion number 119877

119879is defined as

119877119879fl 120588 (119867) (13)

for the periodic epidemic system (2) where 120588(119867) denotes thespectral radius of the matrix119867

Computational and Mathematical Methods in Medicine 5

In the constant case that is 120573(119905) equiv 120573 120574(119905) equiv 120574 120575(119905) equiv 120575forall119905 gt 0 then 119865(119905) equiv 119865 119881(119905) equiv 119881 forall119905 gt 0 in which

119865 = (

0(1 minus 119901) 120573Λ

120583

0119901120573Λ

120583

)

119881 = (

120574 + 120583 0

minus120574 120590 + 120572 + 120583)

(14)

By van den Driessche and Watmough [18] here comes

119877119879= 120588 (119865119881

minus1) =

120573

120583 + 120572 + 120590(119901 +

120574

120583 + 120574(1 minus 119901)) (15)

In the periodic case in order to characterize 119877119879 consider

the following linear 119879-periodic equation

119889119908

119889119905= (minus119881 (119905) +

119865 (119905)

120582)119908 forall119905 isin R (16)

with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) be the evolutionoperator of system (16) on R2 and 119877

119879can be calculated in

numerically according to Lemma 2

Lemma2 (Wang and Zhao [2]Theorem 21) For system (2)the following statements are valid

(i) If 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 120582

0

is an eigenvalue of119867 and hence 119877119879gt 0

(ii) If 119877119879gt 0 then 120582 = 119877

119879is the unique solution of

120588(119882(119879 0 120582)) = 1(iii) 119877

119879= 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0

And the threshold behaviors occur

Lemma 3 (Wang and Zhao [2]Theorem 22) For system (2)the following statements are valid

(i) 119877119879= 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) = 1

(ii) 119877119879gt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) gt 1

(iii) 119877119879lt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) lt 1

Thus the disease-free periodic solution 1198640for system (2) is

locally asymptotically stable if 119877119879lt 1 and unstable if 119877

119879gt 1

Define

[119891] fl1

119879int

119879

0

119891 (119905) 119889119905 (17)

as the average for a continuous periodic function 119891(119905) withthe period 119879 Let [119877

119879] be the basic reproduction number of

the autonomous systems obtained from the average of system(2) that is

1198781015840= Λ minus [120573] 119878119868 minus 120583119878

1198711015840= (1 minus 119901) [120573] 119878119868 minus [120574] 119871 minus 120583119871

1198681015840= 119901 [120573] 119878119868 + [120574] 119871 minus 120590119868 minus [120572] 119868 minus 120583119868

(18)

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 5 For system (2) the graph of the average basic reproductionnumber [119877

119879] and the basic reproduction number 119877

119879with respect

to 1198870which varies from 0003 to 0045 and Λ = 08 120583 = 0008

119901 = 008 1198920= 0003 119896

1= 1198962= 1198963= 1 120590 = 05 119879 = 12 and

1198860= 008

An example is given to show that the basic reproductionnumber of the time-averaged autonomous systems mayunderestimate estimate or overestimate the infection risk

Example 4 Consider the following

120573 (119905) = 1198870 (1 + 1198961 cos(120587 (119905 + 1)

(1198792)))

120574 (119905) = 1198920 (1 + 1198962 cos(120587 (119905 minus 1)

(1198792)))

120572 (119905) = 1198860 (1 + 1198963 cos(120587 (119905 minus 1)

(1198792)))

(19)

Now Lemma 2 is applied to calculate the basic reproduc-tion number 119877

119879of system (2)

Firstly Λ = 08 120583 = 0008 119901 = 008 1198920= 0003

1198961= 1198962= 1198963= 1 120590 = 05 119879 = 12 and 119886

0= 008

in system (2) by numerical computation it can acquire thecurves of the average basic reproduction number [119877

119879] and the

basic reproduction number 119877119879when 119887

0varies respectively

in Figure 5 It can be seen that [119877119879] is always greater than

119877119879as 1198870is ranging from 0003 to 0045 Secondly when

1198870= 0015 in system (2) 119892

0varies from 0003 to 0007

and other parameters are the same as those of Figure 5 thenthe numerical calculations indicate [119877

119879] is greater than 119877

119879

in Figure 6 as 1198920is varying Thirdly 119887

0= 0015 in system

(2) 1198860varies from 001 to 005 and other parameters are the

same as those of Figure 5 then the numerical calculationsindicate [119877

119879] is greater than 119877

119879in Figure 7 as 119886

0is varying

Summing up the above Figures 5 6 and 7 imply that therisk of infection may be overestimated if the average basicreproduction number [119877

119879] is used

6 Computational and Mathematical Methods in Medicine

3 35 4 45 5 55 6 65 705

06

07

08

09

1

11

12

13

14

15

[RT]RT

The trends of RT and [RT] as g0 is changing

g0 times10minus3

Figure 6 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198920which varies from 0003 to 0007 when 119887

0= 0015 and other

parameter values are the same as those of Figure 5

001 0015 002 0025 003 0035 004 0045 00508

085

09

095

1

105

11

115

The trends of RT and [RT] as a0 is changing

[RT]RT

a0

Figure 7 For system (2) the graph of [119877119879] and 119877

119879with respect

to 1198860which varies from 001 to 005 when 119887

0= 00165 and other

parameter values are the same as those of Figure 5

But on the other hand if 119879 = 1 from Figures 8 9 and10 the numerical calculations indicate [119877

119879] is less than 119877

119879

in Figures 8 9 and 10 respectively These imply that therisk of infection may be underestimated if the average basicreproduction number [119877

119879] is used

Especially if 119879 = 5 from Figures 11 12 and 13 thenumerical calculations indicate [119877

119879] is almost equal to 119877

119879in

Figures 11 12 and 13 respectivelyThese imply that the risk ofinfectionmay be estimated by the average basic reproductionnumber [119877

119879] and [119877

119879] can be used in some conditions

0005 001 0015 002 0025 003 0035 004 0045

1

2

3

4

5

6

7

8The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 8 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 1 and other parametervalues are the same as those of Figure 5

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

g0 times10minus3

The trends of RT and [RT] as g0 is changing

Figure 9 For system (2) the graph of [119877119879] and119877

119879with respect to 119892

0

which varies from 0003 to 0007 when 119879 = 1 and other parametervalues are the same as those of Figure 6

If 1198870= 00165 and 119896

1 1198962 1198963vary in [0 1] for system (2)

respectively with other parameters unchanged as Figure 5numerical computation indicates (see Figures 14 15 and 16)the average basic reproduction number overestimates thedisease transmission risk

Finally if 1198870= 0015 and 119879 varies in [0 22] for system (2)

with other parameters unchanged as those shown in Figure 5numerical computation gives the relation between the basicreproduction number 119877

119879and period 119879 in Figure 17 which

indicates the average basic reproduction number [119877119879]may be

Computational and Mathematical Methods in Medicine 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

Submit your manuscripts athttpwwwhindawicom

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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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ObesityJournal of

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Research and TreatmentAIDS

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 5

In the constant case that is 120573(119905) equiv 120573 120574(119905) equiv 120574 120575(119905) equiv 120575forall119905 gt 0 then 119865(119905) equiv 119865 119881(119905) equiv 119881 forall119905 gt 0 in which

119865 = (

0(1 minus 119901) 120573Λ

120583

0119901120573Λ

120583

)

119881 = (

120574 + 120583 0

minus120574 120590 + 120572 + 120583)

(14)

By van den Driessche and Watmough [18] here comes

119877119879= 120588 (119865119881

minus1) =

120573

120583 + 120572 + 120590(119901 +

120574

120583 + 120574(1 minus 119901)) (15)

In the periodic case in order to characterize 119877119879 consider

the following linear 119879-periodic equation

119889119908

119889119905= (minus119881 (119905) +

119865 (119905)

120582)119908 forall119905 isin R (16)

with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) be the evolutionoperator of system (16) on R2 and 119877

119879can be calculated in

numerically according to Lemma 2

Lemma2 (Wang and Zhao [2]Theorem 21) For system (2)the following statements are valid

(i) If 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 120582

0

is an eigenvalue of119867 and hence 119877119879gt 0

(ii) If 119877119879gt 0 then 120582 = 119877

119879is the unique solution of

120588(119882(119879 0 120582)) = 1(iii) 119877

119879= 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0

And the threshold behaviors occur

Lemma 3 (Wang and Zhao [2]Theorem 22) For system (2)the following statements are valid

(i) 119877119879= 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) = 1

(ii) 119877119879gt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) gt 1

(iii) 119877119879lt 1 if and only if 120588(Φ

119865(119905)minus119881(119905)(119879)) lt 1

Thus the disease-free periodic solution 1198640for system (2) is

locally asymptotically stable if 119877119879lt 1 and unstable if 119877

119879gt 1

Define

[119891] fl1

119879int

119879

0

119891 (119905) 119889119905 (17)

as the average for a continuous periodic function 119891(119905) withthe period 119879 Let [119877

119879] be the basic reproduction number of

the autonomous systems obtained from the average of system(2) that is

1198781015840= Λ minus [120573] 119878119868 minus 120583119878

1198711015840= (1 minus 119901) [120573] 119878119868 minus [120574] 119871 minus 120583119871

1198681015840= 119901 [120573] 119878119868 + [120574] 119871 minus 120590119868 minus [120572] 119868 minus 120583119868

(18)

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 5 For system (2) the graph of the average basic reproductionnumber [119877

119879] and the basic reproduction number 119877

119879with respect

to 1198870which varies from 0003 to 0045 and Λ = 08 120583 = 0008

119901 = 008 1198920= 0003 119896

1= 1198962= 1198963= 1 120590 = 05 119879 = 12 and

1198860= 008

An example is given to show that the basic reproductionnumber of the time-averaged autonomous systems mayunderestimate estimate or overestimate the infection risk

Example 4 Consider the following

120573 (119905) = 1198870 (1 + 1198961 cos(120587 (119905 + 1)

(1198792)))

120574 (119905) = 1198920 (1 + 1198962 cos(120587 (119905 minus 1)

(1198792)))

120572 (119905) = 1198860 (1 + 1198963 cos(120587 (119905 minus 1)

(1198792)))

(19)

Now Lemma 2 is applied to calculate the basic reproduc-tion number 119877

119879of system (2)

Firstly Λ = 08 120583 = 0008 119901 = 008 1198920= 0003

1198961= 1198962= 1198963= 1 120590 = 05 119879 = 12 and 119886

0= 008

in system (2) by numerical computation it can acquire thecurves of the average basic reproduction number [119877

119879] and the

basic reproduction number 119877119879when 119887

0varies respectively

in Figure 5 It can be seen that [119877119879] is always greater than

119877119879as 1198870is ranging from 0003 to 0045 Secondly when

1198870= 0015 in system (2) 119892

0varies from 0003 to 0007

and other parameters are the same as those of Figure 5 thenthe numerical calculations indicate [119877

119879] is greater than 119877

119879

in Figure 6 as 1198920is varying Thirdly 119887

0= 0015 in system

(2) 1198860varies from 001 to 005 and other parameters are the

same as those of Figure 5 then the numerical calculationsindicate [119877

119879] is greater than 119877

119879in Figure 7 as 119886

0is varying

Summing up the above Figures 5 6 and 7 imply that therisk of infection may be overestimated if the average basicreproduction number [119877

119879] is used

6 Computational and Mathematical Methods in Medicine

3 35 4 45 5 55 6 65 705

06

07

08

09

1

11

12

13

14

15

[RT]RT

The trends of RT and [RT] as g0 is changing

g0 times10minus3

Figure 6 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198920which varies from 0003 to 0007 when 119887

0= 0015 and other

parameter values are the same as those of Figure 5

001 0015 002 0025 003 0035 004 0045 00508

085

09

095

1

105

11

115

The trends of RT and [RT] as a0 is changing

[RT]RT

a0

Figure 7 For system (2) the graph of [119877119879] and 119877

119879with respect

to 1198860which varies from 001 to 005 when 119887

0= 00165 and other

parameter values are the same as those of Figure 5

But on the other hand if 119879 = 1 from Figures 8 9 and10 the numerical calculations indicate [119877

119879] is less than 119877

119879

in Figures 8 9 and 10 respectively These imply that therisk of infection may be underestimated if the average basicreproduction number [119877

119879] is used

Especially if 119879 = 5 from Figures 11 12 and 13 thenumerical calculations indicate [119877

119879] is almost equal to 119877

119879in

Figures 11 12 and 13 respectivelyThese imply that the risk ofinfectionmay be estimated by the average basic reproductionnumber [119877

119879] and [119877

119879] can be used in some conditions

0005 001 0015 002 0025 003 0035 004 0045

1

2

3

4

5

6

7

8The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 8 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 1 and other parametervalues are the same as those of Figure 5

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

g0 times10minus3

The trends of RT and [RT] as g0 is changing

Figure 9 For system (2) the graph of [119877119879] and119877

119879with respect to 119892

0

which varies from 0003 to 0007 when 119879 = 1 and other parametervalues are the same as those of Figure 6

If 1198870= 00165 and 119896

1 1198962 1198963vary in [0 1] for system (2)

respectively with other parameters unchanged as Figure 5numerical computation indicates (see Figures 14 15 and 16)the average basic reproduction number overestimates thedisease transmission risk

Finally if 1198870= 0015 and 119879 varies in [0 22] for system (2)

with other parameters unchanged as those shown in Figure 5numerical computation gives the relation between the basicreproduction number 119877

119879and period 119879 in Figure 17 which

indicates the average basic reproduction number [119877119879]may be

Computational and Mathematical Methods in Medicine 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

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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

6 Computational and Mathematical Methods in Medicine

3 35 4 45 5 55 6 65 705

06

07

08

09

1

11

12

13

14

15

[RT]RT

The trends of RT and [RT] as g0 is changing

g0 times10minus3

Figure 6 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198920which varies from 0003 to 0007 when 119887

0= 0015 and other

parameter values are the same as those of Figure 5

001 0015 002 0025 003 0035 004 0045 00508

085

09

095

1

105

11

115

The trends of RT and [RT] as a0 is changing

[RT]RT

a0

Figure 7 For system (2) the graph of [119877119879] and 119877

119879with respect

to 1198860which varies from 001 to 005 when 119887

0= 00165 and other

parameter values are the same as those of Figure 5

But on the other hand if 119879 = 1 from Figures 8 9 and10 the numerical calculations indicate [119877

119879] is less than 119877

119879

in Figures 8 9 and 10 respectively These imply that therisk of infection may be underestimated if the average basicreproduction number [119877

119879] is used

Especially if 119879 = 5 from Figures 11 12 and 13 thenumerical calculations indicate [119877

119879] is almost equal to 119877

119879in

Figures 11 12 and 13 respectivelyThese imply that the risk ofinfectionmay be estimated by the average basic reproductionnumber [119877

119879] and [119877

119879] can be used in some conditions

0005 001 0015 002 0025 003 0035 004 0045

1

2

3

4

5

6

7

8The trends of RT and [RT] as b0 is changing

[RT]RT

b0

Figure 8 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 1 and other parametervalues are the same as those of Figure 5

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

g0 times10minus3

The trends of RT and [RT] as g0 is changing

Figure 9 For system (2) the graph of [119877119879] and119877

119879with respect to 119892

0

which varies from 0003 to 0007 when 119879 = 1 and other parametervalues are the same as those of Figure 6

If 1198870= 00165 and 119896

1 1198962 1198963vary in [0 1] for system (2)

respectively with other parameters unchanged as Figure 5numerical computation indicates (see Figures 14 15 and 16)the average basic reproduction number overestimates thedisease transmission risk

Finally if 1198870= 0015 and 119879 varies in [0 22] for system (2)

with other parameters unchanged as those shown in Figure 5numerical computation gives the relation between the basicreproduction number 119877

119879and period 119879 in Figure 17 which

indicates the average basic reproduction number [119877119879]may be

Computational and Mathematical Methods in Medicine 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

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Disease Markers

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BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

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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 7

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4The trends of RT and [RT] as a0 is changing

a0

[RT]RT

Figure 10 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 1 and other parameter

values are the same as those of Figure 7

0005 001 0015 002 0025 003 0035 004 0045

05

1

15

2

25

3

35

4The trends of RT and [RT] as b0 is changing

b0

[RT]RT

Figure 11 For system (2) the graph of [119877119879] and119877

119879with respect to 119887

0

which varies from 0003 to 0045 when 119879 = 5 and other parametervalues are the same as those of Figure 5

superior inferior or equal to the basic reproduction number119877119879 which is up to value of period 119879Furthermore in the next section we will prove some

theoretical results of system (2) in which 119877119879serves as a

threshold parameter if 119877119879lt 1 then there exists a globally

asymptotically stable disease-free periodic state1198640(Λ120583 0 0)

if 119877119879gt 1 then the disease is persistent in the population and

there exists at least one positive periodic solution

3 35 4 45 5 55 6 65 70

05

1

15

2

25

3

35

4

[RT]RT

times10minus3

The trends of RT and [RT] as g0 is changing

g0

Figure 12 For system (2) the graph of [119877119879] and119877

119879with respect to119892

0

which varies from 0003 to 0007 when 119879 = 5 and other parametervalues are the same as those of Figure 6

001 0015 002 0025 003 0035 004 0045 00505

1

15

2

25

3

35

4

[RT]RT

The trends of RT and [RT] as a0 is changing

a0

Figure 13 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198860which varies from 001 to 005 when 119879 = 5 and other parameter

values are the same as those of Figure 7

4 Extinction and Uniform Persistence

The following lemma is useful for our discussion in thissection

Lemma 5 (see [19] Lemma 21) Let 119897 = (1119879) ln(120588(Φ119860(119879)))

and then there exists a positive 119879-periodic function V(119905) suchthat 119890119897119905V(119905) is a solution of (5)

8 Computational and Mathematical Methods in Medicine

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

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Disease Markers

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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

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

0 02 04 06 08 1095

1

105

11

115

[RT]RT

The trends of RT and [RT] as k1 is changing

k1

Figure 14 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198961which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

0 02 04 06 08 11

101

102

103

104

105

106

107

108

109

11

[RT]RT

The trends of RT and [RT] as k2 is changing

k2

Figure 15 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198962which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

Theorem 6 For system (2) the disease-free periodic state1198640(Λ120583 0 0) is globally stable on set Ω if 119877

119879lt 1 and it is

unstable if 119877119879gt 1

Proof By Lemma 3 if 119877119879gt 1 then 119864

0(Λ120583 0 0) is unstable

and if 119877119879lt 1 then 119864

0is locally asymptotically stable Hence

we only need to prove that 1198640is globally attractive for119877

119879lt 1

0 02 04 06 08 11

105

11

115

[RT]RT

The trends of RT and [RT] as k3 is changing

k3

Figure 16 For system (2) the graph of [119877119879] and 119877

119879with respect to

1198963which varies from 0 to 1 when 119887

0= 00165 and other parameter

values are the same as those of Figure 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

[RT]RT

T

The trends of RT and [RT] as T is changing

Figure 17 For system (2) the graph of [119877119879] and 119877

119879with respect to

119879 which varies from 0 to 22 when 1198870= 0015 and other parameter

values are the same as those of Figure 5

Since 119878(119905) 119871(119905) 119868(119905) is a nonnegative solution of system (2)in Ω we have 119878 le Λ120583 and know that

1198711015840le(1 minus 119901) 120573 (119905) Λ

120583119868 minus 120574 (119905) 119871 minus 120583119871

1198681015840le119901120573 (119905) Λ

120583119868 + 120574 (119905) 119871 minus 120590119868 minus 120572 (119905) 119868 minus 120583119868

(20)

for 119905 ge 0

Computational and Mathematical Methods in Medicine 9

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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

Consider an auxiliary system

1015840

=(1 minus 119901) 120573 (119905) Λ

120583 minus 120574 (119905) minus 120583

1015840

=119901120573 (119905) Λ

120583 + 120574 (119905) minus 120590 minus 120572 (119905) minus 120583

(21)

that is

(

)

1015840

= (119865 (119905) minus 119881 (119905)) (

) (22)

It follows from Lemma 5 that there exists a positive 119879-periodic function V

1(119905) such that 1198901198971119905V

1(119905) is a solution of (22)

where 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) Choose 119905

1ge 0 and a

real number 1198861gt 0 such that

(

119871 (1199051)

119868 (1199051)) le 119886

1V1 (0) (23)

By the comparison principle we get

(

119871 (119905)

119868 (119905)) le 119886

1V1(119905 minus 1199051) 1198901198971(119905minus1199051) forall119905 ge 119905

1 (24)

By Lemma 3 it is easy to know that 119877119879lt 1 if and only

if 120588(Φ119865(119905)minus119881(119905)

(119879)) lt 1 thus 1198971= (1119879) ln(120588(Φ

119865(119905)minus119881(119905)(119879))) lt

0 Therefore 119871(119905) rarr 0 119868(119905) rarr 0 and 119878(119905) rarr Λ120583 as 119905 rarrinfin that is 119864

0(Λ120583 0 0) is globally attractive for 119877

119879lt 1 In

conclusion 1198640is globally asymptotically stable if 119877

119879lt 1

Theorem 7 If 119877119879gt 1 system (2) is uniformly persistent and

there exists at least one positive periodic solution

Proof Denote Ω0fl (119878 119871 119868) isin Ω 119871 gt 0 119868 gt 0 and

120597Ω0fl Ω Ω

0 And then 119909

0= (1198780 1198710 1198680) isin Ω

0 Let 119875

Ω rarr Ω be the Poincare map associated with system (2) thatis 119875(119909

0) = 119906(119879 119909

0) forall1199090isin Ω where 120593(119905 119909

0) is the unique

solution of system (2) with 120593(0 1199090) = 1199090

Now it is proved that 119875 is uniformly persistent withrespect to (Ω

0 120597Ω0)

It is easy to see thatΩ andΩ0are positively invariant 120597Ω

0

is a relatively closed set in Ω and 119875 is point dissipative fromTheorem 1

Set 119872120597

= (1198780 1198710 1198680) isin 120597Ω

0 119875119898(1198780 1198710 1198680) isin

120597Ω0 forall119898 ge 0We claim that

119872120597= (119878 0 0) 119878 ge 0 (25)

Obviously (119878 0 0) 119878 ge 0 sube 119872120597 For any (119878

0 1198710 1198680) isin

120597Ω0(119878 0 0) 119878 ge 0 if119871

0gt 0 1198680= 0 then119871(119905) gt 0 forall119905 ge 0

and then 1198681015840 = 120574(119905)119871 gt 0 For the other case 1198710= 0 119868

0gt 0

and then 119868(119905) gt 0 and from the first equation of system (2)thus

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

(1198780+ Λint

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904)

ge Λ119890minusint119905

0(120573(119904)119868(119904)+120583)119889119904

int

119905

0

119890int119904

0(120573(120577)119868(120577)+120583)119889120577

119889119904 gt 0

(26)

for any 119905 gt 0From the second equation of system (2) we have

119871 (119905) = 119890minusint119905

0(120574(119904)+120583)119889119904

(1198710

+ int

119905

0

(1 minus 119901) 120573 (119904) 119878 (119904) 119868 (119904) 119890int119904

0(120574(120577)+120583)119889120577

119889119904) gt 0

forall119905 gt 0

(27)

It then follows that (119878(119905) 119871(119905) 119868(119905)) isin 120597Ω0for 0 lt 119905 ≪ 1Thus

the positive invariance ofΩ0implies (25)

Clearly there is a unique fixed point of 119875 in119872120597 which is

1198640(Λ120583 0 0)For system (2) by the continuity solutions with respect to

the initial values forall120572 gt 0 there exists 120572lowast gt 0 such that for all1199090isin Ω0with 119909

0minus 1198640 ⩽ 120572lowast we have 120601(119905 119909

0) minus 120601(119905 119864

0) lt

120572 forall119905 isin [0 119879]

Then we will show that

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) ge 120572lowast forall119909

0isin Ω0 (28)

If not then

lim sup119898rarrinfin

119889 (119875119898(1199090) 1198640) lt 120572lowast

(29)

for some 1199090isin Ω0

Without loss of generality we can assume that

119889 (119875119898(1199090) 1198640) lt 120572lowast forall119898 ge 0 (30)

Then we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817 lt 120572

forall119898 ge 0 forall119905 isin [0 119879]

(31)

For any 119905 ge 0 let 119905 = 119898119879 + 1199051 where 119905

1isin [0 119879) and 119898 is the

largest integer less than or equal to 119905119879 Therefore we have

1003817100381710038171003817120601 (119905 119875119898(1199090)) minus 120601 (119905 119864

0)1003817100381710038171003817

=1003817100381710038171003817120601 (1199051 119875 (1199090)) minus 120601 (1199051 1198640)

1003817100381710038171003817 lt 120572 forall119905 ge 0

(32)

Note that 119909(119905) fl (119878(119905) 119871(119905) 119868(119905)) = 120601(119905 1199090) It then follows

that 0 le 119878(119905) 119871(119905) 119868(119905) le 120572 forall119905 ge 0 From the first equationof system (2) we have

1198781015840ge Λ minus 120573 (119905) 119878120572 minus 120583119878 (33)

10 Computational and Mathematical Methods in Medicine

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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

Note that the perturbed system

1015840

= Λ minus 120573 (119905) 120572 minus 120583 (34)

admits a unique positive 119879-periodic solution

(119905 120572)

= 119890minusint119905

0(120573(119904)120572+120583)119889119904

( (0 120572) + Λint

119905

0

119890int119904

0(120573(120577)120572+120583)119889120577

119889119904)

(35)

which is globally attractive in R+ where

(0 120572) =

Λint119879

0119890int119904

0(120573(120577)120572+120583)119889120577

119889119904

1 minus 119890int119879

0(120573(119904)120572+120583)119889119904

gt 0 (36)

Applying Lemma 3 we know that 119877119879gt 1 if and only if

120588(Φ119865(119905)minus119881(119905)

(119879)) gt 1 By continuity of the spectrum formatrices ([20] Section II 58) we can choose 120578 which issmall enough such that 120588(120601

119865(119905)minus119881(119905)minus120578119872(119905)(119879)) gt 1 where

119872(119905) = (0 (1 minus 119901) 120573 (119905)

0 119901120573 (119905)) (37)

Since (0 120572) is continuous in 120572 we can fix 120572 gt 0 smallenough that (119905 120572) gt (119905) minus 120578 forall119905 ge 0 Furthermore since thefixed point (0 120572) of the Poincare map associated with (34) isglobally attractive there exists gt 0 such that 119878(119905) gt (119905) minus 120578for 119905 ge As a consequence for 119905 ge it holds that

1198711015840ge (1 minus 119901) 120573 (119905) ( (119905) minus 120578) 119868 minus (120574 (119905) + 120583) 119871

1198681015840ge 119901120573 (119905) ( (119905) minus 120578) 119868 + 120574 (119905) 119871 minus (120590 + 120572 (119905) + 120583) 119868

(38)

Consider another auxiliary system

1015840

= (1 minus 119901) 120573 (119905) ( (119905) minus 120578) minus (120574 (119905) + 120583)

1015840

= 119901120573 (119905) ( (119905) minus 120578) + 120574 (119905) minus (120590 + 120572 (119905) + 120583)

(39)

It follows from Lemma 5 that there exists a positive119879-periodic function ((119905) (119905)) such that ((119905) (119905)) =

119890119897119905((119905) (119905)) is a solution of (39) where

119897 =1

119879ln (120588 (120601

119865(119905)minus119881(119905)minus120578119872(119905) (119879))) (40)

Choose 119905 ge and a small 1205722gt 0 such that (119871(119905) 119868(119905)) ge

((0) (0)) By the comparison principle we get (119871(119905) 119868(119905)) ge1205722((119905 minus 119905) (119905 minus 119905))119890

119897(119905minus119905) forall119905 ge 119905 Since 119877119879

gt 1120588(120601119865(119905)minus119881(119905)minus120578119872(119905)

(119879)) gt 1 And thus 119897 gt 0 which impliesthat 119871(119905) rarr infin and 119868(119905) rarr infin as 119905 rarr infin This leads to acontradiction

So suppose (29) is wrong that is (28) is right Further-more (28) shows that 119864

0is an isolated invariant set inΩ and

119882119904(1198640) cap Ω0= Φ Every orbit in119872

120597converges to 119864

0 and 119864

0

is acyclic in119872120597 By the acyclicity theorem on uniform persis-

tence for maps ([21]Theorem 131 and Remark 131) it fol-lows that 119875 is uniformly persistent with respect to (Ω

0 120597Ω0)

Thus ([21] Theorem 311) implies the uniform persistenceof the solutions of system (2) with respect to (Ω

0 120597Ω0) that

is there exists 120576 gt 0 such that any solution (119878(119905) 119864(119905) 119868(119905))of (2) with initial values (119878(0) 119864(0) 119868(0)) isin Ω

0satisfies

lim119905rarrinfin

119871(119905) ge 120576 and lim119905rarrinfin

119868(119905) ge 120576Moreover by Zhao ([21]Theorem 136) 119875 has a fixed point (119878lowast(0) 119864lowast(0) 119868lowast(0)) isin

Ω0 From the first equation of (2) 119878lowast(119905) satisfies 119878lowast1015840 ge

120583119860 minus 120573(119905)119878lowastΛ120583 minus 120583119878

lowast By the comparison theorem wehave 119878lowast ge 119890

minusint119905

0(120573(119904)Λ120583+120583)119889119904

(119878lowast(0) + Λint

119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904) gt

Λ119890minusint119905

0(120573(119904)Λ120583+120583)119889119904

int119905

0119890int119904

0(120573(120591)Λ120583+120583)119889120591

119889119904 gt 0 forall119905 gt 0 Theseasonality of 119878lowast(119905) implies 119878lowast(0) gt 0 By the second andthird equations of (2) and the irreducibility of the cooperativematrix

(

minus (120574 (119905) + 120583) (1 minus 119901) 120573 (119905) 119878lowast(119905)

120574 (119905) 119901120573 (119905) 119878lowast(119905) minus (120590 + 120572 (119905) + 120583)

) (41)

it follows that (119871lowast(119905) 119868lowast(119905)) isin Int(R2+) forall119905 ge 0 Conse-

quently (119878lowast(119905) 119871lowast(119905) 119868lowast(119905)) is a positive 119879-periodic solutionof (2)

Theorems 6 and 7 have shown that 119877119879is a threshold

parameter which determines whether or not the diseasepersists in the population Now some numerical simulations(Figures 18 and 19) are presented to demonstrate these resultsAnd in these simulations119879 = 12 according to the fact that byone year has 12 months In Figures 18 and 19 the simulationsverify Theorems 6 and 7 respectively

5 Discussion

Monthly pulmonary TB cases from January 2004 to Decem-ber 2012 in Shaanxi province have been analyzed by theseasonal adjustment program It has been found that TB casesshow seasonal variation in Shaanxi the peakmonths (Januaryand March) compare to the lowest trough month (Decem-ber) It is necessary to study the seasonality TB epidemicmodel according to the seasonal component of Shaanxirsquos dataConsidering the regularity of peak TB seasonality may helpallocate resources for the prevention and treatment of TBactivities a periodic TB epidemicmodel has been formulatedand studied

Thebasic reproductionnumber119877119879for the periodicmodel

is given By numerical simulation 119877119879has been compared

to the average basic reproduction number [119877119879] in different

parameter values It has shown that [119877119879] may overestimate

or underestimate 119877119879or be equal to 119877

119879in different cases It is

also up to the value of periodic 119879 Furthermore in theorythe threshold dynamics has been studied for the periodicTB model if 119877

119879lt 1 then the disease-free equilibrium is

globally asymptotically stable that is the TB disease willdisappear eventually if 119877

119879gt 1 then there exists at least one

positive periodic solution and the disease will be uniformlypersistent

Our numerical simulations are in good accordance withthese theoretical results It should be noted that we have

Computational and Mathematical Methods in Medicine 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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 11

0 200 400 600 800 100088

90

92

94

96

98

100

t

S(t)

0 200 400 600 800 10000

10

20

30

40

50

60

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

t

I(t)

0 200 400 600 800 10000

2

4

6

8

10

tR(t)

Figure 18 For system (2) 120573(119905) 120574(119905) and 120572(119905) are listed in Example 4 Λ = 08 120583 = 0008 119901 = 008 1198860= 008 119892

0= 0003 119896

1= 1198962= 1198963= 1

120590 = 05 119879 = 12 and 1198870= 0005 and then 119877

119879= 02814 These figures show that the disease will die out which is the same as Theorem 6

0 200 400 600 800 100020

40

60

80

100

t

S(t)

0 200 400 600 800 100040

50

60

70

80

90

t

L(t)

0 200 400 600 800 10000

02

04

06

08

1

12

14

t

I(t)

0 200 400 600 800 10005

10

15

20

25

t

R(t)

Figure 19 For system (2) 1198870= 004 and other parameter values are the same as those of Figure 18 then 119877

119879= 2251 These figures show that

the disease will be asymptotic to a periodic solution which is the same as Theorem 7

not fit Shaanxirsquos data in these simulations since we cannotaccurately estimate some parametersrsquo values in the periodicmodel according to available data or references Despite lackof comparing the model results with the Shaanxirsquos data our

theoretical results have shown that the basic reproductionnumber with periodic 119877

119879plays a crucial role in determining

dynamics of the seasonality TB disease and could be used forcontrolling the spread of TB epidemic in reality strategies

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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

12 Computational and Mathematical Methods in Medicine

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National Natu-ral Science Foundation of China (nos 11301320 11371369and 11471201) the China Postdoctoral Science FoundationFunded Project (no 2013M532016) the Postdoctoral ScienceFoundation in Shaanxi of China the State Scholarship Fundof China (no 201407820120) and the International Develop-ment Research Centre of Canada

References

[1] WHO Global Tuberculosis Report 2014 World Health Organi-zation Press Geneva Switzerland 2014

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

[3] D Posny and J Wang ldquoModelling cholera in periodic environ-mentsrdquo Journal of Biological Dynamics vol 8 no 1 pp 1ndash192014

[4] J Zhang Z Jin G-Q Sun X-D Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[5] D Behera and P P Sharma ldquoA retrospective study of seasonalvariation in the number of cases diagnosed at a tertiary caretuberculosis hospitalrdquo Indian Journal of Chest Disease andAlliedScience vol 53 no 3 pp 145ndash152 2011

[6] M D Willis C A Winston C M Heilig K P Cain N DWalter and W R Mac Kenzie ldquoSeasonality of tuberculosis inthe United States 1993ndash2008rdquo Clinical Infectious Diseases vol54 no 11 pp 1553ndash1560 2012

[7] R A Atun Y A Samyshkin F Drobniewski S I Kuznetsov IM Fedorin and R J Coker ldquoSeasonal variation and hospitalutilization for tuberculosis in Russia hospitals as social careinstitutionsrdquo European Journal of Public Health vol 15 no 4pp 350ndash354 2005

[8] C M Parrinello A Crossa and T G Harris ldquoSeasonality oftuberculosis inNewYorkCity 1990ndash2007rdquo International Journalof Tuberculosis and Lung Disease vol 16 no 1 pp 32ndash37 2012

[9] C C Leung W W Yew T Y K Chan et al ldquoSeasonalpattern of tuberculosis in Hong Kongrdquo International Journal ofEpidemiology vol 34 no 4 pp 924ndash930 2005

[10] X-X Li L-X Wang H Zhang et al ldquoSeasonal variations innotification of active tuberculosis cases in China 2005ndash2012rdquoPLoS ONE vol 8 no 7 Article ID e68102 2013

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

[12] Tuberculosis ldquoThe data-center of China public health sciencerdquohttpwwwphsciencedatacnShare

[13] A K Janmeja and P R Mohapatra ldquoSeasonality of tuberculo-sisrdquo International Journal of Tuberculosis and Lung Disease vol9 no 6 pp 704ndash705 2005

[14] A R Martineau S Nhamoyebonde T Oni et al ldquoReciprocalseasonal variation in vitamin D status and tuberculosis notifi-cations in Cape Town SouthAfricardquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 108 no47 pp 19013ndash19017 2011

[15] 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

[16] C Castillo-Chavez and B Song ldquoDynamical models of tuber-culosis and their applicationsrdquo Mathematical Biosciences andEngineering vol 1 no 2 pp 361ndash404 2004

[17] C Dye and B G Williams ldquoThe population dynamics andcontrol of tuberculosisrdquo Science vol 328 no 5980 pp 856ndash8612010

[18] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

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

[20] T Kato Perturbation Theory for Linear Operators SpringerBerlin Germany 2nd edition 1976

[21] X-Q ZhaoDynamical Systems in Population Biology SpringerNew York NY USA 2003

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