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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 429567 13 pageshttpdxdoiorg1011552013429567
Research ArticleDynamic of a TB-HIV Coinfection Epidemic Model withLatent Age
Xiaoyan Wang Junyuan Yang and Fengqin Zhang
Department of Applied Mathematics Yuncheng University Yuncheng 044000 Shanxi China
Correspondence should be addressed to Junyuan Yang yangjunyuan00126com
Received 16 November 2012 Accepted 6 January 2013
Academic Editor Jinde Cao
Copyright copy 2013 Xiaoyan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A coepidemic arises when the spread of one infectious disease stimulates the spread of another infectious disease Recently thishas happened with human immunodeficiency virus (HIV) and tuberculosis (TB) The density of individuals infected with latenttuberculosis is structured by age since latency The host population is divided into five subclasses of susceptibles latent TB activeTB (without HIV) HIV infectives (without TB) and coinfection class (infected by both TB and HIV) The model exhibits threeboundary equilibria namely disease free equilibrium TB dominated equilibrium and HIV dominated equilibrium We discussthe local or global stabilities of boundary equilibria We prove the persistence of our model Our simple model of two synergisticinfectious disease epidemics illustrates the importance of including the effects of each disease on the transmission and progressionof the other disease We simulate the dynamic behaviors of our model and give medicine explanations
1 Introduction
Coepidemicsmdashthe related spread of two or more infectiousdiseasesmdashhave afflicted mankind for centuries Worldwidethere were an estimated 137 million coinfected HIV and TBpatients globally in 2007 Around 80 percent of patients livein sub-Saharan Africa 456000 people died of HIV-associatedTB in 2007 HIVAIDS and tuberculosis (TB) are commonlycalled the ldquodeadly duordquo and referred to as HIVTB despitebiological differences HIV is a retrovirus that is transmittedprimarily by homosexual and heterosexual contact needlesharing and from mother to child The disease eventuallyprogresses to AIDS as the immune system weakens HIV canbe treated with highly active antiheroical therapy (HAART)but there is presently no cure [1] Virtually all HIV-infectedindividuals can transmit the virus to others and an infectedindividualrsquos chance of spreading the virus generally increasesas the disease progresses and damages the immune system[2] Tuberculosis is caused by mycobacterium tuberculosisbacteria and is spread through the air Some TB infections areldquolatentrdquo meaning that a person has the TB-causing bacteriabut it is dormant A person with latent TB is not sick and isnot infectious However latent TB can progress to ldquoactiverdquoTB ldquoActiverdquo TB infection means that the TB bacteria are
multiplying and spreading in the body A person with activeTB in their lungs or throat can transmit the bacteria to othersSymptoms of active TB include a cough that lasts severalweeks weight loss loss of appetite fever night sweats andcoughing up blood
HIVweakens the immune system and so people are moresusceptible to catching TB if they are exposed At least one-third of people living with HIV worldwide are infected withTB and are 20ndash30 times more likely to develop TB than thosewithout HIV TB bacteria accelerate the progression of HIVtoAIDS Persons coinfectedwith TB andHIVmay spread thedisease not only to the other HIV-infected persons but alsoto members of the general population who do not participatein any of the high risk behaviors associated with HIVPeople living with HIV and displaying early diagnosis needtreatment in time If TB is present they should receive TBpreventive treatment (IPT)The treatments are not expensiveTherefore it is essential that adequate attention must be paidto study the transmission dynamics of HIV-TB coinfection inthe population Current treatment forHIV is known as highlyactive antiretroviral therapy (HAART) [3]
Some authors have developed simulation models toinvestigate HIV-TB co-epidemic dynamics [4ndash7] West and
2 Journal of Applied Mathematics
Λ
119878
12057321198781198682
120583119878
12057311198781198681
120583119869
119869
12057511986811198682
1205831198681
120574(119886)119890(119886119905)
12057211198681
120583119890(119886119905)
120584119869
1205720(119886)119890(119886119905)
12057221198682
119890
1198681
1198682
1205831198682
Figure 1 The flow diagram of the model (1) Two yellow rectangu-lars denote the individuals infected by latent tuberculosis and activetuberculosisThe red rectangular denotes the individuals infected byHIV The orange rectangular denotes the individuals coinfected byTB and HIV
Thompson [8] developed models which reflect the trans-mission dynamics of both TB and HIV and discussed themagnitude and duration of the effect that the HIV epidemicmay have on TB Naresh and Tripathi [9] presented amodel for HIV-TB coinfection with constant recruitmentof susceptibles and found that TB will be eradicated fromthe population if more than 90 percent TB infectives arerecovered due to effective treatment Naresh et al [10] studiedthe effect of tuberculosis on the spread of HIV infection in alogistically growing human population They found that asthe number of TB infectives decreases due to recovery thenumber of HIV infectives also decreases and endemic equi-librium tends toTB free equilibrium Long et al [11] discussedtwo synergistic infectious disease epidemics illustrating theimportance of including the effects of each disease on thetransmission and progression of the other disease
However these models may exclude coinfection maygreatly simplify infection dynamics and may include fewdisease states (eg the important distinction between latentand active TBwas absent)They assume thatHIV treatment isunavailable Bifurcation will appear in some models [12 13]We perform the additional latent age analyze the coexistenceequilibria and extend the model to include disease recoveryin this paper At the same time we obtain the persistence ofthe system and global stabilities of the equilibria under someconditions
This paper is organized as follows In Section 2 weintroduce the TB-HIV coinfection model In Section 3 we
introduce the reproduction numbers of TB and HIV 1198771 119877
2
and discuss the existence of the equilibria The values of thedisease-free equilibrium the two boundary equilibria andthe coexistence equilibrium are given explicitly Section 4focuses on local and global stabilities of the equilibria InSection 5 we discuss the persistence of the system in suitableperiod In Section 6 we simulate and illustrate our resultsWegive some biological explanations in this section In Section 7we conclude our results and discuss the defect of our model
2 The Model Formulation
Two diseases mentioned in the introduction are spreading ina population of total size 119873(119905) We classify the total popu-lation into four classes the susceptible 119878(119905) the individualsinfected by the tuberculosis 119890(119886 119905) and 119868
1(119905) denotes the
latent TB class which has no infectious ability and active TBclass who can infect the susceptible class respectively theindividuals infected by HIV 119868
2(119905) the individuals coinfected
by tuberculosis and HIV 119869(119905) The individuals infected withactive TB infect the susceptible and then develop into thelatent individuals at 120573
1 The individuals infected by HIV
infect the susceptible and become the individuals infected byHIV at 120573
2 An individual already infected with TB can be
coinfected with HIV at 120575 and thus become jointly infectedindividuals 119869(119905)
Figure 1 presents a schematic flow diagram of the mathe-matical model as follows
119889119878
119889119905= Λ minus 120583119878 minus 120573
1119878119868
1minus 120573
2119878119868
2
+ int
infin
0
1205720(119886) 119890 (119886 119905) 119889119886 + 120572
11198681+ 120572
21198682
120597119890 (119886 119905)
120597119886+120597119890 (119886 119905)
120597119905= minus (120583 + 120574 (119886) + 120572
0(119886)) 119890 (119886 119905)
119890 (0 119905) = 1205731119878119868
1
1198891198681
119889119905= int
infin
0
120574 (119886) 119890 (119886 119905) 119889119886minus 1205831198681minus 120575119868
11198682minus 120572
11198681
1198891198682
119889119905= 120573
2119878119868
2minus (120583 + 120572
2) 119868
2
119889119869
119889119905= 120575119868
11198682minus (120583 + 120584) 119869
(1)
where 120583 is natural death rate and Λ is birth rate 120574(119886) israte of endogenous reactivation of latent TB We assume thatthe individuals separately infected by TB and HIV are notlethal but the coinfection can lead to the extra death at 120584Specifically we assume that jointly infected individuals donot recover Individuals infected with latent TB active TB orHIV alone may be potentially treated at rates 120572
0(119886) 120572
1 and
1205722 respectively
Assumption 1 Suppose that
(a) Λ 120583 120575 1205721 120572
2 120573
1 120573
2isin (0 +infin)
Journal of Applied Mathematics 3
(b) 120584 isin [0 +infin)(c) 120574(119886) 120572
0(119886) isin 119862
119861119880([0 +infin) 119877) cap 119862
+([0 +infin) 119877) and
for each 119886 ge 0 where 119862119861119880(119877) denotes the space
of bounded and uniformly continuous map from[0 +infin) into 119877
We assume (1) with the initial conditions
119878 (0) = 1198780ge 0 119890 (119886 0) = 119890
0(119886) isin 119871
1
+(0 +infin)
1198681(0) = 119868
10ge 0 119868
2(0) = 119868
20ge 0
119869 (0) = 1198690ge 0
(2)
Set
119883 = 119877 times 119884 times 1198772 119883
+= 119877
+times 119884
+times 119877
+
2
1198830= 119877 times 119884
0times 119877
2 119883
0+= 119883
0cap 119883
+
(3)
with
119884 = 119877 times 1198711(0 +infin) 119884
+= [0 +infin) times 119871
1
+(0 +infin)
1198840= 0 times 119871
1(0 +infin)
(4)
Let 119863(A) = 119877 times 119885 times 1198772 with 119885 = 0119877 times 119882
11(0infin)
for each 119909 = (119878 0119877 119890 119868
1 1198682) isin 119883
0 DefineA 119863(A) sub 119883
0sub
119883 rarr 119883 andF 1198830rarr 119883 by
A119909
=((
(
minus120583119878
minus119890 (0)
minus119889119890 (sdot)
119889119886minus (120583 + 120572
0(sdot) + 120574 (sdot)) 119890 (sdot)
minus1205831198681minus 120572
11198681
minus1205831198682minus 120572
21198682
minus (120583 + 120584) 119869
))
)
if 119909 isin 119863 (A)
(5)
F (119909)
=
((((
(
120583minus 1205731119878119868
1minus 120573
2119878119868
2+ int
infin
0
1205720(119886) 119890 (119886) 119889119886 + 120572
11198681+ 120572
21198682
1205731119878119868
1
01198711
int
infin
0
120574 (119886) 119890 (119886) 119889119886 minus 12057511986811198682
1205732119878119868
2
12057511986811198682
))))
)
(6)
Rewrite problem (1) as an abstract Cauchy problem
119889119906119909(119905)
119889119905= A119906
119909(119905) +F (119906
119909(119905)) 119905 ge 0 119906
119909(0) = 119909 isin 119883
0+
(7)
It is well known that A is a Hille-Yosida operator Moreprecisely we have (minus120583 +infin) sub 120588(A) and for each 120582 gt minus120583
10038171003817100381710038171003817(120582 minusA)
minus110038171003817100381710038171003817le1
120582 + 120583 (8)
By applying the results in Magal et al [14ndash16] we obtainthe following proposition
Lemma 1 There exists a uniquely determined semiflow119880(119905)
119905ge0on 119883
0+ such that for each 119909 = (119878
0 0 119890
0 11986810
11986820 1198690)119879isin 119883
0+ there exists a unique continuous map 119880 isin
119862([0 +infin)1198830+) which is an integrated solution of the Cauchy
problem (1) that is to say that
int
119905
0
119880 (119904) 119909 119889119904 isin 119863 (A) forall119905 ge 0
119880 (119905) 119909 = 119909 +Aint119905
0
119880 (119904) 119909 119889119904 + int
119905
0
F (119880 (119904) 119909) 119889119904 forallt ge 0
(9)
The total population size 119873(119905) is the sum of all individuals inall classes
119873 = 119878 (119905) + int
+infin
0
119890 (119886 119905) 119889119886 + 1198681(119905) + 119868
2(119905) + 119869 (119905) (10)
The total population size satisfies the equation 1198731015840(119905) = Λ minus
120583119873 minus 120584119869 We introduce the notation
120587 (119886) = exp(minusint119886
0
(1205720(120591) + 120574 (120591)) 119889120591) (11)
To understand the biological meaning of the quantity 120587(119886) wenote that 120587(119886)119890minus120583119886 is the probability to remain infected with TBtime units after infection In addition we define the quantity
119861 = int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 (12)
which gives the probability of treatment since the individualscan leave TB infectious period via treatment
119862 = int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 (13)
which gives the probability of progression since the individualscan leave TB infectious period via progression Since individu-als can only leave the latent TB infected class through treatmentprogression or death the sum of the probabilities of recoveryprogression and death equals one that is
int
infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 + int
infin
0
120574 (a) 120587 (119886) 119890minus120583119886 119889119886
+ 120583int
infin
0
120587 (119886) 119890minus120583119886119889119886 = 1
(14)
It immediately follows that 119861 + 119862 lt 1
3 Equilibria of the Model with Coinfection
We introduce the reproduction numbers of the two diseasesThe reproduction number of TB is
1198771=Λ120573
1119862
120583 (120583 + 1205721) (15)
4 Journal of Applied Mathematics
and the reproduction number of HIV is
1198772=
Λ1205732
120583 (120583 + 1205722) (16)
We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894
1 1198942 119895 as the equilibria of the modelTherefore equilibria
satisfy the following equations
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ int
+infin
0
1205720(119886) 119890 (119886 119905) 119889119886 + 120572
11198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
12057321199041198942minus (120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(17)
The ODE in the system can be solved to result in
119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)
Substituting for 119894 in the integrals one obtains
int
+infin
0
1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 = 119890 (0) 119861
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862
(19)
With this notation the system for the equilibria becomes
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ 119890 (0) 119861 + 12057211198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
119890 (0) 119862 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
1205732119904119894minus(120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(20)
This system has three boundary equilibria
(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)
The disease-free equilibrium always exists
(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the
TB equilibrium is given by
1199041=Λ
1205831198771
119890 = 119890 (0) 120587 (119886) 119890minus120583119886
1198941=
Λ (1 minus 11198771)
(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862
119890 (0) = 120573111990411198941
(21)
Thus the equilibrium is
1198641= (119904
1 119890 119894
1 0 0) (22)
(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by
1198642= (119904
2 0 0 119894
2 0) (23)
where 1199042= 1119877
2 119894
2= 0 119890
2= 0 119894
2= (Λ120583)(1 minus
11198772) 119895
2= 0
(4) The coexistence equilibrium exists if and only if 1198771gt
1198772 and [1 minus (1119877
2+120583(120583+120572
1)Λ120575)][(Λ120573
1120583120572
11198772)(1 minus
119861) minus 1] gt 0 and it is given by
1198642= (119904 119890 119894
1 1198942 119895) (24)
where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 1198941=
(Λ1205721)(1 minus (1119877
2+120583(120583+120572
1)Λ120575))((Λ120573
1120583120572
11198772)(1 minus
119861) minus 1) 1198942= ((120583 + 120575)120575)(119877
1119877
2minus 1) 119895 = 120575119894
11198942(120583 + 120584)
Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0
4 Stability of Equilibria
In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not
To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894
1 1198942 119895 That
is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894
1 119868
2= 119906 + 119894
2 119869 = 119908 + 119895
Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and
Journal of Applied Mathematics 5
119908These satisfy the following linear eigenvalue problem (here119904 119890 119894
1 1198942 and 119895 are the corresponding equilibria)
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
120582119908 = 1205751198942119911 + 120575119894
1119906 minus (120583 + 120584)119908
(25)
In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing
41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894
1= 119894
2= 119895 = 0 and 119904 = Λ120583
Thus the system above simplifies to the following system
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(26)
From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864
0
Theorem 2 If 1198771lt 1 and 119877
2lt 1 then the disease-free
equilibrium 1198640is locally asymptotically stable If 119877
1gt 1 or
1198772gt 1 then the disease-free equilibrium 119864
0is unstable
Proof To see this from the second to last equation we have120582119906 = [120573
2119904 minus (120583 + 120572
2)]119906 where either 120582 = 120573
2119904 minus (120583 + 120572
2) or
119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877
2minus 1) lt 0 if and only
if 1198772lt 1 Thus if 119877
2gt 1 the disease-free equilibrium 119864
0
is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation
119910 (119886) = 119910 (0) 119890minus(120582+120583)119886
120587 (119886) = 1205731119904119911119890
minus(120582+120583)119886120587 (119886) (27)
Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the
characteristic equation 120582119911 = 1205731119904 int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 119911 minus
(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572
1)(120573
1119904(120583 + 120572
1))
intinfin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 minus 1) or 119911 = 0
We denote the left hand side of the equation above byF
1(120582) = 120582 andF
2(120582) = (120583 + 120572
1)(119877
1(120582) minus 1) where 119877
1(120582) =
(1205731119904(120583 + 120572
1)) int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 and
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(28)
IfF1(120582) = F
2(120582) has a root with Re 120582 ge 0 then Re F
1(120582) ge
0 But |F2(120582)| le (120583 + 120572
1)(119877
1minus 1) lt 0 when 119877
1lt 1 This is a
contradictionThus if both 119877
1lt 1 and 119877
2lt 1 all eigenvalues
have negative real part and the disease-free equilibrium 1198640
is locally asymptotically stable If only 1198771gt 1 then if we
consider F1(120582) = F
2(120582) for 120582 is real we see that F
2(120582) is
a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F
2(0) = (120583 + 120572
1)(119877
1minus 1) gt 0 and F
1(0) = 0
that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864
0is unstableThis concludes the
proof
In what follows we show that the diseases vanish if 1198771lt
1 1198772lt 1
Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864
0is a global attractor
that is lim119905rarrinfin
119890(119886 119905) = 0 1198681rarr 0 119868
2rarr 0 119869 rarr 0 as
119905 rarr infin
Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin
le
Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have
119890 (119886 119905) =
B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886
(29)
SinceB(119905) = 1205731119878119868
1le 120573
1Λ119868
1120583 and
1198681015840
1le 120573
1
Λ
120583int
119905
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1198651(119905) minus 120583119868
1minus 120572
11198681
(30)
where 1198651(119905) = int
infin
119905120574(119886)119890
0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890
minus120583119905119889119886 and
lim119905rarrinfin
1198651(119905) = 0 from the equation for 119868
1then we have the
following inequality
1198681le 119868
1(0) 119890
minus(120583+1205721)119905
+ 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)120591int
119905minus120591
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889120591
+ int
119905
0
119890minus(120583+120572
1)1205911198651(119905 minus 120591) 119889120591
(31)
6 Journal of Applied Mathematics
where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero
as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain
119868infin
1le 119877
1119868infin
1 (32)
Since 1198771lt 1 this inequality can only be satisfied if 119868infin
1rarr 0
as 119905 rarr infinSince B(119905) le 120573
1(Λ120583)119868
1 it is easy to obtain B(119905) rarr 0
as 119905 rarr infinFrom the equation 119868
2then we have the following inequal-
ity
1198681015840
2le 120573
2
Λ
1205831198682minus (120583 + 120572
2) 119868
2 (33)
Solving the inequality we get
1198682le 119868
2(0) 119890
minus(120583+1205722)119905+ 120573
2
Λ
120583int
infin
0
1198682(119905 minus 120591) 119890
minus(120583+1205722)120591119889120591 (34)
Consequently taking a limsup of both sides we obtain
119868infin
2le 119877
2119868infin
2 (35)
If 1198772lt 1 then 119868
2(119905) rarr 0 as 119905 rarr infin
From the fluctuations lemma we can choose sequence 119905119899
such that 119878(119905119899) rarr 119878
infinand 1198781015840(119905
119899) rarr 0 when 119905
119899rarr +infin It
follows from the first equation of (1) that we have 119878infinge Λ120583
Then Λ120583 le 119878infinle 119878
infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr
infin This completes the theorem
42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864
1and
derive conditions for domination of TB We show that theequilibrium 119864
1can lose stability and dominance of the TB
is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877
1) 119894
1= Λ(1 minus 1119877
1)((1 minus 119861)120583119862 +
(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886
The eigenvalue problem takes the form
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(36)
From the last equation we have
120582 = (120583 + 1205722) (1198772
1198771
minus 1) or 119906 = 0 (37)
Hence 1198771gt 119877
2the partial characteristic root of (36) is
negative Substituting
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
119910 (0) = 1205731119904119911 + 120573
11198941119909 (38)
in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation
(120582 + 120583 + 1205721minus 120573
1119904119862 (120582)) (120582 + 120583 + 120573
11198941(1 minus 119861 (120582)))
minus 12057311198941119862 (120582) (120572
1+ 120573
1119904 (119861 (120582) minus 1)) = 0
(39)
Substituting the formula
1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)
where
119864 (120582) = int
infin
0
120587 (119886) 119890minus(120582+120583)119886 (41)
into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572
1) (1 + 120573
11198941119864 (120582)) + 120573
11198941119862 (120582)
120583 + 1205721
=119862 (120582)
119862 (42)
It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864
1is locally asymptotically stable
Therefore we are ready to establish the first result
Theorem 4 Let 1198771gt 1 and 119877
1gt 119877
2 Then the TB dominated
equilibrium 1198641is locally asymptotically stable
For all 119909 = (119878 0119877 119890 119868
1 1198682 119869) isin 119883
0 we define 119875
119878 119883
0rarr
1198771198751198681
1198751198682
1198830rarr 119877 and 119875
11986812
1198830rarr 119877 by
119875119878(119909) = 119878 119875
1198681(119909) = 119868
1 119875
1198682(119909) = 119868
2 (43)
Set119872
11986812
= 1198830+ 119872
119868120
= 119909 isin 11987211986812
| 1198751198681
119909 = 0 1198751198682
119909 = 0
120597119872119868120
=
11987211986812
119872119868120
1198721198681
= 120597119872119868120
11987211986810
= 119909 isin 1198721198681
| 1198751198681
119909 = 0 1198751198682
x = 0 12059711987211986810
=
1198721198681
11987211986810
1198721198682
= 12059711987211986810
11987211986820
= 119909 isin 1198721198682
| 1198751198682
119909 = 0 1198751198681
119909 = 0
12059711987211986820
=
1198721198682
11987211986820
119872119878= 120597119872
11986820
(44)
(see Figure 2)
Lemma 5 If 1198771gt 1 lim inf 119868
1(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 119890 0
119877 0
119877) isin 119872
11986810
We assume there is a1199050gt 0 such that 119868
1lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986810
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
1119878 minus 120583119878
119878 (0) = 1198780
(45)
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
Λ
119878
12057321198781198682
120583119878
12057311198781198681
120583119869
119869
12057511986811198682
1205831198681
120574(119886)119890(119886119905)
12057211198681
120583119890(119886119905)
120584119869
1205720(119886)119890(119886119905)
12057221198682
119890
1198681
1198682
1205831198682
Figure 1 The flow diagram of the model (1) Two yellow rectangu-lars denote the individuals infected by latent tuberculosis and activetuberculosisThe red rectangular denotes the individuals infected byHIV The orange rectangular denotes the individuals coinfected byTB and HIV
Thompson [8] developed models which reflect the trans-mission dynamics of both TB and HIV and discussed themagnitude and duration of the effect that the HIV epidemicmay have on TB Naresh and Tripathi [9] presented amodel for HIV-TB coinfection with constant recruitmentof susceptibles and found that TB will be eradicated fromthe population if more than 90 percent TB infectives arerecovered due to effective treatment Naresh et al [10] studiedthe effect of tuberculosis on the spread of HIV infection in alogistically growing human population They found that asthe number of TB infectives decreases due to recovery thenumber of HIV infectives also decreases and endemic equi-librium tends toTB free equilibrium Long et al [11] discussedtwo synergistic infectious disease epidemics illustrating theimportance of including the effects of each disease on thetransmission and progression of the other disease
However these models may exclude coinfection maygreatly simplify infection dynamics and may include fewdisease states (eg the important distinction between latentand active TBwas absent)They assume thatHIV treatment isunavailable Bifurcation will appear in some models [12 13]We perform the additional latent age analyze the coexistenceequilibria and extend the model to include disease recoveryin this paper At the same time we obtain the persistence ofthe system and global stabilities of the equilibria under someconditions
This paper is organized as follows In Section 2 weintroduce the TB-HIV coinfection model In Section 3 we
introduce the reproduction numbers of TB and HIV 1198771 119877
2
and discuss the existence of the equilibria The values of thedisease-free equilibrium the two boundary equilibria andthe coexistence equilibrium are given explicitly Section 4focuses on local and global stabilities of the equilibria InSection 5 we discuss the persistence of the system in suitableperiod In Section 6 we simulate and illustrate our resultsWegive some biological explanations in this section In Section 7we conclude our results and discuss the defect of our model
2 The Model Formulation
Two diseases mentioned in the introduction are spreading ina population of total size 119873(119905) We classify the total popu-lation into four classes the susceptible 119878(119905) the individualsinfected by the tuberculosis 119890(119886 119905) and 119868
1(119905) denotes the
latent TB class which has no infectious ability and active TBclass who can infect the susceptible class respectively theindividuals infected by HIV 119868
2(119905) the individuals coinfected
by tuberculosis and HIV 119869(119905) The individuals infected withactive TB infect the susceptible and then develop into thelatent individuals at 120573
1 The individuals infected by HIV
infect the susceptible and become the individuals infected byHIV at 120573
2 An individual already infected with TB can be
coinfected with HIV at 120575 and thus become jointly infectedindividuals 119869(119905)
Figure 1 presents a schematic flow diagram of the mathe-matical model as follows
119889119878
119889119905= Λ minus 120583119878 minus 120573
1119878119868
1minus 120573
2119878119868
2
+ int
infin
0
1205720(119886) 119890 (119886 119905) 119889119886 + 120572
11198681+ 120572
21198682
120597119890 (119886 119905)
120597119886+120597119890 (119886 119905)
120597119905= minus (120583 + 120574 (119886) + 120572
0(119886)) 119890 (119886 119905)
119890 (0 119905) = 1205731119878119868
1
1198891198681
119889119905= int
infin
0
120574 (119886) 119890 (119886 119905) 119889119886minus 1205831198681minus 120575119868
11198682minus 120572
11198681
1198891198682
119889119905= 120573
2119878119868
2minus (120583 + 120572
2) 119868
2
119889119869
119889119905= 120575119868
11198682minus (120583 + 120584) 119869
(1)
where 120583 is natural death rate and Λ is birth rate 120574(119886) israte of endogenous reactivation of latent TB We assume thatthe individuals separately infected by TB and HIV are notlethal but the coinfection can lead to the extra death at 120584Specifically we assume that jointly infected individuals donot recover Individuals infected with latent TB active TB orHIV alone may be potentially treated at rates 120572
0(119886) 120572
1 and
1205722 respectively
Assumption 1 Suppose that
(a) Λ 120583 120575 1205721 120572
2 120573
1 120573
2isin (0 +infin)
Journal of Applied Mathematics 3
(b) 120584 isin [0 +infin)(c) 120574(119886) 120572
0(119886) isin 119862
119861119880([0 +infin) 119877) cap 119862
+([0 +infin) 119877) and
for each 119886 ge 0 where 119862119861119880(119877) denotes the space
of bounded and uniformly continuous map from[0 +infin) into 119877
We assume (1) with the initial conditions
119878 (0) = 1198780ge 0 119890 (119886 0) = 119890
0(119886) isin 119871
1
+(0 +infin)
1198681(0) = 119868
10ge 0 119868
2(0) = 119868
20ge 0
119869 (0) = 1198690ge 0
(2)
Set
119883 = 119877 times 119884 times 1198772 119883
+= 119877
+times 119884
+times 119877
+
2
1198830= 119877 times 119884
0times 119877
2 119883
0+= 119883
0cap 119883
+
(3)
with
119884 = 119877 times 1198711(0 +infin) 119884
+= [0 +infin) times 119871
1
+(0 +infin)
1198840= 0 times 119871
1(0 +infin)
(4)
Let 119863(A) = 119877 times 119885 times 1198772 with 119885 = 0119877 times 119882
11(0infin)
for each 119909 = (119878 0119877 119890 119868
1 1198682) isin 119883
0 DefineA 119863(A) sub 119883
0sub
119883 rarr 119883 andF 1198830rarr 119883 by
A119909
=((
(
minus120583119878
minus119890 (0)
minus119889119890 (sdot)
119889119886minus (120583 + 120572
0(sdot) + 120574 (sdot)) 119890 (sdot)
minus1205831198681minus 120572
11198681
minus1205831198682minus 120572
21198682
minus (120583 + 120584) 119869
))
)
if 119909 isin 119863 (A)
(5)
F (119909)
=
((((
(
120583minus 1205731119878119868
1minus 120573
2119878119868
2+ int
infin
0
1205720(119886) 119890 (119886) 119889119886 + 120572
11198681+ 120572
21198682
1205731119878119868
1
01198711
int
infin
0
120574 (119886) 119890 (119886) 119889119886 minus 12057511986811198682
1205732119878119868
2
12057511986811198682
))))
)
(6)
Rewrite problem (1) as an abstract Cauchy problem
119889119906119909(119905)
119889119905= A119906
119909(119905) +F (119906
119909(119905)) 119905 ge 0 119906
119909(0) = 119909 isin 119883
0+
(7)
It is well known that A is a Hille-Yosida operator Moreprecisely we have (minus120583 +infin) sub 120588(A) and for each 120582 gt minus120583
10038171003817100381710038171003817(120582 minusA)
minus110038171003817100381710038171003817le1
120582 + 120583 (8)
By applying the results in Magal et al [14ndash16] we obtainthe following proposition
Lemma 1 There exists a uniquely determined semiflow119880(119905)
119905ge0on 119883
0+ such that for each 119909 = (119878
0 0 119890
0 11986810
11986820 1198690)119879isin 119883
0+ there exists a unique continuous map 119880 isin
119862([0 +infin)1198830+) which is an integrated solution of the Cauchy
problem (1) that is to say that
int
119905
0
119880 (119904) 119909 119889119904 isin 119863 (A) forall119905 ge 0
119880 (119905) 119909 = 119909 +Aint119905
0
119880 (119904) 119909 119889119904 + int
119905
0
F (119880 (119904) 119909) 119889119904 forallt ge 0
(9)
The total population size 119873(119905) is the sum of all individuals inall classes
119873 = 119878 (119905) + int
+infin
0
119890 (119886 119905) 119889119886 + 1198681(119905) + 119868
2(119905) + 119869 (119905) (10)
The total population size satisfies the equation 1198731015840(119905) = Λ minus
120583119873 minus 120584119869 We introduce the notation
120587 (119886) = exp(minusint119886
0
(1205720(120591) + 120574 (120591)) 119889120591) (11)
To understand the biological meaning of the quantity 120587(119886) wenote that 120587(119886)119890minus120583119886 is the probability to remain infected with TBtime units after infection In addition we define the quantity
119861 = int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 (12)
which gives the probability of treatment since the individualscan leave TB infectious period via treatment
119862 = int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 (13)
which gives the probability of progression since the individualscan leave TB infectious period via progression Since individu-als can only leave the latent TB infected class through treatmentprogression or death the sum of the probabilities of recoveryprogression and death equals one that is
int
infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 + int
infin
0
120574 (a) 120587 (119886) 119890minus120583119886 119889119886
+ 120583int
infin
0
120587 (119886) 119890minus120583119886119889119886 = 1
(14)
It immediately follows that 119861 + 119862 lt 1
3 Equilibria of the Model with Coinfection
We introduce the reproduction numbers of the two diseasesThe reproduction number of TB is
1198771=Λ120573
1119862
120583 (120583 + 1205721) (15)
4 Journal of Applied Mathematics
and the reproduction number of HIV is
1198772=
Λ1205732
120583 (120583 + 1205722) (16)
We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894
1 1198942 119895 as the equilibria of the modelTherefore equilibria
satisfy the following equations
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ int
+infin
0
1205720(119886) 119890 (119886 119905) 119889119886 + 120572
11198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
12057321199041198942minus (120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(17)
The ODE in the system can be solved to result in
119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)
Substituting for 119894 in the integrals one obtains
int
+infin
0
1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 = 119890 (0) 119861
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862
(19)
With this notation the system for the equilibria becomes
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ 119890 (0) 119861 + 12057211198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
119890 (0) 119862 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
1205732119904119894minus(120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(20)
This system has three boundary equilibria
(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)
The disease-free equilibrium always exists
(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the
TB equilibrium is given by
1199041=Λ
1205831198771
119890 = 119890 (0) 120587 (119886) 119890minus120583119886
1198941=
Λ (1 minus 11198771)
(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862
119890 (0) = 120573111990411198941
(21)
Thus the equilibrium is
1198641= (119904
1 119890 119894
1 0 0) (22)
(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by
1198642= (119904
2 0 0 119894
2 0) (23)
where 1199042= 1119877
2 119894
2= 0 119890
2= 0 119894
2= (Λ120583)(1 minus
11198772) 119895
2= 0
(4) The coexistence equilibrium exists if and only if 1198771gt
1198772 and [1 minus (1119877
2+120583(120583+120572
1)Λ120575)][(Λ120573
1120583120572
11198772)(1 minus
119861) minus 1] gt 0 and it is given by
1198642= (119904 119890 119894
1 1198942 119895) (24)
where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 1198941=
(Λ1205721)(1 minus (1119877
2+120583(120583+120572
1)Λ120575))((Λ120573
1120583120572
11198772)(1 minus
119861) minus 1) 1198942= ((120583 + 120575)120575)(119877
1119877
2minus 1) 119895 = 120575119894
11198942(120583 + 120584)
Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0
4 Stability of Equilibria
In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not
To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894
1 1198942 119895 That
is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894
1 119868
2= 119906 + 119894
2 119869 = 119908 + 119895
Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and
Journal of Applied Mathematics 5
119908These satisfy the following linear eigenvalue problem (here119904 119890 119894
1 1198942 and 119895 are the corresponding equilibria)
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
120582119908 = 1205751198942119911 + 120575119894
1119906 minus (120583 + 120584)119908
(25)
In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing
41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894
1= 119894
2= 119895 = 0 and 119904 = Λ120583
Thus the system above simplifies to the following system
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(26)
From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864
0
Theorem 2 If 1198771lt 1 and 119877
2lt 1 then the disease-free
equilibrium 1198640is locally asymptotically stable If 119877
1gt 1 or
1198772gt 1 then the disease-free equilibrium 119864
0is unstable
Proof To see this from the second to last equation we have120582119906 = [120573
2119904 minus (120583 + 120572
2)]119906 where either 120582 = 120573
2119904 minus (120583 + 120572
2) or
119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877
2minus 1) lt 0 if and only
if 1198772lt 1 Thus if 119877
2gt 1 the disease-free equilibrium 119864
0
is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation
119910 (119886) = 119910 (0) 119890minus(120582+120583)119886
120587 (119886) = 1205731119904119911119890
minus(120582+120583)119886120587 (119886) (27)
Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the
characteristic equation 120582119911 = 1205731119904 int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 119911 minus
(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572
1)(120573
1119904(120583 + 120572
1))
intinfin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 minus 1) or 119911 = 0
We denote the left hand side of the equation above byF
1(120582) = 120582 andF
2(120582) = (120583 + 120572
1)(119877
1(120582) minus 1) where 119877
1(120582) =
(1205731119904(120583 + 120572
1)) int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 and
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(28)
IfF1(120582) = F
2(120582) has a root with Re 120582 ge 0 then Re F
1(120582) ge
0 But |F2(120582)| le (120583 + 120572
1)(119877
1minus 1) lt 0 when 119877
1lt 1 This is a
contradictionThus if both 119877
1lt 1 and 119877
2lt 1 all eigenvalues
have negative real part and the disease-free equilibrium 1198640
is locally asymptotically stable If only 1198771gt 1 then if we
consider F1(120582) = F
2(120582) for 120582 is real we see that F
2(120582) is
a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F
2(0) = (120583 + 120572
1)(119877
1minus 1) gt 0 and F
1(0) = 0
that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864
0is unstableThis concludes the
proof
In what follows we show that the diseases vanish if 1198771lt
1 1198772lt 1
Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864
0is a global attractor
that is lim119905rarrinfin
119890(119886 119905) = 0 1198681rarr 0 119868
2rarr 0 119869 rarr 0 as
119905 rarr infin
Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin
le
Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have
119890 (119886 119905) =
B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886
(29)
SinceB(119905) = 1205731119878119868
1le 120573
1Λ119868
1120583 and
1198681015840
1le 120573
1
Λ
120583int
119905
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1198651(119905) minus 120583119868
1minus 120572
11198681
(30)
where 1198651(119905) = int
infin
119905120574(119886)119890
0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890
minus120583119905119889119886 and
lim119905rarrinfin
1198651(119905) = 0 from the equation for 119868
1then we have the
following inequality
1198681le 119868
1(0) 119890
minus(120583+1205721)119905
+ 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)120591int
119905minus120591
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889120591
+ int
119905
0
119890minus(120583+120572
1)1205911198651(119905 minus 120591) 119889120591
(31)
6 Journal of Applied Mathematics
where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero
as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain
119868infin
1le 119877
1119868infin
1 (32)
Since 1198771lt 1 this inequality can only be satisfied if 119868infin
1rarr 0
as 119905 rarr infinSince B(119905) le 120573
1(Λ120583)119868
1 it is easy to obtain B(119905) rarr 0
as 119905 rarr infinFrom the equation 119868
2then we have the following inequal-
ity
1198681015840
2le 120573
2
Λ
1205831198682minus (120583 + 120572
2) 119868
2 (33)
Solving the inequality we get
1198682le 119868
2(0) 119890
minus(120583+1205722)119905+ 120573
2
Λ
120583int
infin
0
1198682(119905 minus 120591) 119890
minus(120583+1205722)120591119889120591 (34)
Consequently taking a limsup of both sides we obtain
119868infin
2le 119877
2119868infin
2 (35)
If 1198772lt 1 then 119868
2(119905) rarr 0 as 119905 rarr infin
From the fluctuations lemma we can choose sequence 119905119899
such that 119878(119905119899) rarr 119878
infinand 1198781015840(119905
119899) rarr 0 when 119905
119899rarr +infin It
follows from the first equation of (1) that we have 119878infinge Λ120583
Then Λ120583 le 119878infinle 119878
infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr
infin This completes the theorem
42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864
1and
derive conditions for domination of TB We show that theequilibrium 119864
1can lose stability and dominance of the TB
is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877
1) 119894
1= Λ(1 minus 1119877
1)((1 minus 119861)120583119862 +
(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886
The eigenvalue problem takes the form
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(36)
From the last equation we have
120582 = (120583 + 1205722) (1198772
1198771
minus 1) or 119906 = 0 (37)
Hence 1198771gt 119877
2the partial characteristic root of (36) is
negative Substituting
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
119910 (0) = 1205731119904119911 + 120573
11198941119909 (38)
in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation
(120582 + 120583 + 1205721minus 120573
1119904119862 (120582)) (120582 + 120583 + 120573
11198941(1 minus 119861 (120582)))
minus 12057311198941119862 (120582) (120572
1+ 120573
1119904 (119861 (120582) minus 1)) = 0
(39)
Substituting the formula
1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)
where
119864 (120582) = int
infin
0
120587 (119886) 119890minus(120582+120583)119886 (41)
into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572
1) (1 + 120573
11198941119864 (120582)) + 120573
11198941119862 (120582)
120583 + 1205721
=119862 (120582)
119862 (42)
It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864
1is locally asymptotically stable
Therefore we are ready to establish the first result
Theorem 4 Let 1198771gt 1 and 119877
1gt 119877
2 Then the TB dominated
equilibrium 1198641is locally asymptotically stable
For all 119909 = (119878 0119877 119890 119868
1 1198682 119869) isin 119883
0 we define 119875
119878 119883
0rarr
1198771198751198681
1198751198682
1198830rarr 119877 and 119875
11986812
1198830rarr 119877 by
119875119878(119909) = 119878 119875
1198681(119909) = 119868
1 119875
1198682(119909) = 119868
2 (43)
Set119872
11986812
= 1198830+ 119872
119868120
= 119909 isin 11987211986812
| 1198751198681
119909 = 0 1198751198682
119909 = 0
120597119872119868120
=
11987211986812
119872119868120
1198721198681
= 120597119872119868120
11987211986810
= 119909 isin 1198721198681
| 1198751198681
119909 = 0 1198751198682
x = 0 12059711987211986810
=
1198721198681
11987211986810
1198721198682
= 12059711987211986810
11987211986820
= 119909 isin 1198721198682
| 1198751198682
119909 = 0 1198751198681
119909 = 0
12059711987211986820
=
1198721198682
11987211986820
119872119878= 120597119872
11986820
(44)
(see Figure 2)
Lemma 5 If 1198771gt 1 lim inf 119868
1(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 119890 0
119877 0
119877) isin 119872
11986810
We assume there is a1199050gt 0 such that 119868
1lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986810
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
1119878 minus 120583119878
119878 (0) = 1198780
(45)
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
(b) 120584 isin [0 +infin)(c) 120574(119886) 120572
0(119886) isin 119862
119861119880([0 +infin) 119877) cap 119862
+([0 +infin) 119877) and
for each 119886 ge 0 where 119862119861119880(119877) denotes the space
of bounded and uniformly continuous map from[0 +infin) into 119877
We assume (1) with the initial conditions
119878 (0) = 1198780ge 0 119890 (119886 0) = 119890
0(119886) isin 119871
1
+(0 +infin)
1198681(0) = 119868
10ge 0 119868
2(0) = 119868
20ge 0
119869 (0) = 1198690ge 0
(2)
Set
119883 = 119877 times 119884 times 1198772 119883
+= 119877
+times 119884
+times 119877
+
2
1198830= 119877 times 119884
0times 119877
2 119883
0+= 119883
0cap 119883
+
(3)
with
119884 = 119877 times 1198711(0 +infin) 119884
+= [0 +infin) times 119871
1
+(0 +infin)
1198840= 0 times 119871
1(0 +infin)
(4)
Let 119863(A) = 119877 times 119885 times 1198772 with 119885 = 0119877 times 119882
11(0infin)
for each 119909 = (119878 0119877 119890 119868
1 1198682) isin 119883
0 DefineA 119863(A) sub 119883
0sub
119883 rarr 119883 andF 1198830rarr 119883 by
A119909
=((
(
minus120583119878
minus119890 (0)
minus119889119890 (sdot)
119889119886minus (120583 + 120572
0(sdot) + 120574 (sdot)) 119890 (sdot)
minus1205831198681minus 120572
11198681
minus1205831198682minus 120572
21198682
minus (120583 + 120584) 119869
))
)
if 119909 isin 119863 (A)
(5)
F (119909)
=
((((
(
120583minus 1205731119878119868
1minus 120573
2119878119868
2+ int
infin
0
1205720(119886) 119890 (119886) 119889119886 + 120572
11198681+ 120572
21198682
1205731119878119868
1
01198711
int
infin
0
120574 (119886) 119890 (119886) 119889119886 minus 12057511986811198682
1205732119878119868
2
12057511986811198682
))))
)
(6)
Rewrite problem (1) as an abstract Cauchy problem
119889119906119909(119905)
119889119905= A119906
119909(119905) +F (119906
119909(119905)) 119905 ge 0 119906
119909(0) = 119909 isin 119883
0+
(7)
It is well known that A is a Hille-Yosida operator Moreprecisely we have (minus120583 +infin) sub 120588(A) and for each 120582 gt minus120583
10038171003817100381710038171003817(120582 minusA)
minus110038171003817100381710038171003817le1
120582 + 120583 (8)
By applying the results in Magal et al [14ndash16] we obtainthe following proposition
Lemma 1 There exists a uniquely determined semiflow119880(119905)
119905ge0on 119883
0+ such that for each 119909 = (119878
0 0 119890
0 11986810
11986820 1198690)119879isin 119883
0+ there exists a unique continuous map 119880 isin
119862([0 +infin)1198830+) which is an integrated solution of the Cauchy
problem (1) that is to say that
int
119905
0
119880 (119904) 119909 119889119904 isin 119863 (A) forall119905 ge 0
119880 (119905) 119909 = 119909 +Aint119905
0
119880 (119904) 119909 119889119904 + int
119905
0
F (119880 (119904) 119909) 119889119904 forallt ge 0
(9)
The total population size 119873(119905) is the sum of all individuals inall classes
119873 = 119878 (119905) + int
+infin
0
119890 (119886 119905) 119889119886 + 1198681(119905) + 119868
2(119905) + 119869 (119905) (10)
The total population size satisfies the equation 1198731015840(119905) = Λ minus
120583119873 minus 120584119869 We introduce the notation
120587 (119886) = exp(minusint119886
0
(1205720(120591) + 120574 (120591)) 119889120591) (11)
To understand the biological meaning of the quantity 120587(119886) wenote that 120587(119886)119890minus120583119886 is the probability to remain infected with TBtime units after infection In addition we define the quantity
119861 = int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 (12)
which gives the probability of treatment since the individualscan leave TB infectious period via treatment
119862 = int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 (13)
which gives the probability of progression since the individualscan leave TB infectious period via progression Since individu-als can only leave the latent TB infected class through treatmentprogression or death the sum of the probabilities of recoveryprogression and death equals one that is
int
infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 + int
infin
0
120574 (a) 120587 (119886) 119890minus120583119886 119889119886
+ 120583int
infin
0
120587 (119886) 119890minus120583119886119889119886 = 1
(14)
It immediately follows that 119861 + 119862 lt 1
3 Equilibria of the Model with Coinfection
We introduce the reproduction numbers of the two diseasesThe reproduction number of TB is
1198771=Λ120573
1119862
120583 (120583 + 1205721) (15)
4 Journal of Applied Mathematics
and the reproduction number of HIV is
1198772=
Λ1205732
120583 (120583 + 1205722) (16)
We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894
1 1198942 119895 as the equilibria of the modelTherefore equilibria
satisfy the following equations
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ int
+infin
0
1205720(119886) 119890 (119886 119905) 119889119886 + 120572
11198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
12057321199041198942minus (120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(17)
The ODE in the system can be solved to result in
119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)
Substituting for 119894 in the integrals one obtains
int
+infin
0
1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 = 119890 (0) 119861
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862
(19)
With this notation the system for the equilibria becomes
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ 119890 (0) 119861 + 12057211198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
119890 (0) 119862 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
1205732119904119894minus(120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(20)
This system has three boundary equilibria
(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)
The disease-free equilibrium always exists
(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the
TB equilibrium is given by
1199041=Λ
1205831198771
119890 = 119890 (0) 120587 (119886) 119890minus120583119886
1198941=
Λ (1 minus 11198771)
(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862
119890 (0) = 120573111990411198941
(21)
Thus the equilibrium is
1198641= (119904
1 119890 119894
1 0 0) (22)
(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by
1198642= (119904
2 0 0 119894
2 0) (23)
where 1199042= 1119877
2 119894
2= 0 119890
2= 0 119894
2= (Λ120583)(1 minus
11198772) 119895
2= 0
(4) The coexistence equilibrium exists if and only if 1198771gt
1198772 and [1 minus (1119877
2+120583(120583+120572
1)Λ120575)][(Λ120573
1120583120572
11198772)(1 minus
119861) minus 1] gt 0 and it is given by
1198642= (119904 119890 119894
1 1198942 119895) (24)
where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 1198941=
(Λ1205721)(1 minus (1119877
2+120583(120583+120572
1)Λ120575))((Λ120573
1120583120572
11198772)(1 minus
119861) minus 1) 1198942= ((120583 + 120575)120575)(119877
1119877
2minus 1) 119895 = 120575119894
11198942(120583 + 120584)
Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0
4 Stability of Equilibria
In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not
To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894
1 1198942 119895 That
is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894
1 119868
2= 119906 + 119894
2 119869 = 119908 + 119895
Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and
Journal of Applied Mathematics 5
119908These satisfy the following linear eigenvalue problem (here119904 119890 119894
1 1198942 and 119895 are the corresponding equilibria)
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
120582119908 = 1205751198942119911 + 120575119894
1119906 minus (120583 + 120584)119908
(25)
In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing
41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894
1= 119894
2= 119895 = 0 and 119904 = Λ120583
Thus the system above simplifies to the following system
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(26)
From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864
0
Theorem 2 If 1198771lt 1 and 119877
2lt 1 then the disease-free
equilibrium 1198640is locally asymptotically stable If 119877
1gt 1 or
1198772gt 1 then the disease-free equilibrium 119864
0is unstable
Proof To see this from the second to last equation we have120582119906 = [120573
2119904 minus (120583 + 120572
2)]119906 where either 120582 = 120573
2119904 minus (120583 + 120572
2) or
119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877
2minus 1) lt 0 if and only
if 1198772lt 1 Thus if 119877
2gt 1 the disease-free equilibrium 119864
0
is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation
119910 (119886) = 119910 (0) 119890minus(120582+120583)119886
120587 (119886) = 1205731119904119911119890
minus(120582+120583)119886120587 (119886) (27)
Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the
characteristic equation 120582119911 = 1205731119904 int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 119911 minus
(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572
1)(120573
1119904(120583 + 120572
1))
intinfin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 minus 1) or 119911 = 0
We denote the left hand side of the equation above byF
1(120582) = 120582 andF
2(120582) = (120583 + 120572
1)(119877
1(120582) minus 1) where 119877
1(120582) =
(1205731119904(120583 + 120572
1)) int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 and
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(28)
IfF1(120582) = F
2(120582) has a root with Re 120582 ge 0 then Re F
1(120582) ge
0 But |F2(120582)| le (120583 + 120572
1)(119877
1minus 1) lt 0 when 119877
1lt 1 This is a
contradictionThus if both 119877
1lt 1 and 119877
2lt 1 all eigenvalues
have negative real part and the disease-free equilibrium 1198640
is locally asymptotically stable If only 1198771gt 1 then if we
consider F1(120582) = F
2(120582) for 120582 is real we see that F
2(120582) is
a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F
2(0) = (120583 + 120572
1)(119877
1minus 1) gt 0 and F
1(0) = 0
that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864
0is unstableThis concludes the
proof
In what follows we show that the diseases vanish if 1198771lt
1 1198772lt 1
Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864
0is a global attractor
that is lim119905rarrinfin
119890(119886 119905) = 0 1198681rarr 0 119868
2rarr 0 119869 rarr 0 as
119905 rarr infin
Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin
le
Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have
119890 (119886 119905) =
B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886
(29)
SinceB(119905) = 1205731119878119868
1le 120573
1Λ119868
1120583 and
1198681015840
1le 120573
1
Λ
120583int
119905
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1198651(119905) minus 120583119868
1minus 120572
11198681
(30)
where 1198651(119905) = int
infin
119905120574(119886)119890
0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890
minus120583119905119889119886 and
lim119905rarrinfin
1198651(119905) = 0 from the equation for 119868
1then we have the
following inequality
1198681le 119868
1(0) 119890
minus(120583+1205721)119905
+ 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)120591int
119905minus120591
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889120591
+ int
119905
0
119890minus(120583+120572
1)1205911198651(119905 minus 120591) 119889120591
(31)
6 Journal of Applied Mathematics
where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero
as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain
119868infin
1le 119877
1119868infin
1 (32)
Since 1198771lt 1 this inequality can only be satisfied if 119868infin
1rarr 0
as 119905 rarr infinSince B(119905) le 120573
1(Λ120583)119868
1 it is easy to obtain B(119905) rarr 0
as 119905 rarr infinFrom the equation 119868
2then we have the following inequal-
ity
1198681015840
2le 120573
2
Λ
1205831198682minus (120583 + 120572
2) 119868
2 (33)
Solving the inequality we get
1198682le 119868
2(0) 119890
minus(120583+1205722)119905+ 120573
2
Λ
120583int
infin
0
1198682(119905 minus 120591) 119890
minus(120583+1205722)120591119889120591 (34)
Consequently taking a limsup of both sides we obtain
119868infin
2le 119877
2119868infin
2 (35)
If 1198772lt 1 then 119868
2(119905) rarr 0 as 119905 rarr infin
From the fluctuations lemma we can choose sequence 119905119899
such that 119878(119905119899) rarr 119878
infinand 1198781015840(119905
119899) rarr 0 when 119905
119899rarr +infin It
follows from the first equation of (1) that we have 119878infinge Λ120583
Then Λ120583 le 119878infinle 119878
infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr
infin This completes the theorem
42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864
1and
derive conditions for domination of TB We show that theequilibrium 119864
1can lose stability and dominance of the TB
is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877
1) 119894
1= Λ(1 minus 1119877
1)((1 minus 119861)120583119862 +
(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886
The eigenvalue problem takes the form
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(36)
From the last equation we have
120582 = (120583 + 1205722) (1198772
1198771
minus 1) or 119906 = 0 (37)
Hence 1198771gt 119877
2the partial characteristic root of (36) is
negative Substituting
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
119910 (0) = 1205731119904119911 + 120573
11198941119909 (38)
in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation
(120582 + 120583 + 1205721minus 120573
1119904119862 (120582)) (120582 + 120583 + 120573
11198941(1 minus 119861 (120582)))
minus 12057311198941119862 (120582) (120572
1+ 120573
1119904 (119861 (120582) minus 1)) = 0
(39)
Substituting the formula
1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)
where
119864 (120582) = int
infin
0
120587 (119886) 119890minus(120582+120583)119886 (41)
into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572
1) (1 + 120573
11198941119864 (120582)) + 120573
11198941119862 (120582)
120583 + 1205721
=119862 (120582)
119862 (42)
It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864
1is locally asymptotically stable
Therefore we are ready to establish the first result
Theorem 4 Let 1198771gt 1 and 119877
1gt 119877
2 Then the TB dominated
equilibrium 1198641is locally asymptotically stable
For all 119909 = (119878 0119877 119890 119868
1 1198682 119869) isin 119883
0 we define 119875
119878 119883
0rarr
1198771198751198681
1198751198682
1198830rarr 119877 and 119875
11986812
1198830rarr 119877 by
119875119878(119909) = 119878 119875
1198681(119909) = 119868
1 119875
1198682(119909) = 119868
2 (43)
Set119872
11986812
= 1198830+ 119872
119868120
= 119909 isin 11987211986812
| 1198751198681
119909 = 0 1198751198682
119909 = 0
120597119872119868120
=
11987211986812
119872119868120
1198721198681
= 120597119872119868120
11987211986810
= 119909 isin 1198721198681
| 1198751198681
119909 = 0 1198751198682
x = 0 12059711987211986810
=
1198721198681
11987211986810
1198721198682
= 12059711987211986810
11987211986820
= 119909 isin 1198721198682
| 1198751198682
119909 = 0 1198751198681
119909 = 0
12059711987211986820
=
1198721198682
11987211986820
119872119878= 120597119872
11986820
(44)
(see Figure 2)
Lemma 5 If 1198771gt 1 lim inf 119868
1(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 119890 0
119877 0
119877) isin 119872
11986810
We assume there is a1199050gt 0 such that 119868
1lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986810
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
1119878 minus 120583119878
119878 (0) = 1198780
(45)
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
and the reproduction number of HIV is
1198772=
Λ1205732
120583 (120583 + 1205722) (16)
We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894
1 1198942 119895 as the equilibria of the modelTherefore equilibria
satisfy the following equations
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ int
+infin
0
1205720(119886) 119890 (119886 119905) 119889119886 + 120572
11198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
12057321199041198942minus (120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(17)
The ODE in the system can be solved to result in
119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)
Substituting for 119894 in the integrals one obtains
int
+infin
0
1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
1205720(119886) 120587 (119886) 119890
minus120583119886119889119886 = 119890 (0) 119861
int
+infin
0
120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int
+infin
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862
(19)
With this notation the system for the equilibria becomes
Λ minus 12057311199041198941minus 120573
21199041198942minus 120583119904
+ 119890 (0) 119861 + 12057211198941+ 120572
21198942= 0
119889119890
119889119886= minus120572
0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890
119890 (0) = 12057311199041198941
119890 (0) 119862 minus 1205831198941minus 120575119894
11198942minus 120572
11198941= 0
1205732119904119894minus(120583 + 120572
2) 119894
2= 0
12057511989411198942minus (120583 + 120584) 119895 = 0
(20)
This system has three boundary equilibria
(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)
The disease-free equilibrium always exists
(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the
TB equilibrium is given by
1199041=Λ
1205831198771
119890 = 119890 (0) 120587 (119886) 119890minus120583119886
1198941=
Λ (1 minus 11198771)
(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862
119890 (0) = 120573111990411198941
(21)
Thus the equilibrium is
1198641= (119904
1 119890 119894
1 0 0) (22)
(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by
1198642= (119904
2 0 0 119894
2 0) (23)
where 1199042= 1119877
2 119894
2= 0 119890
2= 0 119894
2= (Λ120583)(1 minus
11198772) 119895
2= 0
(4) The coexistence equilibrium exists if and only if 1198771gt
1198772 and [1 minus (1119877
2+120583(120583+120572
1)Λ120575)][(Λ120573
1120583120572
11198772)(1 minus
119861) minus 1] gt 0 and it is given by
1198642= (119904 119890 119894
1 1198942 119895) (24)
where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 1198941=
(Λ1205721)(1 minus (1119877
2+120583(120583+120572
1)Λ120575))((Λ120573
1120583120572
11198772)(1 minus
119861) minus 1) 1198942= ((120583 + 120575)120575)(119877
1119877
2minus 1) 119895 = 120575119894
11198942(120583 + 120584)
Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0
4 Stability of Equilibria
In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not
To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894
1 1198942 119895 That
is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894
1 119868
2= 119906 + 119894
2 119869 = 119908 + 119895
Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and
Journal of Applied Mathematics 5
119908These satisfy the following linear eigenvalue problem (here119904 119890 119894
1 1198942 and 119895 are the corresponding equilibria)
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
120582119908 = 1205751198942119911 + 120575119894
1119906 minus (120583 + 120584)119908
(25)
In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing
41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894
1= 119894
2= 119895 = 0 and 119904 = Λ120583
Thus the system above simplifies to the following system
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(26)
From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864
0
Theorem 2 If 1198771lt 1 and 119877
2lt 1 then the disease-free
equilibrium 1198640is locally asymptotically stable If 119877
1gt 1 or
1198772gt 1 then the disease-free equilibrium 119864
0is unstable
Proof To see this from the second to last equation we have120582119906 = [120573
2119904 minus (120583 + 120572
2)]119906 where either 120582 = 120573
2119904 minus (120583 + 120572
2) or
119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877
2minus 1) lt 0 if and only
if 1198772lt 1 Thus if 119877
2gt 1 the disease-free equilibrium 119864
0
is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation
119910 (119886) = 119910 (0) 119890minus(120582+120583)119886
120587 (119886) = 1205731119904119911119890
minus(120582+120583)119886120587 (119886) (27)
Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the
characteristic equation 120582119911 = 1205731119904 int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 119911 minus
(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572
1)(120573
1119904(120583 + 120572
1))
intinfin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 minus 1) or 119911 = 0
We denote the left hand side of the equation above byF
1(120582) = 120582 andF
2(120582) = (120583 + 120572
1)(119877
1(120582) minus 1) where 119877
1(120582) =
(1205731119904(120583 + 120572
1)) int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 and
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(28)
IfF1(120582) = F
2(120582) has a root with Re 120582 ge 0 then Re F
1(120582) ge
0 But |F2(120582)| le (120583 + 120572
1)(119877
1minus 1) lt 0 when 119877
1lt 1 This is a
contradictionThus if both 119877
1lt 1 and 119877
2lt 1 all eigenvalues
have negative real part and the disease-free equilibrium 1198640
is locally asymptotically stable If only 1198771gt 1 then if we
consider F1(120582) = F
2(120582) for 120582 is real we see that F
2(120582) is
a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F
2(0) = (120583 + 120572
1)(119877
1minus 1) gt 0 and F
1(0) = 0
that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864
0is unstableThis concludes the
proof
In what follows we show that the diseases vanish if 1198771lt
1 1198772lt 1
Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864
0is a global attractor
that is lim119905rarrinfin
119890(119886 119905) = 0 1198681rarr 0 119868
2rarr 0 119869 rarr 0 as
119905 rarr infin
Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin
le
Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have
119890 (119886 119905) =
B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886
(29)
SinceB(119905) = 1205731119878119868
1le 120573
1Λ119868
1120583 and
1198681015840
1le 120573
1
Λ
120583int
119905
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1198651(119905) minus 120583119868
1minus 120572
11198681
(30)
where 1198651(119905) = int
infin
119905120574(119886)119890
0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890
minus120583119905119889119886 and
lim119905rarrinfin
1198651(119905) = 0 from the equation for 119868
1then we have the
following inequality
1198681le 119868
1(0) 119890
minus(120583+1205721)119905
+ 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)120591int
119905minus120591
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889120591
+ int
119905
0
119890minus(120583+120572
1)1205911198651(119905 minus 120591) 119889120591
(31)
6 Journal of Applied Mathematics
where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero
as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain
119868infin
1le 119877
1119868infin
1 (32)
Since 1198771lt 1 this inequality can only be satisfied if 119868infin
1rarr 0
as 119905 rarr infinSince B(119905) le 120573
1(Λ120583)119868
1 it is easy to obtain B(119905) rarr 0
as 119905 rarr infinFrom the equation 119868
2then we have the following inequal-
ity
1198681015840
2le 120573
2
Λ
1205831198682minus (120583 + 120572
2) 119868
2 (33)
Solving the inequality we get
1198682le 119868
2(0) 119890
minus(120583+1205722)119905+ 120573
2
Λ
120583int
infin
0
1198682(119905 minus 120591) 119890
minus(120583+1205722)120591119889120591 (34)
Consequently taking a limsup of both sides we obtain
119868infin
2le 119877
2119868infin
2 (35)
If 1198772lt 1 then 119868
2(119905) rarr 0 as 119905 rarr infin
From the fluctuations lemma we can choose sequence 119905119899
such that 119878(119905119899) rarr 119878
infinand 1198781015840(119905
119899) rarr 0 when 119905
119899rarr +infin It
follows from the first equation of (1) that we have 119878infinge Λ120583
Then Λ120583 le 119878infinle 119878
infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr
infin This completes the theorem
42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864
1and
derive conditions for domination of TB We show that theequilibrium 119864
1can lose stability and dominance of the TB
is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877
1) 119894
1= Λ(1 minus 1119877
1)((1 minus 119861)120583119862 +
(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886
The eigenvalue problem takes the form
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(36)
From the last equation we have
120582 = (120583 + 1205722) (1198772
1198771
minus 1) or 119906 = 0 (37)
Hence 1198771gt 119877
2the partial characteristic root of (36) is
negative Substituting
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
119910 (0) = 1205731119904119911 + 120573
11198941119909 (38)
in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation
(120582 + 120583 + 1205721minus 120573
1119904119862 (120582)) (120582 + 120583 + 120573
11198941(1 minus 119861 (120582)))
minus 12057311198941119862 (120582) (120572
1+ 120573
1119904 (119861 (120582) minus 1)) = 0
(39)
Substituting the formula
1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)
where
119864 (120582) = int
infin
0
120587 (119886) 119890minus(120582+120583)119886 (41)
into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572
1) (1 + 120573
11198941119864 (120582)) + 120573
11198941119862 (120582)
120583 + 1205721
=119862 (120582)
119862 (42)
It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864
1is locally asymptotically stable
Therefore we are ready to establish the first result
Theorem 4 Let 1198771gt 1 and 119877
1gt 119877
2 Then the TB dominated
equilibrium 1198641is locally asymptotically stable
For all 119909 = (119878 0119877 119890 119868
1 1198682 119869) isin 119883
0 we define 119875
119878 119883
0rarr
1198771198751198681
1198751198682
1198830rarr 119877 and 119875
11986812
1198830rarr 119877 by
119875119878(119909) = 119878 119875
1198681(119909) = 119868
1 119875
1198682(119909) = 119868
2 (43)
Set119872
11986812
= 1198830+ 119872
119868120
= 119909 isin 11987211986812
| 1198751198681
119909 = 0 1198751198682
119909 = 0
120597119872119868120
=
11987211986812
119872119868120
1198721198681
= 120597119872119868120
11987211986810
= 119909 isin 1198721198681
| 1198751198681
119909 = 0 1198751198682
x = 0 12059711987211986810
=
1198721198681
11987211986810
1198721198682
= 12059711987211986810
11987211986820
= 119909 isin 1198721198682
| 1198751198682
119909 = 0 1198751198681
119909 = 0
12059711987211986820
=
1198721198682
11987211986820
119872119878= 120597119872
11986820
(44)
(see Figure 2)
Lemma 5 If 1198771gt 1 lim inf 119868
1(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 119890 0
119877 0
119877) isin 119872
11986810
We assume there is a1199050gt 0 such that 119868
1lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986810
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
1119878 minus 120583119878
119878 (0) = 1198780
(45)
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
119908These satisfy the following linear eigenvalue problem (here119904 119890 119894
1 1198942 and 119895 are the corresponding equilibria)
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
120582119908 = 1205751198942119911 + 120575119894
1119906 minus (120583 + 120584)119908
(25)
In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing
41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894
1= 119894
2= 119895 = 0 and 119904 = Λ120583
Thus the system above simplifies to the following system
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894
1119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(26)
From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864
0
Theorem 2 If 1198771lt 1 and 119877
2lt 1 then the disease-free
equilibrium 1198640is locally asymptotically stable If 119877
1gt 1 or
1198772gt 1 then the disease-free equilibrium 119864
0is unstable
Proof To see this from the second to last equation we have120582119906 = [120573
2119904 minus (120583 + 120572
2)]119906 where either 120582 = 120573
2119904 minus (120583 + 120572
2) or
119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877
2minus 1) lt 0 if and only
if 1198772lt 1 Thus if 119877
2gt 1 the disease-free equilibrium 119864
0
is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation
119910 (119886) = 119910 (0) 119890minus(120582+120583)119886
120587 (119886) = 1205731119904119911119890
minus(120582+120583)119886120587 (119886) (27)
Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the
characteristic equation 120582119911 = 1205731119904 int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 119911 minus
(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572
1)(120573
1119904(120583 + 120572
1))
intinfin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 minus 1) or 119911 = 0
We denote the left hand side of the equation above byF
1(120582) = 120582 andF
2(120582) = (120583 + 120572
1)(119877
1(120582) minus 1) where 119877
1(120582) =
(1205731119904(120583 + 120572
1)) int
infin
0120574(119886)120587(119886)119890
minus(120582+120583)119886119889119886 and
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(28)
IfF1(120582) = F
2(120582) has a root with Re 120582 ge 0 then Re F
1(120582) ge
0 But |F2(120582)| le (120583 + 120572
1)(119877
1minus 1) lt 0 when 119877
1lt 1 This is a
contradictionThus if both 119877
1lt 1 and 119877
2lt 1 all eigenvalues
have negative real part and the disease-free equilibrium 1198640
is locally asymptotically stable If only 1198771gt 1 then if we
consider F1(120582) = F
2(120582) for 120582 is real we see that F
2(120582) is
a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F
2(0) = (120583 + 120572
1)(119877
1minus 1) gt 0 and F
1(0) = 0
that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864
0is unstableThis concludes the
proof
In what follows we show that the diseases vanish if 1198771lt
1 1198772lt 1
Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864
0is a global attractor
that is lim119905rarrinfin
119890(119886 119905) = 0 1198681rarr 0 119868
2rarr 0 119869 rarr 0 as
119905 rarr infin
Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin
le
Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have
119890 (119886 119905) =
B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886
(29)
SinceB(119905) = 1205731119878119868
1le 120573
1Λ119868
1120583 and
1198681015840
1le 120573
1
Λ
120583int
119905
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1198651(119905) minus 120583119868
1minus 120572
11198681
(30)
where 1198651(119905) = int
infin
119905120574(119886)119890
0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890
minus120583119905119889119886 and
lim119905rarrinfin
1198651(119905) = 0 from the equation for 119868
1then we have the
following inequality
1198681le 119868
1(0) 119890
minus(120583+1205721)119905
+ 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)120591int
119905minus120591
0
120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889120591
+ int
119905
0
119890minus(120583+120572
1)1205911198651(119905 minus 120591) 119889120591
(31)
6 Journal of Applied Mathematics
where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero
as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain
119868infin
1le 119877
1119868infin
1 (32)
Since 1198771lt 1 this inequality can only be satisfied if 119868infin
1rarr 0
as 119905 rarr infinSince B(119905) le 120573
1(Λ120583)119868
1 it is easy to obtain B(119905) rarr 0
as 119905 rarr infinFrom the equation 119868
2then we have the following inequal-
ity
1198681015840
2le 120573
2
Λ
1205831198682minus (120583 + 120572
2) 119868
2 (33)
Solving the inequality we get
1198682le 119868
2(0) 119890
minus(120583+1205722)119905+ 120573
2
Λ
120583int
infin
0
1198682(119905 minus 120591) 119890
minus(120583+1205722)120591119889120591 (34)
Consequently taking a limsup of both sides we obtain
119868infin
2le 119877
2119868infin
2 (35)
If 1198772lt 1 then 119868
2(119905) rarr 0 as 119905 rarr infin
From the fluctuations lemma we can choose sequence 119905119899
such that 119878(119905119899) rarr 119878
infinand 1198781015840(119905
119899) rarr 0 when 119905
119899rarr +infin It
follows from the first equation of (1) that we have 119878infinge Λ120583
Then Λ120583 le 119878infinle 119878
infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr
infin This completes the theorem
42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864
1and
derive conditions for domination of TB We show that theequilibrium 119864
1can lose stability and dominance of the TB
is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877
1) 119894
1= Λ(1 minus 1119877
1)((1 minus 119861)120583119862 +
(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886
The eigenvalue problem takes the form
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(36)
From the last equation we have
120582 = (120583 + 1205722) (1198772
1198771
minus 1) or 119906 = 0 (37)
Hence 1198771gt 119877
2the partial characteristic root of (36) is
negative Substituting
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
119910 (0) = 1205731119904119911 + 120573
11198941119909 (38)
in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation
(120582 + 120583 + 1205721minus 120573
1119904119862 (120582)) (120582 + 120583 + 120573
11198941(1 minus 119861 (120582)))
minus 12057311198941119862 (120582) (120572
1+ 120573
1119904 (119861 (120582) minus 1)) = 0
(39)
Substituting the formula
1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)
where
119864 (120582) = int
infin
0
120587 (119886) 119890minus(120582+120583)119886 (41)
into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572
1) (1 + 120573
11198941119864 (120582)) + 120573
11198941119862 (120582)
120583 + 1205721
=119862 (120582)
119862 (42)
It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864
1is locally asymptotically stable
Therefore we are ready to establish the first result
Theorem 4 Let 1198771gt 1 and 119877
1gt 119877
2 Then the TB dominated
equilibrium 1198641is locally asymptotically stable
For all 119909 = (119878 0119877 119890 119868
1 1198682 119869) isin 119883
0 we define 119875
119878 119883
0rarr
1198771198751198681
1198751198682
1198830rarr 119877 and 119875
11986812
1198830rarr 119877 by
119875119878(119909) = 119878 119875
1198681(119909) = 119868
1 119875
1198682(119909) = 119868
2 (43)
Set119872
11986812
= 1198830+ 119872
119868120
= 119909 isin 11987211986812
| 1198751198681
119909 = 0 1198751198682
119909 = 0
120597119872119868120
=
11987211986812
119872119868120
1198721198681
= 120597119872119868120
11987211986810
= 119909 isin 1198721198681
| 1198751198681
119909 = 0 1198751198682
x = 0 12059711987211986810
=
1198721198681
11987211986810
1198721198682
= 12059711987211986810
11987211986820
= 119909 isin 1198721198682
| 1198751198682
119909 = 0 1198751198681
119909 = 0
12059711987211986820
=
1198721198682
11987211986820
119872119878= 120597119872
11986820
(44)
(see Figure 2)
Lemma 5 If 1198771gt 1 lim inf 119868
1(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 119890 0
119877 0
119877) isin 119872
11986810
We assume there is a1199050gt 0 such that 119868
1lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986810
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
1119878 minus 120583119878
119878 (0) = 1198780
(45)
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
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Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero
as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain
119868infin
1le 119877
1119868infin
1 (32)
Since 1198771lt 1 this inequality can only be satisfied if 119868infin
1rarr 0
as 119905 rarr infinSince B(119905) le 120573
1(Λ120583)119868
1 it is easy to obtain B(119905) rarr 0
as 119905 rarr infinFrom the equation 119868
2then we have the following inequal-
ity
1198681015840
2le 120573
2
Λ
1205831198682minus (120583 + 120572
2) 119868
2 (33)
Solving the inequality we get
1198682le 119868
2(0) 119890
minus(120583+1205722)119905+ 120573
2
Λ
120583int
infin
0
1198682(119905 minus 120591) 119890
minus(120583+1205722)120591119889120591 (34)
Consequently taking a limsup of both sides we obtain
119868infin
2le 119877
2119868infin
2 (35)
If 1198772lt 1 then 119868
2(119905) rarr 0 as 119905 rarr infin
From the fluctuations lemma we can choose sequence 119905119899
such that 119878(119905119899) rarr 119878
infinand 1198781015840(119905
119899) rarr 0 when 119905
119899rarr +infin It
follows from the first equation of (1) that we have 119878infinge Λ120583
Then Λ120583 le 119878infinle 119878
infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr
infin This completes the theorem
42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864
1and
derive conditions for domination of TB We show that theequilibrium 119864
1can lose stability and dominance of the TB
is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877
1) 119894
1= Λ(1 minus 1119877
1)((1 minus 119861)120583119862 +
(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886
The eigenvalue problem takes the form
120582119909 = minus 1205731119904119911 minus 120573
11198941119909 minus 120573
2119904119906 minus 120583119904
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911 + 120573
11198941119909
120582119911 = int
infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572
1119911
120582119906 = 1205732119904119906 minus (120583 + 120572
2) 119906
(36)
From the last equation we have
120582 = (120583 + 1205722) (1198772
1198771
minus 1) or 119906 = 0 (37)
Hence 1198771gt 119877
2the partial characteristic root of (36) is
negative Substituting
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
119910 (0) = 1205731119904119911 + 120573
11198941119909 (38)
in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation
(120582 + 120583 + 1205721minus 120573
1119904119862 (120582)) (120582 + 120583 + 120573
11198941(1 minus 119861 (120582)))
minus 12057311198941119862 (120582) (120572
1+ 120573
1119904 (119861 (120582) minus 1)) = 0
(39)
Substituting the formula
1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)
where
119864 (120582) = int
infin
0
120587 (119886) 119890minus(120582+120583)119886 (41)
into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572
1) (1 + 120573
11198941119864 (120582)) + 120573
11198941119862 (120582)
120583 + 1205721
=119862 (120582)
119862 (42)
It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864
1is locally asymptotically stable
Therefore we are ready to establish the first result
Theorem 4 Let 1198771gt 1 and 119877
1gt 119877
2 Then the TB dominated
equilibrium 1198641is locally asymptotically stable
For all 119909 = (119878 0119877 119890 119868
1 1198682 119869) isin 119883
0 we define 119875
119878 119883
0rarr
1198771198751198681
1198751198682
1198830rarr 119877 and 119875
11986812
1198830rarr 119877 by
119875119878(119909) = 119878 119875
1198681(119909) = 119868
1 119875
1198682(119909) = 119868
2 (43)
Set119872
11986812
= 1198830+ 119872
119868120
= 119909 isin 11987211986812
| 1198751198681
119909 = 0 1198751198682
119909 = 0
120597119872119868120
=
11987211986812
119872119868120
1198721198681
= 120597119872119868120
11987211986810
= 119909 isin 1198721198681
| 1198751198681
119909 = 0 1198751198682
x = 0 12059711987211986810
=
1198721198681
11987211986810
1198721198682
= 12059711987211986810
11987211986820
= 119909 isin 1198721198682
| 1198751198682
119909 = 0 1198751198681
119909 = 0
12059711987211986820
=
1198721198682
11987211986820
119872119878= 120597119872
11986820
(44)
(see Figure 2)
Lemma 5 If 1198771gt 1 lim inf 119868
1(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 119890 0
119877 0
119877) isin 119872
11986810
We assume there is a1199050gt 0 such that 119868
1lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986810
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
1119878 minus 120583119878
119878 (0) = 1198780
(45)
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
11987211986820
119872119868120
11987211986810
119872119878
Figure 2 Zoning the area
We solve them as follows
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
1120576)119905)
120583≐ 119878
lowast(119905) (46)
And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there
exists 120575 gt 0 and 1198791gt 0 which satisfy 120573
1(Λ120583)
int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 gt 1 From the second
and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) =
1205731119878 (119905 minus 119886) 119868
1(119905 minus 119886) 120587 (119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886
(47)
From the fourth equation of (1) we get
1198681015840
1(119905) = 120573
1int
119905
0
120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
120587 (119886)
120587 (119886 minus 119905)119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(48)
Solving it we have
1198681(119905) ge 120573
1
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(49)
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(50)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 119905
0
(51)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868
1(119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(52)
Assuming that 1199052lt infin it follows that
1198681(1199052) ge 120573
1
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890
minus120583119886119889119886 119889119904
ge 1205731
Λ
120583int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585
(53)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it
follows that there exists an 1205761gt 0 such that 119868
1(119905) ge 120585
for all 119905 isin [1199052 1199052+ 120576
1] which contradicts the definition of
1199052 Therefore 119868
1(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin=
lim inf119905rarr+infin
1198681(119905) ge 120585 gt 0 Using (51) it follows that
1198681infinge 119868
1infin1205731(Λ120583) int
120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)120587(119886)119890
minus120583119886119889119886 119889119904 which
is impossible Thus 1198681(119905) ge 120576
Theorem 6 If 1198771gt 1 system (1) is permanent in 119872
11986810
Moreover there exists 119860
11986810
a compact subset of 11987211986810
which isa global attractor for 119880(119905)
119905ge0in119872
11986810
Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872
11986810
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt 120583 minus (120573
1+ 120583) 119878 (119905) (54)
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
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Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
Consider the comparison equation
1199061015840(119905) = 120583 minus (120573
1+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(55)
Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
1 for 119905 large enough where 119906lowast =
120583(120583 + 1205731) From the second and third equations of (1) and
Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890
minus120583119886≐ 119898
2
for 119905 large enough
43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864
2is locally stable
whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =
Λ1205831198772 1198942= (Λ120583)(1 minus 1119877
2) The linear eigenvalue problem
becomes
120582119909 = minus 1205731119904119911 minus 120573
2119904119906 minus 120573
21198942119909 minus 120583119909
+ int
infin
0
1205720(119886) 119910 (119886) 119889119886 + 120572
1119911 + 120572
2119906
1199101015840(119886) = minus 120582119910 minus 120572
0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910
119910 (0) = 1205731119904119911
120582119911 = int
+infin
0
120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572
1119911
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(56)
From the equation for 119910(119886) we have
119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886
(57)
Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation
120582 + 1205751198942= 120573
1119904 int
infin
0
120574 (119886) 120587 (119886) 119890minus(120582+120583)119886
119889119886 minus (120583 + 1205721)
= (120583 + 1205721) (1198771(120582)
1198772
minus 1)
(58)
Denoting the left hand side of the equation above byF1(120582) =
120582+1205751198942 andF
2(120582) = (120583+120572
1)((119877
1(120582)119877
2)minus1) and also noting
that
1198771015840
1(120582) lt 0 lim
119905rarr+infin
1198771(120582) = 0 lim
119905rarrminusinfin
1198771(120582) = +infin
(59)
it is easy to see that F1(120582) is an increasing function with 120582
Note thatF1(0) = 120575119868
2ge 0 ButF
2(0) le (120583 + 120572
1)((119877
1119877
2) minus
1) lt 0 when 1198771lt 119877
2 Therefore the characteristic roots of
this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0
120582119909 = minus 1205732119904119906 minus 120573
21198942119909 minus 120583119909 + 120572
2119906
120582119906 = 1205732119904119906 + 120573
21198942119909 minus (120583 + 120572
2) 119906
(60)
We express 119909 from the first equation
119909 =(minus120573
2119904 + 120572
2) 119906
120582 + 120583 + 12057321198942
(61)
and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation
120582 + 120583 + 1205722= 120573
2119904 +12057321198942(minus120573
2119904 + 120572
2)
120582 + 120583 + 12057321198942
(62)
that is
(120582 + 120583) (120582 + 120583 + 1205722+ 120573
21198942minus 120573
2119904) = 0 (63)
Noticing that 1205732119904 = 120583 + 120572
2 we obtain the following
eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative
Consequently the equilibrium 1198642is locally asymptotically
stableTherefore we have the following theorem for equilibrium
1198642
Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot
invade the equilibrium of HIV that is 1198771lt 119877
2 Then the
equilibrium 1198642is locally asymptotically stable
Lemma 8 If 1198772gt 1 then lim inf 119868
2(119905) ge 120576 in119872
11986820
Proof Let 119909 = (1198780 0
119877 0
1
119871 0
119877 1198682 0
119877) isin 119872
11986820
We assume thereis a 119905
0gt 0 such that 119868
2lt 120576 for all 119905 ge 119905
0 From the first equation
of (1) in11987211986820
it is easy to get
119889119878
119889119905ge Λ minus 120576120573
2119878 minus 120583119878
119878 (0) = 1198780
(64)
We solve them
119878 (119905) ge
Λ (1 minus 119890minus(120583+120573
2120576)119905)
120583≐ 119878
lowast(119905) (65)
Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905
0gt 0 we have
1198681015840
2(119905) ge 120573
2
Λ
1205831198682(119905) minus (120583 + 120572
2) 119868
2(119905) forall119905 ge 119905
0 (66)
Solving it we have
1198682(119905) ge 120573
2
Λ
120583int
119905
0
119890minus(120583+120572
2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)
We do lim inf on both side of (67) and denote 1198682infin=
lim inf119905rarr+infin
1198682(119905) gt 0 thus
1198682infinge 119877
21198682infin (68)
which is impossible Thus 1198682(119905) ge 120576
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
Theorem 9 If 1198772gt 1 system (1) is permanent in 119872
11986820
Moreover there exists 119860
11986820
a compact subset of 11987211986820
which isa global attractor for 119880(119905)
119905ge0in119872
11986820
Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872
11986820
be anysolution of (1) From the first equation of (1) we have
1198781015840(119905) gt Λ minus (120573
2+ 120583) 119878 (119905) (69)
Consider the comparison equation
1199061015840(119905) = Λ minus (120573
2+ 120583) 119906 (119905)
119906 (0) = 119878 (0)
(70)
Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576
1≐ 119898
2 for 119905 large enough where 119906lowast =
Λ(1205732+ 120583)
44 Stability of Coexistence Equilibrium 119864lowast In this subsec-
tion we establish that the equilibrium 119864lowastis locally sta-
ble whenever it exists In this case 119904 = Λ1205831198772 119890 =
119890(0)120587(119886)119890minus120583119886 119890(0) = 120573
11199041198941 119894
1= (Λ120572
1)((1 minus (1119877
2+ 120583(120583 +
1205721)Λ120575))((Λ120573
1120583120572
11198772)(1minus119861)minus1)) 119894
2= ((120583+120575)120575)(119877
1119877
2minus
1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the
coexistence equilibrium are as follows
(120582 + 120583 + 1205721+ 120575119894
2minus 120573
1119904119862 (120582))
times (120582 + 12057311198941+ 120573
21198942+ 120583 minus 120573
11198941119861 (120582) +
12057321198942120583
120582)
= (1205731119904119861 (120582) minus 120573
1119904) (120573
11198941minus120575120573
211989411198942
120582)
(71)
Theorem 10 If 1198771gt 119877
2 [1 minus (1119877
2+ (120583(120583 + 120572
1)
Λ120575))][((1205821205731120583120572
11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic
equation (71) has only negative real parts roots then thecoexistence equilibrium 119864
lowastis locally asymptotically stable
5 Persistence of the System
In this section we consider persistence of the system in119872
119868120
when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872
119868120
11987211986810
11987211986820
and 119872119904are positively invariant
Define
120597Γ = 11987211986810
cup11987211986820
cup119872119904
120597Γ = 119872119868120
(72)
Assuming the boundary equilibria of (1) are globally asymp-totically stable we have
119860120597119872119868120
= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878
0 1198900(sdot) 119868
1 1198682 119869))
= 1198640 119864
1 119864
2
(73)
By the above conclusions it follows that 119860120597119872119868120
is isolatedand has an acyclic covering119872 = 119864
0 119864
1 119864
2 Since the orbit
of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864
0 119864
1 119864
2are ejective in119872
119868120
if globally stable conditions of 1198640 119864
1 119864
2are not satisfied
Therefore we have the following lemmas
Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally
asymptotically stable If 1198771gt 1 then 119864
0is ejective in119872
119868120
for119880
11986812
(119905)119905ge0
Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy
1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 gt 1 (74)
Let 119909 = (1198780 0
119877 11986810 11986820 1198690) isin 119872
119868120
with 119909 minus 119909119878 lt 120576 Assume
that
1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)
Defining 119878(119905) = 119875119878119880(119905)119909 and 119868
1(119905) = 119875
1198681
119880(119905)119909 for all 119905 ge 0from (75) it follows that
119878 (119905) geΛ
120583minus 120576 forall119905 ge 0 (76)
From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain
119890 (119886 119905) ge
1205731(Λ
120583minus 120576) 119868
1(119905 minus 119886) 120587
1(119886) 119890
minus120583119886 119905 ge 119886
1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905 119905 lt 119886
(77)
where from the fourth equation of (1) we get
1198681015840
1(119905) ge 120573
1
Λ
120583int
119905
0
1198681(119905 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886
+ 1205731int
infin
119905
120574 (119886) 1198900(119886 minus 119905)
1205871(119886)
1205871(119886 minus 119905)
119890minus120583119905119889119886
minus (120583 + 1205721) 119868
1
(78)
Solving it we have
1198681(119905) ge 120573
1(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
119905minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(79)
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
There exist a 1198791gt 0 such that
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
119905+1198791
0
119890minus(120583+120572
1)119904
times int
119905+1198791minus119904
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
119905
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(80)
Thus for 119905 ge 120575 we have
1198681(119905 + 119879
1)
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904 forall119905 ge 0
(81)
By Assumption 1 there exists 1199051ge 0 such that 119868
1(119905+119879
1) ge
0 for all 119905 + 1198791ge 119905
1 Hence there exists 120585 gt 0 such that
1198681(119905 + 119879
1) ge 120585 for all 119905 + 119879
1isin [2119905
1 2119905
1+ 120575] Set
1199052= sup 119905 + 119879
1ge 2119905
1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905
1+ 120575 119905]
(82)
Assume that 1199052lt infin Then
1198681(1199052) ge 120573
1(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587
1(119886) 119890
minus120583119886119889119886 119889119904
ge 1205731(Λ
120583minus 120576)int
120575
0
119890minus(120583+120572
1)119904
times int
1198791
0
120574 (119886) 1205871(119886) 119890
minus120583119886119889119886 119889119904 120585
(83)
Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868
1(119905) it follows
that there exists an 1205762gt 0 such that 119868
1(119905) ge 120585 for all 119905 isin
[1199052 1199052+ 120576
2] which contradicts the definition of 119905
2 Therefore
1198681(119905) ge 120585 for all 119905 gt 2119905
1 Denote 119868
1infin= lim inf
119905rarr+infin1198681(119905) ge
120585 gt 0 Using (81) it follows that 1198681infinge 119868
1infin1205731(Λ120583 minus
120576) int120575
0119890minus(120583+120572
1)119904int1198791
0120574(119886)119868
1(119905 minus 119904 minus 119886)120587
1(119886)119890
minus120583119886119889119886 119889119904 which is
impossible
Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864
2be
globally asymptotically stable Then one has the following
(i) If 1198771gt 1 then 119909
1198681
= 1198641is ejective in119872
119868120
(ii) If 119877
2gt 1 then 119909
1198682
= 1198642is ejective in119872
119868120
Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864
1 and
1198642be globally asymptotically stable Assume 119877
1gt 1 119877
2gt 1
Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120
lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)
Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872
119868120
) = 1198640 cup 119864
1 cup 119864
2 Using Lemma 12 the result
follows
6 Simulation
In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864
0(the disease-free
equilibrium) 1198641(the TB dominated equilibrium) and 119864
2
(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section
Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=
049 1205732= 015 120584 = 09
120574 (119886) = 002 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(85)
1205721= 001 120572
2= 003 We have 119877
1= 007438 lt 1 and
1198772= 079365 lt 1 which satisfy the conditions of Theorems
2 and 3 1198640should be globally asymptotically stable (see
Figure 3(a)) In this case both TB andHIVwill be eliminated
Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573
2=
015 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(86)
1205721= 001 120572
2= 003 We have 119877
1= 3448770 gt 1 gt
1198772= 079365 which satisfy the conditions of Theorem 4 119864
1
should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics
Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049
1205732= 415 120584 = 09
120574 (119886) = 01 119886 ge 149
0 0 le 119886 le 149
1205720(119886) =
01 119886 ge 149
0 0 le 119886 le 149
(87)
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
35
3
25
2
15
1
05
0
minus05
Case
s
0 5 10 15 20 25 30 35 40 45 50
119878
1198681
1198682
119864
119869
Time 119905
(a)
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
119864
119869
07
06
05
04
03
02
01
0
Case
s
Time 119905
(b)
Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864
1is globally asymptotically stable
35
3
25
2
15
1
05
0
Case
s
0 5 10 15 20 25 30 35 40 45 50
1198781198681
1198682
1198682
119869
Time 119905
Figure 4 shows that 1198642is globally asymptotically stable
1205721= 001 120572
2= 003 We have 119877
2= 1137566 gt 1 gt
1198771= 033470 which satisfies the conditions ofTheorem 7119864
2
should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics
61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined
hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572
0= 0 (latent TB treatment) (IPT) 120572
0= 05
(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)
(IPT) no treatment 1205722= 0 120572
2= 05 (HAART) and 120572
2= 1
(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the
eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment
in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new
HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB
Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))
7 Discussion
We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Applied Mathematics
Different latent TB treatment
1205720 = 11205720 = 051205720 = 0
0 1 2 3 4 5 6 7 8 9 10
0807060504030201
0
119864
Time 119905
(a)
0275025
022502
0175015
012501
0075005
00250
1198681
Different latent treatment
1205720 = 11205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
09
030201
0807
0506
04
0
1198682
Different latent treatment
1205720 = 1
1205720 = 05
1205720 = 0
0 1 2 3 4 5 6 7 8 9 10Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205720 = 1 1205720 = 0
1205720 = 05
Different latent treatment
Time 119905
(d)
Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB
1090807060504030201
0
119864
0 1 2 3 4 5 6 7 8 9 10
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
Time 119905
(a)
Different HAART
1205722 = 0
1205722 = 1
1205722 = 05
04035
03025
02015
01005
0
1198681
0 1 2 3 4 5 6 7 8 9 10Time 119905
(b)
Different HAART1205722 = 0
1205722 = 1
1205722 = 05
09
030201
0807
0506
04
00 1 2 3 4 5 6 7 8 9 10
1198682
Time 119905
(c)
119869
0275025
022502
0175015
012501
0075005
00250
0 1 2 3 4 5 6 7 8 9 10
1205722 = 0
1205722 = 11205722 = 05
Different HAART
Time 119905
(d)
Figure 6 We take different HAART while we take no treatment of IPT
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 13
Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug
Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future
Acknowledgments
This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)
References
[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006
[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005
[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006
[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999
[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003
[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003
[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001
[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997
[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005
[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population
with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009
[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008
[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006
[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981
[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010
[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006
[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995
[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993
[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989
[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of