mathematical analysis of a tb transmission model with dots · mathematical analysis of a tb...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 17, Number 1, Spring 2009

MATHEMATICAL ANALYSIS OF A TB

TRANSMISSION MODEL WITH DOTS

S. O. ADEWALE, C. N. PODDER,AND A. B. GUMEL

ABSTRACT. The paper presents a deterministic model forthe transmission dynamics of Mycobacterium tuberculosis (TB)in a population in the presence of Directly Observed TherapyShort-course (DOTS). The model, which allows for the de-tection and treatment of individuals with symptoms and usesstandard incidence function for the infection rate, is rigorouslyanalysed to gain insight into its dynamical features. The anal-ysis reveals that the model undergoes a backward bifurcation,where a stable disease-free equilibrium (DFE) co-exists with astable endemic equilibrium when the associated reproductionthreshold (Rd) is less than unity. This phenomenon resulteddue to the exogenous re-infection property of TB disease. Itis shown that, in the absence of such re-infection, the modelhas a globally-asymptotically stable DFE when Rd is less thanunity. Further, the model has a unique endemic equilibrium,for a special case, whenever the associated threshold quan-tity exceeds unity. This endemic equilibrium is shown to beglobally-asymptotically stable, for a special case, using a non-linear Lyapunov function of Goh-Volterra type. The modelprovides a reasonable fit to a data set for the TB transmissiondata in Nigeria. Additional numerical simulations show thatexogenous re-infection increases the cumulative number of newTB cases regardless of whether or not the DOTS program is im-plemented (but the corresponding cumulative number of newcases decreases for the case with DOTS program, in comparisonto the case without DOTS).

1 Introduction Tuberculosis (TB), an airborne-transmitted bac-terial disease caused by Mycobacterium tuberculosis, remains one of themost important global public health challenges for decades. In additionto affecting at least one-third of the human population (2 billion peo-ple), TB is the second greatest contributor of adult mortality amongst

Keywords: tuberculosis, DOTS, equilibria, reproduction number, stability.Copyright c©Applied Mathematics Institute, University of Alberta.

1

2 S. O. ADEWALE, C. N. PODDER AND A. B. GUMEL

infectious diseases (causing at least 2 million deaths a year globally)[20, 34, 35, 40, 41]. Over 80% of all TB patients live in 22 coun-tries, mostly in sub-saharan Africa and Asia [20]. Owing to the risingTB mortality and infection rates (especially in developing countries),the World Health Organization (WHO) declared TB as a global publichealth emergency in 1993 [29, 30].

Over the years, a number of global initiatives, spear-headed by WHO,were embarked upon with the hope of minimizing the burden of TBworldwide (in particular, to achieve the Millennium Development Goalof halting and beginning to reverse the incidence of TB by 2015). Theseinclude the “Stop TB Partnership”, “International Standards of Tuber-culosis Care and Patient’s Care” and the “Global Plan to Stop TB”[20]. A notable medical contribution in TB control was the introduc-tion of antibiotics, which resulted in significant decrease in mortality (forinstance, a 70% reduction in TB-related mortality was recorded in theUSA between 1945 to 1955 [1, 13, 18, 28]). As a consequence of this de-velopment, TB-infected people can be effectively treated using multipledrugs via the Directly Observed Therapy Short-course (DOTS) strat-egy [5]. However, if not strictly complied to or administered wrongly,such therapy may lead to the evolution and development of multi-drugresistant TB (MDR-TB) [7, 9].

Numerous mathematical modelling studies have been carried out togain insight into the transmission dynamics and control of TB spreadin human population (see, for instance, [1, 2, 3, 8, 11, 12, 17, 19,21, 32, 33, 34, 35]). For instance, Feng et al. [17] presented a modelfor TB transmission dynamics with exogenous re-infection. A numberof studies, notably by Guo and Li [21] and McCluskey [11, 12], haveprovided rigorous global asymptotic stability results of the equilibria ofTB models with mass action incidence in the absence of exogenous re-infection (using Lyapunov function theory). Similarly, Okuonghae andKorobeinikov [31] also gave global stability results for a mass actionmodel of TB in the presence of exogenous re-infection. The purposeof the current study is to provide a rigorous mathematical analysis ofa model for TB spread, which uses standard incidence function for theinfection rate and allows for the exogenous re-infection of latent andrecovered cases, in the presence of DOTS. The model to be designedis an extension of some of the models presented in the aforementionedstudies.

The paper is organized as follows. The model is formulated in Sec-tion 2, and is qualitatively analysed in Section 3.

TB TRANSMISSION MODEL WITH DOTS 3

2 Model formulation The total homogeneously-mixing popula-tion at time t, denoted by N(t), is sub-divided into mutually-exclusivecompartments of susceptible (S(t)), exposed/latent (E(t)), undetectedinfectious (Tu(t)), detected infectious (Td(t)), recovered (R(t)) individ-uals, together with treated individuals who failed treatment (F (t)), sothat

N(t) = S(t) +E(t) + Tu(t) + Td(t) +R(t) + F (t).

The susceptible population is increased by the recruitment of people(either by birth or immigration) into the population (all recruited in-dividuals are assumed to be susceptible), at a rate Π. This populationis decreased by infection, which can be acquired following effective con-tact with infectious individuals in the undetected (Tu), detected (Td),or failed treatment (F ) categories, at a rate λ given by

(1) λ =β(Tu + ηdTd + ηfF )

N.

In (1), β represents the effective contact rate (i.e., contact capableof leading to TB infection), ηd is a modification parameter that com-pares the transmissibility of detected infectious individuals in relationto undetected infectious individuals. Since detected individuals are as-sumed to be taking therapeutic treatment, it is intuitive to assume that0 < ηd ≤ 1. Similarly, 0 < ηF ≤ 1 is a modification parameter thataccounts for the reduced transmissibility of infectious individuals in thefailed treatment class (F ), in comparison to those in the undetected in-fectious class (Tu). Finally, this population decreases by natural death(at a rate µ). Thus, the rate of change of the susceptible population isgiven by

(2)dS

dt= Π − λS − µS.

A fraction, ξ, of the newly-infected individuals are assumed to showno disease symptoms initially. These individuals (known as “slow pro-gressors”) are moved to the exposed class (E). The remaining fraction,1 − ξ, of the newly infected individuals are assumed to immediatelydisplay disease symptoms (“fast progressors”) and are moved to the un-detected infectious class (Tu). The population of exposed individuals isdecreased by the progression of exposed individuals to active TB (at arate κ) and exogenous re-infection (at a rate ψeλ, where ψe < 1 accountsfor the assumption that exposed individuals have reduced infection rate


in comparison to wholly-susceptible individuals (this is to account forthe fact that individuals with latent TB infection have partial immunityagainst exogenous reinfection [25, 37])). It is further reduced by naturaldeath (at the rate µ). Thus,

(3)dE

dt= ξλS − ψeλE − (κ+ µ)E.

The population of undetected infectious individuals is increased bythe infection of fast progressors (at the rate (1 − ξ)λ) and the develop-ment of symptoms by exposed individuals (at the rate (1− ω1)κ, whereω1 is the fraction of exposed individuals who develop symptoms and aredetected). It is further increased by the exogenous re-infection of ex-posed individuals (at the rate (1 − ω2)ψeλ, where ω2 is the fraction ofre-infected exposed individuals who are detected) recovered individuals(at the rate (1−ω3)ψrλ, where ω3 is the fraction of re-infected recoveredindividuals who are detected). This population is decreased by naturalrecovery (at a rate σu), screening and subsequent detection (at a rateγu), natural death (at the rate µ) and disease-induced death (at a rateδu). Hence,

(4)dTudt

= (1 − ξ)λS + (1 − ω1)κE + (1 − ω2)ψeλE

+ (1 − ω3)ψrλR− (σu + γu + µ+ δu)Tu.

The population of detected infectious individuals increases by thedetection of a fraction of exposed individuals who develop disease symp-toms (at the rate ω1κ), exogenous re-infection of exposed and recoveredindividuals (at the rates ω2ψeλ and ω3ψrλ, respectively) and the de-tection of undetected individuals (at the rate γu). The population isdecreased by natural recovery (at a rate σd), treatment failure (at a rateτ), natural death (at the rate µ) and disease-induced death (at a rateδd < δu). This gives

dTddt

= ω1κE + ω2ψeλE + ω3ψrλR + γuTu(5)

− (σd + τ + µ+ δd)Td.

Individuals in the F compartment are those in whom the DOTS treat-ment has failed. The treatment failure could be due to a number of rea-sons, such as incomplete compliance to the specified DOTS treatment


regimen, development of resistance etc. This population is generated bythe failure of treatment in detected infectious individuals (at the rateτ). In addition to natural death (at the rate µ) and disease-inducedmortality (at the rate δf ), individuals who recovered naturally can leavethis (F ) class and move to the recovered class (at a rate σf ). Thus,

(6)dF

dt= τTd − (σf + µ+ δf )F.

Recovery here means recovery from the illness (not from disease; sinceTB-infected individuals do not completely clear the bacteria from theirsystem. They, instead, undergo a long latency period which could lastmany years [26, 27] or even a lifetime). The population of recoveredindividuals is increased by the natural recovery of undetected and de-tected infectious individuals together with those who failed treatment(at the rates σu, σd and σf , respectively). This population is decreasedby exogenous re-infection (at the rate ψrλ) and natural death (at therate µ). Hence,

(7)dR

dt= σuTu + σdTd + σfF − ψrλR− µR.

Thus, in summary, the TB transmission model, in the presence ofDOTS treatment strategy, is given by the following system of non-lineardifferential equations (a flow diagram of the model is depicted in Fig-ure 1; associated variables and parameters are described in Tables 1and 2).

(8)

dS

dt= Π − λS − µS,

dE

dt= ξλS − ψeλE − (κ+ µ)E,

dTudt

= (1 − ξ)λS + (1 − ω1)κE + (1 − ω2)ψeλE

+ (1 − ω3)ψrλR − (γu + σu + µ+ δu)Tu,

dTddt

= ω1κE + ω2ψeλE + ω3ψrλR+ γuTu

− (σd + τ + µ+ δd)Td,

dF

dt= τTd − (σf + µ+ δf )F,

dR

dt= σuTu + σdTd + σfF − ψrλR− µR.


FIGURE 1: Schematic diagram of the model (8)

The essential features of the model (8) are that it:

(i) allows for disease transmission by individuals in the undetected(Tu), detected (Td) and failed treatment (F ) classes;

(ii) allows for the exogenous re-infection of exposed and recovered indi-viduals (at the rates ψeλ and ψrλ, respectively) and the endogenousre-activation of exposed individuals (at the rate κ);

(iii) allows for slow progression (at the rate ξλ) and fast progression (atthe rate (1 − ξ)λ) to active disease;

(iv) allows for the possibility of treatment failure (at the rate τ);(v) distributes the number of exposed individuals who develop symp-

toms (either due to re-activation or re-infection) into the unde-tected and detected infectious classes.

The model (8) extends the models in many of the aforementioned studies(such as those presented in [1, 2, 3, 7, 17, 21]) by including a separatecompartment (F ) for treated individuals who failed treatment. Further-more, it extends the studies in [11, 21], which were based on using mass


Parameter Desciption

Π Recruitment rate into the populationµ Per capita natural mortality rateβ Effective contact rate for TB infectionηd, ηf Modification parametersσu, σd, σf Recovery rate for individuals in Tu, Td and F classes,

respectivelyξ Fraction of newly-infected individuals who are

slow progressorsψe, ψr Exogenous re-infection rate for individuals in E

and R classes, respectivelyκ Endogenous re-activation rate for exposed individualsω1 Fraction of exposed individuals who are detectedω2 Fraction of re-infected exposed individuals who

develop symptoms and are detectedω3 Fraction of re-infected recovered individuals who are

detectedγu Detection rate for undetected infectious individualsτ Treatment failure rate for detected infectious individualsδu, δd, δf TB-induced mortality rate for individuals in Tu,

Td, and F classes, respectively

TABLE 1: Description of parameters of the model (8).

action incidence and without exogenous re-infection (it also extends thework of Okuonghae [31] by using standard incidence). Also, the modeloffers additional extensions to many of the earlier models by incorporat-ing the slow and fast progression aspect of TB disease (by splitting thenumber of new infected individuals into the latent (E) and undetected(Tu) classes) and also allowing for the screening and detection of infec-tious individuals (at the rate γu). The study further contributes to theliterature by carrying out a detailed rigorous analysis of the model (8).

Before carrying out any mathematical analysis of the model (8), it is,first of all, compared with real data (for TB transmission dynamics inNigeria). The results obtained shows a reasonably good fit (Figure 2),suggesting that the model (8) is suitable for use to realistically studyTB transmission dynamics in human populations.

3 Analysis of the model

Lemma 1. The closed set D ={

(S,E, Tu, Td, F,R) ∈ R6+ : N ≤ Π

µ

}

is

positively-invariant and attracting with respect to the model (8).


Parameter Nominal Value (per year) References

π 2000 (per 100000 population) [31]µ 0.02 [4, 31, 33]β 6 [10]ξ 0.7 [33]ψe, ψr 0.85, 0.85 Assumedκ 0.2522 [31]ω1 0.16 [31]ω2 0.7 [33]ω3 0.7 Assumedσu, σd, σf 0.3,0,3,0.3 Assumedγu 0.2 [38, 40]τ 0.9 [10]δu, δd 0.3, 0.1 [31]δf 0.3 Assumedηd, ηf 0.001, 0.8 Assumed

TABLE 2: Parameter values

2001 2002 2003 2004 2005 2006 20072.5

3

3.5

4

4.5

5

5.5

6x 104

Time (years)

Tot

al n

umbe

r of i

nfec

ted

indi

vidu

als

FIGURE 2: Comparison of observed TB data for Nigeria [38, 40] (solidline) and model prediction (dotted line). Parameter values used are asgiven in Table 2, with β = 1.8, ψe = 0.35 and ψr = 0.35.


Proof. Consider the biologically-feasible region, D, defined above.The rate of change of the total population, obtained by adding all theequations of the model (8), is given by

(9)dN

dt= Π − µN − δuTu − δdTd − δfF.

It follows that dNdt

< 0 whenever N > Πµ. Further, since dN

dt≤ Π − µN ,

it is clear that N(t) ≤ Πµ

if N(0) ≤ Πµ. Therefore, all solutions of the

model with initial conditions in D remain in D for all t > 0 (i.e., theω-limits sets of the system (8) are contained in D). Thus, D is positively-invariant and attracting.

In the region D, the model can be considered as being epidemiologi-cally and mathematically well-posed [22].

3.1 Disease-free equilibrium (DFE) The model (8) has a DFE,obtained by setting the right-hand sides of the equations of the modelto zero, given by

(10) E0 = (S∗, E∗, T ∗

u , T∗

d , F∗, R∗) =

(

Π

µ, 0, 0, 0, 0, 0

)

.

The stability of the DFE, E0, will be analysed using the next genera-tion method (see [39]). The non-negative matrix P (of the new infec-tion terms) and the nonsingular M-matrix V (of the remaining transferterms) are given, respectively, by

P =

0 ξβ ξβηd ξβηf 00 (1 − ξ)β (1 − ξ)βηd (1 − ξ)βηf 00 0 0 0 00 0 0 0 00 0 0 0 0

,

V =

K1 0 0 0 0−(1 − ω1)κ K2 0 0 0

−ω1κ −γu K3 0 00 0 −τ K4 00 −σu −σd −σf µ

,

where, K1 = κ+ µ, K2 = γu + σu + µ + δu, K3 = σd + τ + µ+ δd andK4 = σf +µ+ δf . The associated reproduction number, denoted by Rd,


is given by Rd = ρ(PV −1), where ρ denotes the spectral radius (dom-inant eigenvalue in magnititude) of the next generation matrix PV −1.It follows that

Rd =β

K1K2K3K4(A1 +A2 +A3),

with,

A1 = (1 − ξ)[K1K4(K3 + ηdγu) +K1ηf τγu],

A2 = ξκ(1 − ω1)[K4(K3 + ηdγu) + ηf τγu],

A3 = ξκω1K2(ηdK4 + ηf τ).

Hence, the result below follows from Theorem 2 of [39].

Lemma 2. The DFE of the model (8), given by (10), is locally asymp-totically stable (LAS) if Rd < 1, and unstable if Rd > 1.

The threshold quantity, Rd, is the reproduction number for the model.It measures the average number of new TB infections generated by asingle infectious individual in a population where some of the infectedindividuals are offered DOTS treatment. The epidemiological implica-tion of Lemma 2 is that TB spread can be effectively controlled in thecommunity (when Rd < 1) if the initial sizes of the sub-populations ofthe model are in the basin of attraction of the disease-free equilibriumE0.

Since TB models, with exogenous re-infection and standard incidencefunction, are often shown to exhibit the phenomenon of backward bi-furcation (see, for instance, [8, 17, 20, 33] and some of the referencestherein), where the stable DFE co-exists with a stable endemic equilib-rium when the associated reproduction threshold (Rd) is less than unity,it is instructive to determine whether or not the model (8) also exhibitsthis dynamical feature. This is investigated below.

Theorem 1. The model (8) undergoes a backward bifurcation at Rd = 1if the inequality (34) holds.

The proof of Theorem 1, which is based on the use of centre manifoldtheory [6, 8, 14, 20, 33, 39], is given in Appendix A. The backwardbifurcation phenomenon of the model (8) is numerically illustrated in


(A1) (A2)

(A3) (A4)

(A5) (A6)

FIGURE 3: Simulations of the model (31) showing backward bifur-cation diagrams for populations of (A1) susceptible individuals (S(t)),(A2) exposed individuals (E(t)), (A3) undetected active TB individuals(Tu(t)), (A4) detected active TB individuals (Td(t)), (A5) individualswho failed treatment (F (t)) and (A6) recovered individuals (R(t)). Pa-rameter values used are as given in Table 2, with β = 0.5, τ = 0.20619,γu = 0.4, ψe = 10 and ψr = 10 (so that, Rd = 0.5069, a = 3.0996 andb = 2589.7752).


Figure 3, by simulating the model (8) with parameter values such thatthe inequality (34) is satisfied.

It should be noted that, in the absence of re-infection (i.e., whenψe = ψr = 0), the backward bifurcation coefficient, a, given in (33),reduces to

a =−2βµ

π(w3 + ηdw4 + ηfw5)[v3(1 − ξ)

+ ξv2](w2 + w3 + w4 + w5 + w6) < 0,

since all the model parameters and the eigenvectors wi (i = 2, ..., 6)and vi (i = 1, ..., 6) are non-negative and 0 < ξ < 1. Thus, since theinequality (34) does not hold in this case, the model (8) will not undergobackward bifurcation in the absence of exogenous re-infection (this resultis consistent with earlier studies on TB transmission dynamics; see, forinstance, [33]). This result is summarized below.

Lemma 3. The model (8) does not undergo backward bifurcation atRd = 1 in the absence of exogenous re-infection (i.e., when ψe = ψr =0).

3.2 Global stability of the DFE the of model The result givenin Lemma 3 (discounting the possibility of backward bifurcation whenψe = ψr = 0) suggests that the DFE, E0, of the model (8), may beglobally-asymptotically stable (GAS) when Rd < 1 and ψe = ψr = 0.This is explored below.

Theorem 2. The DFE, E0, of the model (8) with ψe = ψr = 0, is GASin D if Rd ≤ 1.

Proof. Consider the Lyapunov function

F = f1E + f2Tu + f3Td + f4F,

where,

f1 = κ {(1 − ω1)[K4(K3 + ηdγu) + ηf τγu] + ω1K2(ηdK4 + ηf τ)} ,

f2 = K1[K4(K3 + ηdγu) + ηf τγu],

f3 = K1K2(ηdK4 + ηf τ),

f4 = K1K2K3K4ηf ,


with Lyapunov derivative given by (where a dot represents differentia-tion with respect to t)

F = f1E + f2Tu + f3Td + f4F ,

= f1

[

ξβ(Tu + ηdTd + ηfF )S

N−K1E

]

+ f3(ω1κE + γuTu −K3Td)

+ f2

[

(1 − ξ)β(Tu + ηdTd + ηfF )S

N+ (1 − ω1)κE −K2Tu

]

+ f4(τTd −K4F ),

= TuK1K2K3K4

(

S

NRd − 1

)

+ TdηdK1K2K3K4

(

S

NRd − 1

)

+ FηfK1K2K3K4

(

S

NRd − 1

)

,

≤ TuK1K2K3K4

(

Rd − 1

)

+ TdηdK1K2K3K4

(

Rd − 1

)

+ FηfK1K2K3K4

(

Rd − 1

)

,

since S ≤ N in D. Thus, F ≤ 0 if Rd ≤ 1 with F = 0 if and onlyif Tu = Td = F = 0. Also, E → 0 as t → ∞ if Tu = Td = F = 0

(since λ =β(Tu+ηdTd+ηfF )

N= 0 in this case). It follows, from LaSalle’s

Invariance Principle [24], that Tu → 0, Td → 0 and F → 0 as t → ∞.Further, substituting Tu = Td = F = 0 in the first and last equations in(8) shows that S → Π

µand R→ 0 as t→ ∞. Thus, (S,E, Tu, Td, F,R) →

(Πµ, 0, 0, 0, 0, 0) as t → ∞. Hence, since D is positively-invariant, it fol-

lows that the DFE, E0, of the model (8) is GAS in D if Rd ≤ 1.

The epidemiological implication of Theorem 2 is that, in the absenceof exogenous re-infection (i.e., ψe = ψr = 0), the disease will be elimi-nated from the community if the DOTS strategy can bring (and main-tain) the reproduction threshold (Rd) to a value less than or equal tounity. In other words, in the absence of exogenous re-infection, theclassic epidemiological requirement of having the reproduction thresh-old less than or equal to unity (Rd ≤ 1) is necessary and sufficient forTB elimination in a community. In the presence of re-infection, however,disease elimination when Rd ≤ 1 is dependent on the initial sizes of the


sub-populations of the model (this is a consequence of the phenomenonof backward bifurcation in this case).

Figure 4A depicts the solution profiles of the model (8) with ψe =ψr = 0 and Rd < 1, showing convergence to the DFE (in line withTheorem 2).

3.3 Existence of endemic equilibrium point (EEP): special caseIn this section, the possible existence and stability of endemic (positive)equilibria of the model (8) (i.e., equilibria where at least one of theinfected components of the model is non-zero) will be considered for thespecial case where exogenous re-infection does not occur (i.e., ψe = ψr =0).

Let E1 = (S∗∗, E∗∗, T ∗∗

u , T ∗∗

d , F ∗∗, R∗∗) represents any arbitrary en-demic equilibrium of the model (8) with ψe = ψr = 0. Solving theequations of the system at steady-state gives,

S∗∗ =Π

λ∗∗ + µ,

E∗∗ =ξλ∗∗S∗∗

K1,

T ∗∗

u =(1 − ξ)λ∗∗S∗∗ + (1 − ω1)κE

∗∗

K2,

T ∗∗

d =ω1κE

∗∗ + γuT∗∗

u

K3,

F ∗∗ =τT ∗∗

d

K4,

R∗∗ =σuT

∗∗

u + σdT∗∗

d + σfF∗∗

µ.

(11)

The expression for λ, defined in (1), at the endemic steady-state, denotedby λ∗∗, is given by

(12) λ∗∗ =β(T ∗∗

u + ηdT∗∗

d + ηfF∗∗)

N∗.

For mathematical convenience, the expressions in (11) are re-written


(A)

0 10 20 30 40 500

2000

4000

6000

8000

10000

12000

14000

Time (years)

Tota

l num

ber o

f inf

ecte

d in

divi

dual

s

(B)

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5x 104

Time (years)

Tota

l num

ber o

f inf

ecte

d in

divi

dual

s

FIGURE 4: Simulations of the model (8) showing the total number ofinfected individuals (E + Tu + Td + F ) as a function of time, using theparameters in Table 2 with Π = 4000, µ = 0.2, σu = 0.5, σd = 0.5,σf = 0.5 and γu = 2. (A) β = 1.5 and ψe = ψr = 0, (so that,Rd = 0.62). (B) β = 3, ψe = ψr = 0, δu = δd = δf = 0, ω1 = 0and ξ = 1 (so that, Rd2 = 1.4172).


in terms of λ∗∗S∗∗ as follows:

(13)

E∗∗ =ξλ∗∗S∗∗

K1,

T ∗∗

u =(1 − ξ)λ∗∗S∗∗

K2+

(1 − ω1)ξκλ∗∗S∗∗

K1K2= P1λ

∗∗S∗∗,

T ∗∗

d =ω1κξλ

∗∗S∗∗

K1K3+γuP1λ

∗∗S∗∗

K3= P2λ

∗∗S∗∗,

F ∗∗ =τ

K4

[

ω1κξλ∗∗S∗∗

K1K3+γuP1λ

∗∗S∗∗

K3

]

= P3λ∗∗S∗∗,

R∗∗ =σuP1λ

∗∗S∗∗ + σdP2λ∗∗S∗∗ + σfP3λ

∗∗S∗∗

µ

= P4λ∗∗S∗∗,

where,

(14)

P1 =(1 − ξ)

K2+

(1 − ω1)ξκ

K1K2,

P2 =ω1κξ

K1K3+γuK3

[

(1 − ξ)

K2+

(1 − ω1)ξκ

K1K2

]

,

P3 =τ

K4

{

ω1κξ

K1K3+γuK3

[

(1 − ξ)

K2+

(1 − ω1)ξκ

K1K2

]}

,

P4 =σuP1λ

∗∗S∗∗ + σdP2λ∗∗S∗∗ + σfP3

µ.

Substituting the expressions in (13), with (14), into (12) gives

(15) λ∗∗(

S∗∗ +ξλ∗∗S∗∗

K1+ P1λ

∗∗S∗∗ + P2λ∗∗S∗∗ + P3λ

∗∗S∗∗

+ P4λ∗∗S∗∗

)

= βλ∗∗S∗∗(P1 + ηdP2 + ηfP3).

Dividing each term in (15) by λ∗∗S∗∗ (and noting that, at the endemicsteady-state, λ∗∗S∗∗ 6= 0) gives

1 + P5λ∗∗ = β(P1 + ηdP2 + ηfP3),


where,

P5 =ξ

K1+ P1 + P2 + P3 + P4 > 0.

Thus,

1 + P5λ∗∗ =

β

K1K2K3K4(A1 + A2 +A3) = Rd.

Hence,

1 + P5λ∗∗ = Rd,

so that,

(16) λ∗∗ =Rd − 1

P5> 0, whenever Rd > 1.

The components of the unique endemic equilibrium (E1) can then beobtained by substituting the unique value of λ∗∗, given in (16), into theexpressions in (11). Thus, the following result has been established.

Lemma 4. The model (8) with ψe = ψr = 0 has a unique endemicequilibrium, given by E1, whenever Rd > 1.

3.4 Local stability of EEP: Special case The local stability of theunique EEP, E1, will now be explored for the special case where thedisease-induced mortality is negligible (i.e., δu = δd = δf = 0), no fastprogression to active disease (i.e., ξ = 1) and exogenous re-infection doesnot occur (so that, ψe = ψr = 0). Setting δu = δd = δf = ψe = ψr = 0in the model (8) shows that

(17)dN(t)

dt= Π − µN(t).

Hence, it follows from (17) that N(t) → Π/µ = N ∗ as t→ ∞. Further,using the substitution S = N∗ −E − Tu − Td − F −R (and noting thatδu = δd = δf = ψe = ψr = 0) and ξ = 1 in the model (8) gives the


following reduced model

(18)

dE

dt=β(Tu + ηdTd + ηfF )(N∗ −E − Tu − Td − F −R)

N∗

−K1E,

dTudt

= (1 − ω1)κE −K12Tu,

dTddt

= ω1κE + γuTu −K13Td,

dF

dt= τTd −K14F,

dR

dt= σuTu + σdTd + σfF − µR,

where, K12 = γu + σu + µ, K13 = σd + τ + µ and K14 = σf + µ. Forthe reduced model (18), the associated reproduction number, denotedby Rd1, is given by

Rd1 =β

K1K12K13K14(A11 +A12),

with,

A11 = κ(1 − ω1)[K14(K13 + ηdγu) + ηf τγu],

A12 = κω1K12(ηdK14 + ηf τ).

It can be shown, using the approach in Section 3.3, that the system (18)has a unique endemic equilibrium, given by E2 = E1|δu=δd=δf =ψe=ψr=0,ξ=1,whenever Rd1 > 1.

Theorem 3. The unique endemic equilibrium, E2, of the reduced model(18), is LAS whenever Rd1 > 1.

Proof. The proof is based on using the technique in [22] (see also[15, 16]), which employs a Krasnoselskii sub-linearity trick. The ap-proach essentially entails showing that the linearization of the system(18), around the equilibrium E2, has no solutions of the form

(19) Z(t) = Z0 eτt,


with Z0 ∈ Cn \{0} , τ ∈ C, Zi ∈ C and Re (τ) ≥ 0. The consequence of

this is that the eigenvalues of the characteristic polynomial associatedwith the linearized method will have negative real part (in which case,the unique endemic equilibrium, E2, is LAS).

Linearizing the model (18) around the endemic equilibrium, E2, gives

(20)

dE

dt= (−P11 −K1)E + (P12 − P11)Tu + (ηdP12 − P11)Td

+ (ηfP12 − P11)F − P11R,

dTudt

= (1 − ω1)κE −K12Tu,

dTddt

= ω1κE + γuTu −K13Td,

dF

dt= τTd −K14F,

dR

dt= σuTu + σdTd + σfF − µR,

where,

(21)

P11 =β

N∗(T ∗∗

u + ηdT∗∗

d + ηfF∗∗),

P12 =β

N∗S∗∗.

Substituting a solution of the form (19) into the linearized system of(18), around the equilibrium E2, gives the following linear system

(22)

τZ1 = (−P11 −K1)Z1 + (P12 − P11)Z2 + (ηdP12 − P11)Z3

+ (ηfP12 − P11)Z4 − P11Z5,

τZ2 = (1 − ω1)κZ1 −K12Z2,

τZ3 = ω1κZ1 + γuZ2 −K13Z3,

τZ4 = τZ3 −K14Z4,

τZ5 = σuZ2 + σdZ3 + σfZ4 − µZ5.

Firstly, all the negative terms in the last three equations of system (22)are moved to their respective left-hand sides. Each of the last three


equations is then re-written in terms of Z1 and Z2, and the resultssubstituted into the remaining equations of the system (22). Finally,all the negative terms of the remaining (first two) equations are movedto the right-hand sides. These algebraic manipulations result in thefollowing system:

(23)

Z1[1 + F1(τ)] + Z2[1 + F2(τ)] = (MZ)1 + (MZ)2,

Z3[1 + F3(τ)] = (MZ)3,

Z4[1 + F4(τ)] = (MZ)4,

Z5[1 + F5(τ)] = (MZ)5,

where,

(24)

F1(τ) =τ + P11

K1+P11T1

K1,

F2(τ) = 1 +τ

K12+

(1 + T2)P11

K1,

F3(τ) =τ

K13, F4(τ) =

τ

K14, F5(τ) =

τ

µ,

with,

M =

0 βS∗∗

N∗K1

ηdβS∗∗

N∗K1

ηfβS∗∗

N∗K10

(1−ω1)κK12

0 0 0 0

ω1κK13

γu

K13

0 0 0

0 0 τK14

0 0

0 σu

µσd

µ

σf

µ0

,

and,

T1 =ω1κ

τ +K13+

τω1κ

(τ +K13)(τ +K14)+

σdτ + µ

ω1κ

τ +K13

+σfτ + µ

τω1κ

(τ +K13)(τ +K14),

T2 =γu

τ +K13+

τγu(τ +K13)(τ +K14)

+σuτ + µ

+σd

τ + µ

γuτ +K13

+σfτ + µ

τγu(τ +K13)(τ +K14)

.


In the above calculations, the notation M(Z)i (with i = 1, 2, 3, 4, 5)denotes the ith coordinate of the vector M(Z). It should further benoted that the matrix M has non-negative entries, and the equilibriumE2 satisfies E2 = ME2. Furthermore, since the coordinates of E2 are allpositive, it follows then that if Z is a solution of (23), then it is possibleto find a minimal positive real number s such that

(25) ‖Z‖ ≤ sE2,

where, ‖Z‖ = (‖Z1‖, ‖Z2‖, ‖Z3‖, ‖Z4‖, ‖Z5‖) with the lexicographic or-der, and ‖ · ‖ is a norm in C.

The main goal is to show that Re (τ) < 0. Assume the contrary(i.e., Re (τ) ≥ 0). We then need to consider two cases: τ = 0 andτ 6= 0. Assume the first case (i.e., τ = 0). Then, (22) is a homogeneouslinear system in the variables Zi (i = 1, 2, 3, 4, 5). The determinant ofthe system (22) corresponds to that of the Jacobian of the system (18)evaluated at E2, which is given by

M = −T ∗∗

u + ηdT∗∗

d + ηfF∗∗

N∗B1

−K1K12K13K14µ

(

1 −S∗∗

N∗Rd1

)

,

(26)

where,

B1 = γuκ(1 − ω1)[σuK14 + τµ+ τσf +K14µ+K13K14(σu + µ)]

+ ψK12[ω1κ(τσf + σdK14 +K14µ+ τµ) +K13K14µ].

By solving (18) at the endemic steady-state E2, and using the first equa-tion of (18), it can be shown that

S∗∗

N∗=

1

Rd1.

Thus, M< 0. Consequently, the system (22) can only have the trivialsolution Z = 0 (which corresponds to the DFE, E0).

Consider next the case τ 6= 0. In this case, Re (Fi(τ)) ≥ 0 (i =1, 2, 3, 4, 5) since, by assumption, Re (τ) ≥ 0. It is easy to see that thisimplies |1 + Fi(τ)| > 1 for all i. Define F (τ) = min |1 + Fi(τ)| (for i =1, 2, 3, 4, 5), then F (τ) > 1, and hence s

F (τ) < s. The minimality of s

implies that ‖Z‖ > sF (τ)E2. On the other hand, taking norms of both


sides of the second equation of (22), and using the fact that the matrixM is non-negative, gives

(27) F (τ)‖Z3‖ ≤M(‖Z‖)3 ≤ s(M‖E2‖)3 ≤ sT ∗∗

u .

Then, it follows from the above inequality that ‖Z3‖ ≤ sF (τ)T

∗∗

u , which

contradicts Re (Fi(τ)) ≥ 0 . Hence, Re (τ) < 0, so that the endemicequilibrium, E2, is LAS if Rd1 > 1.

The epidemiological implication of Theorem 3 is that the disease willpersist in the community if the reproduction threshold (Rd1) exceedsunity.

3.5 Global stability of EEP: Special case The global asymptoticstability of EEP, E2, of the reduced model (18) is considered for the casewhen there is no detection of exposed individuals (i.e., ω1 = 0). Settingω1 = 0 into the reduced model (18), and using the substitution β1 = β

N∗,

it can be shown that the associated reproduction number of the reducedmodel (18) with ξ = 1 and ω1 = 0, denoted by Rd2, is given by

(28) Rd2 =β1κ(K13K14 + ηdγuK14 + ηf τγu)

K1K12K13K14.

Furthermore, using the approach in Section 3.3, it can be shown thatthe reduced system (18), with ω1 = 0, has a unique EEP, of the form

E3 = E2|ω1=0 = (S∗∗, E∗∗, T ∗∗

u , T ∗∗

d , F ∗∗, R∗∗),

where, S∗∗ > 0, E∗∗ > 0, T ∗∗

u > 0, T ∗∗

d > 0 and R∗∗ > 0, when-ever Rd2 > 1.

Let,

D0 = {(S,E, Tu, Td, F,R) ∈ D : E = Tu = Td = F = R = 0}.

Theorem 4. The unique EEP, E3, of the reduced model (18) withω1 = 0, is GAS in D \ D0 whenever Rd2 > 1.


Proof. Let ω1 = 0 and Rd2 > 1, so that the EEP, E3, exists. Considerthe following non-linear Lyapunov function:

L =

(

S − S∗∗ − S∗∗lnS

S∗∗

)

+

(

E −E∗∗ −E∗∗lnE

E∗∗

)

+β1S

∗∗[(σf + µ)(τ + σd + µ) + ηd(µ+ σf )γu + ηf τγu]

(σf + µ)(τ + σd + µ)(γu + σu + µ)

×

(

Tu − T ∗∗

u − T ∗∗

u lnTuT ∗∗

u

)

+β1S

∗∗(ηd(µ+ σf ) + ηf τ)

(σf + µ)(τ + σd + µ)

×

(

Td − T ∗∗

d − T ∗∗

d lnTdT ∗∗

d

)

+β1S

∗∗ηf(σf + µ)

(

F − F ∗∗ − F ∗∗lnF

F ∗∗

)

,

(29)

with Lyapunov derivative given by,

L =

(

1 −S∗∗

S

)

S +

(

1 −E∗∗

E

)

E

+β1S

∗∗[(σf + µ)(τ + µ) + ηd(µ+ σf )γu + ηf τγu]

(σf + µ)(τ + σd + µ)(γu + σu + µ)

×

(

1 −T ∗∗

u

Tu

)

Tu

+β1S

∗∗[ηd(µ+ σf ) + ηf τ ]

(σf + µ)(τ + σd + µ)

(

1 −T ∗∗

d

Td

)

Td +β1S

∗∗ηfµ+ σf

×

(

1 −F ∗∗

F

)

F ,

=

(

1 −S∗∗

S

)

[β1(T∗∗

u + ηdT∗∗

d + ηfF∗∗)S∗∗ + µS∗∗

− β1(Tu + ηdTd + ηfF )S − µS]

+

(

1 −E∗∗

E

)

[β1(Tu + ηdTd + ηfF )S − (κ+ µ)E]


+β1S

∗∗[(σf + µ)(τ + σd + µ) + ηd(µ+ σf )γu + ηf τγu]

(σf + µ)(τ + σd + µ)(γu + σu + µ)

×

(

1 −T ∗∗

u

Tu

)

[κE − (γu + σu + µ)Tu]

+β1S

∗∗[ηd(σf + µ) + ηf τ ]

(σf + µ)(τ + σd + µ)

×

(

1 −T ∗∗

d

Td

)

[γuTu − (τ + σd + µ)Td]

+β1S

∗∗ηfµ+ σf

(

1 −F ∗∗

F

)

[τTd − (σf + µ)F ],

= µS∗∗(2 −S

S∗∗−S∗∗

S) + 2β1(T

∗∗

u + ηdT∗∗

d + ηfF∗∗)S∗∗

− β1(T∗∗

u + ηdT∗∗

d + ηfF∗∗)

S∗∗2

S

+ β1(Tu + ηdTd + ηfF )S∗∗

− β1(T∗∗

u + ηdT∗∗

d + ηfF∗∗)

S∗∗E

E∗∗

− β1(Tu + ηdTd + ηfF )SE∗∗

E+ β1S

∗∗T ∗∗

u

E

E∗∗

− β1S∗∗Tu − β1S

∗∗E

E∗∗

T ∗∗

u2

Tu+ β1S

∗∗T ∗∗

u

+ β1S∗∗ηdT

∗∗

d

E

E∗∗− β1S

∗∗ηdT∗∗

d

TuT ∗∗

u

− β1S∗∗ηdT

∗∗

d

E

E∗∗

T ∗∗

u

Tu

+ β1S∗∗ηdT

∗∗

d + β1S∗∗ηfF

∗∗E

E∗∗− β1S

∗∗ηfF∗∗TuT ∗∗

u

− β1S∗∗ηfF

∗∗E

E∗∗

T ∗∗

u

Tu+ β1S

∗∗ηfF∗∗T ∗∗

u

Tu+ β1S

∗∗ηdT∗∗

d

TuT ∗∗

u

− β1S∗∗ηdTd − β1S

∗∗ηdTuT ∗∗

u

T ∗∗

d2

Td+ β1S

∗∗ηdT∗∗

d β1S∗∗ηf

TuT ∗∗

u

F ∗∗

− β1S∗∗ηf

TdT ∗∗

d

F ∗∗ − β1S∗∗ηf

TuT ∗∗

u

T ∗∗

d

TdF ∗∗ + β1S

∗∗ηfF∗∗

+ β1S∗∗ηf

TdT ∗∗

d

F ∗∗ − β1S∗∗ηfF − β1S

∗∗ηfTdT ∗∗

d

F ∗∗2

F+ β1S

∗∗ηfF∗∗


= µS∗∗

(

2 −S

S∗∗−S∗∗

S

)

+ β1S∗∗T ∗∗

u

(

3 −S∗∗

S−

E

E∗∗

T ∗∗

u

Tu

−S

S∗∗

E∗∗

E

TuT ∗∗

u

)

+ β1S∗∗ηdT

∗∗

d

(

4 −S∗∗

S−

E

E∗∗

T ∗∗

u

Tu−T ∗∗

d

Td

TuT ∗∗

u

−S

S∗∗

E∗∗

E

TdT ∗∗

d

)

+ β1S∗∗ηfF

∗∗

(

5 −S∗∗

S−

E

E∗∗

T ∗∗

u

Tu−T ∗∗

d

Td

TuT ∗∗

u

−TdT ∗∗

d

F ∗∗

F−

S

S∗∗

E∗∗

E

F

F ∗∗

)

.

Since the arithmetic mean exceeds the geometric mean, it follows thenthat

2 −S

S∗∗−S∗∗

S≤ 0,

3 −S∗∗

S−

E

E∗∗

T ∗∗

u

Tu−

S

S∗∗

E∗∗

E

TuT ∗∗

u

≤ 0,

4 −S∗∗

S−

E

E∗∗

T ∗∗

u

Tu−T ∗∗

d

Td

TuT ∗∗

u

−S

S∗∗

E∗∗

E

TdT ∗∗

d

≤ 0,

5 −S∗∗

S−

E

E∗∗

T ∗∗

u

Tu−T ∗∗

d

Td

TuT ∗∗

u

−TdT ∗∗

d

F ∗∗

F−

S

S∗∗

E∗∗

E

F

F ∗∗≤ 0,

so that L ≤ 0 for Rd2 > 1. Hence, L is a Lyapunov function of thesystem (18) with ω1 = 0, on D \ D0. In other words, lim

t→∞

(Tu, Td, F ) =

(T ∗∗

u , T ∗∗

d , F ∗∗). Setting Tu = T ∗∗

u , Td = T ∗∗

d and F = F ∗∗ in theequation for dR/dt in (18) shows that R → R∗∗ as t → ∞. Thus,by the Lyapunov function L and LaSalle’s Invariance Principle [24],every solution to the equations in the reduced model (18), with ω1 = 0,approaches E3 as t→ ∞ for Rd2 > 1.

Figure 4B depicts the solution profiles of the reduced model (18)converging to the EEP when Rd2 > 1 (in line with Theorem 4).

The effect of exogenous re-infection on TB transmission dynamics ismonitored by simulating the model (8) with the parameters in Table 2 inthe presence and absence of re-infection. The results obtained, depictedin Figure 5, show that re-infection increases the cumulative number ofnew cases regardless of whether or not DOTS is implemented. It should


be mentioned that the cumulative number of new TB cases is decreasedif DOTS program is implemented (see Figure 5A, in comparison to theresults depicted in Figure 5B).

(A)

0 5 10 15 200

2

4

6

8

10

12

14x 104

Time (years)

Cum

ulat

ive

num

ber o

f new

infe

ctio

ns with reinfection and DOTS

without reinfection and DOTS

(B)

0 5 10 15 200

2

4

6

8

10

12

14

16

18x 104

Time (years)

Cum

ulat

ive

num

ber o

f new

infe

ctio

ns

without reinfection and no DOTS

with reinfection and no DOTS

FIGURE 5: Simulations of the model (8) showing the cumulative num-ber of new TB infections as a function of time, using the parameters inTable 2. (A) in the presence and absence of re-infection with DOTS and(B) in the presence and absence of re-infection without DOTS.


Figure 6 shows a decrease in the cumulative number of new TB casesif the undetected TB-infected individuals (in the Tu class) are detectedwithin 6 months (i.e., γu = 2), the cumulative number of new caseswill be about 113,000 within 20 years. However, if the detection rate isfurther decreased to a month (i.e., γu = 12), the cumulative number ofnew cases decreases to about 85,000 (Figure 6).

0 5 10 15 200

2

4

6

8

10

12

14

16

18x 104

Time (years)

Cum

ulat

ive

num

ber o

f new

infe

ctio

ns

γu = 0, 2, 4, 6 and 12 respectively

(from top to bottom)

FIGURE 6: Simulations of the model (8) showing the cumulative num-ber of new TB infections as a function of time for various values of γu.

Other parameter values used are as in Table 2.

Conclusions A deterministic model for TB transmission dynamics,in the presence of DOTS, is presented and rigorously analysed. Some ofthe main findings of the study include:

(i) The model reasonably fits data for TB transmission in Nigeria;(ii) The model undergoes the phenomenon of backward bifurcation,

which is shown to arise due to the exogenous re-infection propertyof TB disease;

(iii) In the absence of re-infection, the model has a globally-asymptoti-cally stable disease-free equilibrium whenever the associated repro-duction threshold (Rd) is less than unity;


(iv) In the absence of re-infection and disease-induced mortality, themodel has a unique endemic equilibrium whenever the associatedreproduction threshold (Rd1) exceeds unity. This equilibrium isshown to be locally-asymptotically stable whenever there is no fastprogression to active disease (i.e., ξ = 1) in the community. Fur-thermore, if, additionally, there is no detection of exposed individ-uals (ω1 = 0), this endemic equilibrium is globally-asymptoticallystable whenever the associated reproduction threshold (Rd2) isgreater than unity;

(v) Exogenous re-infection increases the cumulative number of new TBcases regardless of whether or not the DOTS program is imple-mented (but the number of new cases decreases for the case whenDOTS program is implemented);

(vi) The cumulative number of new TB cases decreases with increasingdetection rate.

Appendix A: Proof of Theorem 1 To prove Theorem 1, it isconvenient to let S = x1, E = x2, Tu = x3, Td = x4, F = x5 and R = x6,so thatN = x1+x2+x3+x4+x5+x6. Further, by introducing the vectornotation x = (x1, x2, x3, x4, x5, x6)

T , the model (8) can be written in theform dx/dt = F (x), where F = (f1, f2, f3, f4, f5, f6)

T , as follows:

(30)

dx1

dt= f1 = Π −

β(x3 + x4ηd + x5ηf )

(x1 + x2 + x3 + x4 + x5 + x6)x1 − µx1,

dx2

dt= f2 =

ξβ(x3 + x4ηd + x5ηf )

(x1 + x2 + x3 + x4 + x5 + x6)x1

− ψeβ(x3 + x4ηd + x5ηf )

(x1 + x2 + x3 + x4 + x5 + x6)x2 − (κ+ µ)x2,

dx3

dt= f3 =

(1 − ξ)β(x3 + x4ηd + x5ηf )

(x1 + x2 + x3 + x4 + x5 + x6)x1 + (1 − ω1)κx2

+(1 − ω2)ψeβ(x3 + x4ηd + x5ηf )

x1 + x2 + x3 + x4 + x5 + x6x2

+(1 − ω3)ψrβ(x3 + x4ηd + x5ηf )

x1 + x2 + x3 + x4 + x5 + x6x6

− (σu + γu + µ+ δu)x3,


(30)

dx4

dt= f4 = ω1κx2 +

ω2ψeβ(x3 + x4ηd + x5ηf )

x1 + x2 + x3 + x4 + x5 + x6x2

+ω3ψrβ(x3 + x4ηd + x5ηf )

x1 + x2 + x3 + x4 + x5 + x6x6 + γux3

− (σd + τ + µ+ δd)x4,

dx5

dt= f5 = τx4 − (σf + µ+ δf )x5,

dx6

dt= f6 = σux3 + σdx4 + σfx5

+ψrβ(x3 + x4ηd + x5ηf )

x1 + x2 + x3 + x4 + x5 + x6x6 − µx6.

The Jacobian of the system (30), or equivalently (8), at the DFE (E0)is given by

J(E0) =

−µ 0 −β −βηd −βηf 00 −K1 ξβ ξβηd ξβηf 00 (1 − ω1)κ (1 − ξ)β −K2 (1 − ξ)βηd (1 − ξ)βηf 00 ω1κ γu −K3 0 00 0 0 τ −K4 00 0 σu σd σf −µ

,

from which it can be shown (as before) that

(31) Rd = ρ(PV −1) =β

K1K2K3K4(A1 +A2 +A3),

where, Ai (i = 1, 2, 3) andKi (i = 1, 2, 3, 4) are as defined in Section 3.1.The above linearized system, of the transformed system (30) with β =β∗, has a zero eigenvalue which is simple. Hence, the Center Manifoldtheory [6] can be used to analyze the dynamics of (30) near β = β∗. Inparticular, Theorem 4.1 in [8], reproduced below for convenience, willbe used.

Theorem 5 (Castillo-Chavez and Song [8]). Consider the following gen-eral system of ordinary differential equations with a parameter φ,dx/dt = f(x, φ), f : R

n × R → Rnandf ∈ C

2(Rn × R), where 0 isan equilibrium point of the system (that is, f(0, φ) ≡ 0 for all φ) and


1. A = Dxf(0, 0) = ( ∂fi

∂xj(0, 0)) is the linearization matrix of the system

around the equilibrium 0 with φ evaluated at 0;2. Zero is a simple eigenvalue of A and all other eigenvalues of A have

negative real parts;3. Matrix A has a right eigenvector w and a left eigenvector v corre-

sponding to the zero eigenvalue.

Let fk be the kth component of f and

(32)

a =n∑

k,i,j=1

vkwiwj∂2fk∂xi∂xj

(0, 0),

b =

n∑

k,i=1

vkwi∂2fk∂xi∂φ

(0, 0).

Then the local dynamics of the system around the equilibrium point 0 istotally determined by the signs of a and b. Particularly, if a > 0 andb > 0, then a backward bifurcation occurs at φ = 0.

Theorem 5 is applied as follows. First of all, suppose β is chosen asa bifurcation parameter. Setting Rd = 1, from (31), and solving for βgives:

β = β∗ =K1K2K3K4

A1 +A2 +A3.

Secondly, the following computations are carried out.

Eigenvectors of J(E0)|β=β∗

It can be shown that the Jacobian of the system (30) at β = β∗

(denoted by J(E0)|β=β∗ = Jβ∗) has a right eigenvector (correspondingto the zero eigenvalue) given by

w = (w1, w2, w3, w4, w5, w6)T ,

where

w1 =−(βw3 + βηdw4 + βηfw5)

µ,

w2 = free,

w3 = free,

w4 =ω1κw2 + γuw3

K3,


w5 =τw4

K4,

w6 =σuw3 + σdw4 + σfw5

µ.

Further, the Jacobian Jβ∗ has a left eigenvector (associated with thezero eigenvalue) given by

v = (v1, v2, v3, v4, v5, v6),

with

v1 = 0,

v2 =(1 − ω1)κv3 + ω1κv4

K1,

v3 = free,

v4 = free,

v5 =ξβηfv2 + (1 − ξ)βηf v3

K4,

v6 = 0.

Computations of a and bFor the system (30), the associated non-zero second partial derivatives

of F (at the DFE (E0)) are given by

∂2f2

∂x2∂x3

=−βµ(ξ + ψe)

Π,

∂2f2

∂x2∂x4

=−βηdµ(ξ + ψe)

Π,

∂2f2

∂x2∂x5

=−βηfµ(ξ + ψe)

Π,

∂2f2

∂x3∂x3

=−2ξβµ

Π,

∂2f2

∂x3∂x4

=−ξβµ(1 + ηd)

Π,

∂2f2

∂x3∂x5

=−ξβµ(1 + ηf )

Π,

∂2f2

∂x3∂x6

=−ξβµ

Π,

∂2f2

∂x4∂x2

=−βηdµ(ξ + ψe)

Π,

∂2f2

∂x4∂x3

=−ξβµ(1 + ηd)

Π,

∂2f2

∂x4∂x4

=−2ξβµηd

Π,


∂2f2

∂x4∂x5

=−ξβµ(ηd + ηf )

Π,

∂2f2

∂x4∂x6

=−ξβµηd

Π,

∂2f2

∂x5∂x2

=−βηfµ(ξ + ψe)

Π,

∂2f2

∂x5∂x3

=−ξβµ(ηf + 1)

Π,

∂2f2

∂x5∂x4

=−ξβµ(ηd + ηf )

Π,

∂2f2

∂x5∂x5

=−2ξβµηf

Π,

∂2f2

∂x5∂x6

=−ξβµηf

Π,

∂2f3

∂x2∂x3

=−βµ(1 − ξ − ψe + ψeω2)

Π,

∂2f3

∂x2∂x4

=−βηdµ(1 − ξ − ψe + ψeω2)

Π,∂2f3

∂x2∂x5

=−βηfµ(1 − ξ − ψe + ψeω2)

Π,

∂2f3

∂x3∂x3

=−2βµ(1 − ξ)

Π,

∂2f3

∂x3∂x4

=−βµ(1 − ξ)(1 + ηd)

Π,

∂2f3

∂x3∂x5

=−βµ(1 − ξ)(1 + ηf )

Π,

∂2f3

∂x3∂x6

=−βµ(1 − ξ − ψr + ψrω3)

Π,

∂2f3

∂x4∂x4

=−2βηdµ(1 − ξ)

Π,

∂2f3

∂x4∂x5

=−βµ(1 − ξ)(ηd + ηf )

Π,

∂2f3

∂x4∂x6

=−βηdµ(1 − ξ − ψr + ψrω3)

Π,∂2f3

∂x5∂x5

=−2βηfµ(1 − ξ)

Π,

∂2f3

∂x5∂x6

=−βηfµ(1 − ξ − ψr + ψrω3)

Π,∂2f4

∂x2∂x3

=ω2ψeβµ

π,

∂2f4

∂x2∂x4

=ω2ψeβηdµ

π,

∂2f4

∂x2∂x5

=ω2ψeβηfµ

π,

∂2f4

∂x3∂x6

=ω3ψrβµ

π,

∂2f4

∂x4∂x6

=ω3ψrβηdµ

π,

∂2f4

∂x5∂x6

=ω3ψrβηfµ

π,

∂2f6

∂x3∂x6

=−ψrβµ

Π,

∂2f6

∂x4∂x6

=−ψrβηdµ

Π,

∂2f6

∂x5∂x6

=−ψrβηfµ

Π,

Using the above expressions, for the partial derivatives in the equationfor a in (32) gives:

a =6∑

k,i,j=1

vkwiwj∂2fk∂xi∂xj

,(33)

=2βµ

π(w3 + ηdw4 + ηfw5)[(ξv3 + v1 − ξv2 − v3)


× (w2 + w3 + w4 + w5 + w6)

+ (w2ψeω2 + w6ψrω3)(v4 − v3)

+ w2ψe(v3 − v2) + w6ψr(v3 − v6)],

from which it can be shown that a > 0 if

(34) B1 > B2,

where,

B1 = (w3 + ηdw4 + ηfw5)[(w2ψeω2 + w6ψrω3)v4 + w2ψev3 + w6ψrv3],

B2 = (w3 + ηdw4 + ηfw5){[v3(1 − ξ) + ξv2](w2 + w3 + w4 + w5 + w6)

+ v3(w2ψeω2 + w6ψrω3) + w2ψev2}.

For the sign of b, it can be shown that the associated nonvanishingderivatives of F are

∂2f2∂x3∂β∗

= ξ,∂2f2

∂x4∂β∗= ξηd,

∂2f2∂x5∂β∗

= ξηf ,

∂2f3∂x3∂β∗

= (1 − ξ),∂2f3

∂x4∂β∗= ηd(1 − ξ),

∂2f3∂x5∂β∗

= ηf (1 − ξ),

so that,

b =

6∑

k,i=1

vkwi∂2fk∂xi∂β∗

= [v3(1 − ξ) + v2ξ](w1 + w2 + w3 + w4 + w5 + w6)

× (w3 + ηdw4 + ηfw5) > 0.

Thus, the transformed system (30), or equivalently, system (8) undergoesbackward bifurcation at Rd = 1, as required.

Acknowledgements One of the authors (ABG) acknowledges, withthanks, the support in part of the Natural Sciences and Engineering Re-search Council (NSERC) and Mathematics of Information Technologyand Complex Systems (MITACS) of Canada. SOA acknowledges thesupport, in part, of the Department of Mathematics, University of Man-itoba, during his research visit in 2009. CNP acknowledges the supportof the Manitoba Health Research Council. The authors are grateful tothe referees and the handling editors for their constructive comments.


REFERENCES

1. J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Long term dynamicsand reemergence of tuberculosis. In: Mathematical Approaches for Emergingand Reemerging Infectious Disease: An Introduction (C. Castillo-Chavez withS.M. Blower, P. Van den Driessche, D. Kirschner, A.A. Yakubu, eds.), IMA125 (2002), Springer-Verlag, New York, 351–360.

2. J. P. Aparicio, A. F. Cappurro and C. Castillo-Chavez, Transmission anddynamics of tuberculosis on generalized households, J. Theor. Biol. 206 (2000),327–341.

3. J. P. Aparicio and J. C. Hernandez, Preventive treatment of tuberculosis throughcontact tracing, in: Mathematical Studies on Human Disease Dynamics: Emerg-ing Paradigms and Challenges, (Abba B. Gumel, Carlos Castillo-Chavez, RonaldE. Mickens , Dominic P. Clemence, eds.) AMS Contemporary Math. Ser. 410

(2006).4. M. C. Becerra, I. F. Pachao-Torreblanca, J. Bayona, R. Celi, S. S. Shin, J. Y.

Kim, P. E. Farmer and M. Murray, Expanding tuberculosis case detection byscreening household contacts, Public Health Report. 120 (2005).

5. W. R. Bishai, N. M. Graham, S. Harrington, D. S. Pope, N. Hooper, J. Astem-borski, L. Sheely, D. Vlahov, G. E. Glass and R. E. Chaisson, Molecular andgeographic patterns of tuberculosis transmission after 15 years of directly ob-served therapy J. Amer. Med. Assoc. 208 (1998), 1679–1684.

6. J. Carr, Application of Centre Manifold Theory, Springer-Verlag, New York,1981.

7. C. Castillo-Chavez and Z. Feng, To treat or not to treat: the case of tubercu-losis, J. Math. Biol. 35 (1997), 629–656.

8. C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and theirapplications, Math. Biosci. Eng. 2(1) (2004), 361–404.

9. T. Cohen and M. Murray, Modeling epidemics of multi drug-resistant M. tu-berculosis of heterogeneous fitness, Nature Med. 10 (2004), 1117–1121.

10. C. Colijn, T. Cohen and M. Murray, Emergent heterogeneity in declining tu-berculosis epidemics, J. Theor. Biol. 247 (2007), 765–774.

11. C. C. McCluskey, Lyapunov functions for tuberculosis models with fast andslow progression, Math. Biosci. Eng. 3(4) (2006), 603–614.

12. C. C. McCluskey, Global stability for a class of mass action systems allowingfor latentecy in tuberculosis, J. Math. Anal. Appl. 338(1) (2008), 518–535.

13. R. Dubos and J. Dubos (1952). The White Plague: Tuberculosis, Man andSociety. Little and Brown, Boston.

14. J. Dushoff, W. Huang and C. Castillo-Chavez, Backward bifurcations andcatastrophe in simple models of fatal diseases, J. Math. Biol. 36 (1998), 227–248.

15. L. Esteva, A. B. Gumel and C. Vargas de Leon, Qualitative study of transmis-sion dynamics of drug-resistant malaria, Math. Comput. Modelling 50(3,4)(2009), 611–630.

16. L. Esteva and C. Vargas de Leon, Influence of vertical and mechanical trans-mission on the dynamics of dengue disease, Math. Biosci. 167 (2000), 51–64.

17. Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for the tuberculosiswith exogeneous reinfection, Theor. Pop. Biol. 57 (2000), 235–247.

18. Fort-forth World Health Assembly, Resolution WHA 44.8, World Health Or-ganization, 1991.


19. M. G. M. Gomes, A. O. Franco, M. C. Gomes and G. F. Medley, The rein-fection threshold promotes variability in tuberculosis epidemiology and vaccineefficacy, Proc. R. Soc. 271 (2004), 617–623.

20. A. B. Gumel and B. Song, Existence of multiple-stable equilibria for a multi-drug-resistant model of mycobacterium tuberculosis, Math. Biosci. Eng. 5(3)(2008), 437–455.

21. H. Guo and M. Y. Li, Global stability in a mathematical model of tuberculosis,Canad. Appl. Math. Q. 14(2) (2006), 185–197.

22. H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium inepidemic models with subpopulations, Math. Biosci. 75 (1985), 205–227.

23. V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Non-linear Systems, Marcel Dekker, Inc., New York and Basel, 1989.

24. J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Ser.Appl. Math. SIAM, Philadelphia, 1976.

25. M. Lipsitch and M. B. Murray, Multiple equilibria: Tuberculosis transmissionrequire unrealistic assumptions, Theoret. Population Biol. 63 (2003), 169–170.

26. G. Lay, Y. Poquet, P. Salek-Peyron, M. P. Puissegur, C. Botanch, H. Bon,F. Levillain, J. L. Duteyrat, J. F. Emile and F. Altare, Langhans giant cellsfrom M. tuberculosis-induced human granulomas cannot mediate mycobaterialuptake, J. Pathology 211(1) (2006), 76–85.

27. A. M. J. A. Lenaerts, S. E. Chase, A. J. Chmielewski and M. H. Cynamon,Evaluation of rifapentine in long-term treatment regimens for tuberculosis inmice, Antimicrobial Agents Chemotherapy 43(10) (1999), 2356–2360.

28. A. Lowell et al. Tuberculosis, Havard University Press, Cambridge, MA, 1996.29. G. Magombedze, W. Garira, and E. Mwenje, Modelling the human immune

response mechanisms to mycobacterium tuberculosis infection in the lungs,Math. Biosci. Eng. 3(4) (2006), 661–682.

30. C. J. Murray and J. A. Salomon, Expanding the WHO tuberculosis controlstrategy: rethinking the role of active case-finding, Int. J. Tuberc. Lung. Dis.2(9) (1998), S9–15.

31. D. Okuonghae and A. Korobeinikov, Dynamics of tuberculosis: The effect ofDirect Observation Therapy Strategy (DOTS) in Nigeria, Math. ModellingNatural Phenomena 2(1) (2007), 101–113.

32. P. Rodrigues, M. G. Gomes and C. Rebelo, Drug resistance in tuberculosis—areinfection model, Theor. Pop. Biol. 71 (2007), 196–212.

33. O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analy-sis of the transmission dynamics of HIV/TB co infection in the presence oftreatment, Math. Biosci. Eng. 5(1) (2008), 145–174.

34. B. H. Singer and D. E. Kirschner, Influence of backward bifurcation on in-terpretation of R0 in a model of epidemic tuberculosis with reinfection, MathBiosci. Eng. 1(1) (2004), 81–93.

35. D. Snider Jr., M. Raviglione and A. Kochi, Global Burden of Tuberculosis, inTuberculosis: Pathogenesis, Protection and Control (B. Bloom, ed.), Washing-ton D.C., ASM Press, 1994.

36. A. Ssematimba, J. Y. T. Mugisha and L. S. Luboobi, Mathematical modelsfor the dynamics of tuberculosis in density-dependent populations: The caseof internally displaced peoples’ camps (IDPCs) in Uganda, J. Math. Stat. 1(3)(2005), 217–224.

37. P. G. Smith and A. R. Moss, Epidemiology of Tuberculosis, in: Tuberculosis:Pathogenis, Protection, and Control (B. R. Bloom, ed), ASM Press, Washing-ton, 1994, 47–59.

38. USAID, Infectious Disease, USAID Report, 2008. (http://www.usaid. gov/our work/global health/id/tuberculosis /countries/africa/nigeria profile. html).


39. P. van den Driessche and J. Watmough, Reproduction numbers and sub-thresh-old endemic equilibria for compartmental models of disease transmission, Math.Biosci. 180 (2002), 29–48.

40. WHO, Global Tuberculosis Control: Surveillance, Planning, Financing, WHOReport/WHO/HTM/TB/2006. 362, 2006. (http://www.who.int/tb/publications/global report/2006/en/).

41. WHO, WHO Anti-tuberculosis Drug Resistance in the World, WHO Report,2008. (http://catalogue.nla.gov.au/Record/3454918).

Corresponding author: A. B. GumelDepartment of Mathematics, University of Manitoba,Winnipeg, Manitoba, R3T 2N2, CanadaE-mail address: [email protected]

Department of Mathematics, University of Manitoba,Winnipeg, Manitoba, R3T 2N2, Canada

Permanent address: (S. O. Adewale)Department of Pure and Applied Mathematics,Ladoke Akintola University of Technology, P.M.B 4000,LAUTECH Ogbomoso, Nigeria

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