Research Journal of Mathematics and Statistics 4(1): 14-20, 2012
ISSN:2040-7505
© Maxwell Scientific Organization, 2012
Submitted: December 09, 2011
Accepted: February 15, 2012
Published: February 25, 2012
Global Stability Results for a Tuberculosis Epidemic Model
1
S.A. Egbetade and 2M.O. Ibrahim
Department of Mathematics and Statistics, The Polytechnic, Ibadan, Nigeria
2
Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria
1
Abstract: In this study, we analyse models of TB dynamics in the literature and present a model of our own.
We conduct global stability analysis of equilibrium states of the model. Our results show that the basic
reproduction number R0 is a threshold parameter of the disease dynamics. In particular, either all positive
solutions approach the disease-free equilibrium (R0 #1) or a unique endemic equilibrium (R0 >1) . The DiseaseFree Equilibrium (DFE) is shown to be Globally Asymptotically Stable (GAS) if R0 #1 and we investigate the
endemic global stability using Lyapunov functions and Volterra-Lyapunov matrix properties.
Keywords: Basic reproduction number, epidemic, equilibrium, global stability, infectious disease, tuberculosis
model, uniform persistence
INTRODUCTION
Salpeter, 1997; Cohen and Murray, 2004; Gomes et al.,
2004). In many such models, there is sharp threshold
behaviour and the asymptotic dynamics are determined by
a parameter R0 the average number of new infections
caused by an infectious individual introduced into a fully
susceptible population (Diekmann and Heesterbeck,
2000). In the next section, we highlight some
mathematical models of TB epidemics and their
contributions to the spread and control of the disease.
Tuberculosis is an airborne disease produced by
Mycobacterium tuberculosis (Brennan and Nikaido,
1995). It is primarily transmitted through the lungs. The
main TB disease is the pulmonary tuberculosis and is
responsible for the largest health burdens (CastilloChavez and Feng, 1997). Individuals with active disease
may infect others if the airborne particles they product
when they cough, talk or sing are inhaled by others.
Untreated individuals suffer severely from loss of energy,
poor appetite, fever, loss of weight, night sweats and chest
pain (Colijn et al., 2006). Once infected, an individual
enters a period of latency which can be of extremely
variable length and the great majority of those infected
(-90%) will never have clinical tuberculosis (World
Health Organization, 2009). Others progress to disease
relatively rapidly falling ill within months or several years
after infection.
Despite many decades of study, the widespread
availability of a vaccine and more recently a highly
visable WHO efforts to promote a unified global control
strategy, TB remains a leading cause of infectious
mortality (World Health Organization, 2010). Recent data
indicate that the overall global incidence of TB is rising as
a result of resurgence of the disease in Africa, parts of
Eastern Europe and Asia (Dye, 2006). In these regions,
the emergence of drug-resistant TB and the convergence
of HIV and TB epidemics have resulted in high TB
incidence and created substantial new challenges for
disease control (Porco et al., 2001).
Many mathematical models have already been
proposed for the disease dynamics with a view to assisting
in the formulation of TB control strategies and the
establishment of interim goals for prevention and
intervention programs (Blower et al., 1995; Salpeter and
METHODOLOGY
Mathematical models of tuberculosis epidemics: It is
not unknown that mathematical modeling has had an
important role in the understanding of TB transmission
dynamics. Mathematical models of TB have played a key
role in the formulation of tuberculosis prevention and
control strategies and the establishment of interim goals
for intervention programs. These models are based on the
underlying transmission mechanism of TB to help public
health administrative workers and other stakeholders in
TB control understand better how the disease is spread.
During the last years, many TB models have been
developed with the inclusion of explicit elements of
demography, biology and behavior (Bailey, 1975). These
elements can affect the future course of the epidemic and
the effects will be highly nonlinear functions of the
parameter values (Ma and Xia, 2008).
At this juncture, we review some past and recent
works on tuberculosis. For clarity, we group models into
two categories: ordinary differential equation models
(SEIR models) and age-structured and delayed models.
Ordinary differential equation models: The first
deterministic mathematical model describing the
dynamics of TB was proposed by Waaler (1968a). He
Corresponding Author: S.A. Egbetade, Department of Mathematics and Statistics, The Polytechnic, Ibadan, Nigeria
14
Res. J. Math. Stat., 4(1):14-20, 2012
progressors to active TB, detection and treatment rates of
active TB and rate at which susceptible individuals
recover. We show global asymptotic stability of the DFE
when R0 # 1 Also, we investigate the endemic global
stability using Lyapunov functions and the results of
Volterra-Lyapunov stable matrices Redheffer
(1985a,1985b).
continued his study in Waaler (1968b), Waaler (1968c),
Waaler and Piot (1969), Waaler (1970a) and Waaler and
Piot (1970b). Following the approach in Waaler (1968a),
Revelle et al. (1969) formulated a TB model with one
progression rate and various latent classes representing
different treatment and control strategies. They argued
that vaccination was cost effective in countries with high
TB burdens.
Blower et al. (1995) described an SEIR-type
population model of tuberculosis epidemics. The model is
calibrated to TB data and R0 is used to derive a threshold
value below which the disease cannot take hold CastilloChavez and Feng (1997) present a TB model with one
form of latency and one class of active TB. They find an
R0 for the model and show global asymptotic stability of
the disease-free equilibrium when R0 < 1 and local
asymptotic stability of the endemic equilibrium for
R0 > 1.
The model of castillo-chavez and feng: Considering a
three dimensional model which consists of Susceptible
(S), Latently infected (L) and Infectious (I), CastilloChavez and Feng (1997) proposed the following
mathematical model:
Age-structured and delayed model: From the models of
Vynnycky and Fine (1997, 1999, 2000), reinfection and
vaccination is assumed to completely protect vast
majority of those vaccinated. They argued that a
reinfected individual move into the faster progressing
latent class but their previous reinfection affords them
some protection not from transmission of the new
infection. Dye et al. (1998) also developed an agestructured model of TB control under DOTS strategy and
improved case detection and cure. They conclude that
DOTS has greater potential today than it did 50 years ago
in developing countries. Salomon et al. (2006), calibrated
the previous model to TB data in South-East Asia. They
argue that if more rapid treatments for TB were available,
with reduced infectious periods, this could significantly
reduce TB mortality and incidence.
Furthermore, other researchers have developed and
studied a big number of spatial and delayed models of TB
dynamics with similar objectives in mind. These works
can be found in Cohen and Murray (2003, 2004, 2005),
Colijn et al. (2006), Debanne et al. (2000), Eames and
Keeling (2003), Pourbohloul and Brunham (2004),
Salpeter and Salpeter (1997) and Song et al. (2002),
These papers revealed that reinfection could possibly play
a role in TB dynamics when disease burden is high. The
authors also reported that reinfection is possible and
previous infection partially protects reinfected individuals
from progression to active disease. When one of the
neighbours progresses, other members of the clusters are
likely to be infected. The stochastic model of Colijn et al.
(2006) argue that if disease transmission is locally
asymptotically stable, this may reduce the effectiveness of
the widely applied therapy.
In this study, we consider the model of CastilloChavez and Feng (1997). We shall extend this work by
incorporating into the model the effect of immigration and
other important factors such as the proportion of fast
dS/dt = v!F IS/N+r1L+r2 L!:S
(1)
dL/dt = F IS/N!(: +v + r1 )L
(2)
d1/dt = vL!(: + :T+ r2)l
(3)
where,
v
F
:
:T
v
r1
r2
N
= Recruitment rote of the popultion
= Transmission rate of the disease
= Natural death rate
= Mortality rate due to TB infection
= Progression rate from latency to active TB
= Cure rate of latent TB infection
= Cure rate of active TB infection
= S + I + is the population size.
In this model, individuals can only move to the
infectious class from the latent class, so there is only one
progression rate (v) and there is recovery from latency
and active disease back to the susceptible class. The
authors find an R0 for their model and show global
asymptotic stability of the disease-free equilibrium when
R0 < 1 and local asymptotic stability of the endemic
equilibrium for R0 > 1.
The extended model: Following the ideas and
terminology in Castillo-Chavez and Feng (1997), our
model equations are as follows:
N
(
$
d
15
dS/dt = (1!() v + sI - $ sI/N!:S
(4)
dE/dt = (1!D) $IS/N!(: + v)E + gE
(5)
dI/dt = dD$IS/N+ dvE!( :+ :T + s)l
(6)
dR/dt = sgE + sl - $IR/N!:R
(7)
= S + E +I+ R is the total size of the population
= Proportion of recruitment due to immigration
= Transmission coefficient
= Detection rate of active TB
Res. J. Math. Stat., 4(1):14-20, 2012
s
g
D
= Treatment rate of active TB
= Rate at which susceptible individuals recover
= Proportion of fast progressors to active TB
Proof: Let
X = {(S, V, E, I): S $ 0,V $ 0, E $ 0, I $ 0}
X0 = {(S, V, E, I) 0 X: I > 0}
MX0 = X\X0 = {(S, V, E, I) 0 X :I = 0}
Other terms are the same as that of Castillo-Chavez and
Feng (1997).
It suffices to prove that MX0 repels uniformly the
solutions of system (5)-(7) in X0 . Since MX0 is relatively
closed in X, then it implies that X and X0 are positively
invariant.
Let:
Proposition 1: The basic reproduction number of the
model (4) - (7) is:
R0 = D$v/:(: + D)(: + s)
(8)
KM = {x0 0 MX0 :$J{x0}0 MX0 for t $ 0}
D = {x0 0 X:I = 0}
If R0 #1, there is only one non-negative equilibrium
point P0(


, 0, 0, 0 )
which is the disease-free equilibrium
Clearly, D d KM and I(0) > 0 for any x0 0 MX0\D.
Since, P0 is globally stable in KM , it follows that {P0} is an
isolated invariant set and acyclic. By Theorem 4.6 in
Thieme (1993), model (5)-(7) is uniformly persistent with
respect to (X0, MX0). Also, by Theorem 2.4 in Zhao
(2003), system (5)-(7) has an equilibrium P* = (S*, E*,
I*, R*) 0 X0. The first Eq. of (5)-(7) ensures that S* > 0
This means that P* is an endemic equilibrium of system
(5)-(7).
We prove the global stability of the endemic
equilibrium by the theorem below.
(DFE) and it is globally asymptotically stable (GAS). If
R0 >1, there exists a unique positive endemic
equilibrium P = (S*, E*, I*, R*) of the system (5) - (7)
which is globally asymptotically stable.
Theorem 1: The disease-free equilibrium DFE, P0, of
model (5)-(7) is globally asymptotically stable (GAS) if
R0 # 1 and unstable if R0 > 1.
Proof: From Theorem 2 in van den Driessche and
Watmough (2002), P0 is locally asymptotically stable if R0
< 1 but unstable if R0 > 1 Now, it suffices to prove that all
solutions converge to the DFE when R0 # 1 The
inequalities S/N# 1 and $IS/N # $ yield:
Theorem 3: The endemic equilibrium, P* of the model
(5)-(7) is globally asymptotically stable if R0 > 1.
Proof: We follow the same approach as in the prove of
Theorem 4.13 in Tian and Wang (2011).
At the endemic equilibrium:
dI/dt = dD$IS/N + dvE!( : + :T + s)I
By using the algorithm in Kamgang and Sallet
(2008), or by a standard comparison theorem (Sun et al.,
2011), we have that the DFE is GAS whenever R0 #1.
Remark 1: The global stability of the DFE of system (4)(7) can also be proved in a manner similar to the proof for
Theorem 2 in Sharomi et al. (2007).
Existence and stability of endemic equilibrium: Using
the techniques of persistence theory (Zhang et al., 1992;
Thieme, 1993; Zhao, 2003), we can show the uniform
persistence of the disease and the existence of the
(1!( ) B + Si* - $I*S*/N - :S* = 0
(9)
(1!D) $I*S*/N – (: + v)E*!gE* = 0
(10)
dD$I*S*/N +dvE*! (: + :T + s)I* = 0
(11)
sgE* + sI* - $I*R*/N!:R = 0
(12)
Dropping R i.e., Eq. (7), we consider the domain:
) = {(S, E, I)/S $ 0, 1 > 0, E > 0, S + I # N}
~
endemic equilibrium P of system (5) - (7).
(13)
which is clearly a positively invariant set in R3
We construct a Lyapunov function in the form:
Theorem 2: Gao and Ruan (2011). For model system (5)(7), if R0 > 1 then the disease is uniformly persistent i.e.,
there exists a constant h > 0 such that every solution
$J(x0) / (S(t), E(t) , I(t), I(R(t)) of system (5)-(7) with x0
/{(S(0), E(0), I(0), R(0)) 0 R4+ × R4+\0} satisfies limJ64 inf
I(t) > h, h >0 and (5)-(7) admit at least one endemic
equilibrium.
F(S, E, I) = "1(S – S*)2 + "2(E – E*)2 + "3(I – I*)2 (14)
where, "1, "2, "3 are positive constants. Then, we have
dF/dt = LF.dX/dt = 2"1(S – S*) [(1!() B +s1!$IS/Nw
:S]
16
Res. J. Math. Stat., 4(1):14-20, 2012
Definition: For any n × n matrix K, let K~ denote the (n 1) × (n - 1) matrix obtained from K by deleting its last
row and last column. This notation will be frequently used
in the steps which follow.
+2a2(E – E*)[(1!D) $IS/N – (: + v)E ]
+2a3(I – I*)[d D$IS/N + dvE – (: + :T+s)I]
(15)
Clearly, when F = F*dF/dt = 0; when F is on the S
!axis (i.e., I = E = 0 ), the sign of dF/dt is indefinite. We
want to show that when F …F* and (I, E) … 0, then dF/dt
< 0 holds everywhere.
Substituting (9)-(12) into (14), we get:
 d 11
Lemma 1: (Cross, 1978). Let D  
 d 21
dF/dt = 2a1(S–S*){-s(I–I*)!$(I – I*)(S –S*)!:(S – S*)}
+2 a2(E – E*){(1 - D) $(I – I*)(S – S*) $D (I, E)(I – I*)(E
–E*) – (: + v)(E – E*)}
+2a3(I – I*){dD $(I – I*)(S –S*)+dv(E – E*) – ( :+
:T+s)(I – I*)}
= !2"1[: + $(I – I*) ](S –S*)2!2a1s(I – I*)(S – S*)
-2a2[- D$(I – I*) – (: + v)](E – E*)2
+ 2a2(1!D) $(I – I*)(S – S*)(E – E*)!2a2D$(I !E)(E –E
*)2
-2a3(:+ :T +s)(I – I*)2 +2a3dD$(S – S*)(I – I*)2
+2a3dv(I – I*)(E – E*)
(16)
Lemma 2: (Redheffer, 1985a, 1985b). Let D = [dij] be a
non-singular n × n matrix (n $ 2) and M = diag (m1,…
,mn) be a positive diagonal n × n matrix. Let U = DG1
~~
~~
~~
~~ T
Then, if dnn > 0 MD
 ( MD) T  0 and MD   MD it is
possible to choose mn > 0 such that MD +DTMT > 0.
matrix. The D is Volterra-Lyapunov stable matrix if and
only if d11 < 0,d22 < 0 and det (D) = d11 d22 – d12 d21 > 0
~
Lemma 3: For the matrix A defined in Eq. (18), A is
Volterra-Lyapunov stable.
Proof:
~  a11
A 
 a 21
Now, expanding D(I, E) and using bilinear
assumption, we obtain:
D(I , E) = D(I*!E*) + $(E!E*)+$(I!I*)
- (: + :r + s) - $S* >0
Setting I = 0 and E = E* in (17), we get:
0 < D(0, E*) = D(I*, E*) - $I*
Thus, D(I*, E*) > $I*. But, at the endemic
equilibrium, we have:
into (15), we have:
(19)
S*D(I*, E*) = (: + :T + s)I* . Hence
where the matrices G and A are given by:
 g1

G  0

 0
0
g2
0
(20)
(18)
~
for some I between I and I* Substituting (16) and (17)
dF/dt = (F – F*)(GA + ATGT(F – F*)T
a12    [   ( I , E )]
 S *

 

a 22  
 (   T  s)  S *
( I , E )
It is clear that "11 < 0. Next, we show that "22 < 0 i.e.,
(17)
Applying mean value theorem to s(I), we obtain:
S(I)!S(I*) = s( I~ )(I – I*)
d 12 
 be a 2 ×2
d 22 
:+:T + s = S*D(I*, E*)/I* >$S*
~
0

0

g3 
Finally, it can be seen that det ( A = a11, a22! a12 a21)
~ is a
> 0 since a12 < 0, a21 > 0. Hence, from Lemma 1, A
Volterra-Lyapunov stable matrix.
Lemma 4: When (I, E) …(0,0), the determinant of - A is
positive, where matrix A is as defined in (18).
  ( I , E )
 S
  S *


A   ( I , E )  (   T  s   S *)  S * 


0
S (I )
   v

Proof. Expanding matrix -A by the 1st column, we get
det
(-A) = [: + D(I, E)][v(: + :T+s )-v$S* - $S(I)S*]
~
+ D(I,E)[v$S* + $As( I )S*]
Assume F …F* and F is not on the S-axis. We show that
there exists g1> 0, g2 > 0, g3 > 0 such that matrix GA +
ATGT is negative definite. For convenience, we write a
matrix D > 0 (< 0) if D is positive (negative) definite.
The second part of det (-A) is clearly positive. Now,
we want to show that:
17
Res. J. Math. Stat., 4(1):14-20, 2012
v(:+:T + s) – v$S* - $s( I~ )S* > 0
Proof: Observe that A-1- U and using (23) we have:
(21)
Now:
~ s I   s I * s I * vE *
s ( I ) 
,


I I*
I*
I*
I > 0,I…I*
~
A 1 
(22)
 c21 

 c22 
~
Hence:
I*s(I*) – vE* < 0 and E1 ×E*-
s
~
(I )
v
By Lemma 1, it can be easily verified that A 1 is
Volterra-Lyapunov stable. Therefore, there exists a
~
positive 2×2 diagonal matrix G such that
~~
~ ~
GA 1  G  A 1 ) T  0 . Since
I* > 0
Substituting (I, E) = (0, E) into (16), we have that:
~
s ( I )
0  (0, E *)  ( I *, E *)  I *  
I*
v
~~
 c11(  ) c21(  ) c31(  ) 


1
(  A) 1 
 c12 (  ) c22 (  ) c32 (  )
det (  A) 

 c31(  ) c23 (  ) c33 (  )
(23)
2 g1c11

1

det(  A)  g1c12  g 2 c12
g1g2c11c22 – (g1c11 + g2c12)2 > 0
(24)
Substituting the expressions for cjk (j, k = 1, 2) and
further manipulation gives:
0 <4g1g2c11c22 – (g1c21 +g2c12)2
~
~
= J – 2g1g2(2µ+ D(I, E)) s  ( I ) $S* - (G1s*)2 $ s  ( I ) [2$+
We have the following:
~
s  ( I ) ]
~
c11     T  s  S *  s' ( I ) S *  0

~
c21   S *  s'  I S *  0
J = 4g1g2 : + D(I, E)[: + :T+s - $S*] – [g2D(I,E) – g1$S*]2
2

Clearly, J > 0 for (25) to hold. Now:

c22      I , E   0
c32  S *  0
g1 S * ( I , E )
 2 g1 [    ( I , E )]

~~
~~
GD  (GD)T  

 g1 S *  g 2  ( I , E ) 2 g 2 (   T  s  S *)
~
c13   I , E s'  I   0

Observe that the (1, 1) and (2, 2) entries of matrix (26) are
positive and that its determinant is exactly J. Hence,

~
c23  s'  I     I , E  0

~~
~~
GD  (GD) T  0 .
c33      I , E       s  S *  S *   I , E   0
Theorem 4: The matrix A defined in (18) is VolterraLyapunov.
Note that we have applied (4.18) to show c11 > 0 and
(19) to obtain c11 < 0 and c33 > 0.
Proof: According to Lemmas (2) and (5), there exists a 3
× 3 diagonal matrix G such that:
Lemma 5: Let D = - A and U = (- A)-1 where A is defined
in (18). Then, there exists a positive 2 × 2 diagonal
~  g1
matrix G  
 0
0

g2 
(25)
where,
c31    2  S * S *      s  S *  0
c12   I , E   0
g1c21  g 2 c12 

2 g 2 c22 
whose determinant:
where, cjk denotes the cofactor of the (j, k) entry of matrix
-A and + or-in the parenthesis indicates the sign of cjk.
Obviously, det(-A) > 0 according to Lemma 4.

~~
U = (-A)-1, it follows that GU  (GU ) T  0 i.e.,
According to Eq. (21), it is clear that (20) holds.
Hence, det(-A) > 0.
We write the inverse of -A as:

  c11
1

det (  A)   c12
G(!A) + (!A)TGT > 0 and GA + ATGT < 0
~~ ~~ T
such that GD  (GD)  0 and
Hence, we obtain dF/dt < 0 when X …X* and X is not
on the S-axis. Theorem 3 is hereby established.
~~
~~
GU  (GU ) T  0 .
18
Res. J. Math. Stat., 4(1):14-20, 2012
Numerical calculation of R0 :Consider Eq. (8) and take
parameters in the model system (5)-(7) as D = 0.004, F =
0.0238, v = 0.60, µ = 0.01425, s = 0.37, For these
parameter values, the basic reproduction number for the
Disease-Free Equilibrium (DFE) is R0 = 0.5716 < 1. This
shows that the DFE is GAS. Hence, infection is temporal
and the disease dies out in time. If we keep the value of µ
unchanged and let = 0.0856, v = 0.80, D = 0.0088, s =
0.14 , then the basic reproduction number is calculated as
R0 = 1.894 > 1 and the disease becomes endemic. In this
situation, an average infectious individual is able to
replace itself and the number of infected rises and an
epidemic results.
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DISCUSSION AND CONCLUSION
We have reviewed and analyzed the literature on
tuberculosis epidemic models. Multiple models exist
about TB transmission mechanism from latent infection
to active disease. These models exhibit remarkably
similar qualities and quantitative dynamics, indicating that
the observed dynamical properties are robust. The
continued emergence of drug-resistant TB and the deadly
combination of HIV and TB epidemics have made
mathematical modeling of TB very challenging and
rewarding. Presently, the modeling literature is moving in
this direction.
In the analysis of our model, threshold values are
obtained in terms of model parameter values. We estimate
the model’s R0 for disease-free state and endemic
equilibrium state. It is shown that the DFE is GAS if R0#1
and that there exists a unique endemic equilibrium which
is also Globally Asymptotically Stable (GAS).
Numerical experiments suggest that a country must
detect and treat TB over a long course because TB
transients can be very long. However, if more rapid
treatments for TB were available, with reduced infectious
periods, this could significantly reduce TB mortality and
incidence and thus lowers R0. When R0 > 1 , it means that
the likelihood of new infections is high and this may lead
to a wiping out of the total susceptible population.
Furthermore, our mathematical analysis suggests that if
the infected reproduction ratio is larger one, then a unique
susceptible extinction equilibrium may be reached and the
infected subpopulation approach a constant positive value.
In conclusion, our work here provides useful insights in
understanding the transmission dynamics of TB with a
view to helping medical and scientific community manage
the disease by different control programs.
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