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Journal of Tuberculosis Research, 2016, 4, 191212
http://www.scirp.org/journal/jtr
ISSN Online: 23298448
ISSN Print: 2329843X
DOI: 10.4236/jtr.2016.44022 December 15, 2016
Effects of the Cytotoxic TCells on the Dynamics of
CoInfection of HIV1 and Mycobacterium
tuberculosis
Chipo Mufudza, Senelani D. HoveMusekwa, Edward T. Chiyaka
Department of Applied Mathematics, Modelling Biomedical Systems Research Group, National University of Science and Technology,
Bulawayo, Zimbabwe
Abstract
Enhancement of the Human Immunodeficiency Virus (HIV) specific cytotoxic T

cells mechanisms in an HIV1 and
Mycobacterium tuberculosis
(Mtb) co
infected
individual seems to improve the clinical picture of an individual by reducing A
c
quired Immuno Deficiency Syndrome (AIDS) state progression rate. In this paper,
we develop a system of deterministic differential e
quations representing the immune
cells involved in an HIV1 and Mtb coinfected individual. Results show that a
l
though the nonlytic arm of the HIV1 cytotoxic Tcells affects the coinfection d
y
namics more than the lytic factors, a combination of both fac
tors results in a more
positive reduced progression to the AIDS state. This is due to the increased prote
c
tion of the CD4+ Tcells by the CTL mechanisms by further
reducing infections and
replications by the HIV. Thus, HIV1 specific CTLs mechanisms’ invol
vement is
here recommended to be part of a solution to the HIV and Mtb coinfection pro
b
lems.
Keywords
Mycobacterium tuberculosis
, HIV, CoInfection, Cytotoxic TCells, Lytic and
NonLytic Factors
1. Introduction
1.1. HIV and Mycobacterium tuberculosis (Mtb) CoInfection
Since the resurgence of HIV, Mtb and HIV has been closely linked, the HIV1, Mtb
coinfection causes a twoinfection disease endangering human immune response
which may also differ depending on the genetic background [1]. Approximately one
How to cite this paper:
Mufudza, C.,
Hove
Musekwa, S.D. and Chiyaka, E.T.
(2016)
Effects of the Cytotoxic T
Cells on
the Dynamics of Co
Infection of HIV
1
and
Mycobacterium tuberculosis
.
Journal
of Tu
berculosis Research
,
4
, 191212.
http://dx.doi.org/10.4236/jtr.2016.44022
Received:
October 28, 2016
Accepted:
December 12, 2016
Published:
December 15, 2016
Copyright © 201
6 by authors and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
C. Mufudza et al.
192
third of the world’s population is latently infected with Mtb [2]. The rate of progression
from infection to full blown disease varies greatly with at least 10% of Mtb infected
individuals developing clinical disease and about half of them developing disease more
than two years after infection, commonly named reactivation or postprimary TB [3].
Thus, the lifetime risk of developing active TB in immunocompetent adults is estimated
to be 5%  10%, but in HIVpositive individuals this risk is increased to 5%  15%
annually [4]. The depletion of CD4+ Tcells, which is a main feature of AIDS, is
certainly an important contributor to the increased risk of reactivation of latent TB and
susceptibility to new Mtb infection. HIV also manipulates macrophage bactericidal
pathways [5], deregulates chemotaxis [6], and tips Th1/Th2 balance [7] and may impair
Tumor Necrosis Factor (TNF)mediated macrophage apoptotic response to Mtb and
thus facilitates bacterial survival [8]. Specifically, TB patients with AIDS present a
dominant granulocytic infiltrate and necrosis without the typical necrosis seen in non
HIVinfected TB granulomas due to the killing of CD4+ Tcells in the granuloma by the
HIV [9].
HIV produces a progressive decline in the cell mediated immunity by virtue of the
pathogen targeting the CD4+ Tcells eventually disabling them and making them dys
functional. The decline impacts negatively on the control of the Mtb by the immune
system since the CD4+ Tcells are also needed for the cell immune mediated response
for the Mtb infection to keep TB in check. HIV also alters the pathogenesis of TB,
greatly increasing the risk of developing active disease in a coinfected individual lead
ing to more extra pulmonary involvement and radiographic manifestations.
The function of many immune cells, including macrophages and Dendritic cells
(DCs), is modulated by both HIV and Mtb, with the presence of Mtb increasing repli
cation of the virus in a coinfected individual [10]. Mtb has been reported to upregu
late HIV1 replication in chronically or acutely infected Tcells or macrophages [11]
[12]. The primary target for Mtb, the alveolar macrophage, can also be infected with
HIV exacerbating HIV replication in macrophages and lung cells obtained by bron
choalveolar lavage from coinfected individuals [10] [12] [13].
In

vitro
Mtb infection
can upregulate both HIV infection and replication within monocytederived macro
phages (MDMs), increase the efficiency of virus transmission from infected MDMs to
Tcells, and favor replication of X4 HIV variants by upregulation of CXCR4 [14]. It is
therefore evident that the presence of each of these diseases has a profound effect on the
other due to the interactions with the immune system. This is because the frequency at
which HIV and Mtb occurs together is determined by the epidemiology of each of the
disease in a given population. Thus, coinfection has also become one of the main com
plications worldwide with TB being the cause of death for one out of every three people
with AIDS.
1.2. Why Cytotoxic Cells?
The challenges that the world is facing in HIV and Mtb coinfection can possibly be
solved through a combined vaccine. However, design of candidate vaccines is a parti
C. Mufudza et al.
193
cularly difficult task since laboratory correlates of protection have not been defined for
Mtb and HIV infections. Vaccination efforts have shown that Mtb infection diminish
HIVspecific Immunoglobulin A (IgA) responses at mucosal surfaces which help
prevent HIV infection or reduce the viral load [15]. In addition, vaccineinduced
immune responses need to be tipped towards protection, avoiding those that may result
in immunopathology, this requires meticulous study of appropriate adjuvants, antigens,
and vaccination regimens for the novel vaccines [16]. Even with treatment, it has been
noted with serious concern regarding current recommendations for treatment of HIV
Mtb coinfection since data suggest that at least 8 months duration of rifapentine
(RPT), or rifampin (RIF) therapy, initial daily dosing, and concurrent antiretroviral
therapy might be associated with better outcomes, but adequately powered randomized
trials are urgently needed to confirm [17]. The criteria for deciding between strategies
to treat a coinfected individual still remains a challenge although a more robust quan
titative measure could help by incorporating rate of change of CD4+ count as a measure
of Integrated Risk Information System (IRIS) risk, as well as viral and bacterial load,
drug toxicity, and improved measures of drug interaction [18]. The presence of the
CTLs cannot be sidelined in an effort to come up with solutions to HIVMtb co
infections. Studies have shown that the presence of Mtb affect the C38 expression on
the CD8+ cells responsible for CTL differentiation. This negatively impacts the HIV
progression which may also depend on the ethnic group (genes) [19]. There have been
several hypotheses set regarding the relationship between HIV specific CTLs and viral
load. The findings strongly support the involvement of CTLs in the control of HIV
infection [20] but does it have the same effect in the case of coinfection? The CD8+
cellular responses like CTLs are potential mediators of protection against HIV [21].
Understanding the vital effects of Cytotoxic Tcells as an immune system component
might be of paramount importance for vaccine development. The tropism of the
coinfection between HIV1, Mtb and the immune system studied by Kirshner [22] can
be bridged by including the role of the Cytotoxic Tcells (CTLs: a special type of the
CD8+ Tcells) in the immune system. CTLs play an important role in controlling both
HIV and Mtb. This is made possible by incorporating the lytic and nonlytic responses
of the CTLs. The lytic response involves the destruction of infected Tcells by the
effector cells. The CTLs are antigen specific as they possess receptor molecules on their
surface that can recognise the antigen epitope (portion of antigen which which
specifically interacts with the immune system). These receptors are designed in such a
way that they recognise antigen only when they are bound to a cell surface molecule
called the major histocompatibility complex (MHC). The MHC presents peptides to
CTLs which then destroy the infected cell if it can bind the peptideMHC class1
complex [23]. In a coinfected individual, the CTLs are responsible for killing infected
Tcells by both HIV and Mtb. This is made possible when naive CD8+ Tcells are
converted into CTLs which are either HIV specific CTLs or Mtb specific CTLs through
the Bcells. The CTLs are essential in many ways in the control of Mtb infection
through their different effector mechanisms including activation of macrophages when
they induce the release of the Interferon gamma (INF
γ
) thereby by activating infected
C. Mufudza et al.
194
macrophages to induce reactive nitrogen intermediates [24]. They also mediate lysis of
infected macrophages through the FasFasL pathway predominantly achieved by the
CD4+ Tcells and through the granule exocytosis pathway [24]. CTLs also induce target
cell apoptosis and kill the bacteria directly by the granulysin which result in the
alteration of bacterial cell membrane.
In this paper, we focus on the HIV specific CTLs only and how they help to reduce
the two infection disease progression. Nonlytic response of the CTLs involves the
prevention of infection of new cells and inhibition of replication by soluble mediators
secreted by the immune system. The CTLs then caters for the nonlytic response by
producing chemokines like the
β
chemokine that helps to reduce viral load at the
early stages of the HIV infection (the asymptomatic stage) as they block the entry of
virions into the CD4+ Tcells [25]. They also produce cytokines. The HIV specific CTLs
principally produce the INF
γ
, Interleukin6 (IL6) and Interleukin10 (IL10) which
suppress the rate of viral production [26]. The INF
γ
cytokines produced by the CTLs
are also responsible for the activation of the macrophages so that they clear the Mtb
bacteria and the HIV virus. Although we are not going to incorporate the Mtb specific
CTLs, it should be noted that they also produce the INF
γ
cytokine and the TNF
which again are responsible for the inhibition of new infection by the Mtb. We will also
consider the dynamics looked at by Kirshner [22] and incorporate the HIV specific
CTLs mechanisms in the immune system and propose a mathematical model describ
ing the dynamics of uninfected Tcells, virally infected CD4+ Tcells, macrophages, HIV
pathogen, Mtb pathogen and the HIV specific CTLs with the immune system.
2. Model Formulation
The model by Kirshner [22] is here introduced and used as our basic model. This
model focuses on the study of the hypothesis that the presence of infection of HIV with
Mtb in the body worsens the clinical picture of HIV and that the presence of HIV can
activate the Mtb infection. This was done incorporating the interactions of the immune
system's key players which include the lymphocytes or Tcells, macrophages, HIV and
Mtb. Four population groups of cells and pathogens are here used namely: the armed
CD4+ and CD8+ Tcell population at time
t
represented by
( )
Tt
, the macrophage
population at a given time represented by
( )
Mt
, the HIV population is represented by
( )
Vt
and the Mtb population at a time
t
, represented by
( )
b
Tt
. The model is
explained in detail in [22]. The following system of equations were proposed:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
() () () ()
( )
( ) ( ) ( )
()
( )
() ( ) ( )
( )
( ) ( ) ( )
( )
1
02 2 1
11 2 3 4
56
,
,
,
.
b
TT t b
M m mb
vv
b Tb b Tb b
bb
Vt T t
Tt s Tt r kVtTt
C Vt T t
Mt M Mt kMtVt rMtVt rMtT t
Vt Vt NkTt NgMt Vt kTt kMt Mt
Tt rTtKTt Tt TkTt kMt
µ
µ
µ
µ
+
=−+ −
++
= −− + +
= + − +−
= −− − +
(1)
C. Mufudza et al.
195
The interactions of both pathogens with the immune system are explored in [22]
using system (1) and results show that the Tcells population is lower in the presence of
both pathogens than in the case of HIV alone. The results also showed that viral load is
higher in a coinfected patient than in a single infected patient and the same trend was
observed for the Mtb. Thus, since progression to AIDS is based on the CD4+ Tcell
count and the viral load, the presence of Mtb in an HIV infected individual worsens the
clinical picture of the AIDS state.
Model with CTL Mechanisms
The lytic and nonlytic factors of the HIV specific CTLs are here incorporated to
system (1), in order to analyse their effects to the dynamics of disease progression in a
coinfected individual. In particular, HIV specific CTL Tcells response in HIV
infection and its effects to the whole immune system dynamics. Thus CTLs are here
modelled with their HIV specific function and not just as general Tcells as done by
[27] and by monitoring the dynamics of six groups of population cell densities: the
population density for the uninfected Tcells at a time
t
(a pool of the CD4+ Tcells,
Mtb specific CTLs and naive CD8+ Tcells)
( )
Tt
; the virally infected CD4+ Tcells at a
time,
( )
*
Tt
; the free viral particles population at a time,
( )
Vt
; the density of the
resting, activated and infected macrophages,
()
Mt
; the population density of the Mtb
pathogen at a time,
( )
b
Tt
and the density of the HIV1 specific CTLs population at a
time
t
,
( )
Ct
. We also assumed that the immune cells proliferation has different limit
values since their production is controlled by the source from which they are coming
from either the thymus for the CD4+ Tcells and the CD8+ Tcells and the bone marrow
for the macrophages. Proliferation of cells is treated differently in the model depending
on whether infection is by the virus or the Mtb bacteria.
Uninfected CD4+ Tcells are produced from the thymus at a rate of
T
s
and die
naturally at a rate
T
µ
. Proliferation of the Tcells due to infection by both the Mtb and
HIV pathogens are modelled by the factors
b
b
mbT
T
rTD
+
and
tv
V
rVD
+
respec
tively, where
,
tm
rr
are maximum proliferation rates and
,
b
Tv
DD
are parameters that
determine the amount of pathogen needed for half maximum stimulation of CD4+
Tcells by the Mtb and HIV respectively [22]. Parameter
v
k
represents the apoptotis
with
T
B
being the level of apoptotic engagement by the receptors [27]. Infection of
the CD4+ Tcells and the macrophages by HIV occurs at rates
1
k
and
2
k
, respectively,
with the infection haboured due to the presence of the CTLs by a factor of
( )
0
1
1aC t
+
, where
0
a
is the efficiency of each CTL in reducing CD4+ Tcell in
fection [27]. Infected CD4+ Tcells are also directly killed by the CTLs at a rate
h
and
lost due to bursting and natural death at a rate of
α
releasing
T
N
α
and
mm
N
α
virions from the infected CD4+ Tcells and macrophages, respectively. Again, this viral
replication is limited due to the presence of the CTLs as they produce chemokines by a
C. Mufudza et al.
196
factor of
( )
0
1
1bC t
+
, where
0
b
is the rate at which CTLs suppresses viral pro
duction. Macrophage proliferation due to infection by the HIV and Mtb are modeled at
the rates
1m
r
and
2m
r
, respectively. Mtb pathogen can also be eliminated from the
system by phagocytosis action of Mtb specific CTLs (here just considered as Tcells) at
a rate
3
k
or by the macrophages at rate
4
k
. The logistic growth rate of the Mtb
bacteria is accounted for by
b
Tb
r KT
, where
K
as carrying capacity and the pathogen
dies naturally at a rate of
0
b
T
µ
>
. The macrophages natural death rate is represented
by m
µ
. Finally, the HIV1 cytotoxic Tcells themselves are produced from the source
c
s
of HIV specific CTLs [28] [29]. The source represent the new CD8+ Tcells from the
thymus and those from the precursors. The naive CD8+ Tcells are then differentiated
into HIV specific CTLs which can also proliferate at rate
T
ρ
due to infection by the
HIV. A process dependent on the help by the CD4+ Tcells and antigen present from
the Bcells [28] [29]. The CTLs can also die naturaly at a rate
c
µ
. Thus these immune
dynamics facts can be represented by the following system of equations:
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )
1
0
1
* **
0
*2
00
02
,
1
,
1
,
11
b
b
TT t m
v bT
v
T
T mm v
M
Vt T t
Tt s Tt r Tt r Tt
VtD TtD
kT t kV t
Vt Tt
T t B aC t
kV t
Tt Tt hCtTt Tt
aC t
N T t N kM tV t
Vt t Vt
bC t bC t
Mt M Mt k
µ
α
αα µ
µ
=−+ +
++
−−
++
= −−
+
=+−
++
= −−
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
21
34
,
,
.
bb
m mb
b Tb b Tb b
cT c
MtVt rMtVt rMtT t
Tt rTt KTt Tt TtkTt kMt
Ct s CtTtVt Ct
µ
ρµ
++
= −− − +
=+−
(2)
3. Initial Conditions and Reproductive Number, R0
3.1. Disease Free State (E0)
The disease free equilibrium is given by
0
E
where the abundance of the uninfected
CD4+ Tcells, infected CD4+ Tcells, HIV, macrophages, Mtb, and CTL Tcells are
given by
*
, , , , ,
c c c c bc c
TTVM T C
, respectively. This is found when all pathogen popu
lations are zero. It is given by:
( )
*
00
, , , , , ,0,0, ,0,
c
T
c c c c bc c Tc
s
s
E TTVM T C M
µµ
= =
(3)
3.2. Reproductive Number, R0
The basic reproductive number (R0) measures the number of new secondary infections
generated by a single coinfected individual cell in the presence of CTLs mechanisms. It
C. Mufudza et al.
197
is a dimensionless parameter which can be used to investigate the potency of the
immune system. It can be used to control infections and disease progression so that
they are kept at low levels. R0 is here given as the dominant eigenvalue of the next
generation matrix of the Jacobian matrix by Watmough’s method [30]. There are
several
0
Rs
′
for system (2), which can be denoted as follows:
•
0v
R
denoting the reproductive ratio given only when there is only viral infection.
This can be defined as the average number of viruses produced by one virus in a
mostly susceptible population of CD4+ Tcells.
•
0b
R
denoting the reproductive ratio given only when there is Mtb infection. It is
defined as the average number macrophages that can be infected from a single Mtb
bacteruim in a mostly susceptible population of macrophages.
•
0
R
denoting the reproductive ratio when there is coinfection and given as
( )( )
( )()( )
()
1 0 20 40
000 3
,,
,.
b
c T T c c c c mm T
Tvc cc cc c T T T
b
ov ob
N ks as s h N kM kM
Rs h a s b s k s Kr
RR
µα µ α µα µ
µµ α µ µ µ µ
++ +
=
+ + + ++
=
(4)
Diseases are not competing, therefore an individual can die of either HIV or Mtb or
both and so they both contribute to overall
0
R
rather than considering the maximum
value of the next generation matrix.
4. Endemic States Analysis
The model system (2) has three endemic states: the virally infected state (where only the
HIV pathogen is present), the Mtb state (which represents the presence of the Mtb
pathogen only) and the coinfected state (where both pathogens are present). The
stability analysis of these states is here discussed and analysed using the Jacobian
method.
4.1. Virally Infected State
( )
W
1
The population density of the uninfected CD4+ Tcells, infected CD4+ Tcells, Virus,
Mtb, Macrophages and the HIV specific CTLs at the virally infected endemic state are
respectively given by
1
W
as:
( )
*
1 11 1 1 1 1
, ,, , , ,
b
W TT VM T C
=
(5)
where
10
b
T=
. The uninfected CD4+ Tcells population density is given by
( )
()
( )
01 1 01 2 1
11
0
11,
.
v mm
T
c
v
a C hC b C N k M
TNk
T
R
αµ α
α
+ + +−
=
=
(6)
1
T
is positive when
()
01 2 1
1
v mm
bC N kM
µα
+>
. That is, the cells only exist when the
condition is satisfied. The population increases with increase in the inhibition of viral
entry, replication rate and direct killings by the CTLs attribute which again shows a
C. Mufudza et al.
198
protective effect by the CTLs. The population of uninfected CD4+ Tcells decreases with
increase in the HIV infection rate and increase in viral production (due to replication
and bursting) as explained in system (2), which attributes to increase in
0v
R
. Thus
increase in
v
R0
reduces
1
T
population and vice versa.
The abundance of the infected CD4+ Tcells at the virally infected state is given by
( )
( )
*111
101 1
.
1kVT
Ta C hC
α
=++
(7)
This is effectively reduced by increase in the inhibition of viral infection and direct
killing of the virus by the CTLs. The population however, increases with increase in the
viral population which inturn increases the reproductive ratio.
The viral load at this endemic steady state is given by the expression:
( )
*
1
101 2 1
.
1
T
v mm
NT
VbC N kM
α
µα
=+−
(8)
The equilibrium value of the viral load is dependent on the average number of the
new viruses produced by the infected macrophages as well as infected CD4+ Tcells.
Increase in the viral production attribute to an increase in the viral load. This viral
population is however, reduced by the effectiveness of the CTLs to inhibit viral pro
duction and replication, a mechanism of the CTLs that reduces viral multiplication and
protects the CD4+ Tcells. Thus, inclusion of these mechanisms in the new model helps
to understand how the virus can be reduced. Positivity of the viral population is
possible when
( )
01 2 1
1
v mm
bC N kM
µα
+>
.
The macrophage population is given by
( )
0
112 2
.
M
Mm
M
MVk r
µ
µ
=+−
(9)
This population is sustained when
( )
12 2Mm
Vr k
µ
>−
and it is reduced with increase
in the viral load.
Finally, the HIV specific CTLs population density is given by:
0
101
.
cc v
cc v T
sTR
CTR V
µρ
=−
(10)
Increase in the viral population density increases the
R
0
v
which then stimulate more
CTL to be released to either directly kill the virus via their phagocytic nature or inhibit
replication and more infections although it also depends on the rate of production of
the CTLs from the source. This is possible when the condition
1
0T
vcc
V
RT
ρ
µ
>
is satisfied.
4.2. Mtb Equilibrium State
( )
2
W
The second endemic equilibrium state is the Mtb infected steady state where the viral
load and the infected Tcell populations are all zeros and is given by:
( )
*
2 22 2 2 2 2
,,, , , ,
b
W TTVMT C
=
(11)
C. Mufudza et al.
199
where
*
22
0TV= =
and
2
T
,
2
M
,
2b
T
,
2
C
are the equilibrium values showing the
abundance of the uninfected Tcells, macrophages, Mtb population and the HIV
specific CTLs, respectively, and the values are given by expressions (12)(15). The
infected Tcell equilibrium value at the Mtb endemic steady state is given by:
2
2
2
.
b
T
b
Tm
bT
s
TT
rTD
µ
=
−
+
(12)
The uninfected CD4+ Tcells density exist only if
( )
22
b
T b T mb
T D rT
µ
+>
. It depends
mainly on the rate at which they are being produced, their natural death and the
proliferation rate due to infection by the Mtb. More Mtb infections trigger release of
more CD4+ Tcells. The protective effect of the CTLs is not exihited at this equilibrium
state since they are HIV specific.
The macrophages are given by:
0
212
.
M
M mb
M
MrT
µ
µ
=−
(13)
This value is dependent on the interactions of the macrophages with the Mtb patho
gen and the natural death rate for the macrophages. When Mtb approaches zero then
the macrophage population goes to the uninfected steady state value of
0
M
. The value
of the CTLs at this endemic state is the same as the one at the uninfected steady state
since the CTLs are not involved in the controlling of Mtb as they are HIV specific. Thus
2.
c
cc
s
CC
µ
= =
(14)
The Mtb population at the Mtb endemic steady state is given by the following ex
pression:
32 4 2
2
.
b
b
TT
b
bT
Kr k T k M
Tr
µ
−− −
=
(15)
The equilibrium value depends totally on the interaction of the Mtb with the macro
phages and the killing by the CD8+ Tcells together with the Mtb natural death rate and
the Mtb carrying capacity. Mtb exists only if the following condition is satisfied
3 42
.
b
b
T
T
kT kM
Kr
µ
++
>
(16)
4.3. CoInfected State
The last endemic state is the coinfected endemic state where both pathogens are
present given by
( )
*
3 33 3 3 3 3
,,, , , ,
b
W TTVM T C=
(17)
where
3
T
,
*
3
T
,
3
V
,
3
M
,
3b
T
,
3
C
are the equilibrium values at the coinfected steady
state. The abundance of the viral load and the CTL equilibrium values are given by (8)
C. Mufudza et al.
200
and (14) respectively, same values as the ones at the virally infected steady state except
that the uninfected Tcell values are replaced by
3
T
. The Mtb population again is still
the same as the one at the Mtb endemic state, expression (15). The abundance of the
uninfected CD4+ Tcells at the coinfected steady state is given by
( )
( )
( )
( )
( )
( )
30
2
0
1
1 4,
bT
bT T
T KR Q B A B
KR QBAB ABBQ
= ++− +
± ++− + + +
(18)
where
,,QAB
are given by equations in Appendix (10.1). This shows that the
uninfected CD4+ Tcells are reduced more at the coinfected steady state due to the
virus and also due to the Mtb infection. The macrophage density at the coinfected
steady state is given by
( )
0
322 1
.
M
M m mb
M
MVk r rT
µ
µ
=+ −−
(19)
where
( )
32 2 1M m mb
V r k rT
µ
> −+
for the macrophages to be feasible at the coinfected
equilibrium state. Thus, the value of the macrophages are seriously reduced in a
coinfected individual due to the presence of both pathogens which reduce the macro
phage population. The uninfected Tcells steady state value for the coinfected endemic
state cannot be explicitly found since it involves a polynomial of higher order. However,
it can be noted that the abundance of uninfected CD4+ Tcells is reduced due to the
presence of both pathogens. This is only possible when we have both the
0v
R
and the
0b
R
being greater than unity for the coinfected state to be established.
5. Stability Analysis
5.1. Disease Free State,
E0
Stability analysis of all the steady states are here done using the characteristic poly
nomial of the Jacobian matrix at each equilibrium state for all our equilibrium states.
The stability analysis of the uninfected steady is analysed using
1
J
in Appendix (10.2).
The eigenvalues are calculated by solving the characteristic polynomial
( )( )
( )
( ) ( )( )
2
0,
TMc
b a d bc
λµ λµ λµ λ λ λ
− − − − − −− =
where
2
,, ,,abb c
and
d
are given in Appendix (10.2). All the four eigenvalues are
negative when
3 40 *
.
b
b
TT
T
ks kM
KK
r
µµ
++
<=
(20)
The other two eigenvalues are obtained from the quadratic equation
( )( )
( )
a d bc
λλ
− −−
which has negative roots if and only if
( )
10ad−+ <
and
0ad bc−>
. This is only possible if
( )( )( )( )
0 0 20
1
*
.
T c c c c c c v mm
TTc
a s b s hs N k M
Nks
N
µ µ µ αµ µ α
αµ
+ + −− −
<
=
(21)
C. Mufudza et al.
201
Stability of
0
E
requires that the Mtb carrying capacity
*
KK<
is satisfied by
condition (20). The same condition is required for system (1) that
c
KK
<
[22], where
*
c
KK=
are the critical values for the Mtb carrying capacity in system (1) and (2),
respectively. This is due to the fact that the CTLs incorporated in system (2) are only
HIV specific and therefore does not change mechanisms in the Mtb dynamics. Thus
with or without the CTL factors, stability of the disease free equilibrium is achievable
when
*c
KK K
<=
. Condition given by 18 is true when
0
1
b
R<
. The second con
dition for stability of
0
E
is dependent on the number of produced by infected CD4+
Tcells, T
N
. Stability depends on
*
N
which is dependent on the CTL’s ability to
directly kill the virus, inhibit viral infection and replication over and above the same
other factors as system (1). Also of interest is the reduction by the CTL factor to
production of more viruses by macrophages
mm
N
α
. Therefore
*
0N→
as either one
or a combination of
( )
00
,,abh→∞
. Condition (21) is only possible when
01
v
R<
.
Thus
0
E
is stable when both
00
,
vb
RR
are less than unity. Unlike in system (1), where
stability of uninfected state depends solely on the CD4+ Tcells production and macro
phages ability to produce more viruses only as shown by
c
N
[22], in system (2)
stability of
0
E
happens o when
( )
*0
1
Tv
N NR<<
, and *
N
is reduced as we increase
CTL mechanisms. This limits disease progression, suppose we are to increase these
mechanisms since
0v
R
is reduced., new/secondary infections by the virus is reduced.
Therefore increasing these CTL mechanisms results in reducing the overal
0
R
, which
also have an impact of protecting the CD4+ Tcells since new infections are reduced. If
a vaccine can be improvised, it will be helpful to enhance these CTLs factors to reduces
viral replication and new infections.
0
E
loose its stability to either
1
W
,
3
W
or
2
W
through the critical values of
*
K
and *
N
.
5.2. Virally Infected State
Stability of
1
W
cannot be done explicitly from the eigenvalues of
2
J
in appendix
(10.3) since the steady state values for the viral load are not explicitly found. However,
one of the eigenvalues, can be easily read from
2
J
and is equal to one of the eigen
values from
1
J
in (10.2) corresponding to condition (20). This implies that for the
virally infected state to be stable we need to have
01
b
R<
same condition established
by Kirshner [22] for system (1). When
01
b
R>
then we have the virally endemic
steady state loosing stability to either
2
W
or
3
W
. This leads to a state where by the
virus establishes itself fast and hence quick progression to AIDS. It is therefore
necessary to keep
01
b
R<
so that the clinical picture of the HIV infection is
maintained at low levels by the immune system. Any changes in the Mtb carrying capa
city
*
K
have a profound effect on the stability of the Stability of
1
W
. Thus, distur
bances in the Mtb natural death rate and Mtb interaction with both the Tcells and the
macrophages then brings qualitative changes to the stability of the virally infected
steady state and hence the clinical picture of HIV infected individual in particular.
Therefore
*
K
can also be used to control infection.
C. Mufudza et al.
202
5.3. Mtb Infectected State
At Mtb infected state stability is characterised by the characteristic polynomial of
3
0JI
λ
−=
given by
()()( )( )( ) ()
0
cf
a m c j a i b m bj
dg
λλ λ λ λλ λ
λ
−− − −− − −+ −− =
−
where
3,,,,, , ,,,,, ,,J abcd f ghi jkmn p
are all given in Appendix (10.4). One of the
eigenvalues can be read directly from the Jacobian that is
1c
λµ
= −
the other two
eigenvalues are calculated from the characteristic polynomial of the following deter
minant
0
cf
dg
λ
λ
−=
−
The roots are negative if and only if
0df cg−+ >
which implies giving the same
condition as in requirement condition (21). This condition is true when
*
T
NN<
that
is (
01
v
R<
) and when
c
TT=
and
c
MM=
given by Equation (4). Stability of the
Mtb state state arises when the reproductive ratio for the HIV is kept below unity.
However, it must also be noted that
0v
R
mainly depends on the effector mechanisms
of the CTLs such that any slight changes in these CTL mechanisms significantly
changes condition (21) hence the stability of the Mtb infected endemic steady state. The
state of the last three eigenvalues can be determined from the following theorem.
Theorem 1
Let
()
23
01 2 3 n
Hn
P s b bs bs bs bs=+ + + ++
with
i
b
real and
00b≥
which is assumed if
00b<
then scale the whole expression with
−1
which does not
change the zeros of
H
P
.
Then
H
P
have roots with negative real parts if and only if the
routh determinants of the polynomial
12
,,,
in
b∆= ∆ ∆
are all positive where
1 3 21
0 2 22
1 3 23
0 2 24
0
i
i
ii
i
i
bb b
bb b
bb b
bb b
b
−
−
−
−
−−
−−
∆= −
−−
− − −−
Thus, from the theorem we have the Mtb infected endemic state stable since the
other roots of the Jacobian matrix have negative real parts since the routh determinants
of the polynomial
( )
23
3 01 2 3
P bb b b
λ λλ λ
=++ +
are all positive where
3
1 0,b= >
( )
11
0,b am ac cm aj ij b∆= = + + − − + >
( )
2 12 03
0,bb bb∆= − >
( )
0
,b bm bj acm aij= −+ −
( )
2.b jamc= −− −
The Mtb infected steady state can only be stable also when the
0
b
T>
which is
given by the condition
C. Mufudza et al.
203
34
.
b
b
T
T
kT kM
Kr
µ
++
>
(20)
Therefore, stability of the Mtb infected steady state is stable when
01
b
R>
and
01
v
R<
. The same critical points
*
K
and
*
N
determine stability for both the Mtb
infected and the virally infected endemic states. Thus, there exists a switching behavior
on stability between
1
W
and
2
W
via these parameters.
1
W
looses its stability via
*
K
and to
2
W
and also
2
W
to
1
W
via both
*
K
and
*
N
. This implies transcritical
bifurcations at these points although they are not analysed in detail in this paper.
5.4. CoInfected State
The coinfected endemic steady state can only be stable when the number of pathogens
for both diseases are greater than zero and this corresponds to the condition when
*
T
NN>
and when
*
KK>
. Thus both
0v
R
and
0b
R
are greater than unity, imply
ing that as long as both pathogens exist, an individual always remains in the coinfected
state which makes it very difficult to control the disease.
6. Numerical Simulations
Matlab 6.5 version was used for all our simulations for both models using ODE45
solver. Simulations on this model give us a portrait of the general behaviour of immune
cells population in the presence of the HIV and Mtb pathogens. We are also concerned
about the parameters which are of importance in stabilising the model and the ranges
in which the system is stable and unstable. Table 1 shows all the parameter values used
for all our simulations are here in used. The initial condition for systems (1) and (2) are
given as
( )
3
0 1500 mm
T
−
=
,
( )
*3
0 = 0.01 mmT−
,
( )
3
0 = 0.001 mmV
−
,
( )
0M=
3
100 mm
−
,
( )
3
0 0.001 mm
b
T
−
=
and
( )
3
0 = 250 mmC
−
adopted from [22] [27].
7. Effects of CTL Mechanisms
7.1. Effects of CTLS on R0
The presence of the lytic factor is almost insignificant in a coinfected state since it
depicts the severe stage of HIV and Mtb. The different mechanisms affect the repro
ductive ratio differently. Figure 1(a) shows how
ev
R
changes when there is only
inhibition of infection, replication and direct killing by CTLs, given by
a
R
,
b
R
, and
h
R
, respectively. The ratios generally decrease with CTL mechanisms. It shows that the
lytic factors are more essential than the nonlytic factors in the order of
0
,ha
and
0
b
.
It also shows that although the reproductive ratio is reduced with increase in CTL
mechanisms, it will never be reduced to zero. This is because of the nature of HIV
which is a retrovirus thus we will always have the virus once a person is infected. A
combination of the different mechanisms here not shown was also done numerically
and observed that the reproductive ratio is reduced more when all the mechanisms are
enhanced [31]. Removing any one of the CTL mechanisms will result in a high ratio
which depends on
T
N
and
K
. Thus condition
hC
α
means the rate of CTL
C. Mufudza et al.
204
Table 1. Numerical values for the simulations.
Parameter
Symbol Value Units Ref.
Rate CD4
+ Tcells becomes infected by free virus
1
k
5
2.4 10
−
×
31
mm d
−−
⋅
[22]
Rate of macrophage infection by HIV
2
k
6
2 10
−
×
31
mm d
−−
⋅
[22]
Rate at which uninfected T
cells kill Mtb
3
k
0.5
31
mm d
−−
⋅
[22]
Rate at which macrophages kill Mtb
4
k
0.5
31
mm d
−−
⋅
[22]
HIV apoptosis rate
v
k
3
2 10
−
×
31
mm d
−−
⋅
[27]
Cytotoxic T cell death
c
µ
1.5
1
d−
[27]
Rate new CTLs are produced
c
s
10
31
mm d
−−
⋅
[27]
Production of T
cells
T
s
20
3
mm d
[22]
Death rate of uninfected T cell population
T
µ
0.02
1
d−
[22]
Macrophage death rate
M
µ
0.03 1
d− [22]
Natural death of HIV
v
µ
2.4 1
d− [22]
Natural death of Mtb
b
T
µ
0.5 1
d− [22]
Stimulation constant
T
B
350
3
mm
−
[27]
Virus cytopathic rate
α
0.25 1
d− [27]
Rate of viral production by HIV burst
m
α
0.25 1
d− [27]
CTL proliferation rate
T
ρ
1 × 10−5
31
mm d
−−
⋅
[27]
Proliferation of uninfected T cell by virus
T
r
0.02 1
d− [22]
Proliferation of uninfected T cell due to Mtb infection
m
r
0.01 1
d− est
Maximal proliferation of the M tuberculosis population
b
T
r
1.0 1
d− [22]
HIV infection reduction rate by CTLs
0
a
0.025
3
mm
[27]
HIV replication reduction rate by the CTLs
0
b
0.05
3
mm
[27]
Rate of HIV infected are killed by CTLs
h
3
2 10
−
×
31
mm d
−−
⋅
[27]
Number of viruses produced by infected T
cells
T
N
1000
3
mm
−
[22]
Number of viruses produced by infected macrophages
m
N
500
3
mm
−
[27]
Stimulation constant for the HIV
v
D
400
3
mm
−
est
Stimulation constant for the Mtb
b
T
D
450
3
mm
−
[22]
Mtb carrying capacity
K
800
3
mm
−
[22]
Equilibrium value for the macrophage population
0
M
100
3
mm
−
[22]
mediated killings is low relative to the rate of cellmediated killings and
hC
α
means that the rate of virus induced CD4+ Tcells killings is very high relative to the
rate of CTL mediated killing [27]. The reproductive ratio here shows that for both high
and low values of CTLs (
C≈∞
and
0C≈
) and
hC
α
, then the order of the ratios
is
01 03 04 02
RRRR>>>
[31]. An indication that the nonlytic arm of the CTLs is more
essential than the lytic arm when
hC
α
for both low and high levels of the CTLs,
since the ratios are generally low when nonlytic arm are in operation than when the
C. Mufudza et al.
205
(a) (b)
(c) (d)
Figure 1. Effects of different CTL mechanisms on the reproductive ratio and Uninfected CD4+ Tcells. (a) Graph of reproductive ratios,
(b) Cell populations propagations without CTLs, (c) Cell populations propagations with CTLs mechanisms incorporated, (d) Effects of
CTLs on viral load.
lytic arm is in operation. Here the
0
, 1:4
i
Ri=
are defined in terms of
i
given by
system (23). Hence, chances of controlling infection by the immune system are
increased when the nonlytic factors are in operation. If
hC
α
, then the order of the
reproductive ratios is given by
02 01 04 03
RRRR>>>
for both low and high levels of the
CTLs. This implies that for this condition, we have the lytic factors being more effective
than the non lytic factors as ratios are lower when the lytic factors are in operation
than when only non lytic factors is involved. In general, we also have lower values of
the reproductive ratios for higher values of CTLs being lower than for low CTL values
indicating the importance of increased involvement of the effector mechanisms of the
C. Mufudza et al.
206
CTLs. Therefore the CTLs have a very vital role to play in the control of HIV infection
in a coinfected individual as it lowers significantly the reproductive ratio hence protect
CD4+ T cell infection by keeping a low viral load and hence keeping the Mtb carrying
capacity lower than
*
K
.
7.2. Cell Population
There is also evidence of the uninfected CD4+ Tcell depletion due the presence of CTL
mechanisms as shown by Figure 1(b) and Figure 1(c). The viral population can never
reach out to higher levels above
2
10
in the presence of CTLs as shown in Figure 1(c)
but reach up to levels of
4
10
in the model by Kirshner [22] as in Figure 1(b). Thus
the presence of these mechanisms limits the viral population growth and hence limit
the depletion of CD4+ Tcells. This has an effect of reducing the rate of progression to
the AIDS state. The different combinations of the CTL mechanisms can be analysed on
the equilibrium values of all the variables. Each of the equilibrium values is then plotted
against time for the different combinations. In this paper, only the different combi
nations under viral population is analysed although all the other populations were done
but not represented as in Figure 1(d). It must also be noted that the other results agree
with those on the viral load. Figure 1(d) shows that a combination of direct killing of
the virus and hinderance of viral replication will go a long way in reducing the viral
load.
i
VS
′
represent the following scenarios:
00
00
00
00
1, represents , 0
2, represents 0, 0, 0
3, represents 0 , 0, 0
4, represent 0, 0, 0
i a bh
i abh
i a bh
i abh
= = ≠
= ≠ ≠=
= ≠≠ = ≠
= = ≠≠
(23)
8. Conclusions
The HIV specific CTLs have a very big role to play in limiting viral spread and con
trolling infection by the virus in the immune system, which means that they have
profound effects to the dynamics of the coinfection of Mtb and HIV in general. In the
early stages of infection when the HIV specific CTLs are still in small amounts, we have
noticed that the lytic factors are more important than the nonlytic factors and reaching
the AIDS state is accelerated when the lytic factors are limited in the immune system, a
weakness of our basic model by Kirshner [22] since it had no mechanisms of the CTL
effects. CTLs are believed to be in abundance at the chronic stage of HIV infection and
hence when
C≈∞
, the nonlytic factors become more effective than the lytic ones.
This implies that both arms of the CTLs are essential for the immune system to control
infection. The same conditions were deduced by other authors [27].
The study shows that both the lytic factors and the nonlytic factors are important in
the controlling of HIV infection in a coinfected individual. The nonlytic mechanisms
are more effective in controlling infection in both cases when rate of CTL mediated
killings is low relative to the rate of cellmediated killings. The lytic factors are more
C. Mufudza et al.
207
important when rate of virus induced CD4+ Tcells killings is very high relative to the
rate of CTL mediated killing regardless of the CTL levels. However, the lytic arm is
more important when it comes to control infection than replication since it is believed
that CTLs will remain high and nonlytic arms are more important in controlling
replication of the virus as they are believed to be more effective at the chronic stage of
the infection [27] as also confirmed by Figure 1(a). The CTLs’ ability to hinder viral
replication protects the clinical picture of a coinfected individual hence protecting the
individual from fast AIDS stage as witnessed by Figure 1(a), Figure 1(b). Effectively
the presence of CTLs reduces the viral load levels and protects the CD4+ T cell levels.
Thus, we can conclude that the effector mechanisms of the HIV specific CTLs are
relevant in controlling infection by the immune system although there is need to deter
mine which onlytic factors are more important at the chronic phase. The increase in
the CTLs mechanisms reduces viral multiplication, which results on the reduction of
viral load, with the nonlytic factors being more effective in viral load reduction.
Reduction of the viral load protects the macrophages too, thereby slowing down the
rate of progression to AIDS. Thus, if drugs can be put in place to enhance these CTL
mechanisms either as a vaccine or treatment, then rate of depletion of CD4 count is
reduced and macrophages are protected. Ultimately, the progression to AIDS state is
reduced thereby increasing the life span of individual coinfected by HIV and Mtb.
Acknowledgements
This research was made possible due to the support and insights from Dr G. Magom
bedze. Thank you very much.
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Appendix
10.1. A1
( ) ( )( ) ( )
0 0 00 0 0
=1 1 1 1
b
T v v b bT
Q s aC aC bC R D K R R D
+ + + + ++
( ) ( )( ) ( )
0 0 00 0 0
1 11 1 b
T v v b bT
A aC aC bC R D K R R D
µ
= + + + + ++
( )( )()
()
( )()
( )
[ ]
0 0 001 0 00 1
111 11
v v vT
B aC aC bC R aC bC R D k k r
=+++ ++ ++−
10.2. A2
( )
( )
1
1
11
01
2
1
00
22 11
2
11
0 00
0 000
1
0 000
11
00 0
00 0 0 0
0 0 00
m
TTb
mm
Tv
m Mm
Tc
rT
bD
kT
hC aC
N kM
NbC bC
Mk r rM
b
TC
µ
α
α
αµ
µ
ρµ
−
+−
+
=−−
++
−−
J
where
1
10
1
vT
Tv
krk
bT
T B D aC
= −−
++
2 3 41
bb
TT
b k T k M Kr
µ
=++ −
The eigenvalues are calculated by solving the characteristic polynomial of the
Jacobian matrix
1
0JI
λ
−=
which can be expanded to become
( )
( )
( )( ) ( ) ( )
20
TcM
b a d bc
λµ λµ λµ λ λ λ
− − − − − −− =
where
( )
,a hC
α
= +
0
,
1T
N
bbC
α
=+
1
0
,
1kT
caC
=+
2
0
.
1
m
v
N kM
dbC
α
µ
= − +
10.3. A3
The Jacobian matrix evaluated at the virally infected endemic steady state is given by
C. Mufudza et al.
211
( )
( )
( )
( )
1
1 23
2
02
1
45
02
222
67 8
02
22 2 9 1 2
10
22 22 2
00
1
00
1
00
1
00 0
00 0 0 0
0 00
mm
mm
T T Tc
k
w ww
aC
kT
w hC w
aC
N kV
ww w
C
M r k w rM
w
VC TC V
α
α
ρ ρ ρµ
−
+
−+ +
=
+
−
−
J
where
( )
21
1 22
2
2 02
2
1
vT
Tt vT
kB
Vk
wr V V
V D aC
TB
µ
=−+ + −
++
+
()
21
2 22
202 2
21
Tv v
T
v
rTD k
k
w TT
aC T B
VD
=−−
++
+
12
3
b
m
T
rT
wD
=
12
402
1kV
waC
=+
( )
012
52
2
02
1
akV
w hC
aC
= −
+
602
1T
N
wbC
α
=+
22
702
1
mm v
N kM
wbC
αµ
= −
+
( )
()
*
0 2 1 22
82
02
1
T mm
b N T N kMV
wbC
αα
+
=+
( )
9 22 2Mm
w Vr k
µ
=−+ −
10 3 2 4 2TT
bb
w r K kT kM
µ
= −− +
10.4. A4
( )
( )
( )
( )
1
12
*
1
0
*
320
2
00 0
22 1 1
5 42
0 00
0 00
1
0 00
11 1
00 0
00 0
00 0 0
b
b
mT
t
vbT
mm T
Tv
m M mb m
bb
Tc
rD
rT
vDTD
kT
hC hT
aC
N kM N bT
NbC bC bC
Mr k rT rM
kT kT v
TC
α
αα
αµ
µ