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Research Article
Stochastic Model for Langerhans Cells and
HIV Dynamics In Vivo
Waema R. Mbogo,1Livingstone S. Luboobi,2and John W. Odhiambo1
1Center for Applied Research in Mathematical Sciences, Strathmore University, P.O. Box 59857, Nairobi 00200, Kenya
2DepartmentofMathematics,MakerereUniversity,P.O.Box7062,Kampala,Uganda
Correspondence should be addressed to Waema R. Mbogo; rmbogo@strathmore.edu
Received September ; Accepted October ; Published January
Academic Editors: D. Haemmerich and X. Liu
Copyright © Waema R. Mbogo et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Many aspects of the complex interactionb etweenHIV and the human immune system remain elusive. Our objective is to study these
interactions, focusing on the specic roles of Langerhans cells (LCs) in HIV infection. In patients infected with HIV, a large amount
of virus is associated with LCs in lymphoid tissue. To assess the inuence of LCs on HIV viral dynamics during antiretroviral therapy,
wepresentandanalyseastochasticmodeldescribingthedynamicsofHIV,CD
+
4T cells, and LCs interactions under therapeutic
intervention in vivo and show that LCs play an important role in enhancing and spreading initial HIV infection. We perform
sensitivity analyses on the model to determine which parameters and/or which interaction mechanisms strongly aect infection
dynamics.
1. Introduction
HIV is a devastating human pathogen that causes serious
immunological diseases in humans around the world. e
virusisabletoremainlatentinaninfectedhostformany
years, allowing for the long-term survival of the virus and
inevitably prolonging the infection process []. e location
and mechanisms of HIV latency are under investigation and
remain important topics in the study of viral pathogenesis.
Given that HIV is a blood-borne pathogen, a number of
cell types have been proposed to be the sites of latency,
including resting memory CD+
4+ T cells, peripheral blood
monocytes, dendritic cells (including Langerhans cells) and
macrophages in the lymph nodes, and haematopoietic stem
cells in the bone marrow []. is study updates the latest
advances in the study of HIV interactions with Langerhans
cells and highlights the potential role of these cells as viral
reservoirs and the eects of the HIV-host-cell interactions on
viral pathogenesis.
Despite advances in our understanding of HIV and the
human immune response in the last years, much of this
complex interaction remains elusive. CD+
4T cells are targets
of HIV and are also important for the establishment and
maintenance of an adaptive immune response []. e skin
andmucosaaretherstlineofdefenseoftheorganism
against external agents, not only as a physical barrier between
the body and the environment but also as the site of initiation
of immune reactions. e immunocompetent cells which
act as antigen-presenting cells are Langerhans cells (LCs).
InfectionofLCsbyHIVisrelevanttoseveralreasons.Firstly,
LCs of mucosal epithelia may be among the rst cells to be
infected following mucosal HIV exposure and, secondly, LCs
may serve as a reservoir for continued infection of CD+
4T
cells, especially in lymph nodes where epidermal LCs migrate
following antigenic activation [].
Many indirect and/or direct experimental data have
shown that LCs may be a privileged target, reservoir, and
vector of dissemination for the HIV from the inoculation
sites (mucosa) to lymph nodes where the emigrated infected
LCs could infect T lymphocytes []. Originated from the
bone marrow, LCs migrate to the peripheral epithelia (skin,
mucous membranes) where they play a primordial role in
the induction of an immune response and are especially
active in stimulating naive T lymphocytes in the primary
Hindawi Publishing Corporation
ISRN Applied Mathematics
Volume 2014, Article ID 594617, 10 pages
http://dx.doi.org/10.1155/2014/594617
ISRN Applied Mathematics
response through a specic cooperation with CD+
4-positive
lymphocytes aer migration to proximal lymph nodes [].
Apart from many plasma membrane determinants, LCs also
express CD+
4molecules which make them susceptible targets
and reservoirs for HIV []. Once infected, these cells due to
their localization in areas at risk (skin, mucous membranes),
their capacity to migrate from the epidermal compartment
to lymph nodes, and their ability to support viral replication
without major cytopathic eects could play a role of vector
in the dissemination of virus from the site of inoculation to
the lymph nodes and thereby contribute to the infection of T
lymphocytes [].
Langerhans cells (LCs), which are members of the den-
dritic cells family and are professional antigen-presenting
cells, reside in epithelial surfaces such as the skin and act
as one of the primary, initial targets for HIV infection [].
ey specialize in antigen presentation and belong to the
skin immune system (SIS) and play a major role in HIV
pathogenesis. As part of the normal immune response, LCs
capture virions at the site of transmission in the mucosa
(peripheral tissues) and migrate to the lymphoid tissue where
they present to naive T cells and hence are responsible for
large-scale infection of CD+
4Tlymphocytes[]. ese cells
play an important role in the transmission of HIV to CD+
4
cells []; thus, LC-CD+
4cell interactions in lymphoid tissue,
which are critical in the generation of immune responses, are
also a major catalyst for HIV replication and expansion. is
replication independent mode of HIV transmission, known
as trans-infection, greatly increases T cell infection in vitro
and is thought to contribute to viral dissemination in vivo
[].
e Langerhans cell is named aer Paul Langerhans, a
German physician and anatomist, who discovered the cells
attheageofinwhilehewasamedicalstudent[].
e uptake of HIV by professional antigen-presenting cells
(APCs) and subsequent transfer of virus to CD+
4Tcellscan
result in explosive levels of virus replication in the T cells.
is could be a major pathogenic process in HIV infection
and development of AIDS. is process of trans- (Latin; to
the other side) infection of virus going across from the APC
to the T cell is in contrast to direct, cis- (Latin; on this side)
infection of T cells by HIV []. Langerhans cell results in a
burst of virus replication in the T cells that is much greater
than that resulting from direct, cis infection of either APC or
Tcells,ortrans-infection between T cells. is consequently
shows that Langerhans cells may be responsible for the quick
spread of HIV infection.
e individual cells of the immune system are highly
interactive, and the overall function of the system is a product
of this multitude of interactions. e interplay between HIV
and the immune system is particularly complicated, as HIV
directly interacts with many immune cells, altering their
functions, ultimately subverting the system at its core [].
Because of this complexity, the immune response and its
interaction with HIV are naturally suited to a mathematical
modelling approach. Elucidating the mechanisms of LC-
HIV-CD+
4T cells interactions is crucial in uncovering more
details about host-HIV dynamics during HIV infection.
To explore the role of LCs in HIV infection, we rst
T : Variables for the stochastic model.
Variabl e Description
() e concentration of healthy (susceptible) CD+
4cells at
time t
𝐼() e concentration of infected CD+
4cells at time t
() e concentration of healthy (susceptible) Langerhans
cells at time t
𝑇() e concentration of latently infected Langerhans cells
at time t
𝐼() e concentration of infected Langerhans cells at time t
() e concentration of virus particles at time t
develop a stochastic model of HIV dynamics in vivo before
therapy. Next, we introduce therapeutic intervention and
nally investigate which parameters and/or which interaction
mechanisms strongly aect the infection dynamics.
e organization of this paper is as follows. In Section ,
we formulate our stochastic model describing the interaction
of HIV and the immune system and obtain a partial dier-
ential equation for the joint probability generating function
of the numbers of healthy immune cells, the HIV infected
immune cells, and the free HIV particles at any time .
e marginal probability distributions for the variables and
the population measures which include the expectation of
thevariablesarealsoderivedinSection .HIVdynamic
model incorporating therapy is derived and model analysis
is discussed in Section . Some concluding remarks follow in
Section .
2. The Role of LC in HIV Infection In Vivo
InHIVinfection,LCsplayadualroleinpromotingimmunity
while also facilitating infection. During antigen presentation,
LC-associated viruses migrate to the lymphoid tissue where
they present to naive T cells and hence facilitate infection
of CD+
4Tcells[]. Taken together, these interactions
suggest that LC dynamics are particularly important to HIV
infection. Several mechanisms have been proposed to trigger
progression from the chronic phase of infection to AIDS.
Many have hypothesized that progressive alteration of the
immune system results in the transition to AIDS [,].
TostudytherolesofLCsduringHIVinfection,wepresent
a mathematical model of HIV infection and accompanying
immune response. Specically, we develop a stochastic model
focusing on the HIV-LC-CD+
4cell dynamics in vivo.Our
modelanalysisallowsustopredicttheimportanceofLC
mechanisms and their role in triggering progression to AIDS.
In particular, our model predicts which mechanisms of LC
dysfunction are most signicant in the transition to AIDS. A
typical life cycle of HIV virus and immune system interaction
is shown in Figure .
2.1. Variables and Parameters for the Model. e variables and
parameters in the model are described as in Tables and .
ISRN Applied Mathematics
𝜆T𝜆L
𝜇V
𝜎L
𝜎LT
𝜋LT
𝛿T
T++
Virus-TC interaction Virus-LC interaction
TC-TC interaction LC-LC interaction
LC-TC interaction 𝛾MLI
𝜅NTI
𝜅TI
𝛽LL
𝛽TT
VL
LT
LI
TI
𝜎LI
𝛾LI
𝛿TI
F : e interaction of HIV virus and the immune system.
T : Parameters for the stochastic model.
Parameter Description
𝑇etotalrateofproductionofhealthyCD
+
4cells per
unit time
𝐿etotalrateofproductionofLangerhanscellsper
unit time
e per capita death rate of healthy CD+
4cells
e per capita death rate of healthy Langerhans cells
1e transmission coecient between uninfected
CD+
4cells and infective virus particles
2e transmission coecient between uninfected
CD+
4cells and IV infected Langerhans cells
3e transmission coecient between uninfected
Langerhans cells and infective virus particles
4e transmission coecient between uninfected
Langerhans cells and HIV infected Langerhans cells
Per capita death rate of infected CD+
4cells
Intracellular delay time
Per capita death rate of infected Langerhans cells
epercapitadeathrateofinfectivevirusparticles
e average number of infective virus particles
produced by an infected CD+
4cell in the absence of
treatment during its entire infectious lifetime
e average number of infective virus particles
produced by an infected Langerhans cell in the
absence of treatment during its entire infectious
lifetime
From Table and by using the population change scenar-
ios and parameters in Tab l e and applying probability argu-
ments, we now summarize the events that occur during the
interval (, + ) together with their transition probabilities
in Table .
e change in population size during the time interval ,
whichisassumedtobesucientlysmalltoguaranteethat
only one such event can occur in (, + ),isgovernedby
the following conditional probabilities:
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V(+
)
=1−
𝑇 + 𝐿 + +
+
1V + 2𝐼 + 3V
+
4𝐼 +𝐼 + 𝐼 + V + ()
×
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
+𝑇 + ()
𝑥−1,𝑥𝐼,𝑦,𝑦𝐼,V()
+𝐿 + ()
𝑥,𝑥𝐼,𝑦−1,𝑦𝐼,V()
+{(+1
) + ()}𝑥+1,𝑥𝐼,𝑦,𝑦𝐼,V()
++1+()
𝑥,𝑥𝐼,𝑦+1,𝑦𝐼,V()
+
1(+1
)(V+1
)+
2(+1
)𝐼
ISRN Applied Mathematics
×
−𝜌𝜏 + ()
𝑥+1,𝑥𝐼−1,𝑦,𝑦𝐼,V+1 ()
+
3 + 1 (V+1
)+
4 + 1 𝐼 + ()
×
𝑥,𝑥𝐼,𝑦+1,𝑦𝐼−1,V+1 ()
+ 𝐼+1+()
𝑥,𝑥𝐼+1,𝑦,𝑦𝐼,V−1 ()
+
𝐼+1+()
𝑥,𝑥𝐼,𝑦,𝑦𝐼+1,V−1 ()
+(V+1
) + ()
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V+1 ().
()
Rearranging (), then dividing through by , and taking the
limit as ⇒ 0 we have the following dierence dierential
equation:
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
=−
𝑇+𝐿+++
1V+
2𝐼
+
3V+
4𝐼+
𝐼+
𝐼+V
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
+𝑇𝑥−1,𝑥𝐼,𝑦,𝑦𝐼,V()+𝐿𝑥,𝑥𝐼,𝑦−1,𝑦𝐼,V()
+(+1
)𝑥+1,𝑥𝐼,𝑦,𝑦𝐼,V()++1
𝑥,𝑥𝐼,𝑦+1,𝑦𝐼,V()
+
1(+1
)(V+1
)+
2(+1
)𝐼
×
−𝜌𝜏𝑥+1,𝑥𝐼−1,𝑦,𝑦𝐼,V+1 ()
+
3 + 1 (V+1
)+
4 + 1 𝐼
×
𝑥,𝑥𝐼,𝑦+1,𝑦𝐼−1,V+1 ()
+
𝐼+1
𝑥,𝑥𝐼+1,𝑦,𝑦𝐼,V−1 ()
+
𝐼+1
𝑥,𝑥𝐼,𝑦,𝑦𝐼+1,V−1 ()
+(V+1
)𝑥,𝑥𝐼,𝑦,𝑦𝐼,V+1 ().
()
is is also called the Master equation or the forward
Kolmogorov partial dierential equation for 𝑥,𝑥𝐼,𝑦,𝑦𝐼,V(),
with initial condition
0,0,0,0,0 ()=−
𝑇+𝐿
0,0,0,0,0 ()
+
1,0,0,0,0 ()+
0,0,1,0,0 ()+
0,0,0,0,1 ().
()
For detailed description and derivation of the model we refer
the reader to [].
2.2. e Probability Generating Function. Now we apply the
generating function method. Multiplying ()by𝑥
1𝑦
2𝑥𝐼
3𝑦𝐼
4V
5
and summing over ,, 𝐼,
𝐼,andV, then applying the
properties of generating function, we obtain
=−
𝑇+𝐿−
1
1−
2
2−
3
3
−
4
4−
5
5−
1152
15
−
2142
14−
3252
25
−
4242
24+𝑇1+𝐿2+
1
+
2+
5
3+
5
4+
5
+
31−𝜌𝜏 2
15+
32−𝜌𝜏 2
14
+
432
25+
442
24.
()
On simplication we have
=
1−1𝑇+
2−1𝐿+1−
1
1
+1−
2
2+
5−
3
3
+
5−
4
4+1−
5
5
+
13−𝜌𝜏 −
152
15
+
23−𝜌𝜏 −
142
14
+
34−
252
25+
44−
242
24.
()
isiscalledLagrangepartialdierentialequationforthe
probability generating function (pgf) .
2.3. e Marginal Generating Functions. Recall that
1,1,1,= ∞
𝑥=0
∞
𝑦=0
∞
V=0𝑥,𝑦,V()𝑥
1.()
ISRN Applied Mathematics
T : In-host interaction of HIV.
Possible transitions in-host interaction of HIV and immune system cells and corresponding probabilities
Event Population components
(, 𝐼,,𝐼,)att
Population components
(, 𝐼,,𝐼,)at(, + ) Probability of transition
Production of healthy
CD+
4cell ( − 1, 𝐼,,
𝐼,V)(,
𝐼,,
𝐼,V)𝑇
Death of healthy CD+
4
cell ( + 1, 𝐼,,
𝐼,V)(,
𝐼,,
𝐼,V)( + 1)
Infection of healthy
CD+
4cell ( + 1, 𝐼−1,,
𝐼,V+1) (,
𝐼,,
𝐼,V)1( + 1)(V+1)
−𝜌𝜏 + 2( + 1)(𝐼)−𝜌𝜏
Production of
Langerhans cell (,𝐼,−1,
𝐼,V)(,
𝐼,,
𝐼,V)𝐿
Death of Langerhans
cell (,𝐼,+1,
𝐼,V)(,
𝐼,,
𝐼,V)( + 1)
Infection of
Langerhans cell (,𝐼,+1,
𝐼−1,V+1) (,
𝐼,,
𝐼,V)3( + 1)(V+1)+
4( + 1)(𝐼)
Production of virions
from the infected cell (,𝐼+1,,
𝐼+1,V−1) (,
𝐼,,
𝐼,V)(𝐼+ 1) + (𝐼+1)
Death of virions (,𝐼,,
𝐼,V+1) (,
𝐼,,
𝐼,V)(V+1)
Assuming 2=
3=
4=
5=1and solving (), we
obtain the marginal partial generating function for healthy
CD+
4cells:
1,1,1,1,1;
=
1−1𝑇+1−
1
1
+
1−𝜌𝜏 −
12
15+
2−𝜌𝜏 −
12
14.
()
Assuming 1=
3=
4=
5=1and solving (), we
obtain the marginal partial generating function for healthy
Langerhans cells:
1,2,1,1,1;
=
2−1𝐿+1−
2
2
+
31 − 22
25+
41 − 22
24.
()
Assuming 1=
2=
4=
5=1and solving (), we
obtain the marginal partial generating function for infected
CD+
4cells:
1,1,3,1,1;
=−
3
3+
13−𝜌𝜏 −1 2
15
+
23−𝜌𝜏 −1 2
14.
()
Assuming 1=
2=
3=
5=1and solving (), we
obtain the marginal partial generating function for infected
Langerhans cells:
1,1,1,
4,1;
=−
4
4+
34−1 2
25.
()
Assuming 1=
2=
3=
4=1and solving (),
we obtain the marginal partial generating function for virus
population:
1,1,1,1,
5;
=
5−1
3+
5−1
4+1−
5
5
+
1−𝜌𝜏 −
52
15+
31 − 52
25.
()
2.4. Numbers of Cells and the Virions. As we know from
probability generating function,
𝑥
=∞
𝑥=0𝑥()𝑥−1.()
Letting =1,wehave
𝑥
𝑧=1 =∞
𝑥=0𝑥()𝑥−1 =[].()
Dierentiating the partial dierential equation of the pgf ,
we get the moments of (), (),𝐼(),𝐼(),and().
ISRN Applied Mathematics
Dierentiating ()withrespectto1and setting =1,we
have
[()]=
𝑇−[()]
−
1[()()]−
2()𝐼().
()
Dierentiating ()withrespectto2and setting =1,we
have
[()]=
𝐿−[()]−
3[()()]
−
4()𝐼().
()
Dierentiating ()withrespectto3and setting =1,we
have
𝐼()=−
𝐼()+
1−𝜌𝜏[()()]
+
2−𝜌𝜏()𝐼().
()
Dierentiating ()withrespectto4and setting =1,we
have
𝐼()=−
𝐼()+
3[()()].()
Dierentiating ()withrespectto5and setting =1,we
have
[()]=
𝐼()+𝐼()
−[()]−
1[()()]
−
3[()()].
()
erefore the moments of (), (),𝐼(),𝐼(),and()
from the pgf before introduction of treatment are
[()]=
𝑇−[()]−
1[()()]
−
2()𝐼(),
[()]=
𝐿−[()]−
3[()()]
−
4()𝐼(),
𝐼()=−
𝐼()+
1−𝜌𝜏[()()]
+
2−𝜌𝜏()𝐼(),
𝐼()=−
𝐼()+
3[()()],
[()]=
𝐼()+𝐼()−[()]
−
1[()()]−
3[()()].
()
3. HIV Dynamics under Therapeutic
Intervention
Now we introduce the treatment eect on the HIV-immune
cell interaction. If we let 1−be the reverse transcriptase
inhibitor and let 1−be the protease inhibitor drug eects,
then introducing these treatment eects on our model, we
have the following forward Kolmogorov partial dierential
equations (Master equation) for 𝑥,𝑥𝐼,𝑦,𝑦𝐼,V():
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
=−
𝑇+𝐿+++(1−
)1V
+(1−
)2𝐼+(1−
)3V+(1−
)4𝐼
+(1−
)𝐼+(1−
)𝐼+V
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
+𝑇𝑥−1,𝑥𝐼,𝑦,𝑦𝐼,V()+𝐿𝑥,𝑥𝐼,𝑦−1,𝑦𝐼,V()
+(+1
)𝑥+1,𝑥𝐼,𝑦,𝑦𝐼,V()++1
𝑥,𝑥𝐼,𝑦+1,𝑦𝐼,V()
+
1(+1
)(V+1
)+
2(+1
)𝐼
×(1−
)−𝜌𝜏𝑥+1,𝑥𝐼−1,𝑦,𝑦𝐼,V+1 ()
+
3 + 1 (V+1
)+
4 + 1 𝐼
×(1−
)𝑥,𝑥𝐼,𝑦+1,𝑦𝐼−1,V+1 ()
+(1−
)𝐼+1
𝑥,𝑥𝐼+1,𝑦,𝑦𝐼,V−1 ()
+(1−
)𝐼+1
𝑥,𝑥𝐼,𝑦,𝑦𝐼+1,V−1 ()
+(V+1
)𝑥,𝑥𝐼,𝑦,𝑦𝐼,V+1 (),
()
with initial condition
0,0,0,0,0 ()=−
𝑇+𝐿
0,0,0,0,0 ()+
1,0,0,0,0 ()
+
0,0,1,0,0 ()+
0,0,0,0,1 ().()
Solving the Master equation using generating function, the
Lagrange partial dierential equation becomes
=
1−1𝑇+
2−1𝐿
+1−
1
1+1−
2
2
+
5−
3(1−
)
3
+
5−
4(1−
)
4
+1−
5
5+(1−
)13−𝜌𝜏 −
152
15
ISRN Applied Mathematics
T : Parameters for the stochastic model.
Parameter Parameter description Estimate
(1 − ) e reverse transcriptase inhibitor drug eect .
(1 − ) e protease inhibitor drug eect .
𝑇Rate of production of healthy CD+
4cells per unit time . day−1mm−3
𝐿Rate of production of LCs per unit time day−1mm−3
Death rate of healthy CD+
4cells . day−1
Death rate of healthy LCs . day−
1Infection between uninfected CD+
4cells and free virus particles . day−1mm−3
2Infection between uninfected CD+
cells and HIV infected LCs . day−1mm−3
3Infection between uninfected LCs and free virus particles . day−1mm−3
4Infection between uninfected LCs and HIV infected LCs . day−1mm−3
Death rate of infected CD+
4cells . day−1
∗Viral induced death of infected CD+
4cells . day−1
Death rate of infected but not yet virus producing cell . day−1
Intracellular delay time .
Death rate of infected LCs . day−1
∗Viral induced death rate of infected LCs . day−1
Death rate of infective virus particles . day−1
Viral production rate by infected CD+
4cells day−1
Viral production rate by infected LCs . day−1
Where 𝜅=𝜅
∗+𝛿and 𝛾=𝛾
∗+𝜎,valuesadaptedfrom[,].
T : Variables for the stochastic model.
Variable Description Initial condition
() e concentration of healthy (susceptible) CD+
4cells at time t cells mm−3
𝐼() e concentration of infected CD+
4cells at time tcellsmm
−3
() e concentration of healthy (susceptible) Langerhans cells at time t cells mm−3
𝐼() e concentration of infected Langerhans cells at time tcellsmm
−3
() e concentration of virus particles at time t. virion mm−3
+(1−
)23−𝜌𝜏 −
142
14
+(1−
)34−
252
25
+(1−
)44−
242
24.
()
Solvingthepgfofthein-hostHIVdynamicswith
therapeutic intervention, we have the moments of
(),(),𝐼(),𝐼(),and():
[()]=
𝑇−[()]−(1−
)1[()()]
−(1−
)2()𝐼(),
[()]=
𝐿−[()]−(1−
)3[()()]
−(1−
)4()𝐼(),
𝐼()=−
𝐼()+(1−
)1−𝜌𝜏[()()]
+(1−
)2−𝜌𝜏()𝐼(),
𝐼()=−
𝐼()+(1−
)3[()()],
[()]=(1−
)𝐼()+(1−
) 𝐼()
−[()]−(1−
)1[()()]
−(1−
)3[()()].
()
3.1. Simulation of the Models
3.1.1. Variables and Parameter Values. e initial variable
values and parameter values for the model are described in
Tables and .
Using the parameter values and initial conditions dened
in Tables and ,weillustratethegeneraldynamicsofthe
ISRN Applied Mathematics
Populations
Time (days)
𝜏=0
T(t)
L(t)
TI(t)
LI(t)
V(t)
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0 100 200 300 400
F : Cells and HIV population dynamics before therapeutic
intervention. e population dynamics are shown when =0.
𝜏=2
1600
1400
1200
1000
800
600
400
200
0
Time (days)
T(t)
L(t)
TI(t)
LI(t)
V(t)
0100 200 300 400 500
Populations
F : Cells and HIV population dynamics before therapeutic
intervention. e population dynamics are shown when =2.
CD+
4T cells and HIV virus by performing sensitivity analysis
on the eect of intracellular delay and drug ecacy.
From the simulations in Figures and ,itisclearthat,
in the primary stage of HIV infection, drastic decrease in the
levels of healthy immune cells occurs but the number of free
virions and infected LCs increases with time.
With 60%drugecacy,theviruspopulationdropsand
stabilizes aer some time ,butthenumberofinfectedLCs
increases with time; see Figures and .
With an increase in ecacy for the current HIV drugs, a
patient will have low, undetectable viral load levels, but the
population of infected LCs is still at large; see Figures and
.
Time (days)
T(t)
L(t)
TI(t)
LI(t)
V(t)
0200 400 600 800 1000
Populations
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
𝜏=0,with 60% drug ecacy
F : Cells and HIV population dynamics under therapeutic
intervention. e population dynamics are shown with drug ecacy
of % and when =0.
𝜏=2,with 60% drug ecacy
5000
4000
3000
2000
1000
0
0 200 400 600 800 1000
Populations
Time (days)
T(t)
L(t)
TI(t)
LI(t)
V(t)
F : Cells and HIV population dynamics under therapeutic
intervention. e population dynamics are shown with drug ecacy
of % and when =2.
4. Discussion
In this study, we derived and analysed a stochastic model
describing the dynamics of HIV, CD+
4Tcells,andLCs
interactions under therapeutic intervention in vivo.is
model included dynamics of ve compartments—the num-
ber of healthy CD+
4cells, the number of infected CD+
4
cells, the number of healthy Langerhans cells, the number
of infected Langerhans cells, and the HIV virions. e
model describes HIV infection before and during therapy.
We derived equations for the joint probability generating
function of the numbers of healthy immune cells, the HIV
infected immune cells, and the free HIV particles at any time
and obtained the moment structures of the healthy immune
cells, infected immune cells, and the virus particles over time
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85% drug ecacy
5000
4000
3000
2000
1000
0
Populations
0 200 400 600 800 1000
Time (days)
T(t)
L(t)
TI(t)
LI(t)
V(t)
F : Cells and HIV population dynamics under therapeutic
intervention. e population dynamics are shown with drug ecacy
of %.
𝜏=1,with 85% drug ecacy
6000
0 200 400 600 800 1000
Time (days)
T(t)
L(t)
TI(t)
LI(t)
V(t)
5000
4000
3000
2000
1000
0
Populations
F : Cells and HIV population dynamics under therapeutic
intervention. e population dynamics are shown with drug ecacy
of % and when =1.
.Wesimulatedthemeannumberofthehealthyimmune
cells,theinfectedimmunecells,andthevirusparticlesbefore
and aer combined therapeutic treatment at any time .
Our analysis shows that eradication of HIV is not possible
without clearance of latently infected Langerhans cells. ere-
fore, understanding HIV therapeutic treatment dynamics in
vivo is critical to eliminate the virus, which shows that LCs
are important in determining the disease progression. Our
model analysis suggests that therapies should be developed
to block the binding of HIV onto Langerhans cells. Such
therapies will have the potential to dramatically accelerate
viral decay. We conclude that, to control the concentrations
of the virus and the infected cells in HIV infected person, a
strategy should aim to improve the drug ecacy; hence the
ecacyoftheproteaseinhibitorandthereversetranscriptase
inhibitor and also the intracellular delay play crucial role
in preventing the progression of HIV []. ese ndings
illustratetheroleofLCsasacentralhubofinteraction
and information exchange during HIV infection. Our model
produces interesting feature that classication of HIV disease
states should not be based on CD+
4cellsastheonlyimmune
cells infected by the virus, but the most reliable HIV state
classication criteria should be classication using clinical
signs (the CDC/WHO classication).
Inourwork,thedynamicsofmutantviruswerenot
considered and also our study only included dynamics of only
ve compartments (healthy immune cells, infected immune
cells, and free virus particles ignoring the latency of infected
immune cells) of which extensions are recommended for
further extensive research. In a follow-up work, we intend to
obtain real data in order to test the ecacy of our models as
we have done here with parameter values from the literature.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
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