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Our analysis shows that eradication of HIV is not possible without clearance of latently infected Langerhans cells. Therefore, 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 efficacy; hence the efficacy of the protease inhibitor and the reverse transcriptase inhibitor and also the intracellular delay play crucial role in preventing the progression of HIV [13]. These findings illustrate the role of LCs as a central hub of interaction and information exchange during HIV infection. Our model produces interesting feature that classification of HIV disease states should not be based on cells as the only immune cells infected by the virus, but the most reliable HIV state classification criteria should be classification using clinical signs (the CDC/WHO classification).
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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 specic 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 inuence 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 aect 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 eects 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
andmucosaaretherstlineofdefenseoftheorganism
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 specic cooperation with CD+
4-positive
lymphocytes aer 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 eects 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 aer Paul Langerhans, a
German physician and anatomist, who discovered the cells
attheageofinwhilehewasamedicalstudent[].
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 aect 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 dier-
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. Specically, 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 signicant 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 .
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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
1e transmission coecient between uninfected
CD+
4cells and infective virus particles
2e transmission coecient between uninfected
CD+
4cells and IV infected Langerhans cells
3e transmission coecient between uninfected
Langerhans cells and infective virus particles
4e transmission coecient 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 + 2𝐼 + 3V
+
4𝐼 +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 dierence dierential
equation:
󸀠
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
=−
𝑇+𝐿+++
1V+
2𝐼
+
3V+
4𝐼+
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 dierential 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
15
−
2142
14−
3252
25
−
4242
24+𝑇1+𝐿2+
1
+
2+
5 
3+
5 
4+
5
+
31−𝜌𝜏 2
15+
32−𝜌𝜏 2
14
+
432
25+
442
24.
()
On simplication we have

 =
1−1𝑇+
2−1𝐿+1
1
1
+1−
2
2+
5−
3
3
+
5−
4
4+1−
5
5
+
13−𝜌𝜏 −
152
15
+
23−𝜌𝜏 −
142
14
+
34−
252
25+
44−
242
24.
()
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;

=
1−1𝑇+1−
1
1
+
1−𝜌𝜏 −
12
15+
2−𝜌𝜏 −
12
14.
()
Assuming 1=
3=
4=
5=1and solving (), we
obtain the marginal partial generating function for healthy
Langerhans cells:
1,2,1,1,1;

=
2−1𝐿+1−
2
2
+
31 − 22
25+
41 − 22
24.
()
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+
13−𝜌𝜏 −12
15
+
23−𝜌𝜏 −12
14.
()
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+
34−12
25.
()
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
15+
31 − 52
25.
()
2.4. Numbers of Cells and the Virions. As we know from
probability generating function,
𝑥
 =
𝑥=0𝑥()𝑥−1.()
Letting =1,wehave
𝑥

𝑧=1 =
𝑥=0𝑥()𝑥−1 =[].()
Dierentiating the partial dierential equation of the pgf ,
we get the moments of (), (),𝐼(),𝐼(),and().
ISRN Applied Mathematics
Dierentiating ()withrespectto1and setting =1,we
have
[()]=
𝑇−[()]
−
1[()()]−
2()𝐼().
()
Dierentiating ()withrespectto2and setting =1,we
have
[()]=
𝐿−[()]−
3[()()]
−
4()𝐼().
()
Dierentiating ()withrespectto3and setting =1,we
have

𝐼()=−
𝐼()+
1−𝜌𝜏[()()]
+
2−𝜌𝜏()𝐼().
()
Dierentiating ()withrespectto4and setting =1,we
have
𝐼()=−
𝐼()+
3[()()].()
Dierentiating ()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 eect on the HIV-immune
cell interaction. If we let 1−be the reverse transcriptase
inhibitor and let 1−be the protease inhibitor drug eects,
then introducing these treatment eects on our model, we
have the following forward Kolmogorov partial dierential
equations (Master equation) for 𝑥,𝑥𝐼,𝑦,𝑦𝐼,V():
󸀠
𝑥,𝑥𝐼,𝑦,𝑦𝐼,V()
=−
𝑇+𝐿+++(1−
)1V
+(1−
)2𝐼+(1−
)3V+(1−
)4𝐼
+(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 dierential equation becomes

 =
1−1𝑇+
2−1𝐿
+1−
1
1+1−
2
2
+
5−
3(1−
)
3
+
5−
4(1−
)
4
+1−
5
5+(1−
)13−𝜌𝜏 −
152
15
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T : Parameters for the stochastic model.
Parameter Parameter description Estimate
(1 − ) e reverse transcriptase inhibitor drug eect .
(1 − ) e protease inhibitor drug eect .
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 tcellsmm
−3
() e concentration of healthy (susceptible) Langerhans cells at time t cells mm−3
𝐼() e concentration of infected Langerhans cells at time tcellsmm
−3
() e concentration of virus particles at time t. virion mm−3
+(1−
)23−𝜌𝜏 −
142
14
+(1−
)34−
252
25
+(1−
)44−
242
24.
()
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 dened
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 eect of intracellular delay and drug ecacy.
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 aer some time ,butthenumberofinfectedLCs
increases with time; see Figures and .
With an increase in ecacy 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 ecacy
F : Cells and HIV population dynamics under therapeutic
intervention. e population dynamics are shown with drug ecacy
of % and when =0.
𝜏=2,with 60% drug ecacy
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 ecacy
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
ISRN Applied Mathematics
85% drug ecacy
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 ecacy
of %.
𝜏=1,with 85% drug ecacy
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 ecacy
of % and when =1.
.Wesimulatedthemeannumberofthehealthyimmune
cells,theinfectedimmunecells,andthevirusparticlesbefore
and aer 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 ecacy; 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 classication of HIV disease
states should not be based on CD+
4cellsastheonlyimmune
cells infected by the virus, but the most reliable HIV state
classication criteria should be classication using clinical
signs (the CDC/WHO classication).
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 ecacy of our models as
we have done here with parameter values from the literature.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
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... Mbogo et al. (2014) studied a stochastic model for Langerhans cells and HIV dynamics in vivo. In their study, they derived and analysed a stochastic model explaining the dynamics of HIV, CD4 + T-cells and Langerhans cells interactions under therapeutical interventions in vivo. ...
Thesis
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Found in the genital mucosa are antigen presenting Langerhans cells that have characteristics and features that attract R5 HIV towards them. Some characteristics include their ability to capture and degrade R5 HIV and the loss of their antigen presenting properties when they are overwhelmed. Hence, they play both inhibitory and mediatory roles for R5 HIV infection. The project investigates the benefits of modelling R5 HIV dynamics in the Langerhans cells during early HIV infection within the host incorporating treatment with maraviroc as an entry inhibitor. Critical thresholds, the reproduction number (R 0), important in determining the invasion of R5 HIV in the Langerhans cells were derived. The results in this study showed that there exist two equilibrium points. These are the disease free equilibrium (DFE) point and the endemic equilibrium point. Stability analysis of the equilibrium points using the Routh-Hurwitz and the center manifold theory techniques was carried out. Our results also showed that there is a supercritical bifurcation that ensures the transition of stability between the two equilibrium points when R 0 = 1. The numerical simulation results suggest that entry inhibitors may need to have high efficacies for them to be effective in reducing the infection of Langerhans cells by R5 HIV.
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In this paper, we propose detailed and reasonable viral dynamics by using a multi-compartment model that incorporating the age since the infection of multiple infected cells, multiple target cells (Langerhans-cells and CD4[Formula: see text] T-cells), multiple viral strains (CCR5 and CXR4 HIV) and multiple infection routes (cell-to-cell and cell-to-virus). The basic reproduction number, [Formula: see text], of the whole model is derived from two transmission mechanisms: one is the potential trigger from the infection routes for a single target cell and other is the joint effect of multiple viral infections for multi-target cells. Accordingly, we study the global stability of the steady states for the single target model. For the whole model, we prove that the infection-free steady state is globally asymptotically stable if [Formula: see text], whereas viruses persist uniformly if [Formula: see text]. Numerical simulations are carried out to illustrate the theoretical results. Sensitive analyses expound the effect of model parameters on the comprehensive reproduction number. It is remarkable to find that simultaneous control of HIV infection for two target cells can effectively reduce the viral loads within-host. Finally, our work suggests that the synergetic mechanism of multi-target cells and multi-strain cannot be ignored during treatment.
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Most existing models have considered the immunological processes occurring within the host and the epidemiological processes occurring at population level as decoupled systems. We present a new model using continuous systems of non linear ordinary differential equations by directly linking the within host dynamics capturing the interactions between Langerhans cells, CD4\(^+\) T-cells, R5 HIV and X4 HIV and the without host dynamics of a basic compartmental HIV/AIDS model. The model captures the biological theories of the cells that take part in HIV transmission. The study incorporates in its analysis the differences in time scales of the fast within host dynamics and the slow without host dynamics. In the mathematical analysis, important thresholds, the reproduction numbers, were computed which are useful in predicting the progression of the infection both within the host and without the host. The study results showed that the model exhibits four within host equilibrium points inclusive of three endemic equilibria whose effects translate into different scenarios at the population level. All the endemic equilibria were shown to be globally stable using Lyapunov functions and this is an important result in linking the within host dynamics to the population dynamics, because the disease free equilibrium point ceases to exist. The effects of linking were observed on the endemic equilibrium points of both the within host and population dynamics. Linking the two dynamics was shown to increase in the viral load within the host and increase in the epidemic levels in the population dynamics.
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A within-host mathematical model to describe the dynamics of target cells and viral load in early HIV-1 infection was developed, which incorporates a combination of RTI and PI treatments by using a pharmacokinetics model. The local stability of uninfected steady state for the model was determined using an alternative threshold. The pharmacokinetics model was employed to estimate drug efficacy in multiple drug dosing. The effect of periodic drug efficacy of pharmacokinetic type on outcome of HIV-1 infection was explored under various treatment interruptions. The effectiveness of treatment interruption was determined according to the time period of the drug holidays. The results showed that long drug holidays lead to therapy failure. Under interruption of treatments combining RTI and PI therapy, effectiveness of the treatment requires a short duration of the drug holiday.
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This article discusses the mathematical and numerical modeling of the immune system of the course of HIV infection without treatment. Presently a significant number of scientific papers are devoted to the study of this problem. However, HIV infection is highly volatile and there is no effective drug, in that HIV has the ability to mutate and reproduce itself in the presence of chemical substances that are meant to inhibit or destroy it. The mathematical models used in this paper are conceptual and exploratory in nature. The proposed mathematical model allow us to obtain a complete description of the dynamics of HIV infection, and also an understanding of the progression to AIDS. Thus, the results of the numerical solution of differential equations in this work show that: the disease develops, and at low concentration of the virus, a certain level of stability does not depend on the initial concentration of infestation. In the absence of treatment, for interesting competition between virus and the loss of virus caused by immune response should be strictly greater than the rate of multiplication of the virus in the blood; the reproduction rate of the uninfected cells should be strictly greater than the mortality rate of the uninfected cells.
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Since the 1990s we have known of the fascinating ability of a complex set of professional antigen presenting cells (APCs; dendritic cells, monocytes/macrophages, and B lymphocytes) to mediate HIV-1 trans infection of CD4(+) T cells. This 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 T cells, or trans infection between T cells. Such APC-to-T cell trans infection first involves a complex set of virus subtype, attachment, entry, and replication patterns that have many similarities among APC, as well as distinct differences related to virus receptors, intracellular trafficking, and productive and nonproductive replication pathways. The end result is that HIV-1 can sequester within the APC for several days and be transmitted via membrane extensions intracellularly and extracellularly to T cells across the virologic synapse. Virus replication requires activated T cells that can develop concurrently with the events of virus transmission. Further research is essential to fill the many gaps in our understanding of these trans infection processes and their role in natural HIV-1 infection.
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Mathematical models are used to provide insights into the mechanisms and dynamics of the progression of viral infection in vivo. Untangling the dynamics between HIV and CD4+ cellular populations and molecular interactions can be used to investigate the effective points of interventions in the HIV life cycle. With that in mind, we develop and analyze a stochastic model for In-Host HIV dynamics that includes combined therapeutic treatment and intracellular delay between the infection of a cell and the emission of viral particles. The unique feature is that both therapy and the intracellular delay are incorporated into the model. We show the usefulness of our stochastic approach towards modeling combined HIV treatment by obtaining probability generating function, the moment structures of the healthy CD4+ cell, and the virus particles at any time t and the probability of virus clearance. Our analysis show that, when it is assumed that the drug is not completely effective, as is the case of HIV in vivo, the predicted rate of decline in plasma HIV virus concentration depends on three factors: the initial viral load before therapeutic intervention, the efficacy of therapy and the length of the intracellular delay.
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Untangling the dynamics between HIV and CD4 cellular populations and molecular interactions can be used to investigate the effective points of interventions in the HIV life cycle. With that in mind,we propose and show the usefulness of a stochastic approach towards modeling HIV and CD4 cells Dynamics in Vivo by obtaining probability generating function, the moment structures of the healthy CD4 cell and the virus particles at any time t and the probability of HIV clearance. The unique feature is that both therapy and the intracellular delay are incorporated into the model. Our analysis show that, when it is assumed that the drug is not completely effective, as is the case of HIV in vivo, the probability of HIV clearance depends on two factors: the combination of drug efficacy and length of the intracellular delay and also education to the infected patients. Comparing simulated data for before and after treatment in-dicates the importance of combined therapeutic intervention and intra-cellular delay in having low, undetectable viral load in HIV infected person.
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HIV-1 Nef enhances dendritic cell (DC)-mediated viral transmission to CD4(+) T cells, but the underlying mechanism is not fully understood. It is also unknown whether HIV-1 infected DCs play a role in activating CD4(+) T cells and enhancing DC-mediated viral transmission. Here we investigated the role of HIV-1 Nef in DC-mediated viral transmission and HIV-1 infection of primary CD4(+) T cells using wild-type HIV-1 and Nef-mutated viruses. We show that HIV-1 Nef facilitated DC-mediated viral transmission to activated CD4(+) T cells. HIV-1 expressing wild-type Nef enhanced the activation and proliferation of primary resting CD4(+) T cells. However, when co-cultured with HIV-1-infected autologous DCs, there was no significant trend for infection- or Nef-dependent proliferation of resting CD4(+) T cells. Our results suggest an important role of Nef in DC-mediated transmission of HIV-1 to activated CD4(+) T cells and in the activation and proliferation of resting CD4(+) T cells, which likely contribute to viral pathogenesis.
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Even though the treatment of human immunodeficiency virus (HIV)-infected individuals with highly active antiretroviral therapy (HAART) provides a complete control of plasma viremia to below detectable levels (<40 copies/mL plasma), there is an unequal distribution of all antiretroviral drugs across diverse cellular and anatomic compartments in vivo. The main consequence of this is the acquisition of resistance by HIV to all known classes of currently prescribed antiretroviral drugs and the establishment of HIV reservoirs in vivo. HIV has a distinct advantage of surviving in the host via both pre-and postintegration latency. The postintegration latency is caused by inert and metabolically inactive provirus, which cannot be accessed either by the immune system or the therapeutics. This integrated provirus provides HIV with a safe haven in the host where it is incessantly challenged by its immune selection pressure and also by HAART. Thus, the provirus is one of the strategies for viral concealment in the host and the provirus can be rekindled, through unknown stimuli, to create progeny for productive infection of the host. Thus, the reservoir establishment remains the biggest impediment to HIV eradication from the host. This review provides an overview of HIV reservoir sites and discusses both the virtues and problems associated with therapies/strategies targeting these reservoir sites in vivo.
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Dendritic cells initiate and sustain immune responses by migrating to sites of pathogenic insult, transporting antigens to lymphoid tissues and signaling immune specific activation of T cells through the formation of the immunological synapse. Dendritic cells can also transfer intact, infectious HIV-1 to CD4 T cells through an analogous structure, the infectious synapse. This replication independent mode of HIV-1 transmission, known as trans-infection, greatly increases T cell infection in vitro and is thought to contribute to viral dissemination in vivo. This review outlines the recent data defining the mechanisms of trans-infection and provides a context for the potential contribution of trans-infection in HIV-1 disease.
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HIV is a devastating human pathogen that causes serious immunological diseases in humans around the world. The virus is able to remain latent in an infected host for many years, allowing for the long-term survival of the virus and inevitably prolonging the infection process. The 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 CD4+ T cells, peripheral blood monocytes, dendritic cells and macrophages in the lymph nodes, and haematopoietic stem cells in the bone marrow. This review updates the latest advances in the study of HIV interactions with monocytes and dendritic cells, and highlights the potential role of these cells as viral reservoirs and the effects of the HIV-host-cell interactions on viral pathogenesis.
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Langerhans cells (LCs) have long been considered to be the major sensitizing cells in the skin by initiating productive immunity in naive resting T cells. This picture has changed over the past decade. We now know (i) that the skin also harbors other types of dendritic antigen-presenting cells and (ii) that the genetically driven removal of the LC population results in increased T-cell immunity against haptens and infectious agents. It is not clear at present whether the situation in genetically modified mice is in any way indicative of the actual in vivo function of LCs. Exciting and challenging years lie ahead of the LC research community.
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Word processed copy. Thesis (M.Sc. (Mathematics & Applied Mathematics))--University of Cape Town, 2005. Includes bibliographical references.