# Stochastic Model for Langerhans Cells and HIV DynamicsIn Vivo

Authors: Waema R. Mbogo; Livingstone S. Luboobi; John W. Odhiambo
Journal: ISRN Applied Mathematics
 (2014)

---

## Abstract

Many aspects of the complex interaction between HIV and the human immune system remain
elusive. Our objective is to study these interactions, focusing on the specific 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 influence of LCs on HIV viral dynamics during antiretroviral therapy, we present and analyse a stochastic model describing the dynamics of HIV,C
D
4
+
T 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 affect infection dynamics.


---

## Body

## 1. Introduction

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 [1]. 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 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 [2]. This 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 effects of the HIV-host-cell interactions on viral pathogenesis.Despite advances in our understanding of HIV and the human immune response in the last 25 years, much of this complex interaction remains elusive.CD
4
+ T cells are targets of HIV and are also important for the establishment and maintenance of an adaptive immune response [3]. The skin and mucosa are the first line of defense of the organism 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. The immunocompetent cells which act as antigen-presenting cells are Langerhans cells (LCs). Infection of LCs by HIV is relevant to several reasons. Firstly, LCs of mucosal epithelia may be among the first cells to be infected following mucosal HIV exposure and, secondly, LCs may serve as a reservoir for continued infection of CD
4
+ T cells, especially in lymph nodes where epidermal LCs migrate following antigenic activation [4].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 [5]. 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 response through a specific cooperation with CD
4
+-positive lymphocytes after migration to proximal lymph nodes [6]. Apart from many plasma membrane determinants, LCs also express CD
4
+ molecules which make them susceptible targets and reservoirs for HIV [7]. 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 effects 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 [7].Langerhans cells (LCs), which are members of the dendritic 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 [8]. They 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
4
+ T lymphocytes [8]. These cells play an important role in the transmission of HIV to CD
4
+ cells [9]; thus, LC-CD
4
+ cell interactions in lymphoid tissue, which are critical in the generation of immune responses, are also a major catalyst for HIV replication and expansion. This 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 [10].The Langerhans cell is named after Paul Langerhans, a German physician and anatomist, who discovered the cells at the age of 21 in 1868 while he was a medical student [11]. The uptake of HIV by professional antigen-presenting cells (APCs) and subsequent transfer of virus to CD
4
+ T cells can result in explosive levels of virus replication in the T cells. This could be a major pathogenic process in HIV infection and development of AIDS. This 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 [12]. 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 T cells, or trans-infection between T cells. This consequently shows that Langerhans cells may be responsible for the quick spread of HIV infection.The individual cells of the immune system are highly interactive, and the overall function of the system is a product of this multitude of interactions. The 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 [3]. 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
4
+ T 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 first develop a stochastic model of HIV dynamics in vivo before therapy. Next, we introduce therapeutic intervention and finally investigate which parameters and/or which interaction mechanisms strongly affect the infection dynamics.The organization of this paper is as follows. In Section2, we formulate our stochastic model describing the interaction of HIV and the immune system and obtain a partial differential 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 t. The marginal probability distributions for the variables and the population measures which include the expectation of the variables are also derived in Section 2. HIV dynamic model incorporating therapy is derived and model analysis is discussed in Section 3. Some concluding remarks follow in Section 4.


## 2. The Role of LC in HIV InfectionIn Vivo

In HIV infection, LCs play a dual role in promoting immunity 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 ofCD
4
+ T cells [15]. 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 [16, 17].To study the roles of LCs during HIV infection, we present a mathematical model of HIV infection and accompanying immune response. Specifically, we develop a stochastic model focusing on the HIV-LC-CD
4
+ cell dynamics in vivo. Our model analysis allows us to predict the importance of LC mechanisms and their role in triggering progression to AIDS. In particular, our model predicts which mechanisms of LC dysfunction are most significant in the transition to AIDS. A typical life cycle of HIV virus and immune system interaction is shown in Figure 1.Figure 1
The interaction of HIV virus and the immune system.


### 2.1. Variables and Parameters for the Model

The variables and parameters in the model are described as in Tables1 and 2.Table 1
Variables for the stochastic model.
Variable
Description
T
(
t
)
The concentration of healthy (susceptible)C
D
4
+ cells at time t
T
I
(
t
)
The concentration of infectedC
D
4
+ cells at time t
L
(
t
)
The concentration of healthy (susceptible) Langerhans cells at timet
L
T
(
t
)
The concentration of latently infected Langerhans cells at timet
L
I
(
t
)
The concentration of infected Langerhans cells at timet
V
(
t
)
The concentration of virus particles at timetTable 2
Parameters for the stochastic model.
Parameter
Description
λ
T
The total rate of production of healthyC
D
4
+ cells per unit time
λ
L
The total rate of production of Langerhans cells per unit time
δ
The per capita death rate of healthyC
D
4
+ cells
σ
The per capita death rate of healthy Langerhans cells
β
1
The transmission coefficient between uninfectedC
D
4
+ cells and  infective virus particles
β
2
The transmission coefficient between uninfectedC
D
4
+ cells and IV infected Langerhans cells
β
3
The transmission coefficient between uninfected Langerhans cells and  infective virus particles
β
4
The transmission coefficient between uninfected Langerhans cells and HIV infected Langerhans cells
κ
Per capita death rate of infectedC
D
4
+ cells
τ
Intracellular delay time
γ
Per capita death rate of infected Langerhans cells
μ
The per capita death rate of infective virus particles
N
The average number of infective virus particles produced  by an infectedC
D
4
+ cell in the absence of treatment during its entire infectious lifetime
M
The average number of infective virus particles produced by an infected Langerhans cell in the absence of treatment during its entire infectious lifetimeFrom Table1 and by using the population change scenarios and parameters in Table 2 and applying probability arguments, we now summarize the events that occur during the interval (
t
,
t
+
Δ
t
) together with their transition probabilities in Table 3.Table 3
In-host interaction of HIV.
Possible transitions in-host interaction of HIV and immune system cells and corresponding probabilities
Event
Population components(T
,
T
I
,
L
,
L
I
,
V) at t
Population components(T
,
T
I
,
L
,
L
I
,
V) at (
t
,
t
+
Δ
t
)
Probability of transition
Production of healthyC
D
4
+ cell
(
x
-
1
,
x
I
,
y
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
λ
T
Δ
t
Death of healthyC
D
4
+ cell
(
x
+
1
,
x
I
,
y
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
δ
(
x
+
1
)
Δ
t
Infection of healthyC
D
4
+ cell
(
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
β
1
(
x
+
1
)
(
v
+
1
)
e
-
ρ
τ
Δ
t
+
β
2
(
x
+
1
)
(
y
I
)
e
-
ρ
τ
Δ
t
Production of Langerhans cell
(
x
,
x
I
,
y
-
1
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
λ
L
Δ
t
Death of Langerhans cell
(
x
,
x
I
,
y
+
1
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
σ
(
y
+
1
)
Δ
t
Infection of Langerhans cell
(
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
β
3
(
y
+
1
)
(
v
+
1
)
Δ
t
+
β
4
(
y
+
1
)
(
y
I
)
Δ
t
Production of virions from the infected cell
(
x
,
x
I
+
1
,
y
,
y
I
+
1
,
v
-
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
κ
N
(
x
I
+
1
)
Δ
t
+
γ
M
(
y
I
+
1
)
Δ
t
Death of virions
(
x
,
x
I
,
y
,
y
I
,
v
+
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
μ
(
v
+
1
)
Δ
tThe change in population size during the time intervalΔ
t, which is assumed to be sufficiently small to guarantee that only one such event can occur in (
t
,
t
+
Δ
t
), is governed by the following conditional probabilities:
(1)
P
x
,
x
I
,
y
,
y
I
,
v
(
t
+
Δ
t
)
=
{
1
-
(
λ
T
Δ
t
+
λ
L
Δ
t
+
δ
x
Δ
t
+
σ
y
Δ
t
+
β
1
x
v
Δ
t
+
β
2
x
y
I
Δ
t
+
β
3
y
v
Δ
t
+
β
4
y
y
I
Δ
t
+
κ
x
I
Δ
t
+
γ
y
I
Δ
t
+
μ
v
Δ
t
)
+
o
(
Δ
t
)
}
×
P
x
,
x
I
,
y
,
y
I
,
v
(
t
)
+
{
λ
T
Δ
t
+
o
(
Δ
t
)
}
P
x
-
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
{
λ
L
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
-
1
,
y
I
,
v
(
t
)
+
{
δ
(
x
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
+
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
{
σ
(
y
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
+
1
,
y
I
,
v
(
t
)
+
{
[
β
1
(
x
+
1
)
(
v
+
1
)
+
β
2
(
x
+
1
)
(
y
I
)
]
×
e
-
ρ
τ
Δ
t
+
o
(
Δ
t
)
}
P
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
(
t
)
+
{
[
β
3
(
y
+
1
)
(
v
+
1
)
+
β
4
(
y
+
1
)
(
y
I
)
]
Δ
t
+
o
(
Δ
t
)
}
×
P
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
(
t
)
+
{
κ
N
(
x
I
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
+
1
,
y
,
y
I
,
v
-
1
(
t
)
+
{
γ
M
(
y
I
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
,
y
I
+
1
,
v
-
1
(
t
)
+
{
μ
(
v
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
,
y
I
,
v
+
1
(
t
)
.
Rearranging (1), then dividing through by Δ
t, and taking the limit as Δ
t
⇒
0 we have the following difference differential equation:
(2)
P
x
,
x
I
,
y
,
y
I
,
v
′
(
t
)
=
-
{
λ
T
+
λ
L
+
δ
x
+
σ
y
+
β
1
x
v
+
β
2
x
y
I
+
β
3
y
v
+
β
4
y
y
I
+
κ
x
I
+
γ
y
I
+
μ
v
}
P
x
,
x
I
,
y
,
y
I
,
v
(
t
)
+
λ
T
P
x
-
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
λ
L
P
x
,
x
I
,
y
-
1
,
y
I
,
v
(
t
)
+
δ
(
x
+
1
)
P
x
+
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
σ
(
y
+
1
)
P
x
,
x
I
,
y
+
1
,
y
I
,
v
(
t
)
+
{
β
1
(
x
+
1
)
(
v
+
1
)
+
β
2
(
x
+
1
)
(
y
I
)
}
×
e
-
ρ
τ
P
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
(
t
)
+
{
β
3
(
y
+
1
)
(
v
+
1
)
+
β
4
(
y
+
1
)
(
y
I
)
}
×
P
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
(
t
)
+
κ
N
(
x
I
+
1
)
P
x
,
x
I
+
1
,
y
,
y
I
,
v
-
1
(
t
)
+
γ
M
(
y
I
+
1
)
P
x
,
x
I
,
y
,
y
I
+
1
,
v
-
1
(
t
)
+
μ
(
v
+
1
)
P
x
,
x
I
,
y
,
y
I
,
v
+
1
(
t
)
.
This is also called the Master equation or the forward Kolmogorov partial differential equation for P
x
,
x
I
,
y
,
y
I
,
v
(
t
), with initial condition
(3)
P
0,0
,
0,0
,
0
′
(
t
)
=
-
(
λ
T
+
λ
L
)
P
0,0
,
0,0
,
0
(
t
)
+
δ
P
1,0
,
0,0
,
0
(
t
)
+
σ
P
0,0
,
1,0
,
0
(
t
)
+
μ
P
0,0
,
0,0
,
1
(
t
)
.
For detailed description and derivation of the model we refer the reader to [18].


### 2.2. The Probability Generating Function

Now we apply the generating function method. Multiplying (2) by z
1
x
z
2
y
z
3
x
I
z
4
y
I
z
5
v and summing over x
,
y
,
x
I
,
y
I, and v, then applying the properties of generating function, we obtain
(4)
∂
G
∂
t
=
-
(
λ
T
+
λ
L
)
G
-
δ
z
1
∂
G
∂
z
1
-
σ
z
2
∂
G
∂
z
2
-
κ
z
3
∂
G
∂
z
3
-
γ
z
4
∂
G
∂
z
4
-
μ
z
5
∂
G
∂
z
5
-
β
1
z
1
z
5
∂
2
G
∂
z
1
∂
z
5
-
β
2
z
1
z
4
∂
2
G
∂
z
1
∂
z
4
-
β
3
z
2
z
5
∂
2
G
∂
z
2
∂
z
5
-
β
4
z
2
z
4
∂
2
G
∂
z
2
∂
z
4
+
λ
T
z
1
G
+
λ
L
z
2
G
+
δ
∂
G
∂
z
1
+
σ
∂
G
∂
z
2
+
z
5
N
κ
∂
G
∂
z
3
+
z
5
M
γ
∂
G
∂
z
4
+
μ
∂
G
∂
z
5
+
z
3
β
1
e
-
ρ
τ
∂
2
G
∂
z
1
∂
z
5
+
z
3
β
2
e
-
ρ
τ
∂
2
G
∂
z
1
∂
z
4
+
z
4
β
3
∂
2
G
∂
z
2
∂
z
5
+
z
4
β
4
∂
2
G
∂
z
2
∂
z
4
.
On simplification we have
(5)
∂
G
∂
t
=
{
(
z
1
-
1
)
λ
T
+
(
z
2
-
1
)
λ
L
}
G
+
(
1
-
z
1
)
δ
∂
G
∂
z
1
+
(
1
-
z
2
)
σ
∂
G
∂
z
2
+
(
N
z
5
-
z
3
)
κ
∂
G
∂
z
3
+
(
M
z
5
-
z
4
)
γ
∂
G
∂
z
4
+
(
1
-
z
5
)
μ
∂
G
∂
z
5
+
β
1
(
z
3
e
-
ρ
τ
-
z
1
z
5
)
∂
2
G
∂
z
1
∂
z
5
+
β
2
(
z
3
e
-
ρ
τ
-
z
1
z
4
)
∂
2
G
∂
z
1
∂
z
4
+
β
3
(
z
4
-
z
2
z
5
)
∂
2
G
∂
z
2
∂
z
5
+
β
4
(
z
4
-
z
2
z
4
)
∂
2
G
∂
z
2
∂
z
4
.
This is called Lagrange partial differential equation for the probability generating function (pgf) G.


### 2.3. The Marginal Generating Functions

Recall that(6)
G
(
z
1
,
1,1
,
t
)
=
∑
x
=
0
∞
‍
∑
y
=
0
∞
‍
∑
v
=
0
∞
‍
P
x
,
y
,
v
(
t
)
z
1
x
.
Assuming z
2
=
z
3
=
z
4
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for healthy C
D
4
+ cells:
(7)
∂
G
(
z
1
,
1,1
,
1,1
;
t
)
∂
t
=
(
z
1
-
1
)
λ
T
G
+
(
1
-
z
1
)
δ
∂
G
∂
z
1
+
β
1
(
e
-
ρ
τ
-
z
1
)
∂
2
G
∂
z
1
∂
z
5
+
β
2
(
e
-
ρ
τ
-
z
1
)
∂
2
G
∂
z
1
∂
z
4
.
Assuming z
1
=
z
3
=
z
4
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for healthy Langerhans cells:
(8)
∂
G
(
1
,
z
2
,
1,1
,
1
;
t
)
∂
t
=
(
z
2
-
1
)
λ
L
G
+
(
1
-
z
2
)
σ
∂
G
∂
z
2
+
β
3
(
1
-
z
2
)
∂
2
G
∂
z
2
∂
z
5
+
β
4
(
1
-
z
2
)
∂
2
G
∂
z
2
∂
z
4
.
Assuming z
1
=
z
2
=
z
4
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for infected C
D
4
+ cells:
(9)
∂
G
(
1,1
,
z
3
,
1,1
;
t
)
∂
t
=
(
N
-
z
3
)
κ
∂
G
∂
z
3
+
β
1
(
z
3
e
-
ρ
τ
-
1
)
∂
2
G
∂
z
1
∂
z
5
+
β
2
(
z
3
e
-
ρ
τ
-
1
)
∂
2
G
∂
z
1
∂
z
4
.
Assuming z
1
=
z
2
=
z
3
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for infected Langerhans cells:
(10)
∂
G
(
1,1
,
1
,
z
4
,
1
;
t
)
∂
t
=
(
M
-
z
4
)
γ
∂
G
∂
z
4
+
β
3
(
z
4
-
1
)
∂
2
G
∂
z
2
∂
z
5
.Assumingz
1
=
z
2
=
z
3
=
z
4
=
1 and solving (5), we obtain the marginal partial generating function for virus population:
(11)
∂
G
(
1,1
,
1,1
,
z
5
;
t
)
∂
t
=
(
N
z
5
-
1
)
κ
∂
G
∂
z
3
+
(
M
z
5
-
1
)
γ
∂
G
∂
z
4
+
(
1
-
z
5
)
μ
∂
G
∂
z
5
+
β
1
(
e
-
ρ
τ
-
z
5
)
∂
2
G
∂
z
1
∂
z
5
+
β
3
(
1
-
z
5
)
∂
2
G
∂
z
2
∂
z
5
.


### 2.4. Numbers of Cells and the Virions

As we know from probability generating function,(12)
∂
G
x
∂
z
=
∑
x
=
0
∞
‍
x
P
x
(
t
)
z
x
-
1
.
Letting z
=
1, we have
(13)
∂
G
x
∂
z
|
z
=
1
=
∑
x
=
0
∞
‍
x
P
x
(
t
)
z
x
-
1
=
E
[
X
]
.
Differentiating the partial differential equation of the pgf G, we get the moments of T
(
t
)
,
L
(
t
)
,
T
I
(
t
)
,
L
I
(
t
), and V
(
t
).Differentiating (5) with respect to z
1 and setting z
=
1, we have
(14)
∂
∂
t
E
[
T
(
t
)
]
=
λ
T
-
δ
E
[
T
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
2
E
[
T
(
t
)
L
I
(
t
)
]
.
Differentiating (5) with respect to z
2 and setting z
=
1, we have
(15)
∂
∂
t
E
[
L
(
t
)
]
=
λ
L
-
σ
E
[
L
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
-
β
4
E
[
L
(
t
)
L
I
(
t
)
]
.
Differentiating (5) with respect to z
3 and setting z
=
1, we have
(16)
∂
∂
t
E
[
T
I
(
t
)
]
=
-
κ
E
[
T
I
(
t
)
]
+
β
1
e
-
ρ
τ
E
[
T
(
t
)
V
(
t
)
]
+
β
2
e
-
ρ
τ
E
[
T
(
t
)
L
I
(
t
)
]
.
Differentiating (5) with respect to z
4 and setting z
=
1, we have
(17)
∂
∂
t
E
[
L
I
(
t
)
]
=
-
γ
E
[
L
I
(
t
)
]
+
β
3
E
[
L
(
t
)
V
(
t
)
]
.
Differentiating (5) with respect to z
5 and setting z
=
1, we have
(18)
∂
∂
t
E
[
V
(
t
)
]
=
N
κ
E
[
T
I
(
t
)
]
+
M
γ
E
[
L
I
(
t
)
]
-
μ
E
[
V
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
.
Therefore the moments of T
(
t
)
,
L
(
t
)
,
T
I
(
t
)
,
L
I
(
t
), and V
(
t
) from the pgf before introduction of treatment are
(19)
d
d
t
E
[
T
(
t
)
]
=
λ
T
-
δ
E
[
T
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
2
E
[
T
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
L
(
t
)
]
=
λ
L
-
σ
E
[
L
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
-
β
4
E
[
L
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
T
I
(
t
)
]
=
-
κ
E
[
T
I
(
t
)
]
+
β
1
e
-
ρ
τ
E
[
T
(
t
)
V
(
t
)
]
+
β
2
e
-
ρ
τ
E
[
T
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
L
I
(
t
)
]
=
-
γ
E
[
L
I
(
t
)
]
+
β
3
E
[
L
(
t
)
V
(
t
)
]
,
d
d
t
E
[
V
(
t
)
]
=
N
κ
E
[
T
I
(
t
)
]
+
M
γ
E
[
L
I
(
t
)
]
-
μ
E
[
V
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
.


## 2.1. Variables and Parameters for the Model

The variables and parameters in the model are described as in Tables1 and 2.Table 1
Variables for the stochastic model.
Variable
Description
T
(
t
)
The concentration of healthy (susceptible)C
D
4
+ cells at time t
T
I
(
t
)
The concentration of infectedC
D
4
+ cells at time t
L
(
t
)
The concentration of healthy (susceptible) Langerhans cells at timet
L
T
(
t
)
The concentration of latently infected Langerhans cells at timet
L
I
(
t
)
The concentration of infected Langerhans cells at timet
V
(
t
)
The concentration of virus particles at timetTable 2
Parameters for the stochastic model.
Parameter
Description
λ
T
The total rate of production of healthyC
D
4
+ cells per unit time
λ
L
The total rate of production of Langerhans cells per unit time
δ
The per capita death rate of healthyC
D
4
+ cells
σ
The per capita death rate of healthy Langerhans cells
β
1
The transmission coefficient between uninfectedC
D
4
+ cells and  infective virus particles
β
2
The transmission coefficient between uninfectedC
D
4
+ cells and IV infected Langerhans cells
β
3
The transmission coefficient between uninfected Langerhans cells and  infective virus particles
β
4
The transmission coefficient between uninfected Langerhans cells and HIV infected Langerhans cells
κ
Per capita death rate of infectedC
D
4
+ cells
τ
Intracellular delay time
γ
Per capita death rate of infected Langerhans cells
μ
The per capita death rate of infective virus particles
N
The average number of infective virus particles produced  by an infectedC
D
4
+ cell in the absence of treatment during its entire infectious lifetime
M
The average number of infective virus particles produced by an infected Langerhans cell in the absence of treatment during its entire infectious lifetimeFrom Table1 and by using the population change scenarios and parameters in Table 2 and applying probability arguments, we now summarize the events that occur during the interval (
t
,
t
+
Δ
t
) together with their transition probabilities in Table 3.Table 3
In-host interaction of HIV.
Possible transitions in-host interaction of HIV and immune system cells and corresponding probabilities
Event
Population components(T
,
T
I
,
L
,
L
I
,
V) at t
Population components(T
,
T
I
,
L
,
L
I
,
V) at (
t
,
t
+
Δ
t
)
Probability of transition
Production of healthyC
D
4
+ cell
(
x
-
1
,
x
I
,
y
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
λ
T
Δ
t
Death of healthyC
D
4
+ cell
(
x
+
1
,
x
I
,
y
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
δ
(
x
+
1
)
Δ
t
Infection of healthyC
D
4
+ cell
(
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
β
1
(
x
+
1
)
(
v
+
1
)
e
-
ρ
τ
Δ
t
+
β
2
(
x
+
1
)
(
y
I
)
e
-
ρ
τ
Δ
t
Production of Langerhans cell
(
x
,
x
I
,
y
-
1
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
λ
L
Δ
t
Death of Langerhans cell
(
x
,
x
I
,
y
+
1
,
y
I
,
v
)
(
x
,
x
I
,
y
,
y
I
,
v
)
σ
(
y
+
1
)
Δ
t
Infection of Langerhans cell
(
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
β
3
(
y
+
1
)
(
v
+
1
)
Δ
t
+
β
4
(
y
+
1
)
(
y
I
)
Δ
t
Production of virions from the infected cell
(
x
,
x
I
+
1
,
y
,
y
I
+
1
,
v
-
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
κ
N
(
x
I
+
1
)
Δ
t
+
γ
M
(
y
I
+
1
)
Δ
t
Death of virions
(
x
,
x
I
,
y
,
y
I
,
v
+
1
)
(
x
,
x
I
,
y
,
y
I
,
v
)
μ
(
v
+
1
)
Δ
tThe change in population size during the time intervalΔ
t, which is assumed to be sufficiently small to guarantee that only one such event can occur in (
t
,
t
+
Δ
t
), is governed by the following conditional probabilities:
(1)
P
x
,
x
I
,
y
,
y
I
,
v
(
t
+
Δ
t
)
=
{
1
-
(
λ
T
Δ
t
+
λ
L
Δ
t
+
δ
x
Δ
t
+
σ
y
Δ
t
+
β
1
x
v
Δ
t
+
β
2
x
y
I
Δ
t
+
β
3
y
v
Δ
t
+
β
4
y
y
I
Δ
t
+
κ
x
I
Δ
t
+
γ
y
I
Δ
t
+
μ
v
Δ
t
)
+
o
(
Δ
t
)
}
×
P
x
,
x
I
,
y
,
y
I
,
v
(
t
)
+
{
λ
T
Δ
t
+
o
(
Δ
t
)
}
P
x
-
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
{
λ
L
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
-
1
,
y
I
,
v
(
t
)
+
{
δ
(
x
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
+
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
{
σ
(
y
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
+
1
,
y
I
,
v
(
t
)
+
{
[
β
1
(
x
+
1
)
(
v
+
1
)
+
β
2
(
x
+
1
)
(
y
I
)
]
×
e
-
ρ
τ
Δ
t
+
o
(
Δ
t
)
}
P
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
(
t
)
+
{
[
β
3
(
y
+
1
)
(
v
+
1
)
+
β
4
(
y
+
1
)
(
y
I
)
]
Δ
t
+
o
(
Δ
t
)
}
×
P
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
(
t
)
+
{
κ
N
(
x
I
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
+
1
,
y
,
y
I
,
v
-
1
(
t
)
+
{
γ
M
(
y
I
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
,
y
I
+
1
,
v
-
1
(
t
)
+
{
μ
(
v
+
1
)
Δ
t
+
o
(
Δ
t
)
}
P
x
,
x
I
,
y
,
y
I
,
v
+
1
(
t
)
.
Rearranging (1), then dividing through by Δ
t, and taking the limit as Δ
t
⇒
0 we have the following difference differential equation:
(2)
P
x
,
x
I
,
y
,
y
I
,
v
′
(
t
)
=
-
{
λ
T
+
λ
L
+
δ
x
+
σ
y
+
β
1
x
v
+
β
2
x
y
I
+
β
3
y
v
+
β
4
y
y
I
+
κ
x
I
+
γ
y
I
+
μ
v
}
P
x
,
x
I
,
y
,
y
I
,
v
(
t
)
+
λ
T
P
x
-
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
λ
L
P
x
,
x
I
,
y
-
1
,
y
I
,
v
(
t
)
+
δ
(
x
+
1
)
P
x
+
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
σ
(
y
+
1
)
P
x
,
x
I
,
y
+
1
,
y
I
,
v
(
t
)
+
{
β
1
(
x
+
1
)
(
v
+
1
)
+
β
2
(
x
+
1
)
(
y
I
)
}
×
e
-
ρ
τ
P
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
(
t
)
+
{
β
3
(
y
+
1
)
(
v
+
1
)
+
β
4
(
y
+
1
)
(
y
I
)
}
×
P
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
(
t
)
+
κ
N
(
x
I
+
1
)
P
x
,
x
I
+
1
,
y
,
y
I
,
v
-
1
(
t
)
+
γ
M
(
y
I
+
1
)
P
x
,
x
I
,
y
,
y
I
+
1
,
v
-
1
(
t
)
+
μ
(
v
+
1
)
P
x
,
x
I
,
y
,
y
I
,
v
+
1
(
t
)
.
This is also called the Master equation or the forward Kolmogorov partial differential equation for P
x
,
x
I
,
y
,
y
I
,
v
(
t
), with initial condition
(3)
P
0,0
,
0,0
,
0
′
(
t
)
=
-
(
λ
T
+
λ
L
)
P
0,0
,
0,0
,
0
(
t
)
+
δ
P
1,0
,
0,0
,
0
(
t
)
+
σ
P
0,0
,
1,0
,
0
(
t
)
+
μ
P
0,0
,
0,0
,
1
(
t
)
.
For detailed description and derivation of the model we refer the reader to [18].


## 2.2. The Probability Generating Function

Now we apply the generating function method. Multiplying (2) by z
1
x
z
2
y
z
3
x
I
z
4
y
I
z
5
v and summing over x
,
y
,
x
I
,
y
I, and v, then applying the properties of generating function, we obtain
(4)
∂
G
∂
t
=
-
(
λ
T
+
λ
L
)
G
-
δ
z
1
∂
G
∂
z
1
-
σ
z
2
∂
G
∂
z
2
-
κ
z
3
∂
G
∂
z
3
-
γ
z
4
∂
G
∂
z
4
-
μ
z
5
∂
G
∂
z
5
-
β
1
z
1
z
5
∂
2
G
∂
z
1
∂
z
5
-
β
2
z
1
z
4
∂
2
G
∂
z
1
∂
z
4
-
β
3
z
2
z
5
∂
2
G
∂
z
2
∂
z
5
-
β
4
z
2
z
4
∂
2
G
∂
z
2
∂
z
4
+
λ
T
z
1
G
+
λ
L
z
2
G
+
δ
∂
G
∂
z
1
+
σ
∂
G
∂
z
2
+
z
5
N
κ
∂
G
∂
z
3
+
z
5
M
γ
∂
G
∂
z
4
+
μ
∂
G
∂
z
5
+
z
3
β
1
e
-
ρ
τ
∂
2
G
∂
z
1
∂
z
5
+
z
3
β
2
e
-
ρ
τ
∂
2
G
∂
z
1
∂
z
4
+
z
4
β
3
∂
2
G
∂
z
2
∂
z
5
+
z
4
β
4
∂
2
G
∂
z
2
∂
z
4
.
On simplification we have
(5)
∂
G
∂
t
=
{
(
z
1
-
1
)
λ
T
+
(
z
2
-
1
)
λ
L
}
G
+
(
1
-
z
1
)
δ
∂
G
∂
z
1
+
(
1
-
z
2
)
σ
∂
G
∂
z
2
+
(
N
z
5
-
z
3
)
κ
∂
G
∂
z
3
+
(
M
z
5
-
z
4
)
γ
∂
G
∂
z
4
+
(
1
-
z
5
)
μ
∂
G
∂
z
5
+
β
1
(
z
3
e
-
ρ
τ
-
z
1
z
5
)
∂
2
G
∂
z
1
∂
z
5
+
β
2
(
z
3
e
-
ρ
τ
-
z
1
z
4
)
∂
2
G
∂
z
1
∂
z
4
+
β
3
(
z
4
-
z
2
z
5
)
∂
2
G
∂
z
2
∂
z
5
+
β
4
(
z
4
-
z
2
z
4
)
∂
2
G
∂
z
2
∂
z
4
.
This is called Lagrange partial differential equation for the probability generating function (pgf) G.


## 2.3. The Marginal Generating Functions

Recall that(6)
G
(
z
1
,
1,1
,
t
)
=
∑
x
=
0
∞
‍
∑
y
=
0
∞
‍
∑
v
=
0
∞
‍
P
x
,
y
,
v
(
t
)
z
1
x
.
Assuming z
2
=
z
3
=
z
4
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for healthy C
D
4
+ cells:
(7)
∂
G
(
z
1
,
1,1
,
1,1
;
t
)
∂
t
=
(
z
1
-
1
)
λ
T
G
+
(
1
-
z
1
)
δ
∂
G
∂
z
1
+
β
1
(
e
-
ρ
τ
-
z
1
)
∂
2
G
∂
z
1
∂
z
5
+
β
2
(
e
-
ρ
τ
-
z
1
)
∂
2
G
∂
z
1
∂
z
4
.
Assuming z
1
=
z
3
=
z
4
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for healthy Langerhans cells:
(8)
∂
G
(
1
,
z
2
,
1,1
,
1
;
t
)
∂
t
=
(
z
2
-
1
)
λ
L
G
+
(
1
-
z
2
)
σ
∂
G
∂
z
2
+
β
3
(
1
-
z
2
)
∂
2
G
∂
z
2
∂
z
5
+
β
4
(
1
-
z
2
)
∂
2
G
∂
z
2
∂
z
4
.
Assuming z
1
=
z
2
=
z
4
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for infected C
D
4
+ cells:
(9)
∂
G
(
1,1
,
z
3
,
1,1
;
t
)
∂
t
=
(
N
-
z
3
)
κ
∂
G
∂
z
3
+
β
1
(
z
3
e
-
ρ
τ
-
1
)
∂
2
G
∂
z
1
∂
z
5
+
β
2
(
z
3
e
-
ρ
τ
-
1
)
∂
2
G
∂
z
1
∂
z
4
.
Assuming z
1
=
z
2
=
z
3
=
z
5
=
1 and solving (5), we obtain the marginal partial generating function for infected Langerhans cells:
(10)
∂
G
(
1,1
,
1
,
z
4
,
1
;
t
)
∂
t
=
(
M
-
z
4
)
γ
∂
G
∂
z
4
+
β
3
(
z
4
-
1
)
∂
2
G
∂
z
2
∂
z
5
.Assumingz
1
=
z
2
=
z
3
=
z
4
=
1 and solving (5), we obtain the marginal partial generating function for virus population:
(11)
∂
G
(
1,1
,
1,1
,
z
5
;
t
)
∂
t
=
(
N
z
5
-
1
)
κ
∂
G
∂
z
3
+
(
M
z
5
-
1
)
γ
∂
G
∂
z
4
+
(
1
-
z
5
)
μ
∂
G
∂
z
5
+
β
1
(
e
-
ρ
τ
-
z
5
)
∂
2
G
∂
z
1
∂
z
5
+
β
3
(
1
-
z
5
)
∂
2
G
∂
z
2
∂
z
5
.


## 2.4. Numbers of Cells and the Virions

As we know from probability generating function,(12)
∂
G
x
∂
z
=
∑
x
=
0
∞
‍
x
P
x
(
t
)
z
x
-
1
.
Letting z
=
1, we have
(13)
∂
G
x
∂
z
|
z
=
1
=
∑
x
=
0
∞
‍
x
P
x
(
t
)
z
x
-
1
=
E
[
X
]
.
Differentiating the partial differential equation of the pgf G, we get the moments of T
(
t
)
,
L
(
t
)
,
T
I
(
t
)
,
L
I
(
t
), and V
(
t
).Differentiating (5) with respect to z
1 and setting z
=
1, we have
(14)
∂
∂
t
E
[
T
(
t
)
]
=
λ
T
-
δ
E
[
T
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
2
E
[
T
(
t
)
L
I
(
t
)
]
.
Differentiating (5) with respect to z
2 and setting z
=
1, we have
(15)
∂
∂
t
E
[
L
(
t
)
]
=
λ
L
-
σ
E
[
L
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
-
β
4
E
[
L
(
t
)
L
I
(
t
)
]
.
Differentiating (5) with respect to z
3 and setting z
=
1, we have
(16)
∂
∂
t
E
[
T
I
(
t
)
]
=
-
κ
E
[
T
I
(
t
)
]
+
β
1
e
-
ρ
τ
E
[
T
(
t
)
V
(
t
)
]
+
β
2
e
-
ρ
τ
E
[
T
(
t
)
L
I
(
t
)
]
.
Differentiating (5) with respect to z
4 and setting z
=
1, we have
(17)
∂
∂
t
E
[
L
I
(
t
)
]
=
-
γ
E
[
L
I
(
t
)
]
+
β
3
E
[
L
(
t
)
V
(
t
)
]
.
Differentiating (5) with respect to z
5 and setting z
=
1, we have
(18)
∂
∂
t
E
[
V
(
t
)
]
=
N
κ
E
[
T
I
(
t
)
]
+
M
γ
E
[
L
I
(
t
)
]
-
μ
E
[
V
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
.
Therefore the moments of T
(
t
)
,
L
(
t
)
,
T
I
(
t
)
,
L
I
(
t
), and V
(
t
) from the pgf before introduction of treatment are
(19)
d
d
t
E
[
T
(
t
)
]
=
λ
T
-
δ
E
[
T
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
2
E
[
T
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
L
(
t
)
]
=
λ
L
-
σ
E
[
L
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
-
β
4
E
[
L
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
T
I
(
t
)
]
=
-
κ
E
[
T
I
(
t
)
]
+
β
1
e
-
ρ
τ
E
[
T
(
t
)
V
(
t
)
]
+
β
2
e
-
ρ
τ
E
[
T
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
L
I
(
t
)
]
=
-
γ
E
[
L
I
(
t
)
]
+
β
3
E
[
L
(
t
)
V
(
t
)
]
,
d
d
t
E
[
V
(
t
)
]
=
N
κ
E
[
T
I
(
t
)
]
+
M
γ
E
[
L
I
(
t
)
]
-
μ
E
[
V
(
t
)
]
-
β
1
E
[
T
(
t
)
V
(
t
)
]
-
β
3
E
[
L
(
t
)
V
(
t
)
]
.


## 3. HIV Dynamics under Therapeutic Intervention

Now we introduce the treatment effect on the HIV-immune cell interaction. If we let1
-
α be the reverse transcriptase inhibitor and let 1
-
ω be the protease inhibitor drug effects, then introducing these treatment effects on our model, we have the following forward Kolmogorov partial differential equations (Master equation) for P
x
,
x
I
,
y
,
y
I
,
v
(
t
):
(20)
P
x
,
x
I
,
y
,
y
I
,
v
′
(
t
)
=
-
{
λ
T
+
λ
L
+
δ
x
+
σ
y
+
(
1
-
α
)
β
1
x
v
l
+
(
1
-
α
)
β
2
x
y
I
+
(
1
-
α
)
β
3
y
v
+
(
1
-
α
)
β
4
y
y
I
l
+
(
1
-
ω
)
κ
x
I
+
(
1
-
ω
)
γ
y
I
+
μ
v
}
P
x
,
x
I
,
y
,
y
I
,
v
(
t
)
+
λ
T
P
x
-
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
λ
L
P
x
,
x
I
,
y
-
1
,
y
I
,
v
(
t
)
+
δ
(
x
+
1
)
P
x
+
1
,
x
I
,
y
,
y
I
,
v
(
t
)
+
σ
(
y
+
1
)
P
x
,
x
I
,
y
+
1
,
y
I
,
v
(
t
)
+
{
β
1
(
x
+
1
)
(
v
+
1
)
+
β
2
(
x
+
1
)
(
y
I
)
}
×
(
1
-
α
)
e
-
ρ
τ
P
x
+
1
,
x
I
-
1
,
y
,
y
I
,
v
+
1
(
t
)
+
{
β
3
(
y
+
1
)
(
v
+
1
)
+
β
4
(
y
+
1
)
(
y
I
)
}
×
(
1
-
α
)
P
x
,
x
I
,
y
+
1
,
y
I
-
1
,
v
+
1
(
t
)
+
(
1
-
ω
)
κ
N
(
x
I
+
1
)
P
x
,
x
I
+
1
,
y
,
y
I
,
v
-
1
(
t
)
+
(
1
-
ω
)
γ
M
(
y
I
+
1
)
P
x
,
x
I
,
y
,
y
I
+
1
,
v
-
1
(
t
)
+
μ
(
v
+
1
)
P
x
,
x
I
,
y
,
y
I
,
v
+
1
(
t
)
,
with initial condition
(21)
P
0,0
,
0,0
,
0
′
(
t
)
=
-
(
λ
T
+
λ
L
)
P
0,0
,
0,0
,
0
(
t
)
+
δ
P
1,0
,
0,0
,
0
(
t
)
+
σ
P
0,0
,
1,0
,
0
(
t
)
+
μ
P
0,0
,
0,0
,
1
(
t
)
.
Solving the Master equation using generating function, the Lagrange partial differential equation becomes
(22)
∂
G
∂
t
=
{
(
z
1
-
1
)
λ
T
+
(
z
2
-
1
)
λ
L
}
G
+
(
1
-
z
1
)
δ
∂
G
∂
z
1
+
(
1
-
z
2
)
σ
∂
G
∂
z
2
+
(
N
z
5
-
z
3
)
(
1
-
ω
)
κ
∂
G
∂
z
3
+
(
M
z
5
-
z
4
)
(
1
-
ω
)
γ
∂
G
∂
z
4
+
(
1
-
z
5
)
μ
∂
G
∂
z
5
+
(
1
-
α
)
β
1
(
z
3
e
-
ρ
τ
-
z
1
z
5
)
∂
2
G
∂
z
1
∂
z
5
+
(
1
-
α
)
β
2
(
z
3
e
-
ρ
τ
-
z
1
z
4
)
∂
2
G
∂
z
1
∂
z
4
+
(
1
-
α
)
β
3
(
z
4
-
z
2
z
5
)
∂
2
G
∂
z
2
∂
z
5
+
(
1
-
α
)
β
4
(
z
4
-
z
2
z
4
)
∂
2
G
∂
z
2
∂
z
4
.
Solving the pgf of the in-host HIV dynamics with therapeutic intervention, we have the moments of T
(
t
)
,
L
(
t
)
,
T
I
(
t
)
,
L
I
(
t
), and V
(
t
):
(23)
d
d
t
E
[
T
(
t
)
]
=
λ
T
-
δ
E
[
T
(
t
)
]
-
(
1
-
α
)
β
1
E
[
T
(
t
)
V
(
t
)
]
-
(
1
-
α
)
β
2
E
[
T
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
L
(
t
)
]
=
λ
L
-
σ
E
[
L
(
t
)
]
-
(
1
-
α
)
β
3
E
[
L
(
t
)
V
(
t
)
]
-
(
1
-
α
)
β
4
E
[
L
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
T
I
(
t
)
]
=
-
κ
E
[
T
I
(
t
)
]
+
(
1
-
α
)
β
1
e
-
ρ
τ
E
[
T
(
t
)
V
(
t
)
]
+
(
1
-
α
)
β
2
e
-
ρ
τ
E
[
T
(
t
)
L
I
(
t
)
]
,
d
d
t
E
[
L
I
(
t
)
]
=
-
γ
E
[
L
I
(
t
)
]
+
(
1
-
α
)
β
3
E
[
L
(
t
)
V
(
t
)
]
,
d
d
t
E
[
V
(
t
)
]
=
(
1
-
ω
)
N
κ
E
[
T
I
(
t
)
]
+
(
1
-
ω
)
M
γ
E
[
L
I
(
t
)
]
-
μ
E
[
V
(
t
)
]
-
(
1
-
α
)
β
1
E
[
T
(
t
)
V
(
t
)
]
-
(
1
-
α
)
β
3
E
[
L
(
t
)
V
(
t
)
]
.


### 3.1. Simulation of the Models

#### 3.1.1. Variables and Parameter Values

The initial variable values and parameter values for the model are described in Tables4 and 5.Table 4
Parameters for the stochastic model.
Parameter
Parameter description
Estimate
(
1
-
α
)
The reverse transcriptase inhibitor drug effect
0.5
(
1
-
ω
)
The protease inhibitor drug effect
0.5
λ
T
Rate of production of healthyC
D
4
+ cells  per unit time
1.5 day−1mm−3
λ
L
Rate of production of LCsper unit time
5 day−1mm−3
δ
Death rate of healthyC
D
4
+ cells
0.001 day−1
σ
Death rate of healthy LCs
0.005 day−1
β
1
Infection between uninfectedC
D
4
+ cells  and free virus particles
0.00024 day−1mm−3
β
2
Infection between uninfectedC
D
4
+ cells  and HIV infected LCs
0.00004 day−1mm−3
β
3
Infection between uninfected LCs and free virus particles
0.00045 day−1mm−3
β
4
Infection between uninfected LCs and HIV infected LCs
0.000015 day−1mm−3
κ
Death rate of infectedC
D
4
+ cells
0.24 day−1
κ
*
Viral induced death of infectedC
D
4
+ cells
0.239 day−1
ρ
Death rate of infected but not yetvirus producing cell
0.24 day−1
τ
Intracellular delay time
0.5
γ
Death rate of infected LCs
0.03 day−1
γ
*
Viral induced death rate of infected LCs
0.025 day−1
μ
Death rate of infective virus particles
2.4 day−1
N
Viral production rate by infectedC
D
4
+ cells
90 day−1
M
Viral production rate by infected LCs
62.5 day−1
Whereκ
=
κ
*
+
δ and γ
=
γ
*+ σ, values adapted from [13, 14].Table 5
Variables for the stochastic model.
Variable
Description
Initial condition
T
(
t
)
The concentration of healthy (susceptible)C
D
4
+ cells  at time t
1000 cells mm−3
T
I
(
t
)
The concentration of infectedC
D
4
+ cells at time t
0 cells mm−3
L
(
t
)
The concentration of healthy (susceptible) Langerhans cells at timet
1000 cells mm−3
L
I
(
t
)
The concentration of infected Langerhans cells at timet
0 cells mm−3
V
(
t
)
The concentration of virus particles  at timet
0.001 virion mm−3Using the parameter values and initial conditions defined in Tables4 and 5, we illustrate the general dynamics of the C
D
4
+ T cells and HIV virus by performing sensitivity analysis on the effect of intracellular delay and drug efficacy.From the simulations in Figures2 and 3, it is clear that, 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.Figure 2
Cells and HIV population dynamics before therapeutic intervention. The population dynamics are shown whenτ
=
0.Figure 3
Cells and HIV population dynamics before therapeutic intervention. The population dynamics are shown whenτ
=
2.With60
% drug efficacy, the virus population drops and stabilizes after some time t, but the number of infected LCs increases with time; see Figures 4 and 5.Figure 4
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 60% and whenτ
=
0.Figure 5
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 60% and whenτ
=
2.With an increase in efficacy 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 Figures6 and 7.Figure 6
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 85%.Figure 7
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 85% and whenτ
=
1.


## 3.1. Simulation of the Models

### 3.1.1. Variables and Parameter Values

The initial variable values and parameter values for the model are described in Tables4 and 5.Table 4
Parameters for the stochastic model.
Parameter
Parameter description
Estimate
(
1
-
α
)
The reverse transcriptase inhibitor drug effect
0.5
(
1
-
ω
)
The protease inhibitor drug effect
0.5
λ
T
Rate of production of healthyC
D
4
+ cells  per unit time
1.5 day−1mm−3
λ
L
Rate of production of LCsper unit time
5 day−1mm−3
δ
Death rate of healthyC
D
4
+ cells
0.001 day−1
σ
Death rate of healthy LCs
0.005 day−1
β
1
Infection between uninfectedC
D
4
+ cells  and free virus particles
0.00024 day−1mm−3
β
2
Infection between uninfectedC
D
4
+ cells  and HIV infected LCs
0.00004 day−1mm−3
β
3
Infection between uninfected LCs and free virus particles
0.00045 day−1mm−3
β
4
Infection between uninfected LCs and HIV infected LCs
0.000015 day−1mm−3
κ
Death rate of infectedC
D
4
+ cells
0.24 day−1
κ
*
Viral induced death of infectedC
D
4
+ cells
0.239 day−1
ρ
Death rate of infected but not yetvirus producing cell
0.24 day−1
τ
Intracellular delay time
0.5
γ
Death rate of infected LCs
0.03 day−1
γ
*
Viral induced death rate of infected LCs
0.025 day−1
μ
Death rate of infective virus particles
2.4 day−1
N
Viral production rate by infectedC
D
4
+ cells
90 day−1
M
Viral production rate by infected LCs
62.5 day−1
Whereκ
=
κ
*
+
δ and γ
=
γ
*+ σ, values adapted from [13, 14].Table 5
Variables for the stochastic model.
Variable
Description
Initial condition
T
(
t
)
The concentration of healthy (susceptible)C
D
4
+ cells  at time t
1000 cells mm−3
T
I
(
t
)
The concentration of infectedC
D
4
+ cells at time t
0 cells mm−3
L
(
t
)
The concentration of healthy (susceptible) Langerhans cells at timet
1000 cells mm−3
L
I
(
t
)
The concentration of infected Langerhans cells at timet
0 cells mm−3
V
(
t
)
The concentration of virus particles  at timet
0.001 virion mm−3Using the parameter values and initial conditions defined in Tables4 and 5, we illustrate the general dynamics of the C
D
4
+ T cells and HIV virus by performing sensitivity analysis on the effect of intracellular delay and drug efficacy.From the simulations in Figures2 and 3, it is clear that, 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.Figure 2
Cells and HIV population dynamics before therapeutic intervention. The population dynamics are shown whenτ
=
0.Figure 3
Cells and HIV population dynamics before therapeutic intervention. The population dynamics are shown whenτ
=
2.With60
% drug efficacy, the virus population drops and stabilizes after some time t, but the number of infected LCs increases with time; see Figures 4 and 5.Figure 4
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 60% and whenτ
=
0.Figure 5
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 60% and whenτ
=
2.With an increase in efficacy 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 Figures6 and 7.Figure 6
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 85%.Figure 7
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 85% and whenτ
=
1.


## 3.1.1. Variables and Parameter Values

The initial variable values and parameter values for the model are described in Tables4 and 5.Table 4
Parameters for the stochastic model.
Parameter
Parameter description
Estimate
(
1
-
α
)
The reverse transcriptase inhibitor drug effect
0.5
(
1
-
ω
)
The protease inhibitor drug effect
0.5
λ
T
Rate of production of healthyC
D
4
+ cells  per unit time
1.5 day−1mm−3
λ
L
Rate of production of LCsper unit time
5 day−1mm−3
δ
Death rate of healthyC
D
4
+ cells
0.001 day−1
σ
Death rate of healthy LCs
0.005 day−1
β
1
Infection between uninfectedC
D
4
+ cells  and free virus particles
0.00024 day−1mm−3
β
2
Infection between uninfectedC
D
4
+ cells  and HIV infected LCs
0.00004 day−1mm−3
β
3
Infection between uninfected LCs and free virus particles
0.00045 day−1mm−3
β
4
Infection between uninfected LCs and HIV infected LCs
0.000015 day−1mm−3
κ
Death rate of infectedC
D
4
+ cells
0.24 day−1
κ
*
Viral induced death of infectedC
D
4
+ cells
0.239 day−1
ρ
Death rate of infected but not yetvirus producing cell
0.24 day−1
τ
Intracellular delay time
0.5
γ
Death rate of infected LCs
0.03 day−1
γ
*
Viral induced death rate of infected LCs
0.025 day−1
μ
Death rate of infective virus particles
2.4 day−1
N
Viral production rate by infectedC
D
4
+ cells
90 day−1
M
Viral production rate by infected LCs
62.5 day−1
Whereκ
=
κ
*
+
δ and γ
=
γ
*+ σ, values adapted from [13, 14].Table 5
Variables for the stochastic model.
Variable
Description
Initial condition
T
(
t
)
The concentration of healthy (susceptible)C
D
4
+ cells  at time t
1000 cells mm−3
T
I
(
t
)
The concentration of infectedC
D
4
+ cells at time t
0 cells mm−3
L
(
t
)
The concentration of healthy (susceptible) Langerhans cells at timet
1000 cells mm−3
L
I
(
t
)
The concentration of infected Langerhans cells at timet
0 cells mm−3
V
(
t
)
The concentration of virus particles  at timet
0.001 virion mm−3Using the parameter values and initial conditions defined in Tables4 and 5, we illustrate the general dynamics of the C
D
4
+ T cells and HIV virus by performing sensitivity analysis on the effect of intracellular delay and drug efficacy.From the simulations in Figures2 and 3, it is clear that, 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.Figure 2
Cells and HIV population dynamics before therapeutic intervention. The population dynamics are shown whenτ
=
0.Figure 3
Cells and HIV population dynamics before therapeutic intervention. The population dynamics are shown whenτ
=
2.With60
% drug efficacy, the virus population drops and stabilizes after some time t, but the number of infected LCs increases with time; see Figures 4 and 5.Figure 4
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 60% and whenτ
=
0.Figure 5
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 60% and whenτ
=
2.With an increase in efficacy 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 Figures6 and 7.Figure 6
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 85%.Figure 7
Cells and HIV population dynamics under therapeutic intervention. The population dynamics are shown with drug efficacy of 85% and whenτ
=
1.


## 4. Discussion

In this study, we derived and analysed a stochastic model describing the dynamics of HIV,CD
4
+ T cells, and LCs interactions under therapeutic intervention in vivo. This model included dynamics of five compartments—the number of healthy CD
4
+ cells, the number of infected CD
4
+ cells, the number of healthy Langerhans cells, the number of infected Langerhans cells, and the HIV virions. The 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 t and obtained the moment structures of the healthy immune cells, infected immune cells, and the virus particles over time t. We simulated the mean number of the healthy immune cells, the infected immune cells, and the virus particles before and after combined therapeutic treatment at any time t.Our analysis shows that eradication of HIV is not possible without clearance of latently infected Langerhans cells. Therefore, understanding HIV therapeutic treatment dynamicsin 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 CD
4
+ 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).In our work, the dynamics of mutant virus were not considered and also our study only included dynamics of only five 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 efficacy of our models as we have done here with parameter values from the literature.



---

*Source: 594617-2014-01-19.xml*