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International Electronic Journal of Pure and Applied Mathematics
——————————————————————————————
Volume 6 No. 2 2013, 83-103
ISSN: 1314-0744
url: http://www.e.ijpam.eu
doi: http://dx.doi.org/10.12732/iejpam.v6i2.3
MATHEMATICAL MODEL FOR HIV AND
CD4+ CELLS DYNAMICS IN VIVO
Waema R. Mbogo1§, Livingstone S. Luboobi2, John W. Odhiambo3
1,3Center for Applied Research in Mathematical Sciences
Strathmore University
Box 59857 00200, Nairobi, KENYA
2Department of Mathematics
Makerere University
Box 7062, Kampala, UGANDA
Abstract: 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.
AMS Subject Classification: 92D30
Received: February 22, 2013 c
2013 Academic Publications, Ltd.
§Correspondence author
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
84 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
Key Words: intracellular delay, therapeutic intervention, CD4+ T-cells, HIV
dynamics, stochastic model
1. Introduction
Since HIV pandemic first became visible, enormous mathematical models have been
developed to describe the immunological response to infection with human immun-
odeficiency virus (HIV). Mathematical modeling has proven to be valuable in un-
derstanding the dynamics of infectious diseases with respect to host-pathogen inter-
actions. When HIV enters the body, it targets all the cells with CD4 + receptors
including the CD4 + T-cells. The knowledge of principal mechanisms of viral patho-
genesis, namely the binding of the retrovirus to the gp120 protein on the CD4 cell,
the entry of the viral RNA into the target cell, the reverse transaction of viral RNA
to viral DNA, the integration of the viral DNA with that of the host, and the ac-
tion of viral protease in cleaving viral proteins into mature products have led to the
design of drugs (chemotherapeutic agents) to control the production of HIV.
Chronic HIV infection causes gradual depletion of the CD4+ T-cell poll, and thus
progressively compromises the host’s immune response to opportunistic infections,
leading to Acquired Immunodeficiency Syndrome (AIDS), [25]. With the spread
of the HIV-AIDS pandemic and in the absence of an ”effective” vaccine or cure,
therapeutic interventions are still heavily relied on. Several research studies have
been carried out in the recent past, both theoretically and experimentally, to analyse
the impact of therapy on the viral load in HIV infected persons in order to ascertain
the effectiveness of the treatment (see, for example, [17], [7], [18], [11], [14], [19],[3]).
Their utility lies in the ability to predict an infected steady state and examining the
effects that changes in parameters have on the outcome of the system over time, to
determine which parameters are most important in disease progression, and further
determine critical threshold values for these parameters.
In HIV infected individuals, the infection exhibits a long asymptomatic phase
(after the initial high infectious phase) of approximately 10 years on average before
the onset of AIDS. During this incubation period which some call the clinical latency
period, the individuals appear to be well and may contribute significantly to the
spread of the epidemic in a community [26]. Some clinical markers such as the
CD4 cell count and the RNA viral load (viraemia) provide information about the
progression of the disease in infected individuals. Also, the clinical latency period
of the disease may provide a sufficiently long period during which to attempt an
effective suppressive therapeutic intervention in HIV infections. Various biological
reasons lead to the introduction of time delays in models of disease transmission.
Time delays are used to model the mechanisms in the disease dynamics (see for
instance [24] and [2]).
Intracellular delays and the target-cell dynamics such as mitosis are two key
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 85
factors that play an important role in the viral dynamics. Mitosis in healthy or
infected target-cell population are typically modelled by a logistic term [16], [17],
[5], [15], [22], [21]. Intracellular delays have been incorporated into the incidence
term in finite or distributed form [4], [6], [9], [10], [12], [13], [18], [23]. In [21] and
[5], in-host viral models with a logistic growth term without intracellular delays
are investigated, and it is shown that sustained oscillations can occur through Hopf
bifurcation when the intrinsic growth rate increases. It is shown in [4], [23], in
in-host models with both a logistic growth term and intracellular delay, that Hopf
bifurcations can occur when the intracellular delay increases. In [9], using in-host
models with a general form of target-cell dynamics and general distributions for
intracellular delays, it is shown that the occurrence of Hopf bifurcation in these
models critically depends on the form of target-cell dynamics. More specifically, it
is proved in [10] that, if the target-cell dynamics are such that no Hopf bifurcations
occur when delays are absent, introducing intracellular delays in the model will not
lead to Hopf bifurcations or periodic oscillations.
To incorporate the intracellular delay phase of the virus life-cycle, [8] assumed
that virus production occurs after the virus entry by a constant delay τ. They came
up with a basic in-host compartmental model of the viral dynamics containing three
compartments:- x(t), y(t) and v(t) denoting the populations of uninfected target
cells, infected target cells that produce virus, and free virus particles, respectively.
They further assumed Parameters δ,αand µare turnover rates of the x, y and v
compartments, respectively. Uninfected target cells are assumed to be produced at
a constant rate λ. They assumed also that cells infected at time t will be activated
and produce viral materials at time t+τ. In their model, constant s is the death
rate of infected but not yet virus-producing cells, and e−sτ describes the probability
of infected target cells surviving the period of intracellular delay from t−τto t.
Constant κdenotes the average number of virus particles each infected cell produces.
With preceding assumptions lead to the following system of differential equations.
dx
dt =λ−δx(t)−βx(t)v(t),
dy
dt =βx(t−τ)v(t−τ)e−sτ −αy(t),(1)
dv
dt =κy(t)−µv(t).
System (1) can be used to model the infection dynamics of HIV, HBV and other
virus [5], [15], [1], [16], [17], [20]. It can also be considered as a model for the
HTLV-I infection if x(t), y(t), and v(t) are regarded as healthy, latently infected,
and actively infected CD4+ T cells [15];[22]. For detailed description and derivation
of the model, as well as the incorporation of intracellular delays, we refer the reader
to [8].
From the literature, Many researchers have employed deterministic models to
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
86 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
study HIV internal dynamics, ignoring the stochastic effects. The present authors
consider a stochastic model for the interaction of HIV virus and the immune sys-
tem in an HIV-infected individual undergoing a combination-therapeutic treatment.
Our aim in this paper is to use a stochastic model obtained by extending the model
of [8] to determine probability distribution, variance and co-variance structures of
the uninfected CD4+ cells, infected CD4+ cells and the free HIV particles in an
infected individual at any time t by examining the combined antiviral treatment of
HIV. Based on the model, we obtain joint probability distribution,expectations, vari-
ance and co-variance structures of variables representing the numbers of uninfected
CD4 + cells,the HIV infected CD4 T cells, and the free HIV particles at any time
t, and derive conclusions for the reduction or elimination of HIV in HIV-infected
individuals, which is one of the main contributions of this paper.
The organization of this paper is as follows: In Section 2, we formulate our
stochastic model describing the interaction of HIV and the immune system and
obtain a partial differential equation for the probability generating function of the
numbers of uninfected CD4 + cells,the HIV infected CD4 T cells, and the free
HIV particles at any time t, also moments for the variables are derived here. In
Section 3, we derive the moments of the variables in a therapeutic environment
and probability of extinction of HIV virus and also provide a numerical illustration
to demonstrate the impact of intercellular delay and therapeutic intervention in
controlling the progression of HIV. Some concluding remarks follow in Section 4.
2. Formulation of the Model
2.1. Transition Probabilities in the Life Cycle of HIV in Host
Human Immunodeficiency Virus (HIV) is among the most studied viruses in biomed-
ical research and causes a tremendous global problem. A large number of parameters
are available from clinical trials which make HIV suitable for mathematical model-
ing and testing. A typical life-cycle of HIV virus and immune system interaction is
shown in Figure 1.
To study the interaction of HIV virus and the immune system, we therefore
need a stochastic version of the deterministic model presented in the literature. A
stochastic process is defined by the probabilities with which different events happen
in a small time interval ∆t. In our model there are two possible events (production
and death/removal) for each population (uninfected cells, infected cells and the free
virons). The corresponding rates in the deterministic model are replaced in the
stochastic version by the probabilities that any of these events occur in a small time
interval ∆t.
Let X(t) be the size of the healthy cells population at time t,Y(t) be the size
of infected cell at time t and V(t) be the size of the virons population at time t.
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 87
Figure 1: The interaction of HIV virus and the CD4+ T-cells
In the model to be formulated, it is now assumed that instead of rates of births
and deaths, there is a possibility of stochastic births or deaths of the heathy cells,
infected cells and the virus particles. Thus X(t), Y(t) and V(t) are time dependent
random variables. This epidemic process can be modeled stochastically by letting
the nonnegative integer values process X(t), Y(t), and V(t) respectively represent
the number of healthy cells, infected cells and virons of the disease at time t. Then
{(X(t), Y (t), V (t)) : t≥0}can be modeled as continuous time Multivariate markov
chain. Let the probability of there being x healthy cells, y infected cells and v
virons in an infected person at time t be denoted by the following joint probability
function:- Px,y,v (t) = P[X(t) = x, Y (t) = y, V (t) = v], for x,y,z = 0, 1, 2,3,... .
The standard argument using the forward Chapman-Kolmogorov differential
equations is used to obtain the joint probability function Px,y,v (t), by considering
the joint probability Px,y,v(t, t + ∆t). This joint probability is obtained as the sum
of the probabilities of the following mutually exclusive events:
•There were x healthy cells, y infected cells and v virons by time t and nothing
happens during the time interval (t, t + ∆t).
•There were x-1 healthy cells, y infected cells and v virons by time t and one
healthy cell is produced from the thymus during the time interval (t, t + ∆t)
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
88 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
Variable Description Initial con-
dition t= 0
X(t) The concentration of uninfected CD4 cells at time t 100
Y(t) The concentration of infected CD4 cells at time t 0.02
V(t) The concentration of virus particles at time t 0.001
Table 1: Variables for the stochastic model
•There were x+1 healthy cells, y infected cells and v virons by time t and one
healthy cell dies or is infected by HIV virus during the time interval (t, t + ∆t)
•There were x healthy cells, y-1 infected cells and v virons by time t and one
healthy cell is infected by HIV virus during the time interval (t, t + ∆t)
•There were x healthy cells, y+1 infected cells and v virons by time t and one
infected cell dies (HIV-infected cell bursts or undergoes a lysis)during the time
interval (t, τ )
•There were x healthy cells, y infected cells and v-1 virons by time t and one
viron is produced(HIV-infected cell undergoes a lysis or the individual angages
in risky behaviours) during the time interval (t, t + ∆t)
•There were x healthy cells, y infected cells and v+1 virons by time t and one
viron dies during the time interval (t, t + ∆t)
We incorporate a time delay between infection of a cell and production of new
virus particles, we let τto be the time lag between the time the virus contacts a
target CD4 T cell and the time the cell becomes productively infected (including
the steps of successful attachment of the virus to the cell, and penetration of virus
into the cell). this means the recruitment of virus producing cells at time t is given
by the density of cells that were newly infected at time t−τand are still alive at
time t. If we also let ρto be the death rate of infected but not yet virus producing
cell, then the probability that the infected cell will survive to virus producing cell
during the short time interval τwill be given by e−ρτ .
2.1.1. Variables and Parameters for the Model
The variables and parameters in the model are described as follows:
We now summarize the events that occur during the interval (t, t + ∆) together
with their transition probabilities in the table below.
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 89
Parameter symbol Parameter description Estimate
(1 −α) The reverse transcriptase inhibitor
drug effect 0.5
(1 −ω) The protease inhibitor drug effect 0.5
λThe total rate of production of healthy CD4 cells 10
δThe per capita death rate of healthy CD4 cells 0.02
βThe transmission coefficient between
uninfected CD4 cells and infective virus particles 0.000024
κper capita death rate of infected CD4 cells 0.5
γThe virus production rate due to risk behaviors 0.001
µThe per capita death rate of infective virus par-
ticles
3
ρThe death rate of infected but not yet virus pro-
ducing cell
0.5
τTime lag during infection 0.5
NThe average number of infective virus particles
produced by
an infected CD4 cell in the absence of treatment
during its entire infectious lifetime 1000
Table 2: Parameters for the stochastic model
The change in population size during the time interval ∆t, which is assumed to
be sufficiently small to guarantee that only one such event can occur in (t, t + ∆t),
is governed by the following conditional probabilities;
Px,y,v (t+ ∆t) = {1−(λ∆t+δx∆t+βxv∆t
+µv∆t+κy∆t+γ∆t) + o(∆t)}Px,y,v (t)
+{λ∆t+o(∆t)}Px−1,y,v(t)
+{δ(x+ 1)∆t+o(∆t)}Px+1,y,v(t)
+{β(x+ 1)(v+ 1)e−ρτ ∆t+o(∆t)}Px+1,y−1,v+1(t) (2)
+{κN(y+ 1)∆t+o(∆t)}Px,y+1,v−1(t)
+{γ∆t+o(∆t)}Px,y,v−1(t)
+{µ(v+ 1)∆t+o(∆t)}Px,y,v+1 (t)
Simplifying equation (2), we have the following forward Kolmogorov partial differ-
ential equations for Px,y,v(t)
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
90 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
Possible transitions in host interaction of HIV and Immune system Cells
and corresponding probabilities
Event Population
components
Population
components
probability
(X,Y,V) at t (X,Y,V) at
(t, t + ∆)
of transition
Production
of uninfected
cell
(x−1, y, v) (x, y , v)λ∆t
Death of un-
infected cell
(x+ 1, y, v) (x, y, v)δ(x+ 1)∆t
Infection of
uninfected
cell
(x+ 1, y −
1, v + 1)
(x, y, v)β(x+ 1)(v+ 1)e−ρτ ∆t
Production
of virons
from
(x, y + 1, v −
1)
(x, y, v)κN (y+ 1)∆t
the bursting
infected cell
Introduction
of Virons
(x, y, v −1) (x, y, v)γ∆t
due to re-
infection be-
cause
of risky be-
haviour
Death of vi-
rons
(x, y, v + 1) (x, y, v)µ(v+ 1)∆t
Table 3: Transitions of In - Host interaction of HIV
P′
x,y,v (t) = −{λ+δx +βe−ρτ xv +µv +κN y +γ}Px,y,v (t)
+λPx−1,y,v (t)
+δ(x+ 1)Px+1,y,v (t)
+β(x+ 1)(v+ 1)e−ρτ Px+1,y−1,v+1(t) (3)
+Nκ(y+ 1)Px,y+1,v−1(t)
+γPx,y,v−1(t)
+µ(v+ 1)Px,y,v+1 (t)
This will also be referred to as the Master equation or the Differential -Difference
equation.
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 91
With the condition
P′
0,0,0(t) = −(λ+γ)P0,0,0(t) + δP1,0,0(t) + µP0,0,1(t) (4)
2.2. The Probability Generating Function
The probability generating function of (X(t), Y(t), V(t)) is defined by
G(z1, z2, z3, t) =
∞
X
x=0
∞
X
y=0
∞
X
v=0
Px,y,v (t)zx
1zy
2zv
3
Differentiating equation (5) with respect to t yields
∂G(z1, z2, z3, t)
∂t =
∞
X
x=0
∞
X
y=0
∞
X
v=0
P′
x,y,v (t)zx
1zy
2zv
3(5)
Differentiating again equation (5) with respect to z1, z2, z3yields
∂3G(z1, z2, z3, t)
∂z1∂z2∂z3
=
∞
X
x=1
∞
X
y=1
∞
X
v=1
xyvPx,y,v (t)zx−1
1zy−1
2zv−1
3
=
∞
X
x=0
∞
X
y=0
∞
X
v=0
(x+ 1)(y+ 1)(v+ 1)Px+1,y+1,v+1(t)zx
1zy
2zv
3
Multiplying equation (3) by zx
1zy
2zv
3and summing over x, y, and v, then applying
equations (4), (5),(6) and (7) and on simplification we obtain
∂G
∂t ={(z1−1)λ+ (z3−1)γ}G+ (1 −z1)δ∂G
∂z1
+ (z3−z2)κN ∂G
∂z2
+ (1 −z3)µ∂G
∂z3
+βe−ρτ (z2−z1z3)∂2G
∂z1∂z3
.
This is called Lagrange partial differential equation for the probability generating
function (pgf).
Attempt to solve equation (8) gave us the following solution
G(z1, z2, z3, t) = aebt{{(z1−1)λ+ (z3−1)γ}F(z1, z2, z3)
+ (1 −z1)δ∂F (z1, z2, z3)
∂z1
+ (Nz3−z2)κ∂ F (z1, z2, z3)
∂z2
+ (1 −z3)µ∂F (z1, z2, z3)
∂z3
+β(e−ρτ z2−z1z3)∂2F(z1, z2, z3)
∂z1∂z3
}
(6)
Where aand bare constants.
Equation (9) is not solvable, even for any simple form of F(z1, z2, z3). However,
it is possible to obtain the moment-structure of (X(t); Y(t); V(t)) from equation (8).
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
92 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
2.3. The Marginal Generating Functions
Recall that
G(z1,1,1, t) =
∞
X
x=0
∞
X
y=0
∞
X
v=0
Px,y,v (t)zx
1
Assuming z2=z3= 1,z1=z3= 1 and z2=z1= 1 and solving equation(8), we
obtain the marginal partial generating functions for X(t), Y(t) and V(t) respectively.
∂G(z1;t)
∂t =∂G(z1,1,1; t)
∂t
= (z1−1)λG + (1 −z1)δ∂G
∂z1
+βe−ρτ (1 −z1)∂2G
∂z1∂z3
(7)
∂G(z2;t)
∂t =∂G(1, z2,1; t)
∂t
= (1 −z2)κN ∂G
∂z2
+βe−ρτ (z2−1) ∂2G
∂z1∂z3
(8)
∂G(z3;t)
∂t =∂G(1,1, z3;t)
∂t
= (z3−1)γG + (z3−1)κN ∂G
∂z2
+ (1 −z3)µ∂G
∂z3
+βe−ρτ (1 −z3)∂2G
∂z1∂z3
(9)
2.4. Numbers of CD4+ T-Cells and the Virons
As we know from probability generating function
∂Gx
∂z =
∞
X
x=0
xPx(t)zx−1
Letting z= 1, we have
∂Gx
∂z |z=1 =
∞
X
x=0
xPx(t)zx−1=E[X]
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 93
Differentiating the partial differential equations of the pgf, we get the moments of
X(t), Y(t) and V(t).
Differentiating equations (10), (11) and (12) with respect to z1,z2and z3respec-
tively and setting z1=z2=z3= 1 we have
∂
∂t E[X(t)] = λ−δE[X(t)] −βE [X(t)V(t)]
∂
∂t E[Y(t)] = −κE[Y(t)] + βe−ρτ E[X(t)V(t)]
∂
∂t E[V(t)] = γ+NκE[Y(t)] −µE[V(t)] −βE[X(t)V(t)]
Therefore the moments of (X(t),Y(t), V(t)) from the pgf before treatment
∂
∂t E[X(t)] = λ−δE[X(t)] −βE [X(t)V(t)]
∂
∂t E[Y(t)] = −κE[Y(t)] + βe−ρτ E[X(t)V(t)] (10)
∂
∂t E[V(t)] = γ+NκE[Y(t)] −µE[V(t)] −βE[X(t)V(t)]
The corresponding deterministic model of HIV-Host interaction as formulated in [8]
is as given below.
dx
dt =λ−δx(t)−βx(t)v(t)
dy
dt =βe−ρτ x(t−τ)v(t−τ)−κy(t) (11)
dv
dt =γ+Nκy(t)−µv(t)−βx(t)v(t)
Comparing system of equations (13) with the system of equations (14), we see
that expected values of the multivariate markov process [ X(t), Y(t), V(t)] satisfies
the corresponding deterministic model of the HIV-Host interaction dynamics.
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
94 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
Figure 2: Shows the CD4 cells and HIV virus population dynamics before
therapeutic intervention and when τ= 0,1.5,2days. The other parameters
and initial conditions are given in Tables 1 and 2
2.4.1. Simulation
Using the parameter values and initial conditions defined in Tables 1 and 2, we
illustrate the general dynamics of the CD4- T cells and HIV virus for model (13)
during infection and in the absence of treatment.
The second graph in Figure 2 is the population dynamics after taking the loga-
rithm of the cell populations in the first graph.
From the simulations, it is clear that in the primary stage of the infection (period
before treatment), a dramatically decrease in the level of the CD4-T cells occur and
the number of the free virons increase with time. With introduction of intracellular
delay, the virus population drops as well as an increase in the CD4 cells but then
they stabilize at some point and coexist in the host as shown in Figure 2.
3. HIV and Therapeutic Intervention
Assume that at time t = 0, a combination therapy treatment is initiated in an
HIV-infected individual. We assume that the therapeutic intervention inhibits ei-
ther the enzyme action of reverse transcriptase or that of the protease of HIV in
a HIV-infected cell. A HIV-infected cell with the inhibited HIV-transcriptase may
be considered a dead cell as it cannot participate in the production of the copies
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 95
of any type of HIV. On the other hand,an HIV-infected cell in which the reverse
transcription has already taken place and the viral DNA is fused with the DNA
of the host, but the enzyme activity of HIV-protease is inhibited, undergoes a ly-
sis releasing infectious free HIV and non-infectious free HIV. A non-infectious free
HIV cannot successfully infect a CD4 cell. Accordingly, at any time t, the blood of
the infected person contains virus-producing HIV-infected cells, infectious free HIV
and non-infectious free HIV. A typical life-cycle of HIV virus and immune system
interaction with therapeutic intervention is shown in Figure 3.
Figure 3: HIV-Host interaction with treatment
Introducing the effect of treatment, the Lagrange partial differential equation
becomes
∂G
∂t ={(z1−1)λ+ (z3−1)γ}G+ (1 −z1)δ∂G
∂z1
+ ((1 −ω)(z3−z2)κN ∂G
∂z2
+ (1 −z3)µ∂G
∂z3
+ (1 −α)βe−(1−α)ρτ (z2−z1z3)∂2G
∂z1∂z3
(12)
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
96 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
3.1. The Marginal Generating Functions
Recall that
G(z1,1,1, t) =
∞
X
x=0
∞
X
y=0
∞
X
v=0
Px,y,v (t)zx
1
Setting z2=z3= 1,z1=z3= 1 and z2=z1= 1 and solving equation(15), we obtain
the marginal partial generating functions for X(t), Y(t) and V(t) respectively as:
∂G(z1;t)
∂t =∂G(z1,1,1; t)
∂t
= (z1−1)λG + (1 −z1)δ∂G
∂z1
+ (1 −α)βe−(1−α)ρτ (1 −z1)∂2G
∂z1∂z3
(13)
∂G(z2;t)
∂t =∂G(1, z2,1; t)
∂t
= (1 −ω)N−z2)κ∂G
∂z2
+ (1 −α)βe−(1−α)ρτ (z2−1) ∂2G
∂z1∂z3
(14)
∂G(z3;t)
∂t =∂G(1,1, z3;t)
∂t
= (z3−1)γG + (1 −ω)(z3−1)κN ∂G
∂z2
+ (1 −z3)µ∂G
∂z3
+ (1 −α)βe−(1−α)ρτ (1 −z3)∂2G
∂z1∂z3
.
3.2. Numbers of CD4+ T-Cells and Virons under Therapeutic
Intervention
From probability generating function,
∂Gx
∂z =
∞
X
x=0
xPx(t)zx−1
Letting z= 1, we have the expected number of target CD4+ T-Cells X(t) is:
E[X(t)] = ∂Gx
∂z |z=1
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MATHEMATICAL MODEL FOR HIV AND... 97
=
∞
X
x=0
xPx(t)zx−1
Differentiating the partial differential equations of the pgf, we get the moments of
X(t), Y(t) and V(t).
Differentiating equations (16), (17) and (18) with respect to z1,z2and z3respec-
tively and setting z1=z2=z3= 1 we have
∂
∂t E[X(t)] = λ−δE[X(t)] −(1 −α)βE[X(t)V(t)]
∂
∂t E[Y(t)] = −κE[Y(t)] + (1 −α)βe−(1−α)ρτ E[X(t)V(t)]
∂
∂t E[V(t)] = γ+ (1 −ω)NκE[Y(t)] −µE[V(t)] −(1 −α)βE[X(t)V(t)]
Therefore the moments of (X(t),Y(t), V(t)) from the pgf with therapeutic interven-
tion are:-
∂
∂t E[X(t)] = λ−δE[X(t)] −(1 −α)βE[X(t)V(t)]
∂
∂t E[Y(t)] = −κE[Y(t)] + (1 −α)βe−(1−α)ρτ E[X(t)V(t)] (15)
∂
∂t E[V(t)] = γ+ (1 −ω)NκE[Y(t)] −µE[V(t)] −(1 −α)βE[X(t)V(t)]
3.2.1. Simulation
Using the parameter values and initial conditions defined in Tables 1 and 2, we
illustrate the general dynamics of the CD4- T cells and HIV virus for model (19)
with therapeutic intervention and test the effect of intracellular delay and drug
efficacy (perform sensitivity analysis).
With 60% drug efficacy, the virus stabilizes and coexists with the CD4 cells
within the host as shown in in the first grapg in Figure 4.
Introducing Intracellular time delay, τ= 0.5, the CD4 cells increase and the virus
population drops then stabilizes and again coexist within the host as illustrated in
the second graph in Figure 4.
For τ= 1 and with 60% drug efficacy, the dynamics change as shown in the last
graph in Figure 4.
We now analyze drug effectiveness on the cell populations.
Figure 5 shows that, with improved HIV drug efficacy, a patient on treatment
will have low, undetectable viral load with time. It also shows that the protease
inhibitor drug efficacy is very important in clearing the virus.
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
98 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
Figure 4: Shows the CD4 cells and HIV virus population dynamics with
drug efficacy of 60% and τ= 0,0.5,1days. The other parameters and
initial conditions are given in Tables 1 and 2
3.3. The Probability of Virus Clearance
We now calculate the distribution of times to extinction for the Virons reservoir.
If we assume the drug given at fixed intervals to sustain its efficacy is effective
in producing non-infectious virus, then we can clear the infectious virus. Using
Reverse transcriptase (ART), the infected cell can be forced to become latently
infected (non-virus producing cell) hence clearing the virus producing cell reservoir.
If we assume that no newly infected cells become productively infected in the short
time τ, (e−ρτ = 0) or that treatment is completely effective (ω= 1), we can obtain
the extinction probability analytically.In this case, the infectious virus cell dynamics
decouples from the rest of the model and can be represented as a pure birth-and-
death process with master equation
P′
v(t) = −v(γ+κ(1 −ω)N+µ+ (1 −α)β)Pv(t) + (v−1)(γ+κ(1 −ω)N)Pv−1(t)
+ (v+ 1)(µ+ (1 −α)β)Pv+1(t)
(16)
where Pv(t) is the probability that at time t, there are vHIV virus particles. This
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
MATHEMATICAL MODEL FOR HIV AND... 99
Figure 5: Shows the CD4 cells and HIV virus population dynamics with
τ= 0.5days and different levels of drug efficacy. The other parameters and
initial conditions are given in Tables 1 and 2
probability has the conditional probability generating function
Gv(z, t)|V(0)=v0= θ(1 −z) + (ǫz −θ))e(θ−ǫ)t
ǫ(1 −z) + (ǫz −θ)e(θ−ǫ)t!v0
Where v0is the initial virus reservoir size, θ=µ+ (1 −α)βand ǫ=γ+ (1 −ω)κN .
The probability of this population going extinct at t or before time t is given by
pext(t)|V(0)=v0=P(V= 0, t)|V(0)=v0=Gv(0, t) = P0(t)
= θ−θe(θ−ǫ)t
ǫ−θe(θ−ǫ)t!v0
The probability of extinction is the value of Gv(z, t) when t→ ∞, that is
pext =P0(∞)
= lim
t→∞ θ−θe(θ−ǫ)t
ǫ−θe(θ−ǫ)t!v0
International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 6, No. 2 (2013)
100 W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo
To evaluate this limit, we use L’Hospitals’ rule, which gives
P0(∞) = θ(ǫ−θ)−θe−(ǫ−θ)∞
ǫ(ǫ−θ)−θe(θ−ǫ)∞!v0
=θ
ǫv0
,for ǫ > θ
=µ+ (1 −α)β
γ+ (1 −ω)κN v0
(17)
From equation (22), it is evident that probability of clearance of the virus is
affected by the initial viral load v0, education (counseling the HIV patients on the
risk of involvement in risk behaviors) and the drug efficacy for the disease.
4. Discussion and Recommendation
In this study,we derived and analyzed a stochastic model for In-Host HIV dynamics
that included combined therapeutic treatment and intracellular delay between the
infection of a cell and the emission of viral particles. This model included dynamics
of three compartments- the number of healthy CD4 cells, the number of infected
CD4 cells and the HIV virons, it described HIV infection of CD4+ T-cells before
and during therapy. We derived equations for the probability generating function
and using numerical techniques we showed the usefulness of our stochastic approach
towards modeling combined HIV treatment by obtaining moment structures of the
healthy CD4+ cell, and the virus particles over time t. We simulated the mean
number of the healthy CD4+ cell, the infected cells and the virus particles before
and after combined therapeutic treatment at any time t.
Our analysis during treatment 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 death rate of the
virons, the efficacy of therapy and the length of the intracellular delay. Our model
produces interesting feature that successfully treated HIV patients will have low,
undetectable viral load. 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
cure rate (drug efficacy) and also to increase the intracellular delay τ. Therefore 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.
The extinction probability model showed that the time it takes to have low,
undetectable viral load(infectious virus) in an HIV infected patient depends on the
initial viral load v0before therapeutic intervention (the lower the viral load the
better)and the efficacy of protease inhibitor (including education). In our work,
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MATHEMATICAL MODEL FOR HIV AND... 101
the dynamics of mutant virus was not considered and also our study only included
dynamics of only three compartments (healthy CD4+ cell, infected CD4+ cells and
infectious HIV virus particles) 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 simulated data.
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