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Stochastic model for In-Host HIV dynamics
with therapeutic intervention
Waema R. Mbogo1, Livingstone S. Luboobi2and John W. Odhiambo3
1 3Center for Applied Research in Mathematical Sciences
Strathmore University , Box 59857 00200, Nairobi - KENYA
rmbogo@strathmore.edu
2Department of Mathematics
Makerere University , Box 7062, Kampala - UGANDA
luboobi@math.mak.ac.ug
Abstract
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.
Keywords: Intracellular delay, Therapeutic intervention, CD4 cells, HIV,Drug
efficacy, Stochastic model.
1corresponding author’s email: rmbogo@strathmore.edu (Rachel W. Mbogo)
1
2
1 Introduction
Since HIV pandemic first became visible, enormous mathematical models have
been developed to describe the immunological response to infection with hu-
man immunodeficiency virus (HIV). Mathematical modeling has proven to be
valuable in understanding the dynamics of infectious diseases with respect to
host-pathogen interactions. 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 pathogenesis, 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 action of viral protease in cleaving viral
proteins into mature products have led to the design of drugs (chemothera-
peutic 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 opportunis-
tic infections, leading to Acquired Immunodeficiency Syndrome (AIDS), [26].
With the spread of the HIV-AIDS pandemic and in the absence of an ”ef-
fective” vaccine or cure, therapeutic interventions are still heavily relied on.
Several research studies have been carried out in the recent past, both theoret-
ically and experimentally, to analyse the impact of therapy on the viral load
in HIV infected persons in order to ascertain the effectiveness of the treat-
ment (see, for example, [18] , [6] , [17], [10], [13], [19],[2]). 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 con-
tribute significantly to the spread of the epidemic in a community [22]. 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 individu-
als. Also, the clinical latency period of the disease may provide a sufficiently
long period during which to attempt an effective suppressive therapeutic inter-
vention 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 [21] and [1]).
Intracellular delays and the target-cell dynamics such as mitosis are two key
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];
3
[18]; [4] ;[14]; [24]; [23]. Intracellular delays have been incorporated into the
incidence term in finite or distributed form [3]; [5]; [8]; [9]; [11] ;[12]; [17] ;
[25]. In [23] and [4], 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 [3]; [25], in in-host models with both a logistic growth term and
intracellular delay, that Hopf bifurcations can occur when the intracellular
delay increases. In [8], 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 [9] 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, [7] as-
sumed 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 κde-
notes 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 [4]; [14]; [15]; [16]; [18]; [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 [14];[24]. For detailed description
and derivation of the model, as well as the incorporation of intracellular delays,
we refer the reader to [7].
From the literature, many researchers have employed deterministic models
to study HIV internal dynamics, ignoring the stochastic effects. We consider
4
a stochastic model for the interaction of HIV virus and the immune system 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 [7] 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, variance and co-variance structures of variables rep-
resenting 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 re-
duction 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 HIV and CD4 cells dynamics before thera-
peutic intervention
To study the interaction of HIV virus and the immune system, we propose a
stochastic model by extending the deterministic model presented in the liter-
ature. 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.
5
2.0.1 The interaction of HIV virus and the CD4 T-cells
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 CD4+ T-cells
6
Let X(t) be the size of the healthy cells population at time t,Y(t) be the
size of infected cell population at time t and V(t) be the size of the virons
population at time t. 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,v = 0, 1, 2 ,3 .....
The standard argument using the forward Chapman-Kolmogorov differen-
tial 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:
2.0.2 population change scenarios
1. There were x healthy cells, y infected cells and v virons by time t and
nothing happens during the time interval (t, t + ∆t).
2. 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)
3. 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)
4. 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)
5. 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 )
6. 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)
7
7. 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 (includ-
ing 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−τ
and 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 .
2.0.3 Variables and parameters for the model
The variables and parameters in the model are described as follows:
Table 1: Variables for the Stochastic model
Variable Description Initial condition 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
Using the population change scenarios and parameters in table 2 above, we
now summarize the events that occur during the interval (t, t + ∆) together
with their transition probabilities in the table below.
8
Table 2: Parameters for the stochastic model
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 particles 3
ρThe death rate of infected but not yet virus producing 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
Possible transitions in host interaction of HIV and Immune system Cells
and corresponding probabilities
Event Population Population 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 uninfected 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 because
of risky behaviour
Death of virons (x, y, v + 1) (x, y, v)µ(v+ 1)∆t
Table 3: Transitions of In - Host interaction of HIV
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)
9
+{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
differential equations for Px,y,v(t)
P0
x,y,v (t) = −{λ+δx +βe−ρτ xv +µv +κNy +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.
With the condition
P0
0,0,0(t) = −(λ+γ)P0,0,0(t) + δP1,0,0(t) + µP0,0,1(t) (4)
2.1 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(5)
Differentiating equation (5) with respect to t yields
∂G(z1, z2, z3, t)
∂t =
∞
X
x=0
∞
X
y=0
∞
X
v=0
P0
x,y,v (t)zx
1zy
2zv
3(6)
10
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
(7)
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
(8)
This is called Lagrange partial differential equation for the probability gener-
ating 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
}
(9)
Where aand bare constants.
Equation (9) is not solvable, even for any simple form of F(z1, z2, z3). How-
ever, it is possible to obtain the moment-structure of (X(t); Y(t); V(t)) from
equation (8).
2.2 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)
11
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
(10)
∂G(z2;t)
∂t =∂G(1, z2,1; t)
∂t
= (1 −z2)κN ∂G
∂z2
+βe−ρτ (z2−1) ∂2G
∂z1∂z3
(11)
∂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
(12)
2.3 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]
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 z3re-
spectively and setting z1=z2=z3= 1 we have
∂
∂t E[X(t)] = λ−δE[X(t)] −βE[X(t)V(t)]
12
∂
∂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)] (13)
∂
∂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 [7] 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) (14)
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.
2.3.1 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
logarithm 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.
13
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
3 HIV and CD4 cells dynamics under thera-
peutic intervention
Assume that at time t = 0, a combination therapy treatment is initiated in
an HIV-infected individual. We assume that the therapeutic intervention in-
hibits either 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 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 lysis 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
14
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
15
Introducing the effect of treatment, the Lagrange partial differential equa-
tion 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
(15)
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
(16)
∂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
(17)
∂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
(18)
16
3.2 Numbers of CD4+ T-cells and Virons under thera-
peutic 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
=
∞
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 z3re-
spectively 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
intervention 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)] (19)
∂
∂t E[V(t)] = γ+ (1 −ω)NκE[Y(t)] −µE[V(t)] −(1 −α)βE[X(t)V(t)]
17
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).
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 con-
ditions are given in Tables 1 and 2
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.
18
We now analyze drug effectiveness on the cell populations.
Figure 5: Shows the CD4 cells and HIV virus population dynamics with τ= 0.5
days and different levels of drug efficacy. The other parameters and initial
conditions are given in Tables 1 and 2
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.Also the figure
shows that a combination of therapeutic intervention and intracellular delay
is very important in lowering the viral load in an HIV infected person.
19
3.3 The probability of HIV clearance
We now calculate the distribution of times to extinction for the Virons reser-
voir. 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 analyt-
ically.In this case, the infectious virus cell dynamics decouples from the rest
of the model and can be represented as a pure immigration-and-death process
with master equation
P0
v(t) = −(γ+κ(1 −ω)N)Pv(t)−v(µ+ (1 −α)e−ρτ β)Pv(t)+(v−1)(γ+κ(1 −ω)N)Pv−1(t)
+ (v+ 1)(µ+ (1 −α)e−ρτ β)Pv+1(t)
(20)
where Pv(t) is the probability that at time t, there are vHIV particles. This
probability has the conditional probability generating function
Gv(z, t)|V(0)=v0=exp{θ
(z−1)(1 −e−θt)}{ze−θt + 1 −e−θt }v0
(21)
Where v0is the initial virus reservoir size, θ=µ+ (1 −α)βe−ρτ and =
γ+ (1 −ω)κN. We noticed that this is a p.g.f of a random variable with a
product of Poisson distribution exp{θ
(z−1)(1 −e−θt)}with mean θ
(1 −e−θt)
and a binomial distribution {ze−θt + 1 −e−θt}v0
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)
=exp{θ
(e−θt −1)}{1−e−θt }v0
The probability of extinction is the value of Gv(z, t) when t→ ∞, that is
pext =P0(∞)
= lim
t→∞ exp{θ
(e−θt −1)}{1−e−θt }v0!
=e−θ
=e−µ+(1−α)βe−ρτ
γ+(1−ω)κN (22)
20
From equation (22), it is evident that probability of clearance of HIV is affected
by the combination drug efficacy and intracellular delay, death of the virons
and education (counseling the HIV patients on the risk of involvement in risk
behaviors).
4 Discussion and recommendation
In this study,we derived and analyzed a stochastic model for In-Host HIV dy-
namics 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 in-
fection of CD4 T-cells before and during therapy. We derived equations for
the probability generating function and using numerical techniques. In this
paper we showed the usefulness of our stochastic approach towards modeling
HIV dynamics 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. We will emphasize further usefulness of
stochastic models in HIV dynamics in our future research.
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 de-
pends on the combination of drug efficacy and intracellular delay and also
education to the infected patients. In our work, 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.
21
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