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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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