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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

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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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