A Mathematical Model on CTL Mediated Control of HIV Infection in
a Long Term Drug Therapy
Priti Kumar Roy
Centre for Mathematical Biology and Ecology
Department of Mathematics, Jadavpur University
Kolkata 700032, West Bengal, India.
E-mail: pritiju@gmail.com
Fax No. +913324146584, Ph.No. +919432095603
Collaborator : Sonia Chowdhury
Amar Nath Chatterjee
Life cycle of HIV within a host cell
Life cycle of HIV within a host cell
Human Immunodeficiency Virus
Introductory Observations

Over the last several years extensive research has been made
in our understanding of the pathogenesis of HIV-1 infection.

HIV-1 pathogenesis and drugs which act either by blocking
the integration of viral RNA into the host CD4+T cells, or by
inhibiting the proper cleavage of viral proteins inside an
infected cell.

HIV-1 infection is very much associated with an extremely
vigorous virus specific Cytotoxic T- Lymphocyte (CTL)
response that declines disease progression.
Introductory Observations




Retroviral therapy when began to a HIV-1 individual, the main
clinical indicators of that HIV-1 positive patient are in the
follow up both the viral load and the CD4+T cells count in
blood plasma.
When therapy is started, make a portion to the immune cells
to be toxic thereby introducing toxicity in the immune
system of the individual.
Thus qualitative aspects of the HIV-1 specific CTL response
is to be an important determinants of the efficacy of these
response in controlling viral replication.
The main purpose of this study is to develop a mathematical
framework that can be used to understand the various drug
therapy in optimum controlled level.
Introductory Observations

In this paper we build on a mathematical model HIV-1 infection
to CD4+T cell as a host cell including the mentioned inhibitor
drug.

We have also considered that the growth of CD4+T cells is
governed by a logistic equation.

During long term drug therapy some infected cell become
toxic and this Cytotoxic T-lymphocyte (CTL) responses
against virus producing cell. Since CTLs production is
simulated by infected CD4+T cells. We have introduced a
positive feed back function to generate excessive CTLs.
Formulation of HIV-1 Model
Assumption-1
To
generate the model of T-cell infection by HIV, we first
consider the T cell dynamics in the absence of HIV. We
propose
 being the rate of production of infectible CD4+T cells
and d is the natural death rate per cell.
Where
We
assume that the proliferation of T cells is governed by a logistic
function in which p is the maximum proliferation rate and Tm is the
maximum T cell population density at which the proliferation shuts off.
Formulation of the Mathematical Model
Assumption-2
In the presence of HIV ,T cell become infected. The number
of new infection at the steady state is proportional to x(t)y(t)
and  be the constant rate contact.


Based on the above assumption we can write down
Assumption- 3
Now during long term drug therapy some infected cell become toxic and this
Cytotoxic T-lymphocyte (CTL) responses against virus producing cell. Since CTLs
production is simulated by infected CD4+T cells ,we have introduced a positive feed
back function of the form
in our model.


Then in the presence of CTL the model dynamics becomes
Where ρ is the killing rate of virus producing cell by CTL, and k
is the equilibrium constant i.e rate of stimulation of CTL and b is the base line rate
mortality of CTL .

Assumption- 4

Let us introduce the non dimensional quantities
and substituting these in the model equation
Theoretical study of the system
Some basic results:

The model equation has the following positive equilibrium
E 1 (P1,0,0) and E*(P*,Q*,R*) where,
Basic reproductive ratio of the model,
Clearly E 1 always exists and E* equilibrium P* is always
greater than zero, and R* > 0 is also greater than zero.
Local Stability Analysis:
Numerical Simulation:
Table.1 Variables and parameters used in the models
Figure.1
Fig.1 shows that the model variables (x(t),y(t),and z(t)) oscillates initially. But system
moves towards its stable region as time increases. It seems that as n increases from 1 to 6
the Uninfected CD4+T cell and CTL population density increases, where as the Infected
cell population decreases.
Figure.2
Figure 2. Solution Trajectories of the system with different k. other parameter remained unchanged as in
Table1. In Fig.2 The numerical simulation shows that the system approaches to its infection free steady
state as k 0.005 and n 6. As k increases from 0.001 to 0.005 together with n from 2 to 6 the level of
Infected CD4+T cells approaches to zero. It can be easily seen that The level of Uninfected CD4 +T cells
and CTL population increases with increases in k and n.
Figure.3
Figure 3. Solution Trajectories of the system with different ρ. other parameter remained unchanged as in
Table1. In Fig.3 variation of three model variables are plotted against time. We see that as ρ increases from
0.1 to 0.3 the Uninfected CD4+T cell population increases gradually. Where as The Infected CD4+T cell
population and level of CTL decreases rapidly. With the increase of n from 2 to 5 it has been observed that
the level of Infected CD4+T cell approached towards zero whereas the CTL population density decreases
gradually and Uninfected CD4+T cell population density very fast. This figure also shows that when n is n 5,
ρ does not effect the solution trajectories for Uninfected and Infected cell population only.
In Delayed System Observation

l
CTL is secreted from the Infected CD4+T cells
after introducing the mentioned inhibitor drug
and this simulation does not produce
instantaneously. Infected
CD4+T cell has a
positive feedback effect on CTL and there exist
a time lag between this process. So we have
incorporated a delay term τ in the third
equation of our basic model.
Delay Induced system
In our above model we have incorporated delay
Figure.4
Figure 4. solution trajectories of the system for
different values
and n. Keeping all other
parameter s are same as in Table.1. In Fig.4 we see
that as increases the amplitude of oscillation of the
system variables increases. It shows that when n=1
and = 1 the system for uninfected cell population
moves towards its stable region. If increases the
amplitude of oscillation increases but as time
increases the system moves towards its
asymptotically stable region. Fig.4 also shows limit
cycle running in presence delay( > 0) together with
n increases from 2 to 6. It also shows that delay
does not effect on Uninfected CD4+T cell
population when n=1. But when n increases
together with delay factor the limit cycle is running.
We also notice that when n=1 there is an oscillation
in early stage, but as time increases the amplitude
of oscillation decreases and moves towards stability.
If n increases from 2 to 6 together with the level of
CTL forms limit cycle.
Discussion and Conclusions:
We have consider a basic mathematical model representing short term dynamics of HIV-1
infection in response to avail drug therapies.
Our main aim is to find out the threshold values of the system parameter for
which the disease can be controlled.
Analytical study focuses on the qualitative aspects of the model dynamics. Here we show
the existence and uniqueness of the solution of the dynamical variables P,Q and R locally
holds in the positive octant, with the condition that all other parameters assume non
negative values.
Through the stability analysis we obtain the sufficient condition.
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Acknowledgement
Research is supported by the Department of
Science and Technology, Government of
India, Mathematical Science. Office
No.SR/S4/MS: 558/08
and its a pleasure to acknowledge
All India Council for Technical Eduction for
their travel support to present the research
work in WCE 2010, U.K.