In this paper, Human Immunodeficiency Virus (HIV) infected cells is found out using a Simulink model. The Simulink solution is equivalent or very close to the exact solution of the problem. Accuracy of the Simulink solution to this problem is better than the existing numerical methods. The main advantage of Simulink model is that solution of any dynamicalproblem can be obtained by anybody without writing any codes. Anillustrative numerical example is presented for the proposed method

Kumaresan Nallasamy

English

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Evaluation of HIV infected Cells
N. Kumaresan 1
Institute of Mathematical Sciences
Faculty of Science, University of Malaya
Kuala Lumpur 50603, Malaysia.
Abstract: In this paper, Human Immunodeficiency Virus (HIV) infected cells is found out using a
Simulink model. The Simulink solution is equivalent or very close to the exact solution of the prob-
lem. Accuracy of the Simulink solution to this problem is better than the existing numerical methods.
The main advantage of Simulink model is that solution of any problem can be obtained by anybody
without writing any codes. An illustrative numerical example is presented for the proposed method.
Key words: CD4+T Cells, HIV, Immunotherapy, Ordinary differential equations and Simulink.
AMS (MOS) Subject Classifications: 49 K 30, 65 Y 15.
1. Introduction
AIDS-Acquired Immunity Deficiency Syndrome is the disease that has affected the whole world. It
is caused by Human Immunodeficiency Virus (HIV). There is still much work to be completed in
the search for an anti-HIV vaccine. The aim of chemotherapies are to kill or halt the pathogen, but
treatment can boost the immune system, can serve to help the body fight infection on its own [6]. The
new treatments are aimed to reducing viral population and improving the immune response. This
brings new hope to the treatment of HIV infection.
Once HIV enters to a human body, the human immune system tries to get rid of it. The invasion is
reported to CD4+T cells. The CD4+T is a protein marker in the surface of the T cells and the letter
T refers to thymus, the organ responsible for maturing these cells after they migrate from the bone
marrow. The surface of CD4+T possesses a protein that can bind to foreign substances such as HIV.
The HIV needs a host in order to reproduce and the above mentioned protein provides shelter. The
HIV virus is a retrovirus, the RNA of virus is converted into DNA inside the CD4+T cell. Thus, when
infected CD4+T cells begin to multiply to fight this pathogen, they produce more virus [2, 4, 5, 6, 3].
Simulink is a MATLAB add-on package that many professional engineers use to model dynamical
processes in control systems. Simulink allows to create a block diagram representation of a system
and run simulations very easily. Simulink is really translating block diagram into a system of ordinary
differential equations. Simulink is the tool of choice for control system design, digital signal process-
ing (DSP) design, communication system design and other simulation applications [1]. This paper
focuses upon the implementation of Simulink approach for finding HIV affected cells in a human body
by solving differential equation.
This paper is organized as follows. In section 2, the Mathematical model of HIV immunology system
is described. In section 3, simulink method is presented. In section 4, numerical example is given to
evaluate HIV cells. The final conclusion section demonstrates the efficiency of the method.
1E-mail : drnk2008@gmail.com, drnk2008@um.edu.my, Telephone: 00603 7967 4126, Fax: 00603 7967 4143
22. Mathematical model of HIV immunology system
The mathematical model of the HIV immunology system is described as a system of nonlinear differ-
ential equation:
dT (t)
dt
= s1 − s2V (t)
B1 + V (t)
− μT (t) − kV (t)T (t), T (0) = T0, (1)
dV (t)
dt
=
gV (t)
B2 + V (t)
− cV (t)T (t), V (0) = V0, (2)
where
1. T represents the concentration of CD4+T cells.
2. V is the concentration of HIV particles.
3. the term s1 − s2V (t)B1+V (t) is the source / proliferation of unaffected cells.
4. μT (t) is the natural loss of uninfected T cells.
5. kV (t)T (t) is loss by infection.
6. cV (t)T (t) is the viral loss.
7. gV (t)B2+V (t) is viral contribution to plasma.
8. μ is the death rate of T cells.
9. k is the infection rate of T cells.
10. g is the input rate of an external virus source.
11. c is the loss rate of virus and B1, B2 are half saturation constants.
3. Simulink method
Simulink is an interactive tool for modelling, simulating and analyzing dynamic systems. It enables
engineers to build graphical block diagrams, evaluate system performance and refine their designs.
Simulink integrates seamlessly with MATLAB and is tightly integrated with state flow for modelling
event driven behavior. Simulink is built on top of MATLAB. A Simulink model for the given prob-
lem can be constructed using building blocks from the simulink library. The solution curves can be
obtained from the model without writing any codes.
A simulink model is constructed for the following system of two differential equations as shown in the
Figure 1.
x′(t) = −x(t) + 1, x(0) = −1
y′(t) = −y(t) + 1, y(0) = 1.
3Constant
+
+
+
+
1
1
s
s
-1
-1
1
1
Constant 1
Scope
Gain
Gain 1
Integrator
Integrator 1
Figure 1. Simulink model
As soon as the model is constructed, the simulink parameters can be changed according to the prob-
lem. The solution of the system of differential equation can be obtained in the display block by
running the model.
Procedure for simulink solution
Step 1. Select the required number of blocks from the simulink Library.
Step 2. Connect the appropriate blocks.
Step 3. Make the required changes in the simulation parameters.
Step 4. Run the simulink model to obtain the solution.
4. Evaluation of HIV cells
The variables and parameter values of the equations 1 and 2 are given in Table 1.
Figure 2 is the Simulink model for the systems of differential equations 1 and 2. The Simulink curves
for the systems are shown in Figure 2. The Simulink solution is displayed in Table 2.
4Variables Parameters
T (0) 1000/mm3
V (0) 1000/ml
s1 2.0mm3d−1
s2 1.5mm3d−1
μ 0.002d−1
k 2.5 × 104mm3d−1
g 30d−1mm3
c 0.007mm3d−1
B1 14.0mm3
B2 1.0mm3
Table 1. Variables and Parameters
Constant
Constant2
Product2 Math
Function1
Math
Function
Gain
Gain2
2
14
1.5
X
X
30
X
1
-K
-K
-K
+
+
+
+
1
s
-
-
-
-
-
Integrator
Integrator1
Gain1
Gain3
Product1
Constant1
Gain 4
Product
Scope
1
1
1
s
u
u
-
Figure 2. Simulink model for HIV Immunology System
51000
900
800
700
600
500
400
300
200
0
0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1
100
0.5
Figure 3. Simulink curve for HIV Immunology System
t T (t) V (t)
0 1000 1000
0.1 987.7 660.3
0.2 975.8 334.4
0.3 969.8 171.5
0.4 966.6 89.2
0.5 964.9 47.5
0.6 963.9 26.3
0.7 963.4 15.5
0.8 963.0 9.9
0.9 962.7 7.0
0.1 962.6 5.4
Table 2. Simulink Solution of HIV Immunology System
5. Conclusion
HIV affected cells can be found out by Simulink model. The simulink solution is equivalent or close to
the exact solution of the problem. A numerical example is given to illustrate the derived results. In
future, Simulink approach can be used to solve Mathematical model of any scientific and engineering
problems without writing any codes.
References
[1] Albagul, Othman O. Khalifa, Wahyudi, Matlab and Simulink in Mechatranics, Int. J. engng Ed., 21 (2005),
896–905.
[2] K. R. Fister, S. Lenhart, J. S. Mc Nally, Optimizing Chemotherapy in an HIV Model, Electron. J. Diff. Eqns., 32
(1998), 1–12.
[3] M. Balter, New Hopes in HIV Disease, Science, 274 (1996), 1998–1991.
[4] D. Kirschner, S. Lenhart, S. Sebin, Optimal control of the Chemotherapy of HIV, J. Math. Biol., 35 (1997),
775–792.
[5] D. Kirschner, A. Perelson, R. Deboer, The Dynamics of HIV infection of CD4+T Cells, Mathematical Biosciences,
114 (1993), 81–125.
[6] D. Kirschner, G. F. Webb, Immunotherapy of HIV-1 Infection, Journal of Biological Systems, 6 (1998), 71–83.


Evaluation of HIV infected Cells

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