Ain Shams Engineering Journal (2014) xxx, xxx-xxx

Ain Shams University Ain Shams Engineering Journal

www.elsevier.com/locate/asej www.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Numerical approximation for HIV infection of CD4+ T cells mathematical model

Vineet K. Srivastava a *, Mukesh K. Awasthi b, Sunil Kumar c

a ISRO Telemetry, Tracking and Command Network (ISTRAC), Bangalore 560058, India b University of Petroleum and Energy Studies, Dehradun 248007, India

c Department of Mathematics, National Institute of Technology, Jamshedpur 831014, Jharkhand, India Received 12 September 2013; revised 19 November 2013; accepted 26 December 2013

KEYWORDS

HIV CD4+ T cells model;

Euler's method; Numerical simulation

Abstract A dynamical model of HIV infection of CD4+ T cells is solved numerically using an approximate analytical method so-called the differential transform method (DTM). The solution obtained by the method is an infinite power series for appropriate initial condition, without any discretization, transformation, perturbation, or restrictive conditions. A comparative study between the present method, the classical Euler's and Runge-Kutta fourth order (RK4) methods is also carried out.

© 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.

1. Introduction

Consider the dynamic model for HIV infection of CD4+ T cells [1]:

f = k - aT + tt(i 1 --gf) - k*VT,

= k* VT - ßl,

dV = N*ßl - c V,

where T(t), I(t) and V(t) denote concentration of uninfected, infected and virus population of CD4+ T cells by HIV in

the blood, respectively. rT^ 1 — is logistic growth of the

* Corresponding author. Tel./fax: +91 8050682145. E-mail address: vineetsriiitm@gmail.com (V.K. Srivastava). Peer review under responsibility of Ain Shams University.

healthy CD4+ T cells, Tmax is the maximum level of CD4+ T cells in the human body, r is the rate at which T cells multiply through mitosis when stimulated by antigen or mito-gen, k is the constant rate which the body produces CD4+ T cells from precursors in the bone marrow and thymus (i.e. k is the rate of production of CD4+ T cells), a is the natural turnover rate of T cells and. k VT is the incidence of HIV infection of healthy CD4+ T cells, where k > 0 is the rate of infection of T cells by virus. b is the per capita rate of disappearance of infected cells. N b is the rate of production of virions by infected cells, where N is the average number of virus particles produced by an infected T-cell and y is the death rate of virus particles [1-3]. In a normal human body, the level of CD4+ T cells in the peripheral blood is regulated at a level between 800 and 1200 mm—3. CD4+ T cells are also named as T helper cells or leukocytes. These cells are the most abundant white blood cells of the human immune system, which fight against diseases. HIV wreaks most havocilly these cells causing their decline and destruction, thus decreasing the resistance of the human immune system. The dynamic model has proved

2090-4479 © 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2013.12.012

V.K. Srivastava et al.

valuable in understanding the dynamics of HIV-1 infection. The practical and industrial exposures of such type of model can be seen in [4]. This type of models has been solved by various researchers [5-8].

Each equation in system (1) represents the rate of change with respect time with the initial conditions

r(0)=ri,o, /(0) = r2fl, F(0) = r3,o. (2)

The purpose of this work is to solve numerically the dynamic model for HIV infection of CD4+ T cells by using the differential transformation method. The obtained results are compared with those obtained by classical Euler's and RK4 methods.

The remaining part of the paper is organized as follows; in Section 2, the basic idea of the differential transform method is illustrated. The numerical implementation of the method for CD4+ T-cells model and numerical results and comparison between DTM, Euler's and RK4 methods are given in Section 3, and the concluding discussion is given in Section 4.

2. Differential transform method (DTM)

The differential transform method is the semi numerical analytical method developed by Zhou [9] for solving integral equations, ordinary, partial differential equations and differential equation systems that uses the form of polynomials as approximations of the exact solutions that are sufficiently differentiable. The method gives the solution in terms of convergent series with easily computable components. The fundamental operations of the DTM and its applications for various kinds of differential equations are given in [10-13]. In this section, the basic definitions and operations of the differential transformation are reviewed.

Let f(t) be a k-times differentiable function with respect to time t, then the differential transform of the kth derivative of f(t) is defined as

F(k) =

where f(t) is the original function and F(k) is the transformed function. The inverse differential transform of F(k) is defined as:

/« = £F(k)(t - to)'.

Clubbing Eqs. (3) and (4) together, we get /) 1

From Eq. (5), it can be seen that the concept of DTM is derived from Taylor series expansion, but the method does not calculate the derivatives symbolically. However, the relative derivatives are evaluated by an iterative way which is given by the transformed equation of the original function. For the implementation point of view, the original function is expressed by a finite series and so Eq. (4) is written as

Table 1 Fundamental operations of the differential transform method.

Original function Transformed function

/(') = «(') ± v(t)

/(') = MO /(') = "('WO /(t)=df

/w = /; «(s)* /(') = r

/( ') = exp(k ') /( ') = sin (ro ' + a)

F(k) = U(k) ± V(k)

F(k) = bU(k)

F(k) = E î=oU(k)Vk_s(k)

F(k) = (k + 1)U(k + 1)

F(k) = (k + 1)(k + 2).. .(k + m)U(k + m)

F(k)=Uikf11, k P 1

F(k) = d(k - m)

F(k)=Ü

F(k) = ¡kk sin (f + a)

/( ') = cos (x' + a) F(k) = ¡kr c°s (f + c

where N is decided by the convergence of natural frequency. The fundamental operations of DTM have been given in Table 1.

3. Numerical simulation

In this section, the numerical simulation is performed. The initial conditions are given as

T(0) = ri,o = 0.1,1(0) = r2,o = 0, V(0) = r3,o = 0.1, 9 k = 0.1, a = 0.02,b = 0.3,r = 3, V. (7)

C = 2.4, k* = 0.0027, Tmax = 1500, N* = 10 J

Applying the DTM to Eq. (1), the following iterative recurrence relation is obtained as

j I k - aTk + rTk - (t^)X),(k)

Tk+1 - (kpi) < maI s=0

> — rs/k-s(k) — k* FkTk-s] k

4+ = (T+î) ] k*Ë VsTk-s - b4

Vk+1 =(k^{N*b1k — cVkg, _

where Tk+1, /k+1 and Fk+1 are the differential transformation of T(t), 1(t) and V(t) respectively. The differential transformation of the initial conditions is given as

/') = £F(k)(' - 'o)k

To = 0.1, /0 = 0, V = 0.1. (9)

Table 2 Comparison between DTM, Euler and RK4 for T(').

DTM Euler RK4

0.0 0.100000 0.100000 0.100000

0.2 0.211648 0.1 91448 0.208801

0.4 0.422685 0.345502 0.406214

0.6 0.817940 0.605002 0.764351

0.8 1.546211 1.042060 1.413870

1.0 2.854053 1.777990 2.591200

Numerical approximation for HIV infection of CD4+ T cells mathematical model

Table 3 Comparison between DTM, Euler and RK4 for I(t).

t DTM Euler RK4

0.0 0.00000 0.000000 0.000000

0.2 6.36664e—06 5.48760e—06 6.03188e—06

0.4 1.39924e—05 1.11233e-05 1.31565e— 05

0.6 2.26514e—05 1.66243e-05 2.12207e—05

0.8 3.32836e—05 2.18436e—05 3.01728e—05

1.0 4.85399e—05 2.67125e—05 4.00314e—05

Table 4 Comparison between DTM, Euler and RK4 for V(t).

t DTM Euler RK4

0.0 0.100000 0.100000 0.100000

0.2 0.061880 0.057761 0.061881

0.4 0.038309 0.033366 0.038296

0.6 0.023920 0.019279 0.023706

0.8 0.016212 0.011145 0.014681

1.0 0.016050 0.006450 0.009102

The differential transform method series solution for the system (1) can be obtained as

T(t) = £ Tnf,

I(t) = Yjnf ;

V{t)=Y, Vntn;

For the computation convince N = 6 is taken. The convergence of differential transform method is shown in Tables 2-4. DTM solution is compared with Euler's method and RK4 method. From Tables 2-4, it can be deduced that DTM solutions are in good agreement with the RK4 method while solutions by Euler's method are less accurate. The sixth-order solution obtained by DTM for T(t), I(t) and V(t) is depicted

Figure 3

Sixth-order differential transform method solution for

Figure 4 Sixth-order differential transform method solution for T(t), I(t) and V(t).

V.K. Srivastava et al.

Figure 5 Comparison of solutions using DTM, RK4 and Euler's methods for T(t).

in Figs. 1-4. Figs. 5-7 show comparison between the DTM, Euler's and RK4 methods.

4. Conclusions

In this article, the differential transform method has been implemented for a dynamical model of HIV CD4+ T cells. The obtained solution by the method is a power series solution for an appropriate initial condition and finds the solution without any discretization, transformation, perturbation, or restrictive conditions. The solutions obtained by the differential transform method are compared well with those obtained by Euler's and RK4 methods. Additionally, this method, which is a simple and powerful mathematical tool, can be easily applied for solving nonlinear problems arising in systems of nonlinear differential equations and also in dynamical systems.

Acknowledgement

Figure 6 Comparison of solutions using DTM, RK4 and Euler's methods for I(t).

Figure 7 Comparison of solutions using DTM, RK4 and Euler's methods for V(t).

The authors are very grateful to the anonymous referees for

carefully reading the paper and for their constructive comments and suggestions which have improved the paper.

References

[1] Liancheng W, Michael YL. Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. Math Biosci 2006;200:44-57.

[2] Asquith B, Bangham CRM. The dynamics of T-cell fratricide: application of a robust approach to mathematical modeling in immunology. J Theoret Biol 2003;222:53-69.

[3] Nowak M, May R. Mathematical biology of HIV infections: antigenic variation and diversity threshold. Math Biosci 1991;106: 1-21.

[4] Perelson AS, Nelson PW. Mathematical analysis of HIV-I dynamics in vivo. SIAM Rev 1999;41:3-44.

[5] Arafa AAM, Rida SZ, Khalil M. Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection. Nonlinear Biomed Phys 2012;36(12).

[6] Ongun MY. The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4 + T cells. Math Comput Model 2011;53(5-6):597-603.

[7] Yiizba§i §uayip. A numerical approach to solve the model for HIV infection of CD4 + T cells. App Math Model 2012;36(12):5876-90.

[8] Dogan N. Numerical treatment of the model for HIV infection of CD + 4T cells by using multistep Laplace Adomian Decomposition Method. Discrete Dynamics in Nature and Society Volume; 2012. Article ID 976352.

[9] Zhou JK. Differential transform and its application for electrical circuits. Wuhan: Huazhong University Press; 1986 [in Chinese].

[10] Fatma A. Solutions of the system of differential equations by differential transform method. Appl Math Comput 2004;147:547-67.

[11] Fatma A. Application of differential transform method to differential-algebraic equations. Appl Math Comput 2004;152: 649-57.

[12] Arikoglu A, Ozkol I. Solution of boundary value problems for integro-differential equations by using differential transform method. Appl Math Comput 2005;168:1145-58.

[13] Srivastava VK, Awasthi MK. (1+n)-Dimensional Burgers' equation and its analytical solution: A comparative study of HPM, ADM and DTM. Ain Shams Eng J 2013. http://dx.doi.org/ 10.1016/j.asej.2013.10.004.

Numerical approximation for HIV infection of CD4+ T cells mathematical model

Vineet Kumar Srivastava received his M.Sc. (Mathematics) degree from the University of Allahabad in 2007 and M.Tech. (Industrial Mathematics and Scientific Computing) degree from Indian Institute of Technology Madras, Chennai, India in the year 2010. His research interests include Numerical PDE, Mathematical modeling, Ocean Engineering, Computational Biology, Computational Physics, Computational Astronomy, Flight Dynamics, Orbital and Celestial Mechanics. Presently, he is working as a Scientist/Engineer in the Indian Space Research Organization (ISRO), Bangalore, India.

Sunil Kumar is an Assistant Professor in the Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India. He received his M.Phil. from CSJM University, Kanpur, and Ph.D. degree from the Indian Institute of Technology, BHU, Varanasi. He is editor of more than fourty international journals. His current research mainly covers fractional calculus, Homotopy methods, Wavelet methods, analytical and numerical solutions of nonlinear problems arising in applied sciences and engineering phenomena.

Mukesh Kumar Awasthi has done his postgraduation in Mathematics from the University of Lucknow in the year 2007. He has obtained his Ph.D degree in Mathematics from Indian Institute of Technology Roorkee in 2012. His research interests are Fluid Mechanics, Hydrodynamic stability, viscous potential flow, Numerical PDE, Mathematical modeling and Computational Physics. Currently, he is working as an assistant Professor in the Department of Mathematics, University of Petroleum and Energy Studies, Dehradun, India.