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0Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 375094, 9 pages http://dx.doi.org/10.1155/2013/375094
Research Article
A Numerical Comparison for a Discrete HIV Infection of CD4+ T-Cell Model Derived from Nonstandard Numerical Scheme
Mevlude Yakit Ongun1 and ilkem Turhan2
1 Department of Mathematics, Suleyman Demirel University, 32260 Isparta, Turkey
2 Department of Mathematics, Dumlupinar University, 43100 KUtahya, Turkey
Correspondence should be addressed to Mevlude Yakit Ongun; mevludeyakit@sdu.edu.tr Received 8 August 2012; Revised 31 October 2012; Accepted 14 November 2012 Academic Editor: Juan Torregrosa
Copyright © 2013 M. Yakit Ongun and Ilkem Turhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A nonstandard numerical scheme has been constructed and analyzed for a mathematical model that describes HIV infection of CD4+ T cells. This new discrete system has the same stability properties as the continuous model and, particularly, it preserves the same local asymptotic stability properties. Linearized Stability Theory and Schur-Cohn criteria are used for local asymptotic stability of this discrete time model. This proposed nonstandard numerical scheme is compared with the classical explicit Euler and fourth order Runge-Kutta methods. To show the efficiency of this numerical scheme, the simulated results are given in tables and figures.
1. Introduction
Mathematical models are used not only in the natural sciences and engineering disciplines, but also in the social sciences. The differential equations in these mathematical models are usually nonlinear autonomous differential equation systems which have only time-independent parameters. It is not always possible to find the exact solutions of the nonlinear models that have at least two ordinary differential equations. It is sometimes more useful to find numerical solutions of this type systems in order to programme easily and visualize the results. Numerous methods can be used to obtain the numerical solutions of differential equations. By applying a numerical method to a continuous differential equation system, it becomes a difference equation system, in other words discrete time system. While applying these numerical methods, it is necessary that the new difference equation system should provide the positivity conditions and exhibit the same quantitative behaviours of continuous system such as stability, bifurcation, and chaos. It is well known that some traditional and explicit schemes such as forward Euler and Runge-Kutta are unsuccessful at generating oscillation, bifurcations, chaos, and false steady states, despite
using adaptative step size [1-6]. For forward Euler's method, if the step size h is chosen small enough and the positivity conditions are satisfied, it is seen that local asymptotic stability for a fixed point is saved while in some special cases Hopf bifurcation cannot be seen. Instead of classical methods, nonstandard finite difference scheme (NFDS) can be alternatively used to obtain more qualitative results and to remove numerical instabilities. These schemes are developed for compensating the weaknesses that may be caused by standard difference methods, for example, numerical instabilities. Also, the dynamic consistency could be presented well by NFDS [7]. The most important advantage of this scheme is that, choosing a convenient denominator function instead of the step size h, better results can be obtained. If the step size h is chosen small enough, the obtained results do not change significantly but if h gets larger this advantage comes into focus.
The NFDS modeling procedures were given in 1989 by Mickens [8]. It removes the problems discussed above by using the suitable denominator function <p = h + 0(h2). The papers [2, 8-16] show how to choose the denominator function and apply this scheme to many models. Micken's method can be summarized by using [13] as follows.
Let us consider the following ordinary differential equa-
^=F(x,\), at
where A is a parameter. The simplest nonstandard finite difference schemes are
tn = hn,
F(x)-^F(x n),
where 0 depends on the step size At = h and satisfies 0 = h + 0(h ). It should be chosen
where R is calculated from a knowledge of the fixed points of (1), that is,
F (x) = 0.
Assume that the last equation has /-real solutions and denote by
[xf, i = 1,2,..., I}.
Now define R, as
viruses, and free HIV virus particles, respectively. k > 0 is the infection rate. Each infected CD4+ T cell is assumed to produce N virus particles during its life time [21]. Tmax is the maximum level of CD4+ T cell population density in the body. rT(1 - (T + I)/Tmax) is logistic equation, where r is the average specific T-cell growth rate [19]. p, a, p, and y are positive constants and p is the source of CD4+ T cells from precursors, a is the death rate of CD4+ T cells, p is the death rate of infected cells, and finally y is the viral clearancerateconstant[22-24]. Nelson et al. focused on other models of HIV-1 infection in [25,26]. These models deal with dynamics occurring after drug treatment. They analyzed the delay differential equation models of HIV-1 infection. Initially they give a standard model of HIV and then afterwards they give delay model of HIV. They analyze the model and give some lemma and proofs. Culshaw and Ruan consider a delay differential equation model of HIV infection of CD4+ T cells [27].
This paper is organized as follows: in Section 2, in order to obtain explicit solutions of (8), first the model is discretizated in a nonstandard form and this discrete model provides the positivity conditions. In Section 3, some lemmas and Linearized Stability Theorem are given for the local asymptotic stability of the discrete time systems. In Section 4, the theorical results obtained in former section are compared with the other numerical methods and the simulated results are given.
and take R as
R =^ ' dx
2. Discretization of the Model
The nonlinear differential equation system (1) will be dis-cretizated as follows:
R = Max [\Ri\; i = 1,2,..., 1}.
In this paper, an NFDS scheme is applied to human immunodeficiency virus (HIV), which has spread rapidly around the world in recent years and thus it gains importance. In the last decade, the published papers about the epidemiology of HIV are less in number and they are not detailed enough. A few of these models were simulated using numerical methods such as Runge-Kutta or Euler methods. However, explicit methods are generally known to exhibit contrived chaos whenever the discretization parameters exceed certain values [17,18].
A model about HIV infection of CD4+ T cells was presented by Perelson and Nelson [19,20]. This model is given as follows:
dl=p-aT + rT(l-—)-kVT, dt y ( T
^=kVT-ßI, dt 1
dV_ dt
= Nßl - yV,
where T(t), I(t), V(t) denote the concentration of CD4+ T cells, the concentration of infected CD4+ T cells by the HIV
T(t)~ ^ T A re
I(t)- ~> ^n+1,
V(t)- - Vn+1,
T2 (t) - > T T A n+1 A re
T(t)I(t)- _V T T
T(t)V(t)- —> T ,v ±n+1y n'
If Tn+1, In+1, and Vn+1 are explicitly solve from (8), the following iterations will be obtained:
T„+i =
In+1 =
Vn+1 =
(1 + (r - a) (h, a)) Tn + (h, a) p 1+h (h,a)(kVn + r(Tn + In)/TmJ '
In + (h,ß)kVnTn+i 1+ß$2 (h, ß) ' V» + h (h,y)NßIn+i 1 + $3 (h,Y)Y '
where denominator functions are chosen as
fa1 (h, a) = <h (h,ß) = fa3(h y) =
(r-a)h
r-a e?h -1 ß ' eyh - 1
Detailed information about how to find different nonlocal terms to different denominator functions can be read in [9,10,13,15]. Let r1, r2, r3 >0 and p, k, r, Tmax > 0. In order to obtain positive iterations Tn+1, In+1, and Vn+1 we have to require r-a > 0 or if r-a < 0 then fa > \/(Tn(a- 1) + p). If we take the numerical values and initial conditions in [21], for each nonnegative initial conditions r1, r2, and r3, the iterations Tn, In, and Vn and consequently Tn+1, In+1, and Vn+1 are also nonnegative.
3. Stability Analysis of the Model
Some useful lemmas and a theorem should be given for local asymptotic stability of discrete systems. Especially, it is necessary to investigate Schur-Cohn criteria which deal with coefficient matrix of the linearized system as follows:
(i) det B < 1,
(ii) 1- tr B + det B > 0,
(iii) 1 + tr B + det B > 0,
where B and tr B denote coefficient matrix of the linearized system and trace of the matrix, respectively. One can find information in [13, 28-31] about the usage of Schur-Cohn criteria which do not need many process as in continuous models.
The following lemmas and theorem given in citejury, cite-jodar are relevant to the roots of characteristic polynomials.
Lemma 1. For the quadratic equation X2 -aX + b = 0 the roots satisfy \Xt\ <1, i= 1,2, if and only if the following conditions are satisfied:
(i) b<1,
(ii) 1- a + b > 0,
(iii) 1 + a + b > 0.
Lemma 2 (Jury conditions, Schur-Cohn criteria, n = 3). Suppose the characteristic polynomial p(X) is given by p(X) = X3 + a1 X2 + a2X + a3. The solutions Xi, i = 1,2,3, of p(X) = 0 satisfy \Xt\ < 1 if the following three conditions are held:
(i) p(1) = 1 + a1 + a2 + a3 >0,
(ii) (-1)3p(-1) = 1-a1 + a2 - a3 >0,
(iii) 1 - (a3)2 > \a2 - a3a1\.
Theorem 3 (the linearized stability theorem). Let x be an equilibrium point of the difference equation
= F(xn,xn-1,...,xn-k), n = 0,1,..
where the function F is a continuously differentiable function defined on some open neighborhood of an equilibrium point x. Then the following statements are true.
(1) If all the roots of the characteristic polynomial have absolute value less then one, then the equilibrium point x is locally asymptotically stable.
(2) If at least one root ofthe characteristicpolynomial has absolute value greater than one, then the equilibrium point x is unstable.
Equilibrium points of (8) are found as follows:
x[ = (TÏ,IÏ,V*)
Tmax (x-r+ ^(«-r)2 + 4rp/Tmax)
x; = (T;,I;,V;)
Tmax (r-a+ ^(a-r)2 +4rp/Tmax)
x; = (T:,I;,V:) = I-L,-L1-ÊL
-pTmaxk2N2 + aYTmaxkN - ryTmJcN + ry2
k (ry + kNßTm
Only fixed points X2 and XJ have real biological meaning: the uninfected steady state XJ = (T2J,!2^,V2j) and the (positive) infected steady state XJ = (TJ,IJJ,VJ) [21, 22]. Firstly, let us examine the fixed point XJ. Equation (10) is rewritten as follows:
(1 + (r - a) (h, a)) Tn + fa (h, a) p
1 + fa (h (X) (kVn + r(Tn + In) /Tmax) '
In + fa (h,ß)kVnTn+l 1+ßfa2 ' ^n + fa3 (h,y)NßIn+1
1+(^3 (K y)y
By using these equations, Jacobian matrix will be found:
J(T , I ,V) =
Ît„ fi„ fvn 9t„ 9i„ 9v„
±n v n
r = (nTn + faP)fair fT" « u2T '
fi„ = fv„ = _
(1Tn + fap) far ^max '
(yTn + <pip) fak
3t„ =
0i„ =
0v„ =
hr = 1 n
^fa^nU (lT„ + fap) V
(1+faß)* (1+hß)cV2Tmax '
_J__(nTn + <f>ip)hr
1+ßfa (1+$2ß)u2 Tmax '
k2faVn №n + fap) fa kfa (nTn + fap) (1+faß)ü>2 (1+faß) u '
faNßkfa 2Vnti faNßkfaVn (nTn + fap) far
(1 + faß) (1 + faY) w (1 + faß) (1 + far) u2Tm
faNß(1 _ kfaVn (rfCn + fa p) far/w2TmaX) (1 + faß)(1 + fa y) ,
h 1 + faNß
Vn 1+faY (1+faß)(1+faY)
-k2fa (nTn + faip)ynfai +kfa1(^L+fa1p)
y= (1 + (r-a)fai),
(o= ( 1+fa (kVn +
r(Tn + In) T
Firstly, let us find Jacobian matrix of (10) around X2 to analyze the stability of this fixed point. We obtain
j(T;,I;,V2-) '^max -X'Plr
S 1 max
-$irX max 1
1+ß$2 foNß
? fokX
gi + ß^ + hNßhkx
(i + ß$2)(i + Yh) (i + ß$2)(i + Yh)t;
<=(1 +
X = nT*2 + faP-
To analyze the stability of X2, we need to find eigenvalues [32] V^rnax -XfarS
(\2 -aX + b) = 0,
a = Trace B, b = Det B,
1+ßfa faNß
(1 + M<
^(1 + ^fa) + fa3N^fakX (1 + Pfa)(1 + Yfa) (1 + Pfa)(l + Yfa)ï
The first eigenvalue is
tfTmax -xfar
We can find the other eigenvalues from
f(X) = \2-a\ + b = 0, ( (1 + yfa)+ ((1 + ßfa) +faNßfakX
¿;(1 + Yfa)(1 + ßfa) 1
(1 +yfa) (1 + ßfa)'
By considering Lemma 1,when \Xt\ < 1, i= 1,2,thefollo wing conditions are satisfied and then the fixed point X2 is locally asymptotic stable
(i) b=1/(1 + yfa)(1 + pfa) < 1,
(ii) f(-1) = 1+a + b = (C((1 + yfa)(1 + pfa2) + 3 + yfa + Pfa) + kfaNfifax)/Ç(1 + pfa)(1 + Yfa) > 0,
(iii) f(1) = 1 - a + b = (C((1 + yfa)(1 + pfa) -1-Yfa-№2) - kfaNfifax)/Ç(1 + Pfa)(1 + Yfa) > 0.
Finally, let us examine the fixed point XJ. Jacobian matrix around the fixed point XJ is obtained as follows:
j(T;,I;,V;) =
n9Tmax - efa1r Qfair
kfa2V; (n^Tmax -Qfair) 92 (1+fa2ß)Tmax
(fa3Nßkfa2V3;)(n9Tmax - 6fair) _
PTmax (1 + falß) (1 + fa3Y) PTmax (1 + falß) (1 + fa3?)
tfTmax -kfa2QfairV3; PTmax (1+faiß)
kfa2Q(9-v;faik) 92 (1 + fa2ß)
fa3*ß (92Tmax - kfa2QfairV3; ) 92 (1 + fa2ß) + fa3^ßkfa2^ (-¡¿Vf fai + 9)
92 (1+fa2ß)(1+fa3Y)
9 = (1+fa1 (kv; +
. r(T; +i;)
Q = (nT; +faip).
By considering Lemma 2, we write the characteristic polyno-
mial of J(TJ,1?, VJ ) as follows
p (X) = X3 + a1X2 + a2X + a3,
ai =- ((1 + Yfa3) (Oi +02 (1+ ßfa2)) + 92Tmax (1 + ßfa2)
+ Tmaxfa2fa3NßkQd3) Tmax (1 + Yfa3) * ^^
= (O1O2 +fa202kV3;fairQ)(1 + yfa3) a2 T2max94 (1+fa2ß)(1+Yfa3)
Tmax92 (Oi +O2 (1+fa2ß)) ?2m^ (1+fa2ß)(1+Yfa3)
fa3Nßkfa2QTmax02 (V^fai + O3
T2max94 (1+fa2ß)(1+Yfa3) _O2_
9 (1+Yfa3)(1+ßfa2)T2m ax '
a^ = -
01 = 92Tmax -kfa2V3; (fair,
02 = n9Tmax - Q^
03 = 9-v;faik,
04 = 92 (1+fa2ß)(1 + fa3Y)-
If Lemma 2 is satisfied, we can say that the fixed point XJ is locally asymptotically stable. Finally, it is important to say that the stability depends on time step size h as it can be seen in Jacobian.
4. Numerical Results
In this section, we will use the values and the initial conditions in [21]. These values are given as follows:
p = 0.1, a = 0.02, ß = 0.3, Y = 2.4, k = 0.0027, Tmax = 1500, N = 10, ri = 0.1, r2 = 0, r3 =0.1, h = 0.01.
R0 = kNTJ/Y is the basic reproduction number. Wang and Li [21] present that if the basic reproduction number R0 < 1, the HIV infection is cleared from the T-cell population; if R0 > 1, the HIV infection persists. In this section we will calculate R0 and see whether XJ is locally asymptotically stable or not for different values of r. And by using the criterion given in Section 3, we will check the validity of the results. For the fixed point XJ, we will use Lemma 2, and we will also conclude whether fixed point XJ is asymptotically stable or not.
4.1. Analysis of the Fixed Point XJ. For r = 0.05, firstly let us calculate the basic reproduction number
= 10.16236213 > 1.
The first eigenvalue is
= 0.9996978319. From (23), other eigenvalues are found as follows: \\2\ = 1.015636510, \\3\ = 0.9583755908. From Lemma 1, we see that
(i) b = 0.9733612405 < 1,
(ii) f(-1) = 3.947373342 > 0,
(iii) f(1) = -0.6508605 x 10-3 < 0. Therefore, the fixed point XJ is unstable for r = 0.05.
For r = 0.8, the basic reproduction number is; R0 = 16.45456718 > 1, and the first eigenvalue is
X1 = 0.9922289848.
From (23),
\X2\ = 1.022738962, \X3\ = 0.9517201132.
So according to Lemma 1,
(i) b = 0.9733612405 < 1,
(ii) f(-1) = 3.947820316 > 0,
(iii) f(1) = -0.10978345 x 10-2 < 0.
As a result, the fixed point Xj is unstable for r = 0.8. For r = 3; the basic reproduction number is
R0 = 16.76287750 > 1.
The first eigenvalue is
X1 = 0.9706383389,
and other eigenvalues are found as follows:
\X2\ = 1.023053068, \X3\ = 0.9514279092.
By Lemma 1,
(i) b = 0.9733612413 < 1,
(ii) f(-1) = 3.947842218 > 0,
(iii) f(1) = -0.11197357 x 10-2 < 0.
Therefore, Xj fixed point is unstable for r = 3. For r = 0.001, the basic reproduction number is
Table 1: Qualitative results of the fixed point Xj for different time step sizes, r = 0.05, t = 0-5000.
R0 = 0.5919960938 < 1.
We obtain the eigenvalues as X1 = 0.999809947, X2 = 0.9972049810, and by Lemma 1, X3 = 0.9760894290:
(i) b = 0.9733612405 < 1,
(ii) f(-1) = 3.9446655650 > 0,
(iii) f(1) = 0.668305 x 10-4 > 0.
So, the fixed point Xj is locally asymptotically stable for r = 0.001.
h Euler Runge-Kutta NFDS
0.001 Convergence Convergence Convergence
0.01 Convergence Convergence Convergence
0.1 Convergence Convergence Convergence
0.5 Divergence Convergence Convergence
1 Divergence Divergence Convergence
10 Divergence Divergence Convergence
100 Divergence Divergence Convergence
Table 2: Qualitative results of the fixed point XJ for different time
step sizes, r = 0.001, t = 0 -500.
h Euler Runge-Kutta NFDS
0.001 Convergence Convergence Convergence
0.01 Convergence Convergence Convergence
0.1 Convergence Convergence Convergence
0.5 Convergence Convergence Convergence
1 Divergence Convergence Convergence
10 Divergence Divergence Convergence
100 Divergence Divergence Convergence
Table 3: Stability results of the fixed points Xj For different r values.
r Ro Stability
0.001 0.591960938 stable
0.01 0.1117598344 stable
0.02 0.9742785788 stable
0.021 1.433991904 unstable
0.04 8.493379921 unstable
0.05 10.16236213 unstable
0.8 16.45456718 unstable
4.2. Analysis of the Fixed Point Xj. For r = 0.05, let us find characteristic polynomial of J(TJJ, Ij ,Vj):
p (X) = X3 - 2.973320340X2 + 2.946641816X - 0.9733214565.
By using Lemma 2,
(i) p(1) = 0.20 x 10-7 > 0,
(ii) (-1)3p(-1) = 7.893283612 > 0,
\-(a,)2 = 0.526453423 x 10-1 ] . / \2 ^ ,
(iii) , , -l } 1 - (a3) > \a2 - a3a v ' \a2-a3a1\ = 0.52645332 x 10 ^ v 3 1 2 3
Therefore the fixed point X3j is locally asymptotically stable for r = 0.05. For r = 0.8, let us find characteristic polynomial of/(T3M3*,V3*):
p (X) = A3 - 2.972874374X2 + 2.945765581X - 0.9728906881.
500 1000 1500 2000 2500 3000 3500 4000 t
- T(n+1)
- Fixed point r3*
300 250 200 150 100 50 0
0 500 1000 1500 2000 2500 3000 3500 4000 t
- I(n+1)
- Fixed point I3
ll\djflMÂ)iÀJlAÀÀAMAAMAAAflMw
0 500 1000 1500 2000 2500 3000 3500 4000 t
- Fixed point V3*
Figure 1: NFDS solutions for T(t), I(t), and V(t), r = 0.05.
From Lemma 2,
(i) p(1) = 0.519 x 10-6 > 0,
(ii) (-1)3p(-1) = 7.891530643 > 0,
\-(a,)2 = 0.534837090 x 10-1 1 , / x2 1 1
(iii) |™| = 0.53483786 x 10-1 } 1 - ^ > ^ - ^
We obtain that the fixed point XJ is unstable for r = 0.8. For r = 3, let us find characteristic polynomial of J(TJ, IJ ,V3J):
p (X) = A3 - 2.971566609X2 + 2.943214150X - 0.9716455797.
600 500 400 300 200 100 0
100 200
400 500
- NFDS, T(
1000 900 800 700 600 500 400 300 200 100 0
400 500
- NFDS, I(n+V
Figure 2: NFDS solutions for T(t) and I(t), r = 0.8.
By using Lemma 2,
(i) p(1) = 0.1961 x 10-5 > 0,
(ii) (-1)3p(-1) = 7.886426339 > 0,
1-M2 = 0559048674 x 10-1 \ . , s2 ^ , ,
(iii) w , } 1 - (a3) > \a2 -fl,fl,i.
v ' \a2-^a1\ = .55904590 x 10 ^ y3' ' 2 3 11
We have the fixed point XJ locally asymptotically stable for r = 3.
5. Conclusions
In general, it is too hard to analyze the stability of nonlinear three-dimensional systems. In this paper, by using the proposed NFDS scheme, nonlinear ordinary differential equation system which describes HIV infection of CD4+ T cells, is discretizated and the behaviour of the model is investigated. It is seen that the local asymptotic stability results of the fixed points XJ and XJ of the discrete time system satisfying the positivity condition are the same as in [21]. In Tables 1 and 2, for different step size h and for different r values, the qualitative stability results, obtained by NFDS, of the fixed point XJ and XJ are respectively compared to classical methods such as forward Euler and Runge-Kutta. The fixed points XJ and XJ are locally asymptotically stable for the values r given in these two tables. If step size h is chosen small enough, the results of the proposed NFDS are similar with the results of the other two numerical methods. (43) But if the step size is chosen larger, the efficiency of NFDS is clearly seen. In Table 3, stability results for fixed point XJ are
0 100 200 300 400 500 600 t
- NFDS
- 4th order Runge-Kutta
Fixed point Г3*
300 200 100
0 100 200 300 400 500 600
- NFDS - 4th order Runge-Kutta
Fixed point I3
1200 1000 800 S 600
400 200 0
0 100 200 300 400 500 600 t
- NFDS
- Fixed point V3*
4th order Runge-Kutta
Figure 3: Comparison with NFDS and 4th order Runge-Kutta solutions for I(t), V(t), and T(t), r = 3.
given for different r values. It is shown in [21] that if R0 > 1, Xj is unstable and HIV infection persist in T-cell population. If 0.093453 < r < 1.9118, then Xj is unstable. So in case of r = 0.8, neither Xj nor Xj are stable (Figure 2). In Figures 1 and 3, the NFDS solutions of T, I and V converges to fixed point Xj as simulated for r = 0.05 and r = 3, respectively. Also in Figure 3, Runge-Kutta and proposed NFDS scheme are compared graphically. All the numerical calculations and simulations are performed by using Maple programme. In conclusion, the efficiency of the proposed NFDS scheme is investigated and compared with other numerical methods.
Acknowledgments
The authors would like to thank Professor Apostolos Had-jidimos and Professor Michael N. Vrahatis for their valuable suggestions during the preparation of this paper. Also, the authors are grateful to the reviewers for their constructive comments, which have enhanced the paper.
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