# A Numerical Comparison for a Discrete HIV Infection of CD4+ T-Cell Model Derived from Nonstandard Numerical Scheme

Authors: Mevlüde Yakıt Ongun; İlkem Turhan
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

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.


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## Body

## 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 ϕ=h+O(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 equation:(1)dxdt=F(x,λ),
where λ is a parameter. The simplest nonstandard finite difference schemes are
(2)t→tn=hn,x(t)→xn,F(x)→F(xn),dxdt→xn+1-xnϕ,
where ϕ depends on the step size Δt=h and satisfies ϕ=h+O(h2). It should be chosen
(3)ϕ=1-e-RhR,
where R is calculated from a knowledge of the fixed points of (1), that is,
(4)F(x~)=0.
Assume that the last equation has I-real solutions and denote by
(5){x~i;i=1,2,…,I}.
Now define Ri as
(6)Ri=dFdx|x=x~i,
and take R as
(7)R=Max{|Ri|;i=1,2,…,I}.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:
(8)dTdt=p-αT+rT(1-T+ITmax)-kVT,dIdt=kVT-βI,dVdt=NβI-γV,
where T(t), I(t), V(t) denote the concentration of CD4+ T cells, the concentration of infected CD4+ T cells by the HIV 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, α, β, and γ are positive constants and p is the source of CD4+ T cells from precursors, α is the death rate of CD4+ T cells, β is the death rate of infected cells, and finally γ is the viral clearance rate constant [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 Section2, 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.


## 2. Discretization of the Model

The nonlinear differential equation system (1) will be discretizated as follows:
(9)T(t)→Tn,I(t)→In+1,V(t)→Vn+1,T2(t)→Tn+1Tn,T(t)I(t)→Tn+1In,T(t)V(t)→Tn+1Vn.
If Tn+1, In+1, and Vn+1 are explicitly solve from (8), the following iterations will be obtained:
(10)Tn+1=(1+(r-α)ϕ1(h,α))Tn+ϕ1(h,α)p1+ϕ1(h,α)(kVn+r(Tn+In)/Tmax),In+1=In+ϕ2(h,β)kVnTn+11+βϕ2(h,β),Vn+1=Vn+ϕ3(h,γ)NβIn+11+ϕ3(h,γ)γ,
where denominator functions are chosen as
(11)ϕ1(h,α)=e(r-α)h-1r-α,ϕ2(h,β)=eβh-1β,ϕ3(h,γ)=eγh-1γ.
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-α>0 or if r-α<0 then ϕ1>1/(Tn(α-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)
detB<1,(ii)
1-trB+detB>0,(iii)
1+trB+detB>0,where B and trB 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, citejodar are relevant to the roots of characteristic polynomials.Lemma 1.
For the quadratic equationλ2-aλ+b=0 the roots satisfy |λi|<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 polynomialp(λ) is given by p(λ)=λ3+a1λ2+a2λ+a3. The solutions λi, i=1,2,3, of p(λ)=0 satisfy |λi|<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).
Letx~ be an equilibrium point of the difference equation
(12)xn+1=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 pointx~ is locally asymptotically stable.(2)
If at least one root of the characteristic polynomial has absolute value greater than one, then the equilibrium pointx~ is unstable.Equilibrium points of (8) are found as follows:
(13)X1*=(T1*,I1*,V1*)=(-Tmax(α-r+(α-r)2+4rp/Tmax)2r,0,0),X2*=(T2*,I2*,V2*)=(Tmax(r-α+(α-r)2+4rp/Tmax)2r,0,0),X3*=(T3*,I3*,V3*)=(γkN,-τN,-βτγ),
where
(14)τ=-pTmaxk2N2+αγTmaxkN-rγTmaxkN+rγ2k(rγ+kNβTmax).
Only fixed points X2* and X3* have real biological meaning: the uninfected steady state X2*=(T2*,I2*,V2*) and the (positive) infected steady state X3*=(T3*,I3*,V3*) [21, 22]. Firstly, let us examine the fixed point X2*. Equation (10) is rewritten as follows:
(15)f=(1+(r-α)ϕ1(h,α))Tn+ϕ1(h,α)p1+ϕ1(h,α)(kVn+r(Tn+In)/Tmax),g=In+ϕ2(h,β)kVnTn+11+βϕ2,h=Vn+ϕ3(h,γ)NβIn+11+ϕ3(h,γ)γ.
By using these equations, Jacobian matrix will be found:
(16)J(Tn,In,Vn)=[fTnfInfVngTngIngVnhTnhInhVn],
where
(17)fTn=ηω-(ηTn+ϕ1p)ϕ1rω2Tmax,fIn=-(ηTn+ϕ1p)ϕ1rω2Tmax,fVn=-(ηTn+ϕ1p)ϕ1kω2,gTn=kϕ2Vnη(1+ϕ2β)ω-kϕ2Vn(ηTn+ϕ1p)ϕ1r(1+ϕ2β)ω2Tmax,gIn=11+βϕ2-kϕ2Vn(ηTn+ϕ1p)ϕ1r(1+ϕ2β)ω2Tmax,gVn=-k2ϕ2Vn(ηTn+ϕ1p)ϕ1(1+ϕ2β)ω2+kϕ2(ηTn+ϕ1p)(1+ϕ2β)ω,hTn=ϕ3Nβkϕ2Vnη(1+ϕ2β)(1+ϕ3γ)ω-ϕ3Nβkϕ2Vn(ηTn+ϕ1p)ϕ1r(1+ϕ2β)(1+ϕ1γ)ω2Tmax,hIn=ϕ3Nβ(1-kϕ2Vn(ηTn+ϕ1p)ϕ1r/ω2Tmax)(1+ϕ2β)(1+ϕ3γ),hVn=11+ϕ3γ+ϕ3Nβ(1+ϕ2β)(1+ϕ3γ)·(-k2ϕ2(ηTn+ϕ1p)Vnϕ1ω2+kϕ2(ηTn+ϕ1p)ω),η=(1+(r-α)ϕ1),ω=(1+ϕ1(kVn+r(Tn+In)Tmax)).
Firstly, let us find Jacobian matrix of (10) around X2* to analyze the stability of this fixed point. We obtain(18)J(T2*,I2*,V2*)=[ηζTmax-χϕ1rζ2Tmax-ϕ1rχζ2Tmax-ϕ1kχζ2011+βϕ2ϕ2kχ(1+βϕ2)ζ0ϕ3Nβ(1+βϕ2)(1+γϕ3)ζ(1+βϕ2)+ϕ3Nβϕ2kχ(1+βϕ2)(1+γϕ3)ζ],where(19)ζ=(1+ϕ1rT2*Tmax),χ=ηT2*+ϕ1p.
To analyze the stability of X2*, we need to find eigenvalues [32]
(20)(λ-ηζTmax-χϕ1rζ2Tmax)(λ2-aλ+b)=0,
where
(21)a=TraceB,b=DetB,B=[11+βϕ2ϕ2kχ(1+βϕ2)ζϕ3Nβ(1+βϕ2)(1+γϕ3)ζ(1+βϕ2)+ϕ3Nβϕ2kχ(1+βϕ2)(1+γϕ3)ζ].
The first eigenvalue is
(22)λ1=ηζTmax-χϕ1rζ2Tmax.
We can find the other eigenvalues from
(23)f(λ)=λ2-aλ+b=0,
where
(24)a=ζ(1+γϕ3)+ζ(1+βϕ2)+ϕ3Nβϕ2kχζ(1+γϕ3)(1+βϕ2),b=1(1+γϕ3)(1+βϕ2).
By considering Lemma 1, when |λi|<1, i=1,2, the following conditions are satisfied and then the fixed point X2* is locally asymptotic stable(i)
b=1/(1+γϕ3)(1+βϕ2)<1,(ii)
f(-1)=1+a+b=(ζ((1+γϕ3)(1+βϕ2)+3+γϕ3+βϕ2)+kϕ3Nβϕ2χ)/ζ(1+βϕ2)(1+γϕ3)>0,(iii)
f(1)=1-a+b=(ζ((1+γϕ3)(1+βϕ2)-1-γϕ3-βϕ2)-kϕ3Nβϕ2χ)/ζ(1+βϕ2)(1+γϕ3)>0.Finally, let us examine the fixed point X3*. Jacobian matrix around the fixed point X3* is obtained as follows:(25)J(T3*,I3*,V3*)=[ηϑTmax-ϱϕ1rϑ2Tmax-ϱϕ1rϑ2Tmax-ϱϕ1kϑ2kϕ2V3*(ηϑTmax-ϱϕ1r)ϑ2(1+ϕ2β)Tmaxϑ2Tmax-kϕ2ϱϕ1rV3*ϑ2Tmax(1+ϕ2β)kϕ2ϱ(ϑ-V3*ϕ1k)ϑ2(1+ϕ2β)(ϕ3Nβkϕ2V3*)(ηϑTmax-ϱϕ1r)ϑ2Tmax(1+ϕ2β)(1+ϕ3γ)ϕ3Nβ(ϑ2Tmax-kϕ2ϱϕ1rV3*)ϑ2Tmax(1+ϕ2β)(1+ϕ3γ)ϑ2(1+ϕ2β)+ϕ3Nβkϕ2ϱ(-kV3*ϕ1+ϑ)ϑ2(1+ϕ2β)(1+ϕ3γ)],where(26)ϑ=(1+ϕ1(kV3*+r(T3*+I3*)Tmax)),ϱ=(ηT3*+ϕ1p).
By considering Lemma 2, we write the characteristic polynomial of J(T3*,I3*,V3*) as follows:
(27)p(λ)=λ3+a1λ2+a2λ+a3,
where
(28)a1=-((1+γϕ3)(θ1+θ2(1+βϕ2))+ϑ2Tmax(1+βϕ2)+Tmaxϕ2ϕ3Nβkϱθ3)Tmax(1+γϕ3)×(ϑ4)-1a2=(θ1θ2+ϕ2θ2kV3*ϕ1rϱ)(1+γϕ3)Tmax2ϑ4(1+ϕ2β)(1+γϕ3)+Tmaxϑ2(θ1+θ2(1+ϕ2β))Tmax2ϑ4(1+ϕ2β)(1+γϕ3)+ϕ3Nβkϕ2ϱTmaxθ2(V3*kϕ1+θ3)Tmax2ϑ4(1+ϕ2β)(1+γϕ3),a3=-θ2ϑ4(1+γϕ3)(1+βϕ2)Tmax2,
where
(29)θ1=ϑ2Tmax-kϕ2V3*ϱϕ1r,θ2=ηϑTmax-ϱϕ1r,θ3=ϑ-V3*ϕ1k,θ4=ϑ2(1+ϕ2β)(1+ϕ3γ).
If Lemma 2 is satisfied, we can say that the fixed point X3* 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:
(30)p=0.1,α=0.02,β=0.3,γ=2.4,k=0.0027,Tmax=1500,N=10,r1=0.1,r2=0,r3=0.1,h=0.01.R0=kNT2*/γ 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 X2* 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 X3*, we will use Lemma 2, and we will also conclude whether fixed point X3* is asymptotically stable or not.


### 4.1. Analysis of the Fixed PointX2*

Forr=0.05, firstly let us calculate the basic reproduction number
(31)R0=kNT2*γ=10.16236213>1.
The first eigenvalue is
(32)λ1=0.9996978319.
From (23), other eigenvalues are found as follows:
(33)|λ2|=1.015636510,|λ3|=0.9583755908.
From Lemma 1, we see that(i)
b=0.9733612405<1,(ii)
f(-1)=3.947373342>0,(iii)
f(1)=-0.6508605×10-3<0.Therefore, the fixed point X2* is unstable for r=0.05.Forr=0.8, the basic reproduction number is;
(34)R0=16.45456718>1,
and the first eigenvalue is
(35)λ1=0.9922289848.
From (23),
(36)|λ2|=1.022738962,|λ3|=0.9517201132.
So according to Lemma 1,(i)
b=0.9733612405<1,(ii)
f(-1)=3.947820316>0,(iii)
f(1)=-0.10978345×10-2<0.As a result, the fixed point X2* is unstable for r=0.8.Forr=3; the basic reproduction number is
(37)R0=16.76287750>1.
The first eigenvalue is
(38)λ1=0.9706383389,
and other eigenvalues are found as follows:
(39)|λ2|=1.023053068,|λ3|=0.9514279092.
By Lemma 1,(i)
b=0.9733612413<1,(ii)
f(-1)=3.947842218>0,(iii)
f(1)=-0.11197357×10-2<0.Therefore, X2* fixed point is unstable for r=3. For r=0.001, the basic reproduction number is
(40)R0=0.5919960938<1.
We obtain the eigenvalues as λ1=0.999809947, λ2=0.9972049810, and by Lemma 1, λ3=0.9760894290:(i)
b=0.9733612405<1,(ii)
f(-1)=3.9446655650>0,(iii)
f(1)=0.668305×10-4>0.So, the fixed point X2* is locally asymptotically stable for r=0.001.


### 4.2. Analysis of the Fixed PointX3*

Forr=0.05, let us find characteristic polynomial of J(T3*,I3*,V3*):
(41)p(λ)=λ3-2.973320340λ2+2.946641816λ-0.9733214565.
By using Lemma 2,(i)
p(1)=0.20×10-7>0,(ii)
(-1)3p(-1)=7.893283612>0,(iii)
1-(a3)2=0.526453423×10-1|a2-a3a1|=0.52645332×10-1}1-(a3)2>|a2-a3a1|.Therefore the fixed point X3* is locally asymptotically stable for r=0.05. For r=0.8, let us find characteristic polynomial of J(T3*,I3*,V3*):
(42)p(λ)=λ3-2.972874374λ2+2.945765581λ-0.9728906881.
From Lemma 2,(i)
p(1)=0.519×10-6>0,(ii)
(-1)3p(-1)=7.891530643>0,(iii)
1-(a3)2=0.534837090×10-1|a2-a3a1|=0.53483786×10-1}1-(a3)2≯|a2-a3a1|.We obtain that the fixed point X3* is unstable for r=0.8. For r=3, let us find characteristic polynomial of J(T3*,I3*,V3*):
(43)p(λ)=λ3-2.971566609λ2+2.943214150λ-0.9716455797.
By using Lemma 2,(i)
p(1)=0.1961×10-5>0,(ii)
(-1)3p(-1)=7.886426339>0,(iii)
1-(a3)2=0.559048674×10-1|a2-a3a1|=.55904590×10-1}1-(a3)2>|a2-a3a1|.We have the fixed point X3* locally asymptotically stable for r=3.


## 4.1. Analysis of the Fixed PointX2*

Forr=0.05, firstly let us calculate the basic reproduction number
(31)R0=kNT2*γ=10.16236213>1.
The first eigenvalue is
(32)λ1=0.9996978319.
From (23), other eigenvalues are found as follows:
(33)|λ2|=1.015636510,|λ3|=0.9583755908.
From Lemma 1, we see that(i)
b=0.9733612405<1,(ii)
f(-1)=3.947373342>0,(iii)
f(1)=-0.6508605×10-3<0.Therefore, the fixed point X2* is unstable for r=0.05.Forr=0.8, the basic reproduction number is;
(34)R0=16.45456718>1,
and the first eigenvalue is
(35)λ1=0.9922289848.
From (23),
(36)|λ2|=1.022738962,|λ3|=0.9517201132.
So according to Lemma 1,(i)
b=0.9733612405<1,(ii)
f(-1)=3.947820316>0,(iii)
f(1)=-0.10978345×10-2<0.As a result, the fixed point X2* is unstable for r=0.8.Forr=3; the basic reproduction number is
(37)R0=16.76287750>1.
The first eigenvalue is
(38)λ1=0.9706383389,
and other eigenvalues are found as follows:
(39)|λ2|=1.023053068,|λ3|=0.9514279092.
By Lemma 1,(i)
b=0.9733612413<1,(ii)
f(-1)=3.947842218>0,(iii)
f(1)=-0.11197357×10-2<0.Therefore, X2* fixed point is unstable for r=3. For r=0.001, the basic reproduction number is
(40)R0=0.5919960938<1.
We obtain the eigenvalues as λ1=0.999809947, λ2=0.9972049810, and by Lemma 1, λ3=0.9760894290:(i)
b=0.9733612405<1,(ii)
f(-1)=3.9446655650>0,(iii)
f(1)=0.668305×10-4>0.So, the fixed point X2* is locally asymptotically stable for r=0.001.


## 4.2. Analysis of the Fixed PointX3*

Forr=0.05, let us find characteristic polynomial of J(T3*,I3*,V3*):
(41)p(λ)=λ3-2.973320340λ2+2.946641816λ-0.9733214565.
By using Lemma 2,(i)
p(1)=0.20×10-7>0,(ii)
(-1)3p(-1)=7.893283612>0,(iii)
1-(a3)2=0.526453423×10-1|a2-a3a1|=0.52645332×10-1}1-(a3)2>|a2-a3a1|.Therefore the fixed point X3* is locally asymptotically stable for r=0.05. For r=0.8, let us find characteristic polynomial of J(T3*,I3*,V3*):
(42)p(λ)=λ3-2.972874374λ2+2.945765581λ-0.9728906881.
From Lemma 2,(i)
p(1)=0.519×10-6>0,(ii)
(-1)3p(-1)=7.891530643>0,(iii)
1-(a3)2=0.534837090×10-1|a2-a3a1|=0.53483786×10-1}1-(a3)2≯|a2-a3a1|.We obtain that the fixed point X3* is unstable for r=0.8. For r=3, let us find characteristic polynomial of J(T3*,I3*,V3*):
(43)p(λ)=λ3-2.971566609λ2+2.943214150λ-0.9716455797.
By using Lemma 2,(i)
p(1)=0.1961×10-5>0,(ii)
(-1)3p(-1)=7.886426339>0,(iii)
1-(a3)2=0.559048674×10-1|a2-a3a1|=.55904590×10-1}1-(a3)2>|a2-a3a1|.We have the fixed point X3* 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 X2* and X3* 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 X3* and X2* are respectively compared to classical methods such as forward Euler and Runge-Kutta. The fixed points X2* and X3* 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. But if the step size is chosen larger, the efficiency of NFDS is clearly seen. In Table 3, stability results for fixed point X2* are given for different r values. It is shown in [21] that if R0>1, X2* is unstable and HIV infection persist in T-cell population. If 0.093453<r<1.9118, then X3* is unstable. So in case of r=0.8, neither X2* nor X3* are stable (Figure 2). In Figures 1 and 3, the NFDS solutions of T, I and V converges to fixed point X3* 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.Table 1
Qualitative results of the fixed pointX3* for different time step sizes, r=0.05, t = 0–5000.
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
ConvergenceTable 2
Qualitative results of the fixed pointX2* 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
ConvergenceTable 3
Stability results of the fixed pointsX2* For different r values.
r
R
0
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
unstableNFDS solutions forT(t), I(t), and V(t), r=0.05.
(a)
(b)
(c)NFDS solutions forT(t) and I(t), r=0.8.
(a)
(b)Comparison with NFDS and 4th order Runge-Kutta solutions forI(t), V(t), and T(t), r=3.
(a)
(b)
(c)



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*Source: 375094-2013-02-26.xml*