Table Of ContentArafaetal.NonlinearBiomedicalPhysics2012,6:1
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RESEARCH Open Access
+
Fractional modeling dynamics of HIV and CD4
T-cells during primary infection
AAM Arafa1*, SZ Rida1 and M Khalil2
Abstract
In this paper, we introduce fractional-order into a model of HIV-1 infection of CD4+ T cells. We study the effect of
the changing the average number of viral particles N with different sets of initial conditions on the dynamics of
the presented model. Generalized Euler method (GEM) will be used to find a numerical solution of the HIV-1
infection fractional order model.
dx
1. Introduction =s−μx−βxy
At the present time there are several countries, particu- dt
dy
larly in Africa, with up to 35% of their populations =βxy−νy
between the ages of 15 and 50 years infected by human dt
immunodeficiency virus (HIV) [1]. Throughout the where all parameters and variables are non-negative. s
world, already over 16 million deaths have been caused
is the assumed constant rate of production of CD4+ T
bythisvirus.HIVisaretrovirusthattargetstheCD4+T -cells, μ is their per capita death rate, bxy is the rate of
lymphocytes, which are the most abundant white blood infection of CD4+ T -cells by virus, and vy is the rate of
cellsofthe immune system.AlthoughHIVinfects other
disappearance of infected cells. The viral variable has
cells also, it wreaks the most havoc on the CD4+ T cells
been omitted for simplicity as it is here assumed to be
bycausingtheirdeclineanddestruction,thusdecreasing linearly related to y. A more complete model of human
theresistanceoftheimmunesystem[2,3].Mathematical
immunodeficiency virus type 1 (HIV-1) dynamics con-
models have been proven valuable inunderstanding the
siders in addition to the uninfected and infected CD4+
dynamicsofHIVinfection[4-6].In1989,Perelsondevel- T -cells, x and y respectively, the number of virions in
opedasimplemodelforthe primary infection withHIV plasma, z. The following three equations are:
[7].Thismodelhasbeenimportantinthefieldofmathe-
maticalmodelingofHIVinfection,andmanyothermod- dx
=s−μx−βxz,
els have been proposed, which take this model as their dt
inspiration. Perelson et al. extended the model in 1993 dy
and discussed some of the model’s behavior [8]. They =βxz−νy,
dt
defined the model by considering four categories: unin- dz
fectedCD4+ Tcells,latently infectedCD4+T cells, pro- =cy−γz−βxz.
dt
ductivelyinfectedCD4+Tcellsandviruspopulation.
We will consider some models for HIV-1 population The third equation in the last mentioned reference
dynamics below [9]. Here there are two components: x, does not contain the term -bxz to account for the fact
the number of uninfected CD4+ T -cells and y, the that when a virus infects a CD4+ T -cell, z decreases at
number of infected such cells. Then the following two the same time as x decreases. Without this we have:
equations describe the evolution of the system:
*Correspondence:[email protected]
1Departmentofmathematics,FacultyofScience,SouthValleyUniversity,
Qena,Egypt
Fulllistofauthorinformationisavailableattheendofthearticle
©2012Arafaetal;licenseeBioMedCentralLtd.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons
AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductionin
anymedium,providedtheoriginalworkisproperlycited.
Arafaetal.NonlinearBiomedicalPhysics2012,6:1 Page2of7
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dx 2. Fractional calculus
=s−μx−βxz,
dt Fractional calculus has been extensively applied in many
dy fields [13,14]. Many mathematicians and applied
=βxz−νy,
researchers have tried to model real processes using the
dt
dz fractional calculus. Jesus, Machado and Cunha analyzed
=cy−γz. the fractional order dynamics in botanical electrical
dt
impedances [15,16]. Petrovic, Spasic and Atanackovic
Rong et al. further modified the model by incorporat- developed a fractional-order mathematical model of a
ing anti-retroviral effects to study the evolutionofdrug human root dentin. In biology, it has been deduced that
resistance [10]. They considered three classes of CD4+ the membranes of cells of biological organism have frac-
T cells: uninfected cells, infected cells in eclipse phase tional-order electrical conductance [17] and then are
and productively infected cells. The model depends on classified in groups of non-integer order models. Frac-
the observationthat for a virus, whenitentersa resting tional derivatives embody essential features of cell rheo-
CD4+ T cell, viral RNA may not be completely reverse logical behavior and have enjoyed greatest success in the
transcribedintoDNA.In [11],the authorsmodifiedthe field of rheology [18]. Fractional order ordinary differen-
ODE model proposed by Culshaw and Ruan into a sys- tial equations are naturally related to systems with
tem of fractional-order [12]. They showed that the memory which exists in most biological systems. Also,
model established in this paper possesses non-negative they are closely related to fractals, which are abundant
solutions, as desired in any population dynamics. They in biological systems. Hence, we propose a system of
obtained a restriction on the number of viral particles FODE for modeling HIV. We first give the definition of
released per infectious cell, in order for infection to be fractional-order integration and fractional-order differ-
sustained. Following Rong et al., we assume here that a entiation [19]. There are several approaches to the gen-
fraction of infected CD4+ T-cells return to the unin- eralization of the notion of differentiation to fractional
fected class. In view of this, the following model is pro- orders e.g. Riemann-Liouville, Caputo and Generalized
posed: Functions approach. For the concept of fractional deri-
vative, we will adopt Caputo’s definition, which is a
dT
=s−KVT−dT+bI, modification of the Riemann-Liouville definition and has
dt
the advantage of dealing properly with initial value
dI
=KVT−(b+δ)I, (1) problems.
dt Definition 1. The fractional integral of order a > 0 of
dV =NδI−cV. a function f: R+® R is given by
dt
1 x
With initial conditions Jαf(x)= ∫(x−t)α−1f(t)dt, (2)
(cid:7)(α)
0
T(0)=T ,I(0)=I ,V(0)=V .
0 0 0 Where J0f(x) = f(x), a > 0, x > 0.
In this model, T, I and V denote the concentration of Definition 2. Riemann-Liouville and Caputo fractional
uninfected CD4+ T cells, infected CD4+ T cells, and free derivatives of order a where a Î (n-1, n) of a continu-
HIV virus particles in the blood, respectively. δ repre- ous function f: R+® R is given respectively by
(cid:2) (cid:3)
sents death rate of infected T cells and includes the pos- Dαf(x)=Dm Jm−αf(x) , (3)
sibility of death by bursting of infected T cells, hence δ
≥ d. The parameter b is the rate at which infected cells (cid:2) (cid:3)
return to uninfected class while c is death rate of virus Dα∗f(x)=Jm−α Dmf(x) , (4)
and N is the average number of viral particles produced
by an infected cell. The rest of the paper is organized as Where m-1 <a ≤ m, m Î N.
follows. Section 2 gives an idea about fractional calculus. The reason of using fractional order differential equa-
In section 3, we introduce fractional-order into the tions is that they are naturally related to systems with
model that describes HIV infection of CD4+ T cells. memory which exists in most biological systems. Also
Section 4 gives an idea about the generalized Taylor’s they are closely related to fractals which are abundant in
formula while section 5 presents the idea of generalized biological systems. The definition of fractional derivative
Euler’s method for solving FODEs. Section 6 is devoted involves an integration which is non local operator (as it
for the numerical results. is defined on an interval) so fractional derivative is a
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non local operator. In other word, calculating time frac- 5. Generalized Euler method (GEM)
tional derivative of a function f (t) at some time t = t Most nonlinear fractional differential equations do not
1
requires all the previous history, i.e. all f (t) from t = 0 have analytic solutions, so approximations and numeri-
to t = t The results derived of the fractional systems cal techniques must be used [21]. The decomposition
1
are of a more general nature. However, the fundamental method (ADM) and the variational iteration method
solutions of these equations still exhibit useful scaling (VIM) are relatively new approaches to provide an ana-
properties that make them attractive for applications. lytical approximate solution to linear and nonlinear pro-
We would like to put your attention that time fractional blems, and they are particularly valuable as tools for
derivatives change also the solutions we usually get in scientists and applied mathematicians, because they pro-
standard system. The concept of fractional or non-inte- vide immediate and visible symbolic terms of analytic
ger order derivation and integration can be traced back solutions, as well as numerical approximate solutions to
to the genesis of integer order calculus itself. Most of both linear and nonlinear differential equations. In
the mathematical theory applicable to the study of non- recent years, the application of the ADM, VIM, in linear
integer order calculus was developed through the end of and nonlinear problems has been developed. On the
19th century. However it is in the past hundred years other hand, these methods are effective for small time, i.
that the most intriguing leaps in engineering and scien- e. t < < 1, however the standard homotopy perturbation
tific application have been found. The calculation tech- method (HPM) cannot solve the problem for larger time
nique has in some cases had to change to meet the and in fact the solution of the chaotic system using
requirement of physical reality. The derivatives are HPM is an open problem. Nevertheless by chance, there
understood in the Caputo sense. The general response are cases at which these methods give good approxima-
expression contains a parameter describing the order of tion for a large range of time (t). A few numerical meth-
the fractional derivative that can be varied to obtain var- ods for fractional differential equations have been
ious responses. One of the basic reasons of using frac- presented in the literature. However many of these
tional order differential equations is that “Fractional methods are used for very specific types of differential
order differential equations are, at least, as stable as equations, often just linear equations or even smaller
their integer order counterpart.” classes. Odibat and Momani derived the generalized
Euler’s method that we have developed for the numeri-
3. Fractional-order model derivation cal solution of initial value problems with Caputo deri-
Now we introduce fractional-order into the model (1) of vatives [22]. The method is a generalization of the
HIV infection of the CD4+ T -cells. The new system is classical Euler’s method. Consider the initial value pro-
described by the following set of FODEs of order a , a , blem
1 1
a > 0: (cid:2) (cid:3)
3 Dα∗y(t)=f t,y(t) ,y(0)=y0, (7)
Dα1(T)=s−KVT−dT+bI,
Dα2(I)=KVT−(b+δ)I, (5) For 0 <a ≤ 1, t > 0.
Let [0, a] be the interval over which we want to find
Dα3(V)=NδI−cV.
the solution of the problem (7). In actuality, we will not
find a function y(t) that satisfies the initial value pro-
4. Generalized Taylor’s formula blem (7). Instead, a set of points {tj, y(tj)} is generated,
In this section we introduce a generalization of Taylor’s and the points are used for our approximation. For con-
venience we subdivide the interval [0, a] into k subinter-
formula that involves Caputo fractional derivatives. This
generalization is presented in [20]. vals [tj, tj + 1] of equal width h = a/k by using the nodes
Suppose that tj = jh, forj = 0, 1,..., k. Assume that
Dkαf(x)∈ C(0,a],fork=0,1,...,n+1, where 0 <a ≤ y(t),Dα∗y(t),andD2∗αy(t) are continuous on [0, a] and
∗
use the generalized Taylor’s formula (5) to expand y(t)
1. Then we have
about t = t = 0. For each value t there is a value c so
(cid:5) (cid:6) 0 1
f(x)=(cid:4)i=n0 (cid:7)(ixαiα+1)(cid:2)Di∗α(cid:3)(0+)+(cid:7)(D(n∗(n++11)α)fα+(ξ1))x(n+1)α (6) thaty(t)=y(t0)+(Dα∗y(t))(t0)(cid:7)(αtα+1)+(D2∗αy(t))(c1)(cid:7)(2tα2α+1) (8)
With 0 ≤ ξ ≤ x, ∀ x Î (0, a].
In case of a = 1, the generalized Taylor’s formula (6) When (Dα∗y(t))(t0)=f(t0,y(t0)) and h = t1 are substi-
reduces to the classical Taylor’s formula. tuted into equation (8), the result is an expression for y
(t ):
1
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Figure3TheconcentrationofthefreeHIVvirusparticlesatN
Figure1TheconcentrationoftheuninfectedCD4+TcellsatN
=1000inthe1stcase.Graysolidline(a=1),Dottedline(a=
=1000inthe1stcase.Graysolidline(a=1),Dottedline(a=
0.99),Blacksolidline(a=0.95).
0.99),Blacksolidline(a=0.95).
y(t1)=y(t0)+f(cid:2)t0,y(t0)(cid:3)(cid:7)(αhα+1)+(D2∗αy(t))(c1)(cid:7)(2hα2α+1) y(cid:2)tj+1(cid:3)=y(cid:2)tj(cid:3)+ (cid:7)(αhα+1)f(cid:2)tj,y(cid:2)tj(cid:3)(cid:3) (9)
If the step size h is chosen small enough, then we may
neglect the second-order term (involving h2a) and get for j = 0, 1,...,k-1. It is clear that if a = 1, then the
generalized Euler’s method (9) reduces to the classical
hα Euler’s method.
y(t )=y(t )+ f(t ,y(t ))
1 0 (cid:7)(α+1) 0 0
6. Numerical results
The process is repeated and generates a sequence of We will solve the system (5) by using (GEM). Consider
points that approximates the solution y(t). The general that a = a = a = a. We used the following data set: s
1 2 3
formula for generalized Euler’s method (GEM) when tj =10,b=0.2,k=0.000024,d=0.01,δ =0.16,c =3.4,N
+1 = tj + h is varies.ForthissetofdataR0=3.13whenN=1000(Fig-
ures1,2,3,4,5,6)andR =5.01whenN=1600(Figures
0
7,8,9,10,11,12).Theinitialconditionsinthefirstcase
studyareT(0)=1000,I(0)=0,V(0)=0.001whileinthe
Figure2TheconcentrationoftheinfectedCD4+TcellsatN= Figure 4 The concentration of the uninfected CD4+ T at N =
1000inthe1stcase.Graysolidline(a=1),Dottedline(a=0.99), 1000inthe2ndcase.Graysolidline(a=1),Dottedline(a=
Blacksolidline(a=0.95). 0.99),Blacksolidline(a=0.95).
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Figure5TheconcentrationoftheinfectedCD4+TatN=1000 Figure7TheconcentrationoftheinfectedCD4+TcellsatN=
inthe2ndcase.Graysolidline(a=1),Dottedline(a=0.99), 1600inthe1stcase.Graysolidline(a=1),Dottedline(a=0.99),
Blacksolidline(a=0.95). Blacksolidline(a=0.95).
second case are T(0) =1000, I(0)= 10,V(0)= 10. In the cells increase. This assumes that the growth of healthy
twocasesthesystemgoestoinfectedsteadystate. T-cells slows down during the course of HIV infection.
We have to give an attention to the parameter b which
7. Conclusion is called the reverting rate of infected cells to uninfected
In this paper we employed the Generalized Euler class due to non-completion of reverse transcription.
method (GEM) as a reasonable basis for studying the Further, since only small fraction of infected cells will
solution of human T-cell lymphotropic virus (HIV-I) revert back due to incompletion of reverse transcription,
infection of CD4+ T-cells. We modified the integer- we expect the reverting rate b to be small. The basic
order model (1) into a fractional-order model (5). From reproduction number of the presented model (5) is
the obtained results in the presented figures, it is clear given in as:
that in the primary stage of the infection with the (HIV)
virus, a dramatically decrease in the level of the CD4+ NδKs
R =
T-cells occurs because of the death of such infected 0 cd(b+δ)
cells. On the other hand, the number of the free HIV
virus particles and the number of susceptible CD4+ T
Figure6TheconcentrationofthefreeHIVvirusparticlesatN Figure8TheconcentrationoftheinfectedCD4+TcellsatN=
=1000inthe2ndcase.Graysolidline(a=1),Dottedline(a= 1600inthe1stcase.Graysolidline(a=1),Dottedline(a=0.99),
0.99),Blacksolidline(a=0.95). Blacksolidline(a=0.95).
Arafaetal.NonlinearBiomedicalPhysics2012,6:1 Page6of7
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Figure9TheconcentrationofthefreeHIVvirusparticlesatN Figure11TheconcentrationoftheinfectedCD4+TcellsatN
=1600inthe1stcase.Graysolidline(a=1),Dottedline(a= =1600inthe2ndcase.Graysolidline(a=1),Dottedline(a=
0.99),Blacksolidline(a=0.95). 0.99),Blacksolidline(a=0.95).
It represents the average number of secondary infec- CD4+ T-cells and increase the number of free virus
tion caused by a single infected T cell in an entirely substantially, but does not change the stability of the
susceptible T cell population, throughout its infectious steady state. The concentration of susceptible CD4+ T
period. For system (5), if the basic reproduction num- cells T(t), infected CD4+ T cells I(t), and free HIV
ber R0 ≤ 1, the the virus is cleared and no HIV infec- virus particles V(t) in the blood have been obtained,
tion persists. If R0 > 1, the HIV infection persists in therefore when a ® 1 the solution of the fractional
the T-cell population. In the two presented cases, R0 = model (5) Ta (t), Ia(t), Va(t), reduce to the standard
3.13 when N = 1000, (see Figures 1, 2, 3, 4, 5, 6) and solution T(t), I(t), V(t). Finally, the recent appearance
R0 = 5.01 when N = 1600 (see Figures 7, 8, 9, 10, 11, of fractional differential equations as models in some
12), so the system goes to infected steady state. It is fields of applied mathematics makes it necessary to
clear from the definition of R0 that R0 decreases as the investigate methods of solution for such equations
reverting rate, b of infected cells increases, hence R0 (analytical and numerical) and we hope that this work
can be low for a high parametric value of b. Increasing is a step in this direction.
the N value will decrease the numbers of uninfected
Figure10TheconcentrationoftheuninfectedCD4+Tcellsat Figure12TheconcentrationofthefreeHIVvirusparticlesatN
N=1600inthe2ndcase.Graysolidline(a=1),Dottedline(a= =1600inthe2ndcase.Graysolidline(a=1),Dottedline(a=
0.99),Blacksolidline(a=0.95). 0.99),Blacksolidline(a=0.95).
Arafaetal.NonlinearBiomedicalPhysics2012,6:1 Page7of7
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Authordetails 21. HashimI,AbdulazizO,MomaniS:Homotopyanalysismethodfor
1Departmentofmathematics,FacultyofScience,SouthValleyUniversity, fractionalIVPs.CommunicationsinNonlinearScienceandNumerical
Qena,Egypt.2Departmentofmathematics,FacultyofEngineering,Modern Simulation2009,14:674-684.
ScienceandArtsUniversity(MSA),Giza,Egypt. 22. ZaidMO,MomaniS:Analgorithmforthenumericalsolutionof
differentialequationsoffractionalorder.JournalofAppliedMathematics&
Authors’contributions Informatics2008,26:15-27.
AAprojectedandcoordinatedthenumericalexperiments.SRparticipated
themaincomputerprogramming.MKperformedthenumericalsimulation. doi:10.1186/1753-4631-6-1
Allauthorsreadandapprovedthemanuscript. Citethisarticleas:Arafaetal.:FractionalmodelingdynamicsofHIV
andCD4+T-cellsduringprimaryinfection.NonlinearBiomedicalPhysics
20126:1.
Competinginterests
Theauthorsdeclarethattheyhavenocompetinginterests.
Received:2September2011 Accepted:3January2012
Published:3January2012
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