research article numerical modeling of fractional-order biological...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 816803 11 pageshttpdxdoiorg1011552013816803

Research ArticleNumerical Modeling of Fractional-Order Biological Systems

Fathalla A Rihan12

1 Department of Mathematical Sciences College of Science UAE University Al Ain 15551 UAE2Department of Mathematics Faculty of Science Helwan University Cairo 11795 Egypt

Correspondence should be addressed to Fathalla A Rihan frihanuaeuacae

Received 17 May 2013 Accepted 23 June 2013

Academic Editor Ali H Bhrawy

Copyright copy 2013 Fathalla A Rihan This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We provide a class of fractional-order differential models of biological systems with memory such as dynamics of tumor-immunesystem and dynamics of HIV infection of CD4+ T cells Stability and nonstability conditions for disease-free equilibrium andpositive equilibria are obtained in terms of a threshold parameterR

0(minimum infection parameter) for each model We provide

unconditionally stable method using the Caputo fractional derivative of order 120572 and implicit Eulerrsquos approximation to find anumerical solution of the resulting systems The numerical simulations confirm the advantages of the numerical technique andusing fractional-order differential models in biological systems over the differential equations with integer order The results maygive insight to infectious disease specialists

1 Introduction

Mathematical models using ordinary differential equationswith integer order have been proven valuable in understand-ing the dynamics of biological systems However the behav-ior ofmost biological systems hasmemory or aftereffectsThemodelling of these systems by fractional-order differentialequations has more advantages than classical integer-ordermathematical modeling in which such effects are neglectedAccordingly the subject of fractional calculus (ie calculusof integral and derivatives of arbitrary order) has gainedpopularity and importance mainly due to its demonstratedapplications in numerous diverse and widespread fields ofscience and engineering For example fractional calculus hasbeen successfully applied to system biology [1ndash5] physics[6ndash9] chemistry and biochemistry [10] hydrology [11 12]medicine [13 14] and finance [15] In some situations thefractional-order differential equations (FODEs) models seemmore consistent with the real phenomena than the integer-order models This is due to the fact that fractional deriva-tives and integrals enable the description of the memoryand hereditary properties inherent in various materials andprocesses Hence there is a growing need to study and use thefractional-order differential and integral equations

Fractional-order differential equations are naturallyrelated to systems with memory which exists in most

biological systems Also they are closely related to fractals[16] which are abundant in biological systems It has beendeduced in [3] that the membranes of cells of biologicalorganism have fractional-order electrical conductance andthen are classified in groups of noninteger-order modelsFractional derivatives embody essential features of cellrheological behavior and have enjoyed greatest success inthe field of rheology [17] In this paper we propose systemsof FODEs for modeling the interactions of tumor-immunesystem (see Section 2) and HIV infection of CD4+ T cellswith immune system (see Section 3)

However analytical and closed solutions of these types offractional equations cannot generally be obtained As a con-sequence approximate and numerical techniques are playingimportant role in identifying the solution behavior of suchfractional equations and exploring their applications [18ndash21]The Adomian decomposition method [22 23] extrapolationmethod [24] multistep method [25] monotone iterativetechnique [26] and predictor-corrector approach [18 27]are proposed to provide numerical solutions for linear andnonlinear FODEs The monograph of [28] investigates theinterconnection between fractional differential equations andclassical differential equations of integer-order derivatives Inthis paper we provide a numerical approach for the resultingsystem using implicit Eulerrsquos scheme (see Section 4) We alsostudy the stability and convergence of the proposed method

2 Abstract and Applied Analysis

We first give the definition of fractional-order integrationand fractional-order differentiation [29 30]There are severalapproaches to the generalization of the notion of differenti-ation to fractional orders for example Riemann-LiouvilleCaputo and Generalized Functions approaches Let 1198711 =

1198711[119886 119887] be the class of Lebesgue integrable functions on [119886 119887]

119886 lt 119887 lt infin

Definition 1 The fractional integral (or the Riemann-Liouville integral) of order 120573 isin R+ of the function 119891(119905) 119905 gt 0

(119891 R+ rarr R) is defined by

119868120573

119886119891 (119905) =

1

Γ (120573)

int

119905

119886

(119905 minus 119904)120573minus1

119891 (119904) 119889119904 119905 gt 0 (1)

The fractional derivative of order 120572 isin (119899 minus 1 119899) of 119891(119905) isdefined by two (nonequivalent) ways

(i) Riemann-Liouville fractional derivative take frac-tional integral of order (119899 minus 120572) and then take 119899thderivative as follows

119863120572

lowast119891 (119905) = 119863

119899

lowast119868119899minus120572

119886119891 (119905) 119863

119899

lowast=

119889119899

119889119905119899 119899 = 1 2

(2)

(ii) Caputo-fractional derivative take 119899th derivative andthen take a fractional integral of order (119899 minus 120572)

119863120572

119891 (119905) = 119868119899minus120572

119886119863119899

lowast119891 (119905) 119899 = 1 2 (3)

We notice that the definition of time-fractional derivativeof a function 119891(119905) at 119905 = 119905

119899involves an integration and

calculating time-fractional derivative that requires all the pasthistory that is all the values of 119891(119905) from 119905 = 0 to 119905 =

119905119899 For the concept of fractional derivative we will adopt

Caputorsquos definition which is a modification of the Riemann-Liouville definition and has the advantage of dealing properlywith initial value problems The following remark addressessome of the main properties of the fractional derivatives andintegrals (see [8 29ndash31])

Remark 2 Let 120573 120574 isin R+ and 120572 isin (0 1) Then

(i) if 119868120573119886

1198711

rarr 1198711 and 119891(119905) isin 119871

1 then 119868120573

119886119868120574

119886119891(119905) =

119868120573+120574

119886119891(119905)

(ii) lim120573rarr119899

119868120573

119886119891(119909) = 119868

119899

119886119891(119905) uniformly on [119886 119887] 119899 = 1 2

3 where 1198681119886119891(119905) = int

119905

0119891(119904)119889119904

(iii) lim120573rarr0

119868120573

119886119891(119905) = 119891(119905) weakly

(iv) if 119891(119905) is absolutely continuous on [119886 119887] thenlim120572rarr1

119863120572

lowast119891(119905) = 119889119891(119905)119889119905

(v) thus 119863120572

lowast119891(119905) = (119889119889119905)119868

1minus120572

lowast119891(119905) (Riemann-Liouville

sense) and119863120572119891(119905) = 119868

1minus120572

lowast(119889119889119905)119891(119905) (Caputo sense)

The generalized mean value theorem and another propertyare defined in the following remark [32]

Remark 3

(i) Suppose that 119891(119905) isin 119862[119886 119887] and 119863120572

lowast119891(119905) isin 119862(119886 119887] for

0 lt 120572 le 1 and then we have

119891 (119905) = 119891 (119886) +

1

Γ (120572)

119863120572

lowast119891 (120585) (119905 minus 119886)

120572

with 119886 lt 120585 lt 119905 forall119905 isin (119886 119887]

(4)

(ii) If (i) holds and119863120572lowast119891(119905) ge 0 for all 119905 isin [119886 119887] then 119891(119905)

is nondecreasing for each 119905 isin [119886 119887] If 119863120572lowast119891(119905) le 0

for all 119905 isin [119886 119887] then 119891(119905) is nonincreasing for each119905 isin [119886 119887]

We next provide a class of fractional-order differentialmodels to describe the dynamics of tumour-immune systeminteractions

2 Fractional Model of Tumor-Immune System

Immune system is one of the most fascinating schemes fromthe point of view of biology and mathematics The immunesystem is complex intricate and interesting It is known to bemultifunctional andmultipathway somost immune effectorsdo more than one job Also each function of the immunesystem is typically done by more than one effector whichmakes it more robustThe reason of using FODEs is that theyare naturally related to systems with memory which exists intumor-immune interactions

Ordinary and delay differential equations have long beenused in modeling cancer phenomena [33ndash37] but fractional-order differential equations have short history in modelingsuch systems with memory The authors in [1] used a systemof fractional-order differential equations inmodeling cancer-immune system interactionThemodel includes two immuneeffectors 119864

1(119905) 1198642(119905) (such as cytotoxic T cells and natural

killer cells) interacting with the cancer cells 119879(119905) with aHolling function of type III (Holling type III describes asituation inwhich the number of prey consumed per predatorinitially rises slowly as the density of prey increases but thenlevels off with further increase in prey density In other wordsthe response of predators to prey is depressed at low preydensity then levels off with further increase in prey density)The model takes the form

119863120572

119879 = 119886119879 minus 11990311198791198641minus 11990321198791198642

119863120572

1198641= minus11988911198641+

11987921198641

1198792+ 1198961

0 lt 120572 le 1

119863120572

1198642= minus11988921198642+

11987921198642

1198792+ 1198961

(5)

where 119879 equiv 119879(119905) 1198641equiv 1198641(119905) 1198642equiv 1198642(119905) and 119886 119903

1 1199032 1198891

1198892 1198961 and 119896

2are positive constants The interaction terms

in the second and third equations of model (5) satisfy thecrossreactivity property of the immune system It has beenassumed that (119889

11198961(1 minus 119889

1)) ≪ (119889

21198962(1 minus 119889

2)) to avoid

Abstract and Applied Analysis 3

the nonbiological interior solution where both immuneeffectors coexist The equilibrium points of the system (5) are

E0= (0 0 0) E

1= (radic

11988911198961

(1 minus 1198891)

119886

1199031

0)

E2= (radic

11988921198962

(1 minus 1198892)

0

119886

1199032

)

(6)

The first equilibrium E0is the nave the second E

1is the

memory and the third E2is endemic according to the value

of the tumor size Stability analysis shows that the nave state isunstable However thememory state is locally asymptoticallystable if 119889

1lt 1198892and 119889

1lt 1 While the endemic state is

locally asymptotically stable if 1198892lt 1198891and 119889

2lt 1 there

is bifurcation at 1198891= 1 The stability of the memory state

depends on the value of one parameter namely the immuneeffector death rate

Now we modify model (5) to include three populationsof the activated immune-system cells 119864(119905) the tumor cells119879(119905) and the concentration of IL-2 in the single tumor-sitecompartment 119868

119871(119905) We consider the classic bilinear model

that includes Holling function of type I and external effectorcells 119904

1and input of IL-2 119904

2 (holling Type I is a linear

relationship where the predator is able to keep up withincreasing density of prey by eating them in direct proportionto their abundance in the environment If they eat 10 of theprey at low density they continue to eat 10 of them at highdensities) The interactions of the three populations are thengoverned by the fractional-order differential model

1198631205721119864 = 1199041+ 1199011119864119879 minus 119901

2119864 + 1199013119864119868119871

1198631205722119879 = 119901

4119879 (1 minus 119901

5119879) minus 119901

6119864119879 0 lt 120572

119894le 1 119894 = 1 2 3

1198631205723119868119871= 1199042+ 1199017119864119879 minus 119901

8119868119871

(7)

with initial conditions 119864(0) = 1198640 119879(0) = 119879

0 and 119868

119871(0) =

1198681198710

The parameter 1199011is the antigenicity rate of the tumor

(immune response to the appearance of the tumor) and 1199041

is the external source of the effector cells with rate of death1199012 The parameter 119901

4incorporates both multiplication and

death of tumor cells The maximal carrying capacity of thebiological environment for tumor cell is 119901minus1

5 1199013is considered

as the cooperation rate of effector cells with interleukin-2parameter 119901

6is the rate of tumor cells 119901

7is the competition

rate between the effector cells and the tumor cells Externalinput of IL-2 into the system is 119904

2 and the rate loss parameter

of IL-2 cells is 1199018

In the absence of immunotherapy with IL-2 we have

1198631205721119864 = 1199041+ 1199011119864119879 minus 119901

2119864

0 lt 120572119894le 1 119894 = 1 2

1198631205722119879 = 119901

4119879 (1 minus 119901

5119879) minus 119901

6119864119879

(8)

To ease the analysis and stability of the steady states withmeaningful parameters and minimize sensitivity (or robust-ness) of themodel we nondimensionalize the bilinear system(8) by taking the rescaling

119909 =

119864

1198640

119910 =

119879

1198790

120596 =

11990111198790

1199051199041198640

120579 =

1199012

119905119904

120590 =

1199041

1199051199041198640

119886 =

1199014

119905119904

119887 = 11990151198790 1 =

11990161198640

119905119904

120591 = 119905119904119905

(9)

Therefore after the previous substitution into (8) and replac-ing 120591 by 119905 the model becomes

1198631205721119909 = 120590 + 120596119909119910 minus 120579119909

1198631205722119910 = 119886119910 (1 minus 119887119910) minus 119909119910

(10)

21 Equilibria and Local Stability of Model (10) The steadystates of the reduced model (10) are again the intersection ofthe null clines 1198631205721119909 = 0 1198631205722119910 = 0 If 119910 = 0 the tumor-free equilibrium is atE

0= (119909 119910) = (120590120579 0) This steady state

is always exist since 120590120579 gt 0 From the analysis it is easyto prove that the tumor-free equilibriumE

0= (120590120579 0) of the

model (10) is asymptotically stable if threshold parameter (theminimum tumor-clearance parameter) R

0= 119886120579120590 lt 1 and

unstable ifR0gt 1

However if119910 = 0 the steady states are obtained by solving

1205961198861198871199102

minus 119886 (120596 + 120579119887) 119910 + 119886120579 minus 120590 = 0 (11)

In this case we have two endemic equilibria E1and E

2

E1= (1199091 1199101) where 119909

1=

minus119886 (119887120579 minus 120596) minus radicΔ

2120596

1199101=

119886 (119887120579 + 120596) + radicΔ

2119886119887120596

E2= (1199092 1199102) where 119909

2=

minus119886 (119887120579 minus 120596) + radicΔ

2120596

1199102=

119886 (119887120579 + 120596) minus radicΔ

2119886119887120596

(12)

with Δ = 1198862(119887120579 minus 120596)

2

+ 4120590120596119886119887 gt 0 The Jacobian matrix ofthe system (10) at the endemic equilibriumE

1is

119869 (E1) = (

1205961199101minus 120579 120596119909

1

minus1199101

119886 minus 21198861198871199101minus 1199091

) (13)


Proposition 4 Assume that the endemic equilibrium E1

exists and has nonnegative coordinates If R0= 119886120579120590 lt 1

then tr(119869(E1)) gt 0 and E

1is unstable

Proof Since

tr (119869 (E1)) =

1205962minus 120596 (119886119887 + 119887120579) minus 119886119887

2120579

2119887120596

+

120596 minus 119886119887

2119886119887120596

radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(14)

then inequality tr(119869(E1)) gt 0 is true if

119886 [1205962

minus 120596 (119886119887 + 119887120579) minus 1198861198872

120579]

gt (119886119887 minus 120596)radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(15)

Therefore when 119886120579 lt 120590 we have1205962minus120596119887(119886+120579)minus1198861205791198872 gt 0 andhence both sides of the inequality are positiveTherefore if theequilibrium pointE

1exists and has nonnegative coordinates

then tr(119869(E1)) gt 0 and the point (E

1) is unstable whenever

R0= 119886120579120590 lt 1

Similarly we arrive at the following proposition

Proposition 5 If the point E2exists and has nonnegative

coordinates then it is asymptotically stable

Wenext examine a fractional-order differential model forHIV infection of CD4+ T cells

3 Fractional Model of HIV Infection ofCD4+ T Cells

As mentioned before many mathematical models havebeen developed to describe the immunological response toinfection with human immunodeficiency virus (HIV) usingordinary differential models (see eg [38 39]) and delaydifferential models [40 41] In this section we extend theanalysis and replace the original system of Perelson et al [39]by an equivalent fractional-order differential model of HIVinfection of CD4+ T cells that consists of three equations

1198631205721119867 = 119904 minus 120583

119867119867 + 119903119867(1 minus

119867 + 119868

119867max) minus 1198961119881119867

1198631205722119868 = 1198961015840

1119881119867 minus 120583

119868119868

1198631205723119881 = 119872120583

119887119868 minus 1198961119881119867 minus 120583

119881119881

05 lt 120572119894le 1 119894 = 1 2 3

(16)

Here119867 = 119867(119905) represents the concentration of healthy CD4+T cells at time 119905 119868 = 119868(119905) represents the concentration ofinfected CD4+ T cells and 119881 = 119881(119905) is the concentrationof free HIV at time 119905 In this system the logistic growth of

the healthy CD4+ T-cells is (1 minus ((119867 + 119868)119867max)) and theproliferation of infected cells is neglected The parameter 119904 isthe source of CD4+ T cells from precursors 120583

119867is the natural

death rate of CD4+ T cells (120583119867119867max gt 119904 cf [39 page 85])

119903 is their growth rate (thus 119903 gt 120583119867in general) and 119867max is

their carrying capacity The parameter 1198961represents the rate

of infection of T cells with free virus 11989610158401is the rate at which

infected cells become actively infected 120583119868is a blanket death

term for infected cells to reflect the assumption that we do notinitially know whether the cells die naturally or by burstingIn addition 120583

119887is the lytic death rate for infected cells Since

119872 viral particles are released by each lysing cell this term ismultiplied by the parameter119872 to represent the source for freevirus (assuming a one-time initial infection) Finally120583

119881is the

loss rate of virus The initial conditions for infection by freevirus are119867(0) = 119867

0 119868(0) = 119868

0 and 119881(0) = 119881

0

Theorem 6 According to Remark 3 and the fact that 119904 ge 0then the system (16) has a unique solution (119867 119868 119881)

119879 whichremains in R3

+and bounded by119867max [11]

31 Equilibria and Local Stability of Model (16) To evaluatethe equilibrium points of system (16) we put 1198631205721119867(119905) =

1198631205722119868(119905) = 119863

1205723119881(119905) = 0 Then the infection-free equilibrium

point (uninfected steady state) is Elowast0

= (1198670 0 0) and

endemic equilibrium point (infected steady state) is Elowast+

=

(119867lowast 119868lowast 119881lowast) where

119867lowast

=

120583119881120583119868

1198961015840

1119872120583119887minus 1198961120583119868

119868lowast

=

1198961015840

1119867lowast119881lowast

120583119868

119881lowast

=

120583119868[(119904 + (119903 minus 120583

119867)119867lowast)119867max minus 119903119867

lowast2

]

119867lowast[1198961015840

1119903119867lowastminus 1198961120583119868119867max]

(17)

The Jacobian matrix 119869(Elowast0) for system (16) evaluated at the

uninfected steady state Elowast0is then given by

119869 (Elowast

0)

= (

minus120583119867+ 119903 minus 2119903119867

0

119867max

minus1199031198670

119867maxminus11989611198670

0 minus120583119868

1198961015840

11198670

0 119872120583119887

minus (11989611198670+ 120583119881)

)

(18)

Let us introduce the following definition and assumption toease the analysis

Definition 7 The threshold parameter Rlowast0(the minimum

infection-free parameter) is the parameter that has theproperty that ifRlowast

0lt 1 then the endemic infected state does

not exist while ifRlowast0gt 1 the endemic infected state persists

where

Rlowast

0=

1198961015840

11205831198871198721198670

120583119868(120583119881+ 11989611198670)

(19)


We also assume that

1198670=

119903 minus 120583119867+ [(119903 minus 120583

119867)2

+ 4119903119904119867minus1

max]12

2119903119867minus1

max

(20)

The uninfected steady state is asymptotically stable if all of theeigenvalues120582 of the Jacobianmatrix 119869(Elowast

0) given by (18) have

negative real parts The characteristic equation det(119869(E0) minus

119868) = 0 becomes

(120582 + 120583119867minus 119903 +

21199031198670

119867max) (1205822

+ 119861120582 + 119862) = 0 (21)

where 119861 = 120583119868+1198961198681198670+120583119881and119862 = 120583

119868(11989611198670+120583119881)minus1198961015840

11205831198871198721198670

Hence the three roots of the characteristic equation (21) are

1205821= minus120583119867+ 119903 minus

1199031198670

119867maxequiv minusradic(119903 minus 120583

119867)2

+ 4119903119904119867minus1

max lt 0

12058223

=

1

2

[minus119861 plusmn radic1198612minus 4119862]

(22)

Proposition 8 IfRlowast0equiv (1198961015840

11205831198871198721198670)120583119868(120583119881+11989611198670) lt 1 then

119862 gt 0 and the three roots of the characteristic equation (21)willhave negative real parts

Corollary 9 In case of uninfected steady state Elowast0 one has

three cases

(i) If Rlowast0

lt 0 the uninfected state is asymptoticallystable and the infected steady state E

+does not exist

(unphysical)

(ii) IfRlowast0= 1 then 119862 = 0 and from (21) implies that one

eigenvalue 120582 = 0 and the remaining two eigenvalueshave negative real parts The uninfected and infectedsteady states collide and there is a transcritical bifur-cation

(iii) IfRlowast0gt 1 then 119862 lt 0 and thus at least one eigenvalue

will be positive real root Thus the uninfected stateE0is unstable and the endemically infected state Elowast

+

emerges

To study the local stability of the positive infected steadystates Elowast

+for Rlowast

0gt 1 we consider the linearized system of

(16) at Elowast+ The Jacobian matrix at Elowast

+becomes

119869 (Elowast

+) =

(

(

minus119871lowast

minus119903119867lowast

119867maxminus1198961119867lowast

1198961015840

1119881lowast

minus120583119868

1198961015840

1119867lowast

1198961119881lowast

119872120583119887

minus (1198961119867lowast+ 120583119881)

)

)

(23)

Here

119871lowast

= minus[120583119867minus 119903 + 119896

1119881lowast

+

119903 (2119867lowast+ 119868lowast)

119867max] (24)

Then the characteristic equation of the linearized system is

119875 (120582) = 1205823

+ 11988611205822

+ 1198862120582 + 1198863= 0 (25)

1198861= 120583119868+ 120583119881+ 1198961119867lowast

+ 119871lowast

1198862= 119871lowast

(120583119868+ 120583119881+ 1198961119867lowast

) + 120583119868(120583119881+ 1198961119867lowast

)

minus 1198962

1119867lowast

119881lowast

minus 1198961015840

1119867lowast

(119872120583119887minus

119903119881lowast

119867max)

1198863= 1198961015840

1119867lowast

[1198961119872120583119887119881lowast

minus

119903120583119881119881lowast

119867maxminus 119871lowast

119872120583119887]

+ 119871lowast

120583119868(120583119881+ 1198961119867lowast

) minus 1205831198681198962

1119867lowast

119881lowast

(26)

The infected steady state Elowast+is asymptotically stable if all of

the eigenvalues have negative real parts This occurs if andonly if Routh-Hurwitz conditions are satisfied that is 119886

1gt 0

1198863gt 0 and 119886

11198862gt 1198863

4 Implicit Eulerrsquos Scheme for FODEs

Since most of the fractional-order differential equations donot have exact analytic solutions approximation and numer-ical techniques must be used Several numerical methodshave been proposed to solve the fractional-order differentialequations [18 42 43] In additionmost of resulting biologicalsystems are stiff (One definition of the stiffness is that theglobal accuracy of the numerical solution is determined bystability rather than local error and implicit methods aremore appropriate for it) The stiffness often appears due tothe differences in speed between the fastest and slowestcomponents of the solutions and stability constraints Inaddition the state variables of these types of models arevery sensitive to small perturbations (or changes) in theparameters occur in the model Therefore efficient use of areliable numerical method that based in general on implicitformulae for dealing with stiff problems is necessary

Consider biological models in the form of a system ofFODEs of the form

119863120572

119883(119905) = 119865 (119905 119883 (119905) P) 119905 isin [0 119879] 0 lt 120572 le 1

119883 (0) = 1198830

(27)

Here 119883(119905) = [1199091(119905) 1199092(119905) 119909

119899(119905)]119879 P is the set of param-

eters appear in themodel and119865(119905 119883(119905)) satisfies the Lipschitzcondition

119865 (119905 119883 (119905) P) minus 119865 (119905 119884 (119905) P) le 119870 119883 (119905) minus 119884 (119905)

119870 gt 0

(28)

where 119884(119905) is the solution of the perturbed system

Theorem10 Problem (27) has a unique solution provided thatthe Lipschitz condition (28) is satisfied and119870119879

120572Γ(120572 + 1) lt 1

Proof Using the definitions of Section 1 we can apply afractional integral operator to the differential equation (27)


and incorporate the initial conditions thus converting theequation into the equivalent equation

119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904 (29)

which also is a Volterra equation of the second kind Definethe operatorL such that

L119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904

(30)

Then we have

L119883 (119905) minusL119884 (119905)

le

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) minus 119865 (119904 119884 (119904) P) 119889119904

le

119870

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1 sup119904isin[0119879]

|119883 (119904) minus 119884 (119904)| 119889119904

le

119870

Γ (120572)

119883 minus 119884int

119905

0

119904120572minus1

119889119904

le

119870

Γ (120572)

119883 minus 119884119879120572

(31)

Thus for119870119879120572Γ(120572 + 1) lt 1 we have

L119883 (119905) minusL119884 (119905) le 119883 minus 119884 (32)

This implies that our problem has a unique solution

However converted Volterra integral equation (29) iswith a weakly singular kernel such that a regularization isnot necessary any more It seems that there exists only a verysmall number of software packages for nonlinear Volterraequations In our case the kernel may not be continuousand therefore the classical numerical algorithms for theintegral part of (29) are unable to handle the solution of(27) Therefore we implement the implicit Eulerrsquos scheme toapproximate the fractional-order derivative

Given model (27) and mesh points T = 1199050 1199051 119905

119873

such that 1199050= 0 and 119905

119873= 119879 Then a discrete approximation

to the fractional derivative can be obtained by a simplequadrature formula using the Caputo fractional derivative(3) of order 120572 0 lt 120572 le 1 and using implicit Eulerrsquosapproximation as follows (see [44])

119863120572

lowast119909119894(119905119899)

=

1

Γ (1 minus 120572)

int

119905

0

119889119909119894(119904)

119889119904

(119905119899minus 119904)minus120572

119889119904

asymp

1

Γ (1 minus 120572)

119899

sum

119895=1

int

119895ℎ

(119895minus1)ℎ

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)] (119899ℎ minus 119904)minus120572

119889119904

=

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)]

times [(119899 minus 119895 + 1)1minus120572

minus (119899 minus 119895)1minus120572

] ℎ1minus120572

=

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]

+

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]119874 (ℎ2minus120572

)

(33)

Setting

G (120572 ℎ) =

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572 120596

120572

119895= 1198951minus120572


(where 120596120572

1= 1)

(34)

then the first-order approximation method for the compu-tation of Caputorsquos fractional derivative is then given by theexpression

119863120572

lowast119909119894(119905119899) = G (120572 ℎ)

119899

sum

119895=1

120596120572

119895(119909119899minus119895+1

119894minus 119909119899minus119895

119894) + 119874 (ℎ) (35)

From the analysis and numerical approximation we alsoarrive at the following proposition

Proposition 11 The presence of a fractional differential orderin a differential equation can lead to a notable increase inthe complexity of the observed behaviour and the solution iscontinuously depends on all the previous states

41 Stability and Convergence We here prove that thefractional-order implicit difference approximation (35) isunconditionally stable It follows then that the numericalsolution converges to the exact solution as ℎ rarr 0

In order to study the stability of the numerical methodlet us consider a test problem of linear scaler fractionaldifferential equation

119863120572

lowast119906 (119905) = 120588

0119906 (119905) + 120588

1 119880 (0) = 119880

0 (36)

such that 0 lt 120572 le 1 and 1205880lt 0 120588

1gt 0 are constants


0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)

Figure 1 Numerical simulations of the FODEs model (5) when 120572 = 075 and 120572 = 095 with (119886 = 1199031= 1199032= 1 119889

1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889

2= 03 The system converges to the stable steady statesE

1E2 The fractional derivative damps the oscillation

behavior

Theorem 12 The fully implicit numerical approximation (35)to test problem (36) for all 119905 ge 0 is consistent and uncon-ditionally stable

Proof We assume that the approximate solution of (36) is ofthe form 119906(119905

119899) asymp 119880119899equiv 120577119899 and then (36) can be reduced to

(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)

Since (1 minus (1205880G120572ℎ)) ge 1 for allG

120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)

Thus for 119899 = 2 the previous inequality implies

1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)

Using the relation (39) and the positivity of the coefficients1205962 we get

1205772le 1205771 (42)

Repeating the process we have from (40)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)

Figure 2 Numerical simulations of the FODEs model (10) when 119904 = 01181 120596 = 01184 120579 = 01747 119903 = 0636 and 119887 = 0002 1205721= 1205722=

1 09 07The endemic stateE2is locally asymptotically stable whenR

0= 119886120579120590 gt 1The fractional derivative damps the oscillation behavior

since each term in the summation is negative Thus 120577119899

le

120577119899minus1

le 120577119899minus2

le sdot sdot sdot le 1205770 With the assumption that 120577

119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability

Of course this numerical technique can be used both forlinear and for nonlinear problems and it may be extendedto multiterm FODEs For more details about stability andconvergence of the fractional Euler method we refer to [1945]

42 Numerical Simulations We employed the implicit Eulerrsquosscheme (35) to solve the resulting biological systems ofFODEs (5) (10) and (16) Interesting numerical simulationsof the fractional tumor-immune models (5) and (10) withstep size ℎ = 005 and 05 lt 120572 le 1 and parameters valuesgiven in the captions are displayed in Figures 1 2 and 3 The

equilibrium points (for infection-free and endemic cases) arethe same in both integer-order (when 120572 = 1) and fractional-order (120572 lt 1) models We notice that in the endemic steadystates the fractional-order derivative damps the oscillationbehavior From the graphs we can see that FODEs have richdynamics and are better descriptors of biological systemsthan traditional integer-order models

The numerical simulations for the HIV FODEmodel (16)(with parameter values given in the captions) are displayed inFigure 4We note that the solution of themodel with variousvalues of 120572 continuously depends on the time-fractionalderivative but arrives to the equilibriumpointsThe displayedsolutions in the figure confirm that the fractional order ofthe derivative plays the role of time delay in the system Weshould also note that although the equilibrium points arethe same for both integer-order and fractional-order models


025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)


1 09 07 The infection-free steady state E0is locally asymptotically stable whenR

0= 119886120579120590 lt 1

the solution of the fractional order model tends to the fixedpoint over a longer period of time

5 Conclusions

In fact fractional-order differential equations are generaliza-tions of integer-order differential equations Using fractional-order differential equations can help us to reduce the errorsarising from the neglected parameters in modeling biologicalsystems withmemory and systems distributed parameters Inthis paper we presented a class of fractional-order differentialmodels of biological systems with memory to model theinteraction of immune system with tumor cells and withHIV infection of CD4+ T cells The models possess non-negative solutions as desired in any population dynamicsWe obtained the threshold parameterR

0that represents the

minimum tumor-clearance parameter orminimum infectionfree for each model

We provided unconditionally stable numerical techniqueusing the Caputo fractional derivative of order 120572 and implicitEulerrsquos approximation for the resulting system The numer-ical technique is suitable for stiff problems The solutionof the system at any time 119905

lowast is continuously depends onall the previous states at 119905 le 119905

lowast We have seen that thepresence of the fractional differential order leads to a notableincrease in the complexity of the observed behaviour and playthe role of time lag (or delay term) in ordinary differentialmodel We have obtained stability conditions for disease-free equilibrium and nonstability conditions for positiveequilibria We should also mention that one of the basicreasons of using fractional-order differential equations is thatfractional-order differential equations are at least as stableas their integer-order counterpart In addition the presenceof a fractional differential order in a differential equation canlead to a notable increase in the complexity of the observedbehaviour and the solution continuously depends on all


0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104

Figure 4 Numerical simulations of the FODEs model (16) when 119904 = 20 dayminus1mmminus3 120583119867= 002 dayminus1 120583

119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10

minus5mm3 dayminus1 11989610158401= 2 times 10

minus5mm3 dayminus1 119903 = 03 dayminus1119867max = 1500mmminus3 and119872 = 1400 The trajectories ofthe system approach to the steady state Elowast

+

the previous states The analysis can be extended to othermodels related to the immune response systems

Acknowledgments

The work is funded by the UAE University NRF Project noNRF-7-20886 The author is grateful to Professor ChangpinLirsquos valuable suggestions and refereesrsquo constructive com-ments

References

[1] E Ahmed A Hashish and F A Rihan ldquoOn fractional ordercancer modelrdquo Journal of Fractional Calculus and AppliedAnalysis vol 3 no 2 pp 1ndash6 2012

[2] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[3] K S Cole ldquoElectric conductance of biological systemsrdquo ColdSpring Harbor Symposia on Quantitative Biology pp 107ndash1161993

[4] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[5] H Xu ldquoAnalytical approximations for a population growthmodel with fractional orderrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 5 pp 1978ndash19832009

[6] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences no 54 pp 3413ndash3442 2003

[7] A M A El-Sayed ldquoNonlinear functional-differential equationsof arbitrary ordersrdquo Nonlinear Analysis Theory Methods ampApplications vol 33 no 2 pp 181ndash186 1998

[8] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific River Edge NJ USA 2000

[9] G M Zaslavsky ldquoChaos fractional kinetics and anomaloustransportrdquo Physics Reports vol 371 no 6 pp 461ndash580 2002

[10] S B Yuste L Acedo and K Lindenberg ldquoReaction front in anA + B rarr C reaction-subdiffusion processrdquo Physical Review Evol 69 no 3 Article ID 036126 pp 1ndash36126 2004

[11] W Lin ldquoGlobal existence theory and chaos control of fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 332 no 1 pp 709ndash726 2007

[12] H Sheng Y Q Chen and T S Qiu Fractional Processes andFractional-Order Signal Processing Springer New York NYUSA 2012

[13] K Assaleh and W M Ahmad ldquoModeling of speech signalsusing fractional calculusrdquo in Proceedings of the 9th InternationalSymposium on Signal Processing and its Applications (ISSPA rsquo07)Sharjah United Arab Emirates February 2007

[14] Y Ferdi ldquoSome applications of fractional order calculus todesign digital filters for biomedical signal processingrdquo Journalof Mechanics in Medicine and Biology vol 12 no 2 Article ID12400088 13 pages 2012

[15] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons and Fractals vol 36 no5 pp 1305ndash1314 2008

[16] A Rocco and B J West ldquoFractional calculus and the evolutionof fractal phenomenardquo Physica A vol 265 no 3 pp 535ndash5461999

[17] V D Djordjevic J Jaric B Fabry J J Fredberg and DStamenovic ldquoFractional derivatives embody essential featuresof cell rheological behaviorrdquo Annals of Biomedical Engineeringvol 31 no 6 pp 692ndash699 2003

[18] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[19] C Li and F Zeng ldquoThe finite difference methods for fractionalordinary differential equationsrdquo Numerical Functional Analysisand Optimization vol 34 no 2 pp 149ndash179 2013


[20] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[21] X Zhang and Y Han ldquoExistence and uniqueness of positivesolutions for higher order nonlocal fractional differential equa-tionsrdquo Applied Mathematics Letters vol 25 no 3 pp 555ndash5602012

[22] AM A El-Sayed andM Gaber ldquoTheAdomian decompositionmethod for solving partial differential equations of fractal orderin finite domainsrdquo Physics Letters A vol 359 no 3 pp 175ndash1822006

[23] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

[24] K Diethelm and G Walz ldquoNumerical solution of fractionalorder differential equations by extrapolationrdquo Numerical Algo-rithms vol 16 no 3-4 pp 231ndash253 1997

[25] L Galeone and R Garrappa ldquoOn multistep methods for differ-ential equations of fractional orderrdquo Mediterranean Journal ofMathematics vol 3 no 3-4 pp 565ndash580 2006

[26] G Wang R P Agarwal and A Cabada ldquoExistence resultsand the monotone iterative technique for systems of nonlinearfractional differential equationsrdquo Applied Mathematics Lettersvol 25 no 6 pp 1019ndash1024 2012

[27] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[28] V Lakshmikantham S Leela and J Vasundhara Theory ofFractional Dynamic Systems Cambridge Academic PublishersCambridge UK

[29] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[30] S G Samko A A Kilbas and O I Marichev Fractionalintegrals and derivatives Gordon and Breach Science YverdonSwitzerland 1993

[31] C Li and Y Ma ldquoFractional dynamical system and its lineariza-tion theoremrdquo Nonlinear Dynamics vol 71 no 4 pp 621ndash6332013

[32] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[33] N Bellomo A Bellouquid J Nieto and J Soler ldquoMultiscalebiological tissue models and flux-limited chemotaxis for mul-ticellular growing systemsrdquoMathematical Models ampMethods inApplied Sciences vol 20 no 7 pp 1179ndash1207 2010

[34] AGokdogan A Yildirim andMMerdan ldquoSolving a fractionalordermodel ofHIV infection of CD+ T cellsrdquoMathematical andComputer Modelling vol 54 no 9-10 pp 2132ndash2138 2011

[35] D Kirschner and J C Panetta ldquoModeling immunotherapyof the tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998

[36] F A Rihan M Safan M A Abdeen and D Abdel RahmanldquoQualitative and computational analysis of a mathematicalmodel for tumor-immune interactionsrdquo Journal of AppliedMathematics vol 2012 Article ID 475720 19 pages 2012

[37] R Yafia ldquoHopf bifurcation in differential equations with delayfor tumor-immune system competition modelrdquo SIAM Journalon Applied Mathematics vol 67 no 6 pp 1693ndash1703 2007

[38] K A Pawelek S Liu F Pahlevani and L Rong ldquoA model ofHIV-1 infection with two time delays mathematical analysisand comparison with patient datardquo Mathematical Biosciencesvol 235 no 1 pp 98ndash109 2012

[39] A S Perelson D E Kirschner and R De Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol114 no 1 pp 81ndash125 1993

[40] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[41] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[42] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3-6 pp 465ndash475 2003

[43] C Li and F Zeng ldquoFinite difference methods for fractionaldifferential equationsrdquo International Journal of Bifurcation andChaos in Applied Sciences and Engineering vol 22 no 4 28pages 2012

[44] F A Rihan ldquoComputational methods for delay parabolicand time-fractional partial differential equationsrdquo NumericalMethods for Partial Differential Equations vol 26 no 6 pp1556ndash1571 2010

[45] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


We first give the definition of fractional-order integrationand fractional-order differentiation [29 30]There are severalapproaches to the generalization of the notion of differenti-ation to fractional orders for example Riemann-LiouvilleCaputo and Generalized Functions approaches Let 1198711 =

1198711[119886 119887] be the class of Lebesgue integrable functions on [119886 119887]

119886 lt 119887 lt infin

Definition 1 The fractional integral (or the Riemann-Liouville integral) of order 120573 isin R+ of the function 119891(119905) 119905 gt 0

(119891 R+ rarr R) is defined by

119868120573

119886119891 (119905) =

1

Γ (120573)

int

119905

119886

(119905 minus 119904)120573minus1

119891 (119904) 119889119904 119905 gt 0 (1)

The fractional derivative of order 120572 isin (119899 minus 1 119899) of 119891(119905) isdefined by two (nonequivalent) ways

(i) Riemann-Liouville fractional derivative take frac-tional integral of order (119899 minus 120572) and then take 119899thderivative as follows

119863120572

lowast119891 (119905) = 119863

119899

lowast119868119899minus120572

119886119891 (119905) 119863

119899

lowast=

119889119899

119889119905119899 119899 = 1 2

(2)

(ii) Caputo-fractional derivative take 119899th derivative andthen take a fractional integral of order (119899 minus 120572)

119863120572

119891 (119905) = 119868119899minus120572

119886119863119899

lowast119891 (119905) 119899 = 1 2 (3)

We notice that the definition of time-fractional derivativeof a function 119891(119905) at 119905 = 119905

119899involves an integration and

calculating time-fractional derivative that requires all the pasthistory that is all the values of 119891(119905) from 119905 = 0 to 119905 =

119905119899 For the concept of fractional derivative we will adopt

Caputorsquos definition which is a modification of the Riemann-Liouville definition and has the advantage of dealing properlywith initial value problems The following remark addressessome of the main properties of the fractional derivatives andintegrals (see [8 29ndash31])

Remark 2 Let 120573 120574 isin R+ and 120572 isin (0 1) Then

(i) if 119868120573119886

1198711

rarr 1198711 and 119891(119905) isin 119871

1 then 119868120573

119886119868120574

119886119891(119905) =

119868120573+120574

119886119891(119905)

(ii) lim120573rarr119899

119868120573

119886119891(119909) = 119868

119899

119886119891(119905) uniformly on [119886 119887] 119899 = 1 2

3 where 1198681119886119891(119905) = int

119905

0119891(119904)119889119904

(iii) lim120573rarr0

119868120573

119886119891(119905) = 119891(119905) weakly

(iv) if 119891(119905) is absolutely continuous on [119886 119887] thenlim120572rarr1

119863120572

lowast119891(119905) = 119889119891(119905)119889119905

(v) thus 119863120572

lowast119891(119905) = (119889119889119905)119868

1minus120572

lowast119891(119905) (Riemann-Liouville

sense) and119863120572119891(119905) = 119868

1minus120572

lowast(119889119889119905)119891(119905) (Caputo sense)

The generalized mean value theorem and another propertyare defined in the following remark [32]

Remark 3

(i) Suppose that 119891(119905) isin 119862[119886 119887] and 119863120572

lowast119891(119905) isin 119862(119886 119887] for

0 lt 120572 le 1 and then we have

119891 (119905) = 119891 (119886) +

1

Γ (120572)

119863120572

lowast119891 (120585) (119905 minus 119886)

120572

with 119886 lt 120585 lt 119905 forall119905 isin (119886 119887]

(4)

(ii) If (i) holds and119863120572lowast119891(119905) ge 0 for all 119905 isin [119886 119887] then 119891(119905)

is nondecreasing for each 119905 isin [119886 119887] If 119863120572lowast119891(119905) le 0

for all 119905 isin [119886 119887] then 119891(119905) is nonincreasing for each119905 isin [119886 119887]

We next provide a class of fractional-order differentialmodels to describe the dynamics of tumour-immune systeminteractions

2 Fractional Model of Tumor-Immune System

Immune system is one of the most fascinating schemes fromthe point of view of biology and mathematics The immunesystem is complex intricate and interesting It is known to bemultifunctional andmultipathway somost immune effectorsdo more than one job Also each function of the immunesystem is typically done by more than one effector whichmakes it more robustThe reason of using FODEs is that theyare naturally related to systems with memory which exists intumor-immune interactions

Ordinary and delay differential equations have long beenused in modeling cancer phenomena [33ndash37] but fractional-order differential equations have short history in modelingsuch systems with memory The authors in [1] used a systemof fractional-order differential equations inmodeling cancer-immune system interactionThemodel includes two immuneeffectors 119864

1(119905) 1198642(119905) (such as cytotoxic T cells and natural

killer cells) interacting with the cancer cells 119879(119905) with aHolling function of type III (Holling type III describes asituation inwhich the number of prey consumed per predatorinitially rises slowly as the density of prey increases but thenlevels off with further increase in prey density In other wordsthe response of predators to prey is depressed at low preydensity then levels off with further increase in prey density)The model takes the form

119863120572

119879 = 119886119879 minus 11990311198791198641minus 11990321198791198642

119863120572

1198641= minus11988911198641+

11987921198641

1198792+ 1198961

0 lt 120572 le 1

119863120572

1198642= minus11988921198642+

11987921198642

1198792+ 1198961

(5)

where 119879 equiv 119879(119905) 1198641equiv 1198641(119905) 1198642equiv 1198642(119905) and 119886 119903

1 1199032 1198891

1198892 1198961 and 119896

2are positive constants The interaction terms

in the second and third equations of model (5) satisfy thecrossreactivity property of the immune system It has beenassumed that (119889

11198961(1 minus 119889

1)) ≪ (119889

21198962(1 minus 119889

2)) to avoid



E0= (0 0 0) E

1= (radic

11988911198961

(1 minus 1198891)

119886

1199031

0)

E2= (radic

11988921198962

(1 minus 1198892)

0

119886

1199032

)

(6)


1is the



1lt 1198892and 119889



2lt 1 there









1198631205721119864 = 1199041+ 1199011119864119879 minus 119901

2119864 + 1199013119864119868119871

1198631205722119879 = 119901

4119879 (1 minus 119901

5119879) minus 119901

6119864119879 0 lt 120572

119894le 1 119894 = 1 2 3

1198631205723119868119871= 1199042+ 1199017119864119879 minus 119901

8119868119871

(7)


0 and 119868

119871(0) =

1198681198710









7is the competition





1198631205721119864 = 1199041+ 1199011119864119879 minus 119901

2119864

0 lt 120572119894le 1 119894 = 1 2

1198631205722119879 = 119901

4119879 (1 minus 119901

5119879) minus 119901

6119864119879

(8)


119909 =

119864

1198640

119910 =

119879

1198790

120596 =

11990111198790

1199051199041198640

120579 =

1199012

119905119904

120590 =

1199041

1199051199041198640

119886 =

1199014

119905119904

119887 = 11990151198790 1 =

11990161198640

119905119904

120591 = 119905119904119905

(9)


1198631205721119909 = 120590 + 120596119909119910 minus 120579119909

1198631205722119910 = 119886119910 (1 minus 119887119910) minus 119909119910

(10)




0= (120590120579 0) of the


0= 119886120579120590 lt 1 and

unstable ifR0gt 1


1205961198861198871199102

minus 119886 (120596 + 120579119887) 119910 + 119886120579 minus 120590 = 0 (11)


2

E1= (1199091 1199101) where 119909

1=


2120596

1199101=

119886 (119887120579 + 120596) + radicΔ

2119886119887120596

E2= (1199092 1199102) where 119909

2=


2120596

1199102=

119886 (119887120579 + 120596) minus radicΔ

2119886119887120596

(12)

with Δ = 1198862(119887120579 minus 120596)

2


1is

119869 (E1) = (

1205961199101minus 120579 120596119909

1

minus1199101

119886 minus 21198861198871199101minus 1199091

) (13)





1is unstable

Proof Since

tr (119869 (E1)) =

1205962minus 120596 (119886119887 + 119887120579) minus 119886119887

2120579

2119887120596

+

120596 minus 119886119887

2119886119887120596

radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(14)


119886 [1205962

minus 120596 (119886119887 + 119887120579) minus 1198861198872

120579]

gt (119886119887 minus 120596)radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(15)





R0= 119886120579120590 lt 1







1198631205721119867 = 119904 minus 120583

119867119867 + 119903119867(1 minus

119867 + 119868

119867max) minus 1198961119881119867

1198631205722119868 = 1198961015840

1119881119867 minus 120583

119868119868

1198631205723119881 = 119872120583

119887119868 minus 1198961119881119867 minus 120583

119881119881

05 lt 120572119894le 1 119894 = 1 2 3

(16)












119881is the


0 119868(0) = 119868

0 and 119881(0) = 119881

0





1198631205722119868(119905) = 119863



= (1198670 0 0) and


=


119867lowast

=

120583119881120583119868

1198961015840

1119872120583119887minus 1198961120583119868

119868lowast

=

1198961015840


120583119868

119881lowast

=

120583119868[(119904 + (119903 minus 120583

119867)119867lowast)119867max minus 119903119867

lowast2

]

119867lowast[1198961015840

1119903119867lowastminus 1198961120583119868119867max]

(17)



119869 (Elowast

0)

= (

minus120583119867+ 119903 minus 2119903119867

0

119867max

minus1199031198670

119867maxminus11989611198670

0 minus120583119868

1198961015840

11198670

0 119872120583119887

minus (11989611198670+ 120583119881)

)

(18)






where

Rlowast

0=

1198961015840

11205831198871198721198670

120583119868(120583119881+ 11989611198670)

(19)


We also assume that

1198670=

119903 minus 120583119867+ [(119903 minus 120583

119867)2

+ 4119903119904119867minus1

max]12

2119903119867minus1

max

(20)




119868) = 0 becomes

(120582 + 120583119867minus 119903 +

21199031198670

119867max) (1205822

+ 119861120582 + 119862) = 0 (21)

where 119861 = 120583119868+1198961198681198670+120583119881and119862 = 120583

119868(11989611198670+120583119881)minus1198961015840

11205831198871198721198670


1205821= minus120583119867+ 119903 minus

1199031198670


119867)2

+ 4119903119904119867minus1

max lt 0

12058223

=

1

2


(22)


11205831198871198721198670)120583119868(120583119881+11989611198670) lt 1 then



three cases

(i) If Rlowast0


+does not exist

(unphysical)





+

emerges


+for Rlowast



+becomes

119869 (Elowast

+) =

(

(

minus119871lowast



1198961015840

1119881lowast

minus120583119868

1198961015840

1119867lowast

1198961119881lowast

119872120583119887

minus (1198961119867lowast+ 120583119881)

)

)

(23)

Here

119871lowast

= minus[120583119867minus 119903 + 119896

1119881lowast

+

119903 (2119867lowast+ 119868lowast)

119867max] (24)


119875 (120582) = 1205823

+ 11988611205822

+ 1198862120582 + 1198863= 0 (25)

1198861= 120583119868+ 120583119881+ 1198961119867lowast

+ 119871lowast

1198862= 119871lowast

(120583119868+ 120583119881+ 1198961119867lowast

) + 120583119868(120583119881+ 1198961119867lowast

)

minus 1198962

1119867lowast

119881lowast

minus 1198961015840

1119867lowast

(119872120583119887minus

119903119881lowast

119867max)

1198863= 1198961015840

1119867lowast

[1198961119872120583119887119881lowast

minus

119903120583119881119881lowast


119872120583119887]

+ 119871lowast

120583119868(120583119881+ 1198961119867lowast

) minus 1205831198681198962

1119867lowast

119881lowast

(26)



1gt 0

1198863gt 0 and 119886

11198862gt 1198863




119863120572

119883(119905) = 119865 (119905 119883 (119905) P) 119905 isin [0 119879] 0 lt 120572 le 1

119883 (0) = 1198830

(27)

Here 119883(119905) = [1199091(119905) 1199092(119905) 119909



119865 (119905 119883 (119905) P) minus 119865 (119905 119884 (119905) P) le 119870 119883 (119905) minus 119884 (119905)

119870 gt 0

(28)



120572Γ(120572 + 1) lt 1




119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904 (29)


L119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904

(30)

Then we have

L119883 (119905) minusL119884 (119905)

le

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) minus 119865 (119904 119884 (119904) P) 119889119904

le

119870

Γ (120572)

int

119905

0


|119883 (119904) minus 119884 (119904)| 119889119904

le

119870

Γ (120572)

119883 minus 119884int

119905

0

119904120572minus1

119889119904

le

119870

Γ (120572)

119883 minus 119884119879120572

(31)


L119883 (119905) minusL119884 (119905) le 119883 minus 119884 (32)




119873

such that 1199050= 0 and 119905



119863120572

lowast119909119894(119905119899)

=

1

Γ (1 minus 120572)

int

119905

0

119889119909119894(119904)

119889119904

(119905119899minus 119904)minus120572

119889119904

asymp

1

Γ (1 minus 120572)

119899

sum

119895=1

int

119895ℎ

(119895minus1)ℎ

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)] (119899ℎ minus 119904)minus120572

119889119904

=

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)]



] ℎ1minus120572

=

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]

+

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]119874 (ℎ2minus120572

)

(33)

Setting

G (120572 ℎ) =

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572 120596

120572

119895= 1198951minus120572


(where 120596120572

1= 1)

(34)


119863120572

lowast119909119894(119905119899) = G (120572 ℎ)

119899

sum

119895=1

120596120572

119895(119909119899minus119895+1

119894minus 119909119899minus119895

119894) + 119874 (ℎ) (35)





119863120572

lowast119906 (119905) = 120588

0119906 (119905) + 120588

1 119880 (0) = 119880

0 (36)


1gt 0 are constants


0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)


1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889



behavior




(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)


120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)


1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)


1205772le 1205771 (42)


120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E0= (0 0 0) E

1= (radic

11988911198961

(1 minus 1198891)

119886

1199031

0)

E2= (radic

11988921198962

(1 minus 1198892)

0

119886

1199032

)

(6)


1is the



1lt 1198892and 119889



2lt 1 there









1198631205721119864 = 1199041+ 1199011119864119879 minus 119901

2119864 + 1199013119864119868119871

1198631205722119879 = 119901

4119879 (1 minus 119901

5119879) minus 119901

6119864119879 0 lt 120572

119894le 1 119894 = 1 2 3

1198631205723119868119871= 1199042+ 1199017119864119879 minus 119901

8119868119871

(7)


0 and 119868

119871(0) =

1198681198710









7is the competition





1198631205721119864 = 1199041+ 1199011119864119879 minus 119901

2119864

0 lt 120572119894le 1 119894 = 1 2

1198631205722119879 = 119901

4119879 (1 minus 119901

5119879) minus 119901

6119864119879

(8)


119909 =

119864

1198640

119910 =

119879

1198790

120596 =

11990111198790

1199051199041198640

120579 =

1199012

119905119904

120590 =

1199041

1199051199041198640

119886 =

1199014

119905119904

119887 = 11990151198790 1 =

11990161198640

119905119904

120591 = 119905119904119905

(9)


1198631205721119909 = 120590 + 120596119909119910 minus 120579119909

1198631205722119910 = 119886119910 (1 minus 119887119910) minus 119909119910

(10)




0= (120590120579 0) of the


0= 119886120579120590 lt 1 and

unstable ifR0gt 1


1205961198861198871199102

minus 119886 (120596 + 120579119887) 119910 + 119886120579 minus 120590 = 0 (11)


2

E1= (1199091 1199101) where 119909

1=


2120596

1199101=

119886 (119887120579 + 120596) + radicΔ

2119886119887120596

E2= (1199092 1199102) where 119909

2=


2120596

1199102=

119886 (119887120579 + 120596) minus radicΔ

2119886119887120596

(12)

with Δ = 1198862(119887120579 minus 120596)

2


1is

119869 (E1) = (

1205961199101minus 120579 120596119909

1

minus1199101

119886 minus 21198861198871199101minus 1199091

) (13)





1is unstable

Proof Since

tr (119869 (E1)) =

1205962minus 120596 (119886119887 + 119887120579) minus 119886119887

2120579

2119887120596

+

120596 minus 119886119887

2119886119887120596

radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(14)


119886 [1205962

minus 120596 (119886119887 + 119887120579) minus 1198861198872

120579]

gt (119886119887 minus 120596)radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(15)





R0= 119886120579120590 lt 1







1198631205721119867 = 119904 minus 120583

119867119867 + 119903119867(1 minus

119867 + 119868

119867max) minus 1198961119881119867

1198631205722119868 = 1198961015840

1119881119867 minus 120583

119868119868

1198631205723119881 = 119872120583

119887119868 minus 1198961119881119867 minus 120583

119881119881

05 lt 120572119894le 1 119894 = 1 2 3

(16)












119881is the


0 119868(0) = 119868

0 and 119881(0) = 119881

0





1198631205722119868(119905) = 119863



= (1198670 0 0) and


=


119867lowast

=

120583119881120583119868

1198961015840

1119872120583119887minus 1198961120583119868

119868lowast

=

1198961015840


120583119868

119881lowast

=

120583119868[(119904 + (119903 minus 120583

119867)119867lowast)119867max minus 119903119867

lowast2

]

119867lowast[1198961015840

1119903119867lowastminus 1198961120583119868119867max]

(17)



119869 (Elowast

0)

= (

minus120583119867+ 119903 minus 2119903119867

0

119867max

minus1199031198670

119867maxminus11989611198670

0 minus120583119868

1198961015840

11198670

0 119872120583119887

minus (11989611198670+ 120583119881)

)

(18)






where

Rlowast

0=

1198961015840

11205831198871198721198670

120583119868(120583119881+ 11989611198670)

(19)


We also assume that

1198670=

119903 minus 120583119867+ [(119903 minus 120583

119867)2

+ 4119903119904119867minus1

max]12

2119903119867minus1

max

(20)




119868) = 0 becomes

(120582 + 120583119867minus 119903 +

21199031198670

119867max) (1205822

+ 119861120582 + 119862) = 0 (21)

where 119861 = 120583119868+1198961198681198670+120583119881and119862 = 120583

119868(11989611198670+120583119881)minus1198961015840

11205831198871198721198670


1205821= minus120583119867+ 119903 minus

1199031198670


119867)2

+ 4119903119904119867minus1

max lt 0

12058223

=

1

2


(22)


11205831198871198721198670)120583119868(120583119881+11989611198670) lt 1 then



three cases

(i) If Rlowast0


+does not exist

(unphysical)





+

emerges


+for Rlowast



+becomes

119869 (Elowast

+) =

(

(

minus119871lowast



1198961015840

1119881lowast

minus120583119868

1198961015840

1119867lowast

1198961119881lowast

119872120583119887

minus (1198961119867lowast+ 120583119881)

)

)

(23)

Here

119871lowast

= minus[120583119867minus 119903 + 119896

1119881lowast

+

119903 (2119867lowast+ 119868lowast)

119867max] (24)


119875 (120582) = 1205823

+ 11988611205822

+ 1198862120582 + 1198863= 0 (25)

1198861= 120583119868+ 120583119881+ 1198961119867lowast

+ 119871lowast

1198862= 119871lowast

(120583119868+ 120583119881+ 1198961119867lowast

) + 120583119868(120583119881+ 1198961119867lowast

)

minus 1198962

1119867lowast

119881lowast

minus 1198961015840

1119867lowast

(119872120583119887minus

119903119881lowast

119867max)

1198863= 1198961015840

1119867lowast

[1198961119872120583119887119881lowast

minus

119903120583119881119881lowast


119872120583119887]

+ 119871lowast

120583119868(120583119881+ 1198961119867lowast

) minus 1205831198681198962

1119867lowast

119881lowast

(26)



1gt 0

1198863gt 0 and 119886

11198862gt 1198863




119863120572

119883(119905) = 119865 (119905 119883 (119905) P) 119905 isin [0 119879] 0 lt 120572 le 1

119883 (0) = 1198830

(27)

Here 119883(119905) = [1199091(119905) 1199092(119905) 119909



119865 (119905 119883 (119905) P) minus 119865 (119905 119884 (119905) P) le 119870 119883 (119905) minus 119884 (119905)

119870 gt 0

(28)



120572Γ(120572 + 1) lt 1




119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904 (29)


L119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904

(30)

Then we have

L119883 (119905) minusL119884 (119905)

le

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) minus 119865 (119904 119884 (119904) P) 119889119904

le

119870

Γ (120572)

int

119905

0


|119883 (119904) minus 119884 (119904)| 119889119904

le

119870

Γ (120572)

119883 minus 119884int

119905

0

119904120572minus1

119889119904

le

119870

Γ (120572)

119883 minus 119884119879120572

(31)


L119883 (119905) minusL119884 (119905) le 119883 minus 119884 (32)




119873

such that 1199050= 0 and 119905



119863120572

lowast119909119894(119905119899)

=

1

Γ (1 minus 120572)

int

119905

0

119889119909119894(119904)

119889119904

(119905119899minus 119904)minus120572

119889119904

asymp

1

Γ (1 minus 120572)

119899

sum

119895=1

int

119895ℎ

(119895minus1)ℎ

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)] (119899ℎ minus 119904)minus120572

119889119904

=

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)]



] ℎ1minus120572

=

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]

+

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]119874 (ℎ2minus120572

)

(33)

Setting

G (120572 ℎ) =

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572 120596

120572

119895= 1198951minus120572


(where 120596120572

1= 1)

(34)


119863120572

lowast119909119894(119905119899) = G (120572 ℎ)

119899

sum

119895=1

120596120572

119895(119909119899minus119895+1

119894minus 119909119899minus119895

119894) + 119874 (ℎ) (35)





119863120572

lowast119906 (119905) = 120588

0119906 (119905) + 120588

1 119880 (0) = 119880

0 (36)


1gt 0 are constants


0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)


1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889



behavior




(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)


120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)


1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)


1205772le 1205771 (42)


120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1is unstable

Proof Since

tr (119869 (E1)) =

1205962minus 120596 (119886119887 + 119887120579) minus 119886119887

2120579

2119887120596

+

120596 minus 119886119887

2119886119887120596

radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(14)


119886 [1205962

minus 120596 (119886119887 + 119887120579) minus 1198861198872

120579]

gt (119886119887 minus 120596)radic1198862(119887120579 + 120596)

2

minus 4119886119887120596 (119886120579 minus 120590)

(15)





R0= 119886120579120590 lt 1







1198631205721119867 = 119904 minus 120583

119867119867 + 119903119867(1 minus

119867 + 119868

119867max) minus 1198961119881119867

1198631205722119868 = 1198961015840

1119881119867 minus 120583

119868119868

1198631205723119881 = 119872120583

119887119868 minus 1198961119881119867 minus 120583

119881119881

05 lt 120572119894le 1 119894 = 1 2 3

(16)












119881is the


0 119868(0) = 119868

0 and 119881(0) = 119881

0





1198631205722119868(119905) = 119863



= (1198670 0 0) and


=


119867lowast

=

120583119881120583119868

1198961015840

1119872120583119887minus 1198961120583119868

119868lowast

=

1198961015840


120583119868

119881lowast

=

120583119868[(119904 + (119903 minus 120583

119867)119867lowast)119867max minus 119903119867

lowast2

]

119867lowast[1198961015840

1119903119867lowastminus 1198961120583119868119867max]

(17)



119869 (Elowast

0)

= (

minus120583119867+ 119903 minus 2119903119867

0

119867max

minus1199031198670

119867maxminus11989611198670

0 minus120583119868

1198961015840

11198670

0 119872120583119887

minus (11989611198670+ 120583119881)

)

(18)






where

Rlowast

0=

1198961015840

11205831198871198721198670

120583119868(120583119881+ 11989611198670)

(19)


We also assume that

1198670=

119903 minus 120583119867+ [(119903 minus 120583

119867)2

+ 4119903119904119867minus1

max]12

2119903119867minus1

max

(20)




119868) = 0 becomes

(120582 + 120583119867minus 119903 +

21199031198670

119867max) (1205822

+ 119861120582 + 119862) = 0 (21)

where 119861 = 120583119868+1198961198681198670+120583119881and119862 = 120583

119868(11989611198670+120583119881)minus1198961015840

11205831198871198721198670


1205821= minus120583119867+ 119903 minus

1199031198670


119867)2

+ 4119903119904119867minus1

max lt 0

12058223

=

1

2


(22)


11205831198871198721198670)120583119868(120583119881+11989611198670) lt 1 then



three cases

(i) If Rlowast0


+does not exist

(unphysical)





+

emerges


+for Rlowast



+becomes

119869 (Elowast

+) =

(

(

minus119871lowast



1198961015840

1119881lowast

minus120583119868

1198961015840

1119867lowast

1198961119881lowast

119872120583119887

minus (1198961119867lowast+ 120583119881)

)

)

(23)

Here

119871lowast

= minus[120583119867minus 119903 + 119896

1119881lowast

+

119903 (2119867lowast+ 119868lowast)

119867max] (24)


119875 (120582) = 1205823

+ 11988611205822

+ 1198862120582 + 1198863= 0 (25)

1198861= 120583119868+ 120583119881+ 1198961119867lowast

+ 119871lowast

1198862= 119871lowast

(120583119868+ 120583119881+ 1198961119867lowast

) + 120583119868(120583119881+ 1198961119867lowast

)

minus 1198962

1119867lowast

119881lowast

minus 1198961015840

1119867lowast

(119872120583119887minus

119903119881lowast

119867max)

1198863= 1198961015840

1119867lowast

[1198961119872120583119887119881lowast

minus

119903120583119881119881lowast


119872120583119887]

+ 119871lowast

120583119868(120583119881+ 1198961119867lowast

) minus 1205831198681198962

1119867lowast

119881lowast

(26)



1gt 0

1198863gt 0 and 119886

11198862gt 1198863




119863120572

119883(119905) = 119865 (119905 119883 (119905) P) 119905 isin [0 119879] 0 lt 120572 le 1

119883 (0) = 1198830

(27)

Here 119883(119905) = [1199091(119905) 1199092(119905) 119909



119865 (119905 119883 (119905) P) minus 119865 (119905 119884 (119905) P) le 119870 119883 (119905) minus 119884 (119905)

119870 gt 0

(28)



120572Γ(120572 + 1) lt 1




119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904 (29)


L119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904

(30)

Then we have

L119883 (119905) minusL119884 (119905)

le

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) minus 119865 (119904 119884 (119904) P) 119889119904

le

119870

Γ (120572)

int

119905

0


|119883 (119904) minus 119884 (119904)| 119889119904

le

119870

Γ (120572)

119883 minus 119884int

119905

0

119904120572minus1

119889119904

le

119870

Γ (120572)

119883 minus 119884119879120572

(31)


L119883 (119905) minusL119884 (119905) le 119883 minus 119884 (32)




119873

such that 1199050= 0 and 119905



119863120572

lowast119909119894(119905119899)

=

1

Γ (1 minus 120572)

int

119905

0

119889119909119894(119904)

119889119904

(119905119899minus 119904)minus120572

119889119904

asymp

1

Γ (1 minus 120572)

119899

sum

119895=1

int

119895ℎ

(119895minus1)ℎ

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)] (119899ℎ minus 119904)minus120572

119889119904

=

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)]



] ℎ1minus120572

=

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]

+

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]119874 (ℎ2minus120572

)

(33)

Setting

G (120572 ℎ) =

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572 120596

120572

119895= 1198951minus120572


(where 120596120572

1= 1)

(34)


119863120572

lowast119909119894(119905119899) = G (120572 ℎ)

119899

sum

119895=1

120596120572

119895(119909119899minus119895+1

119894minus 119909119899minus119895

119894) + 119874 (ℎ) (35)





119863120572

lowast119906 (119905) = 120588

0119906 (119905) + 120588

1 119880 (0) = 119880

0 (36)


1gt 0 are constants


0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)


1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889



behavior




(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)


120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)


1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)


1205772le 1205771 (42)


120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










We also assume that

1198670=

119903 minus 120583119867+ [(119903 minus 120583

119867)2

+ 4119903119904119867minus1

max]12

2119903119867minus1

max

(20)




119868) = 0 becomes

(120582 + 120583119867minus 119903 +

21199031198670

119867max) (1205822

+ 119861120582 + 119862) = 0 (21)

where 119861 = 120583119868+1198961198681198670+120583119881and119862 = 120583

119868(11989611198670+120583119881)minus1198961015840

11205831198871198721198670


1205821= minus120583119867+ 119903 minus

1199031198670


119867)2

+ 4119903119904119867minus1

max lt 0

12058223

=

1

2


(22)


11205831198871198721198670)120583119868(120583119881+11989611198670) lt 1 then



three cases

(i) If Rlowast0


+does not exist

(unphysical)





+

emerges


+for Rlowast



+becomes

119869 (Elowast

+) =

(

(

minus119871lowast



1198961015840

1119881lowast

minus120583119868

1198961015840

1119867lowast

1198961119881lowast

119872120583119887

minus (1198961119867lowast+ 120583119881)

)

)

(23)

Here

119871lowast

= minus[120583119867minus 119903 + 119896

1119881lowast

+

119903 (2119867lowast+ 119868lowast)

119867max] (24)


119875 (120582) = 1205823

+ 11988611205822

+ 1198862120582 + 1198863= 0 (25)

1198861= 120583119868+ 120583119881+ 1198961119867lowast

+ 119871lowast

1198862= 119871lowast

(120583119868+ 120583119881+ 1198961119867lowast

) + 120583119868(120583119881+ 1198961119867lowast

)

minus 1198962

1119867lowast

119881lowast

minus 1198961015840

1119867lowast

(119872120583119887minus

119903119881lowast

119867max)

1198863= 1198961015840

1119867lowast

[1198961119872120583119887119881lowast

minus

119903120583119881119881lowast


119872120583119887]

+ 119871lowast

120583119868(120583119881+ 1198961119867lowast

) minus 1205831198681198962

1119867lowast

119881lowast

(26)



1gt 0

1198863gt 0 and 119886

11198862gt 1198863




119863120572

119883(119905) = 119865 (119905 119883 (119905) P) 119905 isin [0 119879] 0 lt 120572 le 1

119883 (0) = 1198830

(27)

Here 119883(119905) = [1199091(119905) 1199092(119905) 119909



119865 (119905 119883 (119905) P) minus 119865 (119905 119884 (119905) P) le 119870 119883 (119905) minus 119884 (119905)

119870 gt 0

(28)



120572Γ(120572 + 1) lt 1




119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904 (29)


L119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904

(30)

Then we have

L119883 (119905) minusL119884 (119905)

le

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) minus 119865 (119904 119884 (119904) P) 119889119904

le

119870

Γ (120572)

int

119905

0


|119883 (119904) minus 119884 (119904)| 119889119904

le

119870

Γ (120572)

119883 minus 119884int

119905

0

119904120572minus1

119889119904

le

119870

Γ (120572)

119883 minus 119884119879120572

(31)


L119883 (119905) minusL119884 (119905) le 119883 minus 119884 (32)




119873

such that 1199050= 0 and 119905



119863120572

lowast119909119894(119905119899)

=

1

Γ (1 minus 120572)

int

119905

0

119889119909119894(119904)

119889119904

(119905119899minus 119904)minus120572

119889119904

asymp

1

Γ (1 minus 120572)

119899

sum

119895=1

int

119895ℎ

(119895minus1)ℎ

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)] (119899ℎ minus 119904)minus120572

119889119904

=

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)]



] ℎ1minus120572

=

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]

+

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]119874 (ℎ2minus120572

)

(33)

Setting

G (120572 ℎ) =

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572 120596

120572

119895= 1198951minus120572


(where 120596120572

1= 1)

(34)


119863120572

lowast119909119894(119905119899) = G (120572 ℎ)

119899

sum

119895=1

120596120572

119895(119909119899minus119895+1

119894minus 119909119899minus119895

119894) + 119874 (ℎ) (35)





119863120572

lowast119906 (119905) = 120588

0119906 (119905) + 120588

1 119880 (0) = 119880

0 (36)


1gt 0 are constants


0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)


1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889



behavior




(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)


120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)


1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)


1205772le 1205771 (42)


120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904 (29)


L119883 (119905) = 119883 (0) +

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) 119889119904

(30)

Then we have

L119883 (119905) minusL119884 (119905)

le

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

119865 (119904 119883 (119904) P) minus 119865 (119904 119884 (119904) P) 119889119904

le

119870

Γ (120572)

int

119905

0


|119883 (119904) minus 119884 (119904)| 119889119904

le

119870

Γ (120572)

119883 minus 119884int

119905

0

119904120572minus1

119889119904

le

119870

Γ (120572)

119883 minus 119884119879120572

(31)


L119883 (119905) minusL119884 (119905) le 119883 minus 119884 (32)




119873

such that 1199050= 0 and 119905



119863120572

lowast119909119894(119905119899)

=

1

Γ (1 minus 120572)

int

119905

0

119889119909119894(119904)

119889119904

(119905119899minus 119904)minus120572

119889119904

asymp

1

Γ (1 minus 120572)

119899

sum

119895=1

int

119895ℎ

(119895minus1)ℎ

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)] (119899ℎ minus 119904)minus120572

119889119904

=

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[

119909119895

119894minus 119909119895minus1

119894

ℎ

+ 119874 (ℎ)]



] ℎ1minus120572

=

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]

+

1

(1 minus 120572) Γ (1 minus 120572)

times

119899

sum

119895=1

[119909119895

119894minus 119909119895minus1

119894] [(119899 minus 119895 + 1)

1minus120572


]119874 (ℎ2minus120572

)

(33)

Setting

G (120572 ℎ) =

1

(1 minus 120572) Γ (1 minus 120572)

1

ℎ120572 120596

120572

119895= 1198951minus120572


(where 120596120572

1= 1)

(34)


119863120572

lowast119909119894(119905119899) = G (120572 ℎ)

119899

sum

119895=1

120596120572

119895(119909119899minus119895+1

119894minus 119909119899minus119895

119894) + 119874 (ℎ) (35)





119863120572

lowast119906 (119905) = 120588

0119906 (119905) + 120588

1 119880 (0) = 119880

0 (36)


1gt 0 are constants


0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)


1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889



behavior




(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)


120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)


1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)


1205772le 1205771 (42)


120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

05

1

15

Time0 20 40 60 80 100

0

05

1

15

2

25

Time

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

Time0 05 1 15 2 25

0

02

04

06

08

1

12

14

16

Tumor

T(t)

T(t)

T(t)

E1(t)

E1(t)

E1(t)

E2(t)

T(t)E1(t)E2(t)

T(t)E1(t)E2(t)

E2(t)

E2(t)

T(t)

E1(t)

E2(t)

120572 = 095

120572 = 075 120572 = 075

120572 = 075

Effec

tor c

ellsE1(t)E2(t)


1= 03 119889

2= 07 119896

1= 03

1198962= 07) and when 119889

1= 07 119889



behavior




(1 minus

1205880

G120572ℎ

)120577119899

= 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) +

1205881

G120572ℎ

119899 ge 2

(37)

or

120577119899=

120577119899minus1

+ sum119899

119895=2120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) + 1205881G120572ℎ

(1 minus (1205880G120572ℎ))

119899 ge 2

(38)


120572ℎ then

1205771le 1205770 (39)

120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) 119899 ge 2 (40)


1205772le 1205771+ 120596(120572)

2(1205770minus 1205771) (41)


1205772le 1205771 (42)


120577119899le 120577119899minus1

+

119899

sum

119895=2

120596(120572)

119895(120577119899minus119895

minus 120577119899minus119895+1

) le 120577119899minus1

(43)


0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 1500

05

1

15

2

25

3

35

4

Time0 50 100 150

2

4

6

8

10

12

Time

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

Effec

tor c

ells

E(t)

Effector cells E(t)

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09

120572 = 07

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)





le

120577119899minus1

le 120577119899minus2


119899= |119880119899| le

1205770= |1198800| which entails 119880119899 le 119880

0 we have stability






025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










025 03 035 04 045 05 055 06 065 07 0750

005

01

015

02

025

03

035

Effector cells E(t)

0 20 40 60 80 100 120 140 160 180 200025

03

035

04

045

05

055

06

065

07

075

Time

Effec

tor c

ells

E(t)

120572 = 1

120572 = 09

120572 = 08

120572 = 1

120572 = 09

120572 = 08

0 20 40 60 80 100 120 140 160 180 2000

005

01

015

02

025

03

035

Time

120572 = 1

120572 = 09

120572 = 08

Tum

or ce

llsT

(t)

Tum

or ce

llsT

(t)



0= 119886120579120590 lt 1


5 Conclusions








0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000

1200

Time0 20 40 60 80 100 120 140 160 180 200

0100200300400500600700

Time

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

Time

Initi

al d

ensit

y of

HIV

RN

A

0 200 400 600 800 1000 1200

02004006008000

5

10

H(t)I(t)

V(t)120572 = 1

120572 = 09120572 = 07

120572 = 1

120572 = 09

120572 = 07Uni

nfec

ted

CD4+

T-ce

ll po

pulat

ion

size

120572 = 1

120572 = 09

120572 = 07

120572 = 1

120572 = 09120572 = 07

Infe

cted

CD

4+T-

cell

dens

ity

times104

times104


119868= 026 dayminus1 120583

119887= 024 dayminus1

120583119881= 24 dayminus1 119896

1= 24 times 10



+


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article numerical modeling of fractional-order biological...

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