We study a fractional differential model of HIV infection of CD4+ T-cell, in which the CD4+ T-cell proliferation plays an important role in HIV infection under antiretroviral therapy. An appropriate method is given to ensure that both the equilibria are asymptotically stable for τ≥0. We calculate the basic reproduction
number R0, the IFE E0, two IPEs E1* and E2*, and so on, and judge the stability of the equilibrium. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells.
1. Introduction
Human immunodeficiency virus (HIV) is a lentivirus (a member of the retrovirus family) that causes acquired immunodeficiency syndrome (AIDS) [1], a condition in humans in which the immune system begins to fail, leading to life-threatening opportunistic infections. HIV infects primarily vital cells in the human immune system such as helper T-cell (to be specific, CD4+ T-cell), macrophages, and dendritic cells. When CD4+ T-cell numbers decline below a critical level, cell-mediated immunity is lost and the body becomes progressively more susceptible to opportunistic infections (see [2]).
There are only a few results for dynamics of HIV infection of CD4+ T-cell. In 1992, Perelson et al. [3] examined a model for the interaction of HIV with CD4+ T-cell who considered four populations: uninfected T-cell, latently infected T-cell, actively infected T cells, and free virus, and they also considered effects of AZT on viral growth and T-cell population dynamics. In 2000, Culshaw and Ruan [4] firstly simplified their model into one consisting of only three components: the healthy CD4+ T-cell, infected CD4+ T-cell, and free virus and discussed the existence and stability of the infected steady state, and they studied the effect of the time delay on the stability of the endemically infected equilibrium; criteria were given to ensure that the infected equilibrium is asymptotically stable for all delay.
For backward bifurcations in other disease models, we refer the reader to [5–12]. In [12], this paper analyzed the backward bifurcation sources and application in infectious disease model. HIV/AIDS infection model is a special case of infectious disease model. For recent work on global analysis and persistence of HIV models, we refer the reader to [13–18] and references therein. A discussion on HIV infection and CD4+ T-cell depletion is given in the review paper [19].
In 2012, Shu and Wang [12] considered a new model frame that included full logistic growth terms of both healthy and infected CD4+ T-cell:
(1)ddtT(t)=s-μ1T(t)+rT(t)VI(t)C+VI(t)-k(1-nrt)VI(t)T(t),ddtIi(t)=k(1-nrt)VI(t)T(t)-μ2Ti(t),ddtVI(t)=(1-np)Nμ2Ti(t)-μ3VI(t),ddtVNI(t)=npNμ2Ti(t)-μ3VNI(t).
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, chemical technology, population dynamics, biotechnology, and economics (see e.g., [20–26]). As one of the important topics in the research differential equations, the boundary value problem has attained a great deal of attention (see [27–41] and the references therein).
Many mathematicians and researchers in the field of application are trying to model fractional order differential equations. In biology, the researchers found that biological membranes with fractional order have the nature of electronic conductivity, so it can be classified as a model of the fractional order. Because of the memory property of fractional calculus, we introduce the fractional calculus into HIV model. Both in mathematics and biology, fractional calculus will correspond with objective reality more than ODE. It is particularly important for us to study fractional HIV model.
Furthermore, delay plays an important role in the process of spreading infectious diseases; it can be used to simulate the incubation period of infectious diseases, the period of patients infected with disease, period of patients immune to disease, and so on. The basic fact reflected by the specific mathematical model with time delay is that the change of trajectory about time t not only depends on the t moment itself but also is affected by some certain conditions before, even the reflection of some certain factors before. This kind of circumstance is abundant in the objective world.
Recently, Yan and Kou [2] have introduced fractional-order derivatives into a model of HIV infection of CD4+ T-cell with time delay:
(2)DαT(t)=s-μTT(t)+rT(t)(1-T(t)+I(t)Tmax)-k1T(t)V(t),DαI(t)=k1′T(t-τ)V(t-τ)-μII(t),DαV(t)=NμbI(t)-k1T(t)V(t)-μvV(t),
with the initial conditions:
(3)T(θ)=T0,I(0)=0,V(θ)=V0,θ∈[-τ,0].
Motivated by the works mentioned above, we will consider this model where the CD4+ T-cell proliferation does play an important role in HIV infection under antiretroviral therapy; a more appropriate method is given to ensure that both equilibria are asymptotically stable for τ≥0. We calculate the basic reproduction number R0, the IFE E0, two IPEs E1* and E2*, and so on under certain conditions and judge the stability of the equilibrium. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells. In this paper, we establish mathematical model as follows:
(4)DαT(t)=s-μ1T(t)+rT(t)VI(t)C+VI(t)-k(1-nrt)T(t-τ)VI(t-τ),DαI(t)=k(1-nrt)T(t-τ)VI(t-τ)-μ2I(t),DαV(t)=(1-nP)Nμ2I(t)-μ3VI(t),
with the initial conditions:
(5)T(θ)=T0,I(0)=0,V(θ)=V0,θ∈[-τ,0],
where Dα denotes Caputo's fractional derivative of order α with the lower limit zero. T(t),I(t), and V(t) represent the concentration of healthy CD4+ T-cell at time t, infected CD4+ T-cell at time t, and free HIV virus particles in the blood at time t, respectively. The positive constant τ represents the length of the delay in days. A complete list of the parameter values for the model is given in Table 1.
Parameters and values of model (4).
Parameter
Description
Value
T
Uninfected CD4+ T-cell population size
1000 mm^{−3}
I
Infected CD4+ T-cell density
0
VI
Free infectious virus particles
10-3 mm^{−3}
VNI
Noninfectious virus particles
10-3 mm^{−3}
T0
CD4+ T-cell population for HIV-negative persons
1000 mm^{−3}
μ1
Natural death rate of CD4+ T-cell
0.02 day^{−1}
μ2
Blanket death rate of infected CD4+ T-cell
0.26 day^{−1}
μ3
Death rate of free virus
2.4 day^{−1}
k
Rate of CD4+ T-cell becoming infected with virus
2.4×10-5 mm^{3} day^{−1}
k′
Rate of infected cells becoming active
2×10-5 mm^{3} day^{−1}
s
Source term for uninfected CD4+ T-cell
10 day^{−1} mm^{−3}
N
Number of virions produced by infected CD4+ T-cell
Varies
r
The maximal proliferation rate (r<μ1)
Varies
C
The half saturation constant of the proliferation process
Varies
nrt
The effectiveness of RTIs (nrt = 0 means the therapy is totally ineffective, while nrt = 1 indicates the therapy is 100% effective and the cell-to-cell infection is completely stopped)
Varies
np
The effectiveness of PIs (np = 1 meaning the therapy with PIs is 100% effective and no newly infectious virus particles will be produced [42])
Varies
rTVIC+VI
The stimulation of T-cell to proliferate in the presence of virus [43]
Varies
Furthermore, we assume that T(t)>0,I(t)≥0, and V(t)≥0 for all t≥-τ.
This paper is organized in the following way. In the next section, some necessary definitions and lemmas are presented. In Section 3, the stability of the equilibria is given. In Section 4, we calculate some of the data and judge the stability of the equilibrium. In Section 5, we will give the numerical simulation for the fractional HIV model. Finally, the conclusions are given.
2. Preliminaries
In this section, we introduce definitions and lemmas which will be used later.
Definition 1 (see [<xref ref-type="bibr" rid="B20">20</xref>, <xref ref-type="bibr" rid="B38">25</xref>]).
The fractional (arbitrary) order integral of the function f:[0,∞)→R of order p>0 is defined by
(6)Ipf(x)=1Γ(p)∫0x(x-s)p-1f(s)ds.
Definition 2 (see [<xref ref-type="bibr" rid="B20">20</xref>]).
Let α≥0, n=[α]+1, where [α] denotes the integer part of number α. If f∈ACn[a,b], the Caputo fractional derivative of order α of f is defined by
(7)Dcαf(t)=1Γ(n-α)∫atf(n)(s)(t-s)α+1-nds,t>0,n-1<α<n.
Lemma 3 (see [<xref ref-type="bibr" rid="B21">44</xref>]).
The equilibrium point (xeq,yeq) of the fractional differential system:
(8)Dαx(t)=f1(x,y),Dαy(t)=f2(x,y),α∈(0,1],x(0)=x0,y(0)=y0
is locally asymptotically stable if all the eigenvalues of the Jacobian matrix
(9)A=(∂f1∂x∂f1∂y∂f2∂x∂f2∂y),
evaluated at the equilibrium point satisfying the following condition:
(10)|arg(eig(A))|>απ2.
The stable and unstable regions for 0<α≤1 are shown in Figure 1 [45, 46].
Stability region of system (4) with order 0<α≤1.
Proposition 4 (see [<xref ref-type="bibr" rid="B46">12</xref>]).
Consider model (4).
Assume that (H1): r≤k(1-nrt)C is satisfied.
If R0≤1, then the IFE E0 is the only equilibrium (Table 3).
If R0>1, then there are two equilibria: the IFE E0 and a unique IPE E*.
Assume that (H2): r>k(1-nrt)C is satisfied. Let a=(r-k(1-nrt)C)2/μ1; then 0<a<1 (a<1 follows from the assumption that r<μ1).
If R0<1-a, then the IFE E0 is the only equilibrium.
If 1-a<R0<1, then there are three equilibria: the IFE E0 and two IPEs, denoted by E1*=(T1*,T1i,VI1*) and E2*=(T2*,T2i,VI2*) such that VI2*>VI1*.
If R0=1-a or R0≥1, then there are two equilibria: the IFE E0 and a unique IPE.
Remark 5.
It is similar to [47, 48] to prove the existence of solution of the fractional delay equations, and the one without no delay time is also parallel to [29, 30].
By the next generation matrix method [11], we easily get
(11)ℱ=(k(1-nrt)VIT(1-np)Nμ2Ti),𝒱=(μ2Tiμ3VI);
we obtain the basic reproduction number of model (4)
(12)R0=s(1-np)(1-nrt)kNμ1μ3.
3. The Stability of the Equilibria
In this section, we investigate the existence of equilibria of system (4).
In order to find the equilibria of system (4), we put
(13)s-μTT(t)+rT(t)VI(t)C+VI(t)-k(1-nrt)T(t-τ)VI(t-τ)=0,k(1-nrt)T(t-τ)VI(t-τ)-μ2I(t)=0,(1-NP)Nμ2I(t)-μ3VI(t)=0.
Following the analysis in [12], we get that system (13) has always the infection free equilibrium (IFE) E0=((s/μ1),0,0) and the infection persistent equilibrium (IPE) E*=(T*,Ti*,VI*), where
(14)T*=μ3(1-np)(1-nrt)Nk,I*=μ3(1-np)Nμ2VI*,g(VI)=VI2+(p-rk(1-nrt)+C)VI+pC,p=μ1(k(1-nrt))(1-R0).
Next, we will discuss the stability for the local asymptotic stability of the viral free equilibrium E0 and the infected equilibrium E*.
For the local asymptotic stability of the viral free equilibrium E0, we have the following result.
Lemma 6.
If R0<1, then E0 is locally asymptotically stable for τ≥0. If R0=1, then E0 is locally stable for τ≥0. If R0>1, then E0 is a saddle point with a two-dimensional stable manifold and a one-dimensional unstable manifold.
Proof.
The associated transcendental characteristic equation at E0=(T0,0,0) is given by
(15)(λ+μ1)[λ2+(μ2+μ3)λ+μ2μ3-k(1-nrt)×(1-np)Nμ2T0e-λτ]=0.
Obviously, the above equation has the characteristic root
(16)λ1=-μ1<0.
Next, we consider the transcendental polynomial
(17)λ2+(μ2+μ3)λ+μ2μ3-k(1-nrt)(1-np)Nμ2T0e-λτ=0.
For τ=0, we get
(18)λ2+(μ2+μ3)λ+μ2μ3-R02μ2μ3=0.
We have
(19)λ2,3=-(μ2+μ3)±(μ2+μ3)2-4μ2μ3(1-R02)2.
If R02>1, the characteristic equation has a positive eigenvalue and two negative eigenvalues. E0 is thus unstable with a two-dimensional stable manifold and a one-dimensional unstable manifold. If R02=1, λ2,3=0, then E0 is locally stable. If R02<1, the other two eigenvalues have negative real parts if and only if μ2μ3-R02μ2μ3>0, then E0 is locally asymptotically stable.
For τ≠0, we get
(20)λ2+(μ2+μ3)λ+μ2μ3-k(1-nrt)(1-np)Nμ2T0e-λτ=0.
Assume that the above equation has roots λ=ω(cos(βπ/2)±isin(βπ/2)) for ω>0 and τ>0; we get
(21)ω2(cosβπ2±isinβπ2)2+ω(μ2+μ3)(cosβπ2±isinβπ2)+μ2μ3-k(1-nrt)(1-np)Nμ2T0e-τω(cos(βπ/2)±isin(βπ/2))=0.
Separating the real and imaginary parts gives
(22)ω2(cos2βπ2-sin2βπ2)+ωcosβπ2(μ2+μ3)+μ2μ3-k(1-nrt)(1-np)Nμ2se-τωcos(βπ/2)μ1×cos(-τωsinβπ2)=0,2iω2cosβπ2sinβπ2+iωsinβπ2(μ2+μ3)-k(1-nrt)(1-np)Nμ2se-τωcos(βπ/2)μ1×isin(-τωsinβπ2)=0.
From the second equation of (22), we have
(23)sinβπ2=0,
that is (βπ/2)=kπ, k=0,1,2,….
For (βπ/2)=kπ, k=0,2,4,…, substituting into the first equation of (22), we have
(24)ω2+ω(μ2+μ3)+μ2μ3=k(1-nrt)(1-np)Nμ2se-τωμ1.
For the parameter values given in Table 1, we take any R02<1, then the infected equilibrium E0=((s/μ1),0,0), and we get that the above equation is unequal for ω>0. Therefore, β≥2>α.
According to Lemma 3, the uninfected equilibrium E0 is locally asymptotically stable. The proof is completed.
Next, for the sake of convenience, at E*=(T*,I*,V*), we give the following symbols:
(25)A=μ1+μ2+μ3-rVI*C+VI*,B=k(1-nrt)VI*,C=μ12+μ1μ3-rVI*(μ2+μ3)C+VI*,D=k(1-nrt)VI*(μ2+μ3),E=μ1μ2μ3-rVI*μ2μ3C+VI*,F=k(1-nrt)VI*μ2μ3-VI*μ2μ3rC(C+VI*)2-μ1μ2μ3+μ2μ3rVI*C+VI*.
Then the characteristic equation of the linear system is
(26)λ3+(A+Be-λτ)λ2+(C+De-λτ)λ+E+Fe-λτ=0.
Using the results in [49], we get
(27)D(λ)=λ3+(A+B)λ2+(C+D)+E+F,D′(λ)=3λ2+2(A+B)λ+(C+D).
Denote
(28)D(λ)=-|1A+BC+DE+F001A+BC+DE+F32(A+B)C+D00032(A+B)C+D00032(A+B)C+D|=18(A+B)(C+D)(E+F)+(A+B)2(C+D)2-4(E+F)(A+B)3-4(C+D)3-27(E+F)2.
Lemma 7 (see [<xref ref-type="bibr" rid="B1">49</xref>]).
The infected equilibrium E* is asymptotically stable for any time delay τ≥0 if either
A+B>0,E+F>0,(A+B)(C+D)>E+F if D(λ)>0;
if D(λ)<0, A+B≥0, C+D≥0, E+F>0, α<2/3;
if D(λ)<0, A+B<0, C+D<0, α>2/3, then all roots of D(λ)=0 satisfy |arg(λ)|<απ/2;
if D(λ)<0,A+B>0,C+D>0,(A+B)(C+D)=E+F for all α∈[0,1).
4. Comparison with Some of the Data
In this section, we calculate the basic reproduction number R0, the IFE E0, two IPEs E1* and E2*, D(λ), A+B, C+D, E+F, and (A+B)(C+D). On the basis of these data, we apply all the conditions in Lemma 7 to judge the stability of the equilibrium E1* and E2* (Table 2).
We take r=0.01, C=1, N=100, E0=(500,0,0), and a=(r-k(1-nrt)C)2/μ1, and we get the following.
Line
nrt
np
k(1-nrt)C
1-a
R0
E1*
E2*
1
0.1
0.05
0.00002
0.5454
1.6016
(194.9317, 30.9602, 1911.7950)
not exist
2
0.5
0.25
0.00001
0.5340
1.0607
(444.4444, 21.3511, 1040.8668)
not exist
3
0.5425
0.27125
0.00001
0.5326
1.0001
(499.8953, 19.2137, 910.1286)
not exist
4
0.5426
0.2713
0.00001
0.5326
1.0000
(500.0390, 0.000003, 0.0002)
(500.0389, 19.2082, 909.8039)
5
0.55
0.275
0.00001
0.5323
0.9893
(510.8556, 0.0009, 0.0444)
(510.8557, 18.7911, 885.5296)
6
0.6
0.3
0.000009
0.5305
0.9165
(595.2381, 0.0104, 0.4716)
(595.2381, 15.5354, 706.8618)
7
0.7
0.35
0.000007
0.5265
0.7649
(854.7009, 0.1186, 5.0099)
(854.7009, 5.4462, 230.1012)
8
0.7252
0.3626
0.000007
0.5254
0.7249
(951.5244, 0.8287, 34.3354)
(951.5245, 1.0116, 41.9104)
9
0.7253
0.3627
0.000007
0.5253
0.7247
Not exist
Not exist
10
0.8
0.4
0.000005
0.5217
0.6000
Not exist
Not exist
11
0.82
0.41
0.000005
0.5206
0.5644
Not exist
Not exist
12
0.8446
0.4223
0.000004
0.5191
0.5190
Not exist
Not exist
13
0.85
0.425
0.000004
0.5188
0.5087
Not exist
Not exist
Line
D(λ)
A+B
C+D
E+F
(A+B)(C+D)
Stability
1
−1.7882
2.7113
0.1317
0.0258
0.3570
(ii)
2
−0.5611
2.6825
0.0550
0.0078
0.1477
(ii)
3
−0.4497
2.6800
0.0484
0.0062
0.1297
(ii)
4
0.0165, −0.4495
2.6800, 2.6800
0.0484, 0.0484
−0.000001, 0.0062
0.1297, 0.1297
Unsuited (ii)
5
0.0346, −0.4305
2.6796, 2.6796
0.0473, 0.0473
−0.0003, 0.0060
0.1267, 0.1267
Unsuited (ii)
6
0.1125, −0.3054
2.6768, 2.6768
0.0399, 0.0399
−0.0014, 0.0042
0.1068, 0.1068
Unsuited (ii)
7
0.0681, −0.0707
2.6717, 2.6717
0.0263, 0.0263
−0.0008, 0.0010
0.0703, 0.0703
Unsuited (ii)
8
0.0060, 0.0015
2.6705, 2.6705
0.0232, 0.0232
−0.00003, 0.00003
0.0618, 0.0618
Unsuited (i)
We take r=0.01, nrt=0.5, np=0.25, N=100, E0=(500,0,0), and a=(r-k(1-nrt)C)2/μ1, and we have the following.
Line
C
k(1-nrt)C
1-a
R0
E1*
E2*
14
10
0.0001
0.6035
1.0607
(444.4444, 21.2037, 1033.6821)
Not exist
15
100
0.0012
0.7864
1.0607
(444.4444, 19.7599, 963.2939)
Not exist
16
300
0.0036
0.9200
1.0607
(444.4444, 16.7808, 818.0663)
Not exist
17
500
0.0060
0.9746
1.0607
(444.4444, 14.1982, 692.1614)
Not exist
18
700
0.0084
0.9965
1.0607
(444.4444, 12.0858, 589.1841)
Not exist
19
830
0.0100
1.0000
1.0607
(444.4444, 10.9728, 534.9223)
Not exist
20
900
0.0108
0.9992
1.0607
(444.4444, 10.4534, 509.6013)
Not exist
Line
D(λ)
A+B
C+D
E+F
(A+B)(C+D)
Stability
14
−0.5531
2.6825
0.0550
0.0077
0.1477
(ii)
15
−0.4782
2.6825
0.0551
0.0067
0.1477
(ii)
16
−0.3449
2.6825
0.0550
0.0049
0.1477
(ii)
17
−0.2523
2.6825
0.0550
0.0037
0.1477
(ii)
18
−0.1926
2.6825
0.0551
0.0029
0.1477
(ii)
19
−0.1668
2.6825
0.0550
0.0025
0.1477
(ii)
20
−0.1561
2.6825
0.0550
0.0024
0.1477
(ii)
Remark 8.
(1) If nrt={0.1,0.5,0.5425}, there are two equilibria: the IFE E0 and a unique IPE E1*. In addition, the system at E1* satisfies the second condition in Lemma 7, then the IPE E1* is locally asymptotically stable (Table 5).
(2) If nrt={0.5426,0.55,0.6,0.7,0.7252}, there are three equilibria: the IFE E0 and two IPEs. The system at E1* doesn't satisfy all the conditions in Lemma 7, then the IPE E1* is unstable. The system at E2* satisfies the second condition in Lemma 7, then the IPE E2* is locally asymptotically stable.
(3) If nrt={0.7253,0.8,0.82,0.8446,0.85}, the IFE E0 is the only equilibrium.
Remark 9.
If C={10,100,300,500,700,830,900}, there are two equilibria: the IFE E0 and a unique IPE E1*. In addition, the system at E1* satisfies the second condition in Lemma 7, then the IPE E1* is locally asymptotically stable.
Remark 10.
If r={0,0.004,0.008,0.015,0.019}, there are two equilibria: the IFE E0 and a unique IPE E1*. In addition, the system at E1* satisfies the second condition in Lemma 7, then the IPE E1* is locally asymptotically stable.
Remark 11.
(1) If N={10,30,40,47}, the IFE E0 is the only equilibrium.
(2) If N=48, there are three equilibria: the IFE E0 and two IPEs. The system at E1* and E2* does not satisfy all the conditions in Lemma 7, then the IPEs E1* and E2* are unstable.
(3) If N={60,88}, there are two equilibria: the IFE E0 and a unique IPE E1*. In addition, the system at E1* does not satisfy all the conditions in Lemma 7, then the IPE E1* is unstable. The system at E2* satisfies the second condition in Lemma 7, then the IPE E2* is locally asymptotically stable.
(4) If N={89,100,200,600,1000}, there are two equilibria: the IFE E0 and a unique IPE E1*. The system at E1* satisfies the second condition in Lemma 7, then the IPE E2* is locally asymptotically stable.
(5) If N={10000,20000}, there are two equilibria: the IFE E0 and a unique IPE E1*. The system at E1* satisfies the first condition in Lemma 7, then the IPE E1* is locally asymptotically stable.
5. Numerical Simulations
In this section, we use the Adams-type predictor-corrector method for the numerical solution of nonlinear system (4) with time delay.
Firstly, we will replace system (4) by the following equivalent fractional integral equations:
(29)T(t)=T(0)+Iα[s-μTT(t)+rT(t)VI(t)C+VI(t)hhhhhhh-k(1-nrt)T(t-τ)VI(t-τ)rT(t)VI(t)C+VI(t)],I(t)=I(0)+Iα×[k(1-nrt)T(t-τ)VI(t-τ)-μ2I(t)],V(t)=V(0)+Iα[(1-nP)Nμ2I(t)-μ3VI(t)].
Next, we apply the predict, evaluate, correct, evaluate (PECE) method.
The approximate solution is displayed in (Figures 2(a)–2(d), 3(a)–3(d), 4(a)–4(d), 5(a)–5(d), 6(a)–6(d), and 7(a)–7(d)). When α=1, system (4) is the classical integer-order ODE.
In (a)–(d), α=0.9, N=100, and τ=0.
In (a)–(d), α=0.9, τ=2, C=1, and r=0.01.
In (a)–(d), nrt=0.5, np=0.25, α=0.9, τ=0, C=10, and r=0.01.
In (a)–(d), nrt=0.5, np=0.25, α=0.9, τ=2, C=10, and r=0.01.
In (a)–(d), nrt=0.5, np=0.25, N=100, τ=0, C=10, and r=0.01.
In (a)–(d), nrt=0.5, np=0.25, N=100, τ=2, C=10, and r=0.01.
Remark 12.
Figures 2 and 3 show that, if nrt={0.7253,0.8,0.85} and τ={0,2}, the system at E0=(500,0,0) is locally stable.
Remark 13.
Figures 4 and 5 show that, if N={10,30,40,47} and τ={0,2}, the system at E0=(500,0,0) is locally stable.
Remark 14.
Figures 6 and 7 show that, if α={0.9,0.93,0.96,0.99,1} and τ={0,2}, the system at E0=(500,0,0) is locally stable. As α increases, the trajectory of the system approaches the steady state faster and gets close to the integer-order ODE.
6. Extending the Model
In this section, we add the term -μ3VI in the third equation of model (4) which we consider the loss of virions due to all causes, and we also add the bilinear term -k(1-nrt)T(t-τ)VI(t-τ) which we consider free infectious virions when they enter the target cells. We extend model (4) to the following system of differential equations:
(30)DαT(t)=s-μ1T(t)+rT(t)VI(t)C+VI(t)-k(1-nrt)T(t-τ)VI(t-τ),DαI(t)=k′(1-nrt)T(t-τ)VI(t-τ)-μ2I(t),DαV(t)=(1-nP)Nμ2I(t)-μ3VI(t)-k(1-nrt)T(t-τ)VI(t-τ).
Following the analysis in [12], we get that system (30) has always the uninfected equilibrium E0=((s/μ1),0,0), and the infected equilibrium E**=(T**,I**,VI**), where
(31)T**=μ3(1-np)[(1-nrt)Nk′-k],I**=μ3k′μ2[(1-np)Nk′-k]VI**,g(VI)=VI2+VI[μ1k(1-nrt)(1-R2)-r(1-nrt)k+C1-nrt-R2sμ3]+μ1C(1-nrt)k-sC[(1-np)Nk′-k]kμ3=0.
By the next generation matrix method [11], we obtain the basic reproduction number of model (30)
(32)R2=(1-np)(1-nrt)Nk′sμ1μ3+k(1-nrt)s.
7. Conclusion
In this paper, we modified the ODE model proposed by Shu and Wang [12] and the fractional model proposed by Yan and Kou [2] into a system of fractional order. We study a fractional differential model of HIV infection of CD4+ T cell. We will consider this model where the CD4+ T-cell proliferation does play an important role in HIV infection under antiretroviral therapy. The more appropriate method is given to ensure that both the equilibria are asymptotically stable for τ≥0 under some conditions. We calculate the basic reproduction number R0, the IFE E0, two IPEs E1* and E2*, and so on, under certain conditions and judge the stability of the equilibrium. According to Tables 1 and 4, we get that, if nrt={0.7253,0.8,0.85} and N={10,30,40,47}, there is only the IFE E0 for τ≥0. In addition, if α={0.9,0.93,0.96,0.99,1} under some conditions, there is only the IFE E0 for τ≥0. We describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells.
We take C=10, nrt=0.5, np=0.25, N=100, E0=(500,0,0), and a=(r-k(1-nrt)C)2/μ1, we give the following.
Line
r
k(1-nrt)C
1-a
R0
E1*
E2*
21
0
0.0001
0.9940
1.0607
(444.4444, 4.2735, 208.3333)
Not exist
22
0.004
0.0001
0.8632
1.0607
(444.4444, 10.9858, 535.5567)
Not exist
23
0.008
0.0001
0.6920
1.0607
(444.4444, 17.7929, 867.4018)
Not exist
24
0.015
0.0001
0.3782
1.0607
(444.4444, 29.7389, 1449.7703)
Not exist
25
0.019
0.0001
0.1950
1.0607
(444.4444, 36.5710, 1782.8352)
Not exist
Line
D(λ)
A+B
C+D
E+F
(A+B)(C+D)
Stability
21
−0.0952
2.6825
0.0550
0.0016
0.1477
(ii)
22
−0.2749
2.6825
0.0551
0.0040
0.1477
(ii)
23
−0.4600
2.6825
0.0550
0.0064
0.1477
(ii)
24
−0.7866
2.6825
0.0550
0.0108
0.1477
(ii)
25
−0.9740
2.6825
0.0550
0.0550
0.1477
(ii)
We take C=10, nrt=0.5, np=0.25, r=0.01, E0=(500,0,0), and a=(r-k(1-nrt)C)2/μ1, and we get the following.
Line
N
k(1-nrt)C
1-a
R0
E1*
E2*
26
10
0.00001
0.5340
0.3354
Not exist
Not exist
27
30
0.00001
0.5340
0.5809
Not exist
Not exist
28
40
0.00001
0.5340
0.6708
Not exist
Not exist
29
47
0.00001
0.5340
0.7271
Not exist
Not exist
30
48
0.00001
0.5340
0.7348
(925.9259, 0.6491, 15.1879)
(925.9259, 2.1572, 50.4788)
31
60
0.00001
0.5340
0.8216
(740.7407, 0.0641, 1.8756)
(740.7407, 9.8732, 288.7910)
32
88
0.00001
0.5340
0.9950
(505.0505, 0.0005, 0.0204)
(505.0505, 19.0127, 815.6462)
33
89
0.00001
0.5340
1.0006
(499.3758, 19.2318, 834.4192)
Not exist
34
100
0.00001
0.5340
1.0607
(444.4444, 21.3511, 1040.8668)
Not exist
35
200
0.00001
0.5340
1.5000
(222.2222, 29.9116, 2916.3810)
Not exist
36
600
0.00001
0.5340
2.5981
(74.0741, 35.6123, 10416.5867)
Not exist
37
1000
0.00001
0.5340
3.3541
(44.4444, 36.7520, 17916.6202)
Not exist
38
10000
0.00001
0.5340
10.6066
(4.4444, 38.2906, 186666.6622)
Not exist
39
20000
0.00001
0.5340
15.0000
(2.2222, 38.3761, 374166.6644)
Not exist
Line
D(λ)
A+B
C+D
E+F
(A+B)(C+D)
Stability
30
0.1109, 0.0417
2.6742, 2.6723
0.0328, 0.0278
−0.0014, −0.0005
0.0878, 0.0743
Unsuited, unsuited
31
0.0747, −0.1399
2.6784, 2.6738
0.0443, 0.0319
−0.0008, 0.0020
0.1185, 0.0853
Unsuited (i)
32
0.0173, −0.4352
2.6800, 2.6799
0.0483, 0.0482
−0.00001, 0.0060
0.1296, 0.1291
Unsuited (i)
33
−0.4454
2.6801
0.0487
0.0062
0.1307
(ii)
34
−0.5570
2.6826
0.0553
0.0077
0.1483
(ii)
35
−1.5273
2.7050
0.1150
0.0218
0.3110
(ii)
36
−4.7830
2.7950
0.3543
0.0780
0.9903
(ii)
37
−7.1387
2.8850
0.5937
0.1342
1.7129
(ii)
38
30.8901
4.9100
5.9802
1.3978
29.3628
(i)
39
482.3545
7.1600
11.9652
11.9652
85.6708
(i)
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This project was supported by NNSF of China Grant nos. 11271087 and 61263006.
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