# Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate

Authors: Adnane Boukhouima; Khalid Hattaf; Noura Yousfi
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV) infection. In the model, the infection transmission process is modeled by a specific functional response. First, we show that the model is mathematically and biologically well posed. Second, the local and global stabilities of the equilibria are investigated. Finally, some numerical simulations are presented in order to illustrate our theoretical results.


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## Body

## 1. Introduction

Fractional order differential equations (FDEs) are a generalization of ordinary differential equations (ODEs) and they have many applications in various fields such as mechanics, image processing, viscoelasticity, bioengineering, finance, psychology, and control theory [1–7]. In addition, it has been deduced that the membranes of cells of biological organisms have fractional order electrical conductance [8].Modeling by FDEs has more advantages to describe the dynamics of phenomena with memory which exists in most biological systems, because fractional order derivatives depend not only on local conditions but also on the past. More precisely, calculating the time-fractional derivative of a functionf(t) at some time t=t1 requires all the previous history, that is, all f(t) from t=0 to t=t1. In addition, the region of stability of FDEs is larger than that of ODEs. Moreover, some previous study compared between the results of the fractional order model, the results of the integer model, and the measured real data obtained from 10 patients during primary HIV infection [9]. This study proved that the results of the fractional order model give predictions to the plasma virus load of the patients better than those of the integer model.From the above biological and mathematical reasons, we propose a fractional order model to describe the dynamics of HIV infection that is given by(1)DαTt=λ-dT-βTV1+α1T+α2V+α3TV+ρI,DαIt=βTV1+α1T+α2V+α3TV-a+ρI,DαVt=kI-μV,where T(t), I(t), and V(t) represent the concentrations of uninfected CD4+ T-cells, infected cells, and free virus particles at time t, respectively. Uninfected cells are assumed to be produced at a constant rate λ, die at the rate dT, and become infected by a virus at the rate βTV/(1+α1T+α2V+α3TV), where α1,α2,α3≥0 are the saturation factors measuring the psychological or inhibitory effect. Infected cells die at the rate aI and return to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus at the rate ρI. Free virus particles are produced from infected cells at the rate kI and cleared at the rate μV.The fractional order derivative used in system (1) is in the sense of Caputo. We use this Caputo fractional derivative for two reasons: the first reason is that the fractional derivative of a constant is zero and the second reason is that the initial value problems depend on the integer order derivative only. In addition, we choose 0<α≤1 in order to have the same initial conditions as ODE systems.On the other hand, system (1) generalizes many special cases existing in the literature. For example, when α1=α2=α3=0, we get the model of Arafa et al. [10]. Further, we obtain the model of Liu et al. [11] when α3=0. It is very important to note that when α=1, system (1) becomes a model with an ordinary derivative which is the generalization of the ODE models presented in [12–15].The rest of the paper is organized as follows. In the next section, we give some preliminary results. In Section3, equilibria and their local stability are investigated. In Section 4, the global stability of the two equilibria is established. Numerical simulations of our theoretical results are presented in Section 5. Finally, the paper ends with conclusion in Section 6.


## 2. Preliminary Results

We first recall the definitions of the fractional order integral, Caputo fractional derivative, and Mittag-Leffler function that are given in [16].Definition 1.
The fractional integral of orderα>0 of a function f:R+→R is defined as follows: (2)Iαft=1Γα∫0tt-xα-1fxdx,where Γ(·) is the Gamma function.Definition 2.
The Caputo fractional derivative of orderα>0 of a continuous function f:R+→R is given by (3)Dαft=In-αDnft,where D=d/dt and n-1<α≤n, n∈N.
In particular, when0<α≤1, we have (4)Dαft=1Γ1-α∫0tf′xt-xαdx.Definition 3.
Letα>0. The function Eα, defined by (5)Eαz=∑j=0∞zjΓαj+1,is called the Mittag-Leffler function of parameter α.Letf:Rn→Rn with n≥1. Consider the fractional order system (6)Dαxt=fx,xt0=x0,with 0<α≤1, t0∈R, and x0∈Rn. For the global existence of solution of system (6), we need the following lemma.Lemma 4.
Assume thatf satisfies the following conditions: (i)
f(x) and (∂f/∂x)(x) are continuous for all x∈Rn.(ii)
f(x)≤ω+λx for all x∈Rn, where ω and λ are two positive constants.Then, system (6) has a unique solution on t0,+∞.The proof of this lemma follows immediately from [17]. For biological reasons, we assume that the initial conditions of system (1) satisfy (7)T0=ϕ10≥0,I0=ϕ20≥0,V0=ϕ30≥0.In order to establish the nonnegativity of solutions with initial conditions (7), we need also the following lemmas.Lemma 5 (see [18]).
Suppose thatg(t)∈C[a,b] and Dαg(t)∈C[a,b] for 0<α≤1; then, one has (8)gt=ga+1ΓαDαgξt-aα,a<ξ<t,∀t∈a,b.Lemma 6 (see [18]).
Suppose thatg(t)∈C[a,b] and Dαg(t)∈C[a,b] for 0<α≤1. If Dαg(t)≥0∀t∈[a,b], then g(t) is nondecreasing for each t∈[a,b]. If Dαg(t)≤0∀t∈[a,b], then g(t) is nonincreasing for each t∈[a,b].Theorem 7.
For any initial conditions satisfying (7), system (1) has a unique solution on 0,+∞. Moreover, this solution remains nonnegative and bounded for all t≥0. In addition, one has (i)
N(t)≤N(0)+λ/δ,(ii)
V(t)≤V(0)+k/μI∞,where N(t)=T(t)+I(t) and δ=min{a,d}.Proof.
It is easy to see that the vector function of system (1) satisfies the first condition of Lemma 4. It remains to prove the second condition. Let (9)Xt=TtItVt,ζ=λ00.To this end, we discuss four cases: (i)
Ifα1≠0, then system (1) can be written as follows: (10)DαXt=ζ+A1X+α1T1+α1T+α2V+α3TVA2X,where(11)A1=-dρ00-a+ρ00k-μ,A2=00-βα100βα1000.Moreover, we have(12)DαXt≤ζ+A1+A2X.(ii)
Ifα2≠0, we have (13)DαXt=ζ+A1X+α2T1+α1T+α2V+α3TVA3X,where(14)A3=00-βα200βα2000.Then,(15)DαXt≤ζ+A1+A3X.(iii)
Ifα3≠0, we have (16)DαXt=ζ+A1X+α3TV1+α1T+α2V+α3TVA4,where(17)A4=-βα3βα30.Then,(18)DαXt≤ζ+A4+A1X.(iv)
Ifα1=α2=α3=0, we have(19)DαXt=ζ+A1X+VA5X,where(20)A5=-β00β00000.Then,(21)DαXt≤ζ+VA5+A1X.Thus, the second condition of Lemma 4 is satisfied. Then, system (1) has a unique solution on 0,+∞. Next, we show that this solution is nonnegative. From (1), we have (22)DαTtT=0=λ+ρI≥0,DαItI=0=βTV1+α1T+α2V+α3TV≥0,DαVtV=0=kI≥0.According to Lemmas 5 and 6, we deduce that the solution of (1) is nonnegative.
Finally, we prove that the solution is bounded. By adding the first two equations of system (1), we get (23)DαNt≤λ-δNt.Hence,(24)Nt≤N0Eα-δtα+λδ1-Eα-δtα.Since 0≤Eα(-δtα)≤1, we have (25)Nt≤N0+λδ.The third equation of system (1) implies that (26)Vt=V0Eα-μtα+k∫0tIsαt-sα-1dEαds-μt-sαds.Then, (27)Vt≤V0Eα-μtα+kI∞μ1-Eα-μtα.Consequently,(28)Vt≤V0+kI∞μ.This completes the proof.


## 3. Equilibria and Their Local Stability

It is easy to see that system (1) always has a disease-free equilibrium E0(λ/d,0,0). Therefore, the basic reproduction number of our system (1) is given by (29)R0=kβλμa+ρd+λα1.Biologically, this basic reproduction number represents the average number of secondary infections produced by one infected cell during the period of infection when all cells are uninfected. Further, it is not hard to get the following result.Theorem 8.
(i) IfR0≤1, system (1) has a unique disease-free equilibrium of the form E0(T0,0,0), where T0=λ/d. (ii) If R0>1, the disease-free equilibrium is still present and system (1) has a unique chronic infection equilibrium of the form E1(T1,(λ-dT1)/a,k(λ-dT1)/aμ), where T1=2(a+ρ)(aμ+α2λk)/(akβ+(a+ρ)(α2dk-α1aμ-α3kλ)+δ-) with (30)δ-=akβ+a+ρα2dk-α1aμ-α3kλ2+4α3kda+ρ2aμ+α2λk.Next, we investigate the local stability of equilibria. LetEe(T,I,V) be an arbitrary equilibrium of system (1). Then, the characteristic equation at Ee is given by (31)-d-V∂f∂T-ξρ-V∂f∂V-fT,VV∂f∂T-a+ρ-ξV∂f∂V+fT,V0k-μ-ξ=0,where (32)fT,V=βT1+α1T+α2V+α3TV.We recall that the equilibrium Ee is locally asymptotically stable if all roots ξi of (31) satisfy the following condition [19]: (33)arg⁡ξi>απ2.Theorem 9.
(i) IfR0<1, then E0 is locally asymptotically stable. (ii) If R0>1, then E0 is unstable.Proof.
Evaluating (31) at E0, we have (34)d+ξξ2+a+ρ+μξ+μa+ρ1-R0=0.Obviously, the roots of (34) are (35)ξ1=-d,ξ2=-a+ρ+μ-a+ρ+μ2-4μa+ρ1-R02,ξ3=-a+ρ+μ+a+ρ+μ2-4ua+ρ1-R02.It is clear that ξ1 and ξ2 are negative. However, ξ3 is negative if R0<1 and it is positive if R0>1. Therefore, E0 is locally asymptotically stable if R0<1 and unstable if R0>1.Now, we focus on the local stability of the chronic infection equilibriumE1. It follows from (31) that the characteristic equation at E1 is given by (36)Pξ≔ξ3+a1ξ2+a2ξ+a3=0,where (37)a1=μ+d+a+ρ+βV11+α2V11+α1T1+α2V1+α3T1V12,a2=dμ+a+ρ+βV1a+μ1+α2V1+kT1α2+α3T11+α1T1+α2V1+α3T1V12,a3=βV1aμ1+α2V1+kdT1α2+α3T11+α1T1+α2V1+α3T1V12.It is obvious that a1>0, a2>0, and a3>0. Further, we have (38)a1a2-a3=adμ+a+ρ+βV1a1+α2V1+kT1α2+α3T11+α1T1+α2V1+α3T1V12+ddμ+a+ρ+a+μβV11+α2V11+α1T1+α2V1+α3T1V12+μ+ρ+βV11+α2V11+α1T1+α2V1+α3T1V12h2>0.So, Routh–Hurwitz conditions are satisfied. Let D(P) denote the discriminant of the polynomial P given by (36); then, (39)DP=18a1a2a3+a1a22-4a3a13-4a23-27a32.Using the results in [19], we easily obtain the following result.Theorem 10.
Assume thatR0>1. (i)
IfD(P)>0, then E1 is locally asymptotically stable for all α∈(0,1].(ii)
IfD(P)<0 and α<2/3, then E1 is locally asymptotically stable.


## 4. Global Stability

In this section, we study the global stability of the disease-free equilibriumE0 and the chronic infection equilibrium E1.Theorem 11.
IfR0≤1, then the disease-free equilibrium E0 is globally asymptotically stable.Proof.
Define Lyapunov functionalL0(t) as follows: (40)L0t=T01+α1T0ΦTT0+ρ21+α1T0a+dT0T-T0+I2+a+ρkV,where Φ(x)=x-1-ln(x), x>0. Calculating the derivative of L0(t) along solutions of system (1) and using the results in [20], we get (41)DαL0t≤11+α1T01-T0TDαT+DαIρ1+α1T0a+dT0T-T0+IDαT+DαI+a+ρkDαV.Using λ=dT0, we obtain (42)DαL0t≤-dT-T021+α1T0T-11+α1T01-T0TfT,VV+ρ1+α1T01-T0TI+fT,VV-dρT-T02a+dT01+α1T0-aρI2a+dT01+α1T0+ρT01+α1T0IT0-T-a+ρμkV,DαL0t≤-1T+ρa+dT0dT-T021+α1T0-ρIT-T02TT01+α1T0-aρI2a+dT01+α1T0+a+ρμkR0-1V-βT0α2+α3T1+α1T01+α1T+α2V+α3TVV2.Hence, if R0≤1, then DαL0(t)≤0. Furthermore, it is clear that the largest invariant set of {(T,I,V)∈D:DαL0(t)=0} is the singleton {E0}. Therefore, by LaSalle’s invariance principle [21], E0 is globally asymptotically stable.Theorem 12.
The chronic infection equilibriumE1 is globally asymptotically stable if R0>1 and (43)R0≤1+da+ρλα2k+aμ+ρλ2α3kρμa+ρd+λα1.Proof.
Define Lyapunov functionalL1(t) as follows: (44)L1t=1+α2V11+α1T1+α2V1+α3T1V1T1ΦTT1+I1ΦII1+a+ρkV1ΦVV1+ρ1+α2V12T1a+d1+α1T1+α2V1+α3T1V1T-T1+I-I12.Then, we have (45)DαL1t≤1+α2V11+α1T1+α2V1+α3T1V11-T1TDαT+1-I1IDαIρ1+α2V1T1a+d1+α1T1+α2V1+α3T1V1T-T1+I-I1DαT+DαI+a+ρk1-V1VDαV.
Usingλ=dT1+aI1, f(T1,V1)V1=(a+ρ)I1, μ/k=I1/V1, and 1-fT1,V1/fT,V1=((1+α2V1)/(1+α1T1+α2V1+α3T1V1))1-T1/T, we get(46)DαL1t≤d1-fT1,V1fT,V1T1-T+a+ρI14-fT1,V1fT,V1-I1IVV1fT,VfT1,V1-II1V1V-fT,V1fT,V+a+ρI1-1-VV1+fT,V1fT,V+VV1fT,VfT,V1-dρ1+α2V1T1a+d1+α1T1+α2V1+α3T1V1T-T12-aρ1+α2V1T1a+d1+α1T1+α2V1+α3T1V1I-I12-ρ1+α2V1TT11+α1T1+α2V1+α3T1V1T-T12I-I1.Thus, (47)DαL1t≤-1+α2V1T-T12TT11+α1T1+α2V1+α3T1V1dT1-ρI1+dρTd+a+ρI-a+ρI11+α1Tα2+α3TV-V12V11+α1T+α2V+α3TV1+α1T+α2V1+α3TV1-a+ρI1ΦfT1,V1fT,V1+ΦI1IVV1fT,VfT1,V1+ΦII1V1V+ΦfT,V1fT,V-aρ1+α2V1T1a+d1+α1T1+α2V1+α3T1V1I-I12.It is clear that Φ(x)≥0. Consequently, DαL1(t)≤0 if dT1≥ρI1. In addition, it is easy to see that this condition is equivalent to (43). Further, the largest invariant set of {(T,I,V)∈D:DαL1(t)=0} is the singleton {E1}. By LaSalle’s invariance principle, E1 is globally asymptotically stable.It is important to see that(48)limρ→0⁡da+ρλα2k+aμ+ρλ2α3kρμa+ρd+λα1=+∞.According to Theorem 12, we obtain the following result.Corollary 13.
The chronic infection equilibriumE1 is globally asymptotically stable when R0>1 and ρ is sufficiently small.


## 5. Numerical Simulations

In this section, we give some numerical simulations in order to illustrate our theoretical results. We discretize system (1) by using fractional Euler’s method presented in [22]. Firstly, we take the parameter values as shown in Table 1.Table 1
Parameter values of system (1).
Parameters
Values
λ
10
d
0.0139
β
0.00024
ρ
0.01
a
0.5
k
600
u
3
α
1
0.1
α
2
0.01
α
3
0.00001By calculation, we haveR0=0.9283<1. Then, system (1) has a disease-free equilibrium E0(719.4245,0,0). By Theorem 11, the solution of (1) converges to E0 (see Figure 1). Consequently, the virus is cleared and the infection dies out.Figure 1
Stability of the disease-free equilibriumE0.Now, we chooseβ=0.001 and we keep the other parameter values. In this case, R0=3.8678 and (49)1+da+ρλα2k+aμ+ρλ2α3kρμa+ρd+λα1=415.885.Hence, condition (43) is satisfied. Therefore, the chronic infection equilibrium E1(130.1613,16.3815,3276.3) is globally asymptotically stable. Figure 2 demonstrates this result.Figure 2
Stability of the chronic infection equilibriumE1.


## 6. Conclusion

In this paper, we have proposed a fractional order model of HIV infection with specific functional response and cure rate. This functional response covers the most functional responses used by several authors such as the saturated incidence rate, the Beddington-DeAngelis functional response, and the Crowley-Martin functional response. We have shown that the proposed model has a bounded and nonnegative solution as desired in any population dynamics. By using stability analysis of fractional order system, we have proved that if the basic reproduction numberR0≤1, the disease-free equilibrium E0 is globally asymptotically stable for all α∈(0,1], which means that the virus is cleared and the infection dies out. However, when R0>1, the disease-free equilibrium E0 becomes unstable and there exists another biological equilibrium, namely, chronic infection equilibrium E1, that is globally asymptotically stable provided that condition (43) is satisfied. In this case, the HIV virus persists in the host and the infection becomes chronic. Furthermore, we have remarked that if the cure rate ρ is equal to zero or is sufficiently small, condition (43) is satisfied and the global stability of E1 is only characterized by R0>1.According to the above theoretical analysis, we deduce that the global dynamics of the model are fully determined by the basic reproduction numberR0. In addition, we see that the fractional order parameter α has no effect on the global dynamics of our model, but it can affect the time for arriving at both steady states (see Figures 1 and 2). Moreover, the fractional order model and main results presented by Liu et al. in [11] are generalized and improved.



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