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

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.


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][2][3][4][5][6][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 function () at some time  =  1 requires all the previous history, that is, all () from  = 0 to  =  1 .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 where (), (), and () represent the concentrations of uninfected CD4 + T-cells, infected cells, and free virus particles at time , respectively.Uninfected cells are assumed to be produced at a constant rate , die at the rate , and become infected by a virus at the rate /(1 +  1  +  2  +  3 ), where  1 ,  2 ,  3 ≥ 0 are the saturation factors measuring the psychological or inhibitory effect.Infected cells die at the rate  and return to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus at the rate .Free virus particles are produced from infected cells at the rate  and cleared at the rate .
The fractional order derivative used in system (1) is in the sense of Caputo.We use this Caputo fractional derivative for 2 International Journal of Differential Equations 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 <  ≤ 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  1 =  2 =  3 = 0, we get the model of Arafa et al. [10].Further, we obtain the model of Liu et al. [11] when  3 = 0.It is very important to note that when  = 1, system (1) becomes a model with an ordinary derivative which is the generalization of the ODE models presented in [12][13][14][15].
The rest of the paper is organized as follows.In the next section, we give some preliminary results.In Section 3, 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.

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 order  > 0 of a function  : R + → R is defined as follows: where Γ(⋅) is the Gamma function.
Definition 2. The Caputo fractional derivative of order  > 0 of a continuous function  : R + → R is given by where  = / and  − 1 <  ≤ ,  ∈ N.
In particular, when 0 <  ≤ 1, we have Definition 3. Let  > 0. The function   , defined by is called the Mittag-Leffler function of parameter .
Let  : R  → R  with  ≥ 1.Consider the fractional order system with 0 <  ≤ 1,  0 ∈ R, and  0 ∈ R  .For the global existence of solution of system (6), we need the following lemma.
The proof of this lemma follows immediately from [17].For biological reasons, we assume that the initial conditions of system (1) satisfy In order to establish the nonnegativity of solutions with initial conditions (7), we need also the following lemmas.
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 To this end, we discuss four cases: (i) If  1 ̸ = 0, then system (1) can be written as follows: where Moreover, we have (ii) If  2 ̸ = 0, we have where (iii) If  3 ̸ = 0, we have where (iv) If  1 =  2 =  3 = 0, we have where 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 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 Hence, Since 0 ≤   (−  ) ≤ 1, we have The third equation of system (1) implies that Then, Consequently, This completes the proof.

Equilibria and Their Local Stability
It is easy to see that system (1) always has a disease-free equilibrium  0 (/, 0, 0).Therefore, the basic reproduction number of our system (1) is given by 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.
(ii) If  0 > 1, the disease-free equilibrium is still present and system (1) has a unique chronic infection equilibrium of the form  1 ( where We recall that the equilibrium   is locally asymptotically stable if all roots   of (31) satisfy the following condition [19]: Proof.Evaluating (31) at  0 , we have Obviously, the roots of (34) are It is clear that  1 and  2 are negative.However,  3 is negative if  0 < 1 and it is positive if  0 > 1.Therefore,  0 is locally asymptotically stable if  0 < 1 and unstable if  0 > 1.
Now, we focus on the local stability of the chronic infection equilibrium  1 .It follows from (31) that the characteristic equation at  1 is given by where It is obvious that  1 > 0,  2 > 0, and  3 > 0. Further, we have +  ( ( +  + ) So, Routh-Hurwitz conditions are satisfied.Let () denote the discriminant of the polynomial  given by (36); then, Using the results in [19], we easily obtain the following result.

Global Stability
In this section, we study the global stability of the disease-free equilibrium  0 and the chronic infection equilibrium  1 .
Corollary 13.The chronic infection equilibrium  1 is globally asymptotically stable when  0 > 1 and  is sufficiently small.

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.
International Journal of Differential Equations

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 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 reproduction number  0 ≤ 1, the diseasefree equilibrium  0 is globally asymptotically stable for all  ∈ (0, 1], which means that the virus is cleared and the infection dies out.However, when  0 > 1, the disease-free equilibrium  0 becomes unstable and there exists another biological equilibrium, namely, chronic infection equilibrium  1 , 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  is equal to zero or is sufficiently small, condition (43) is satisfied and the global stability of  1 is only characterized by  0 > 1.
According to the above theoretical analysis, we deduce that the global dynamics of the model are fully determined by the basic reproduction number  0 .In addition, we see

Figure 1 :
Figure 1: Stability of the disease-free equilibrium  0 .

Figure 2 :
Figure 2: Stability of the chronic infection equilibrium  1 .