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.
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 function at some time requires all the previous history, that is, all from to . 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 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 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 , where 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 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 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 , we get the model of Arafa et al. [10]. Further, we obtain the model of Liu et al. [11] when . It is very important to note that when , 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 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.
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 order of a function is defined as follows: where is the Gamma function.
Definition 2. The Caputo fractional derivative of order of a continuous function is given by where and , .
In particular, when , we have
Definition 3. Let . The function , defined by is called the Mittag-Leffler function of parameter .
Let with . Consider the fractional order system with , , and . For the global existence of solution of system (6), we need the following lemma.
Lemma 4. Assume that satisfies the following conditions: (i) and are continuous for all .(ii) for all , where and are two positive constants.Then, system (6) has a unique solution on .
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.
Lemma 5 (see [18]). Suppose that and for ; then, one has
Lemma 6 (see [18]). Suppose that and for . If , then is nondecreasing for each . If , then is nonincreasing for each
Theorem 7. For any initial conditions satisfying (7), system (1) has a unique solution on . Moreover, this solution remains nonnegative and bounded for all . In addition, one has (i),(ii),where and .
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 , then system (1) can be written as follows: where Moreover, we have (ii) If , we have where Then, (iii) If , we have where Then, (iv) If , we have where Then, Thus, the second condition of Lemma 4 is satisfied. Then, system (1) has a unique solution on . 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 , we have The third equation of system (1) implies that Then, Consequently,This completes the proof.
3. Equilibria and Their Local Stability
It is easy to see that system (1) always has a disease-free equilibrium . 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.
Theorem 8. (i) If , system (1) has a unique disease-free equilibrium of the form , where . (ii) If , the disease-free equilibrium is still present and system (1) has a unique chronic infection equilibrium of the form , where with
Next, we investigate the local stability of equilibria. Let be an arbitrary equilibrium of system (1). Then, the characteristic equation at is given by where We recall that the equilibrium is locally asymptotically stable if all roots of (31) satisfy the following condition [19]:
Theorem 9. (i) If , then is locally asymptotically stable. (ii) If , then is unstable.
Proof. Evaluating (31) at , we have Obviously, the roots of (34) are It is clear that and are negative. However, is negative if and it is positive if . Therefore, is locally asymptotically stable if and unstable if .
Now, we focus on the local stability of the chronic infection equilibrium . It follows from (31) that the characteristic equation at is given by where It is obvious that , , and . 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.
Theorem 10. Assume that . (i)If , then is locally asymptotically stable for all .(ii)If and , then is locally asymptotically stable.
4. Global Stability
In this section, we study the global stability of the disease-free equilibrium and the chronic infection equilibrium .
Theorem 11. If , then the disease-free equilibrium is globally asymptotically stable.
Proof. Define Lyapunov functional as follows: where , . Calculating the derivative of along solutions of system (1) and using the results in [20], we get Using , we obtain Hence, if , then . Furthermore, it is clear that the largest invariant set of is the singleton . Therefore, by LaSalle’s invariance principle [21], is globally asymptotically stable.
Theorem 12. The chronic infection equilibrium is globally asymptotically stable if and
Proof. Define Lyapunov functional as follows: Then, we have Using , , , and , we getThus, It is clear that . Consequently, if . In addition, it is easy to see that this condition is equivalent to (43). Further, the largest invariant set of is the singleton . By LaSalle’s invariance principle, is globally asymptotically stable.
It is important to see that According to Theorem 12, we obtain the following result.
Corollary 13. The chronic infection equilibrium is globally asymptotically stable when and 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.
By calculation, we have . Then, system (1) has a disease-free equilibrium . By Theorem 11, the solution of (1) converges to (see Figure 1). Consequently, the virus is cleared and the infection dies out.
Now, we choose and we keep the other parameter values. In this case, and Hence, condition (43) is satisfied. Therefore, the chronic infection equilibrium is globally asymptotically stable. Figure 2 demonstrates this result.
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 number , the disease-free equilibrium is globally asymptotically stable for all , which means that the virus is cleared and the infection dies out. However, when , the disease-free equilibrium becomes unstable and there exists another biological equilibrium, namely, chronic infection equilibrium , 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 is only characterized by .
According to the above theoretical analysis, we deduce that the global dynamics of the model are fully determined by the basic reproduction number . In addition, we see that the fractional order parameter 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.