International Journal of Differential Equations
Volume 2018 (2018), Article ID 1019242, 12 pages
https://doi.org/10.1155/2018/1019242

Research Article

A Fractional Order Model for Viral Infection with Cure of Infected Cells and Humoral Immunity

Adnane Boukhouima,¹ Khalid Hattaf,1,2 and Noura Yousfi¹

¹Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco
²Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco

Correspondence should be addressed to Khalid Hattaf; k.hattaf@yahoo.fr

Received 9 September 2018; Accepted 7 November 2018; Published 2 December 2018

Guest Editor: Nurcan B. Savaşaneril

Copyright © 2018 Adnane Boukhouima et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we study the dynamics of a viral infection model formulated by five fractional differential equations (FDEs) to describe the interactions between host cells, virus, and humoral immunity presented by antibodies. The infection transmission process is modeled by Hattaf-Yousfi functional response which covers several forms of incidence rate existing in the literature. We first show that the model is mathematically and biologically well-posed. By constructing suitable Lyapunov functionals, the global stability of equilibria is established and characterized by two threshold parameters. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

1. Introduction

The immune response plays an important role to control the dynamics of viral infections such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), and human T-cell leukemia virus (HTLV). Therefore, many mathematical models have been developed to incorporate the role of immune response in viral infections. Some of these models considered the cellular immune response mediated by cytotoxic T lymphocytes (CTL) cells that attack and kill the infected cells [1–5] and the others considered the humoral immune response based on the antibodies which are produced by the B-cells and are programmed to neutralize the viruses [6–11]. However, all these models have been formulated by using ordinary differential equations (ODEs) in which the memory effect is neglected while the immune response involves memory [12, 13].

Fractional derivative is a generalization of integer derivative and it is a suitable tool to model real phenomena with memory which exists in most biological systems [14–16]. The fractional derivative is a nonlocal operator in contrast to integer derivative. This means that if we want to compute the fractional derivative at some point , it is necessary to take into account the entire history from the starting point up to the point . For these reasons, modeling some real process by using fractional derivative has drawn attention of several authors in various fields [17–22]. In biology, it has been shown that the fractional derivative is useful to analyse the rheological proprieties of cells [23]. Furthermore, it has been deduced that the membranes of cells of biological organism have fractional order electrical conductance [24]. Recently, much works have been done on modeling the dynamics of viral infections with FDEs [25–31]. These works ignored the impact of the immune response and the majority of them deal only with the local stability.

In some viral infections, the humoral immune response is more effective than cellular immune response [32]. For this reason, we improve the above ODE and FDE models by proposing a new fractional order model that describes the interactions between susceptible host cells, viral particles, and the humoral immune response mediated by the antibodies; that is,

where , , , , and are the concentrations of susceptible host cells, latently infected cells (infected cells which are not yet able to produce virions), productive infected cells, free virus particles, and antibodies at time , respectively. Susceptible host cells are assumed to be produced at a constant rate , die at the rate , and become infected by virus at the rate . Latently 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 . Productive infected cells are produced from latently infected cells at the rate and die at the rate . Free virus particles are produced from productive infected cells at the rate , cleared at the rate , and are neutralized by antibodies at the rate . Antibodies are activated against virus at the rate and die at the rate .

In system (1), represents the Caputo fractional derivative of order defined for an arbitrary function by

with [33]. Further, the infection transmission process in (1) is modeled by Hattaf-Yousfi functional response [34] which was recently used in [35, 36] and has the form , where are the saturation factors measuring the psychological or inhibitory effect and is the infection rate. In addition, this functional response generalizes many common types existing in the literature such as the specific functional response proposed by Hattaf et al. in [37] and used in [2, 31] when ; the Crowley-Martin functional response introduced in [38] and used in [39] when and ; and the Beddington-DeAngelis functional response proposed in [40, 41] and used in [3, 4, 10] when and . Also, the Hattaf-Yousfi functional response is reduced to the saturated incidence rate used in [9] when and and the standard incidence function used in [27] when and , and it was simplified to the bilinear incidence rate used in [5, 6] when and .

On the other hand, system (1) becomes a model with ODEs when , which improves and generalizes the ODE model with bilinear incidence rate [42], the ODE model with saturated incidence rate [43], and the ODE model with specific functional response [44].

The rest of the paper is organized as follows. The next section deals with some basic proprieties of the solutions and the existence of equilibria. The global stability of equilibria is established in Section 3. To verify our theoretical results, we provide some numerical simulations in Section 4, and we conclude in Section 5.

2. Basic Properties and Equilibria

In this section, we will show that our model is well-posed and we discuss the existence of equilibria.

Since system (1) describes the evolution of cells, then we need to prove that the cell numbers should remain nonnegative and bounded. For biological considerations, we assume that the initial conditions of (1) satisfy

Then we have the following result.

Theorem 1. Assume that the initial conditions satisfy (3). Then there exists a unique solution of system (1) defined on . Moreover, this solution remains nonnegative and bounded for all .

Proof. First, system (1) can be written as follows:

where

It is important to note that when , (4) becomes a system with ODEs. In this case, we refer the reader to [45] for the existence of solutions and to the works [46–50] for the stability of equilibria. In the case of FDEs, we will use Lemma 2.4 in [31] to prove the existence and uniqueness of solutions. Hence, we put

We discuss four cases:

(i)If , can be formulated as follows:

 where

 Hence,

(ii)If , we can write in the form

 where

 Moreover, we get

(iii)If , we have

 where

 Further, we obtain

(iv)If , we have

 where

 Then

Hence, the conditions of Lemma 2.4 in [31] are verified. Then system (1) has a unique solution on . Now, we show the nonnegativity of solutions. By (1), we have

As in [31, Theorem 2.7], we deduce that the solution of (1) is nonnegative.
Finally, we prove the boundedness of solutions. We define the function

Then, we have

where . Thus, we obtain

Since , we get

This completes the proof.

Now, we discuss the existence of equilibria. It is clear that system (1) has always an infection-free equilibrium . Then the basic reproduction number of (1) is as follows:

To find the other equilibria, we solve the following system:

From (29), we get or . Then we discuss two cases.

If , by (25)-(28), we have , , , and

Since , , and , then . Consequently, there is no equilibrium when .

We define the function on by

We have , , and .

Hence if , (30) has a unique root . As a result, when there exists an equilibrium satisfying , , , and .

If , then . By (25)-(27), we obtain , , , and

Since , , and , we have . Hence, there is no equilibrium if .

We define the function on by

We have , , and .

Let us introduce the reproduction number for humoral immunity as follows:

which denotes the average life expectancy of antibodies and is the number of free viruses at . For the biological significance, represents the average number of the antibodies activated by virus.

If , we have and

Therefore, there is no equilibrium when .

If , then and

In this case, (32) has one root . Consequently, when , there exists an equilibrium satisfying , , , , and . When ,

We summarize the above discussions in the following theorem.

Theorem 2.  

(i)If , then system (1) has one infection-free equilibrium of the form , where .(ii)If , then system (1) has an infection equilibrium without humoral immunity of the form , where , , , and .(iii)If , then system (1) has an infection equilibrium with humoral immunity of the form , where , , , , and .

3. Global Stability of Equilibria

In this section, we focus on the global stability of equilibria.

Theorem 3. If , then the infection-free equilibrium is globally asymptotically stable and it becomes unstable if .

Proof. The proof of the first part of this theorem is based on the construction of a suitable Lyapunov functional that satisfies the conditions given in [51, Lemma 4.6]. Hence, we define a Lyapunov functional as follows:

where for . It is not hard to show that the functional is nonnegative. In fact, the function has a global minimum at . Consequently, for all .
Calculating the fractional derivative of along solutions of system (1) and using the results in [52], we get

Using , we obtain

Hence if , then . In addition, the equality holds if and only if , and . If , then . If , from (1), we get which implies that . Consequently, the largest invariant set of is the singleton . Therefore, by the LaSalle’s invariance principle [51], is globally asymptotically stable.
The proof of the instability of is based on the computation of the Jacobean matrix of system (1) and the results presented in [53–55]. The Jacobean matrix of (1) at any equilibrium is given by

We recall that is locally asymptotically stable if the all eigenvalues of (40) satisfy the following condition [53–55]:

From (40), the characteristic equation at is given as follows:

where

Obviously, (42) has the roots and . If , we have and . Then, there exists satisfying . In addition, we have . Consequently, when , is unstable.

Theorem 4.  

(i)The infection equilibrium without humoral immunity is globally asymptotically stable if , , and

(ii)When , is unstable.

Proof. Define a Lyapunov functional as follows:

Calculating the fractional derivative of , we get

Using , , , , and , we obtain

Hence,

Using the arithmetic-geometric inequality, we have

Since , we have if . It is easy to see that this condition is equivalent to (44). Furthermore, if and only if , and . We discuss two cases: If , then . If , from (1), we get , and then . Hence, the largest invariant set of is the singleton . By the LaSalle’s invariance principle, is globally asymptotically stable.
At , the characteristic equation of (40) is given as follows:

where

We can easily see that (50) has the root . Then, when , we have . In this case, is unstable.

Theorem 5. The infection equilibrium with humoral immunity is globally asymptotically stable if and

Proof. Consider the following Lyapunov functional:

Computing the fractional derivative of and using , , , , and , we get

From (49), we have when . This condition is equivalent to (52). In addition, if , and . Further, ; then . Consequently, the largest invariant set of is the singleton . By the LaSalle’s invariance principle, is globally asymptotically stable.

It is important to note that when is sufficiently small or is sufficiently large, the two conditions (44) and (52) are satisfied. Then, we have the following corollary.

Corollary 6. Assume that . When is sufficiently small or is sufficiently large, then we have the following:

(i)The infection equilibrium without humoral immunity is globally asymptotically stable if .(ii)The infection equilibrium with humoral immunity is globally asymptotically stable if .

4. Numerical Simulations

In this section, we validate our theoretical results to HIV infection. Firstly, we take the parameter values as shown in Table 1.

Table 1: Parameter values of system (1).

parameters
values
parameters
values
parameters
values

10

0.27

0.2

0.0139

0.01

0.0001

0.00024

800

1

0.01

3

0.1

0.0347

0.01

0.01

0.00001

By calculation, we have . Then system (1) has an infection-free equilibrium . By Theorem 3, the solution of (1) converges to (see Figure 1). Consequently, the virus is cleared and the infection dies out.

Figure 1: Stability of the infection-free equilibrium .

Now, we choose and we keep the other parameter values. Hence, we obtain , , and

Consequently, condition (44) is satisfied. Therefore, the infection equilibrium without humoral immunity is globally asymptotically stable. Figure 2 demonstrates this result. In this case, the infection becomes chronic.

Figure 2: Stability of the infection equilibrium without humoral immunity .

Next, we take and do not change the other parameter values. In this case, we have , , and . Hence, condition (52) is satisfied. Consequently, system (1) has an infection equilibrium with humoral immunity which is globally asymptotically stable. Figure 3 illustrates this result. We can observe that the activation of the humoral immune response increases the healthy cells and decreases the productive infected cells and viral load to a lower levels but it is not able to eradicate the infection.

Figure 3: Stability of the infection equilibrium with humoral immunity .

5. Conclusion

In the present paper, we have studied the dynamics of a viral infection model by taking into account the memory effect represented by the Caputo fractional derivative and the humoral immunity. We have proved that the solutions of the model are nonnegative and bounded which assure the well-posedness. We have shown that the proposed model has three infection equilibriums, namely, the infection-free equilibrium , the infection equilibrium without humoral immunity , and the infection equilibrium with humoral immunity . By constructing suitable Lyapunov functionals, the global stability of these equilibria is fully determined by two threshold parameters and . More precisely, when , is globally asymptotically stable, whereas if , it becomes unstable and another equilibrium point appears, that is, , which is globally asymptotically stable whenever and condition (44) is satisfied. In the case that , becomes unstable and there exists another equilibrium point which is globally asymptotically stable when condition (52) is satisfied. In addition, we remarked that when is sufficiently small or is sufficiently large, conditions (44) and (52) are verified, and then the global stability of and is characterized only by and .

From our theoretical and numerical results, we deduce that the order of the fractional derivative has no effect on the dynamics of the model. However, when the value of decreases (long memory), the solutions of our model converge rapidly to the steady states (see Figures 1–3). This behavior can be explained by the memory term included in the fractional derivative which represents the time needed for the interaction between cells and viral particles and the time needed for the activation of humoral immune response. In fact, the knowledge about the infection and the activation of the humoral immune response in an early stage can help us to control the infection.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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