Global Stability of an HIV-1 Infection Model with General Incidence Rate and Distributed Delays

Abstract

In this work an HIV-1 infection model with nonlinear incidence rate and distributed intracellular delays and with humoral immunity is investigated. The disease transmission function is assumed to be governed by general incidence rate . The intracellular delays describe the time between viral entry into a target cell and the production of new virus particles and the time between infection of a cell and the emission of viral particle. Lyapunov functionals are constructed and LaSalle invariant principle for delay differential equation is used to establish the global asymptotic stability of the infection-free equilibrium, infected equilibrium without cells response, and infected equilibrium with cells response. The results obtained show that the global dynamics of the system depend on both the properties of the general incidence function and the value of certain threshold parameters and which depends on the delays.

1. Introduction

Immunity can be broadly categorized into adaptive immunity and innate immunity. Adaptive immunity is mediated by clonally distributed and lymphocytes, namely, humoral and cellular immunity, and is characterized by specificity and memory. The humoral immunity plays an important role in antiviral defence by attacking virus. A basic mathematical model describing HIV-1 infection dynamics model with humoral immunity was introduced by Murase et al. [1] as where , , , and represent the densities of uninfected cells, infected cells, virus, and cells at time , respectively. and are the birth and death rate constants of uninfected cells. is the infection rate, is the average number of virus particles produced over the lifetime of a single infected cells, and is the death rate of infected cells; is the death rate constant of the virus, and are the recruited rate and death rate constants of cells, and is the cells neutralization rate. Mathematical models for virus dynamics with antibody immune response has drawn much attention of researchers (see, e.g., [1–13] and the reference therein). Recently many studies have been done to improve the model (1) by introducing delays and changing the incidence rate according to different practical background. These studies used different delayed models with different forms of incidence rate; see, for example, [6, 9–11] for discrete delays and [5, 13] for distributed delays.

In the present paper, motivated by the works of [1, 5, 13], we propose the following model with a general incidence rate and distributed delays and humoral immunity: where the parameters in system (2) have the same meanings as in system (1). is the general incidence rate. It is assumed in (2) that the uninfected cells that are contacted by the virus particles at time become infected cells at time , where is distributed according to over the interval , where is the limit superior of this delay. The constant    is assumed to be the death rate for infected cells during time period but not yet virus-producing cells and the term denotes the surviving rate of infected cells during the delay period. On the other hand, it is assumed in (2) that a cell infected at time starts to yield new infectious virus at time , where is distributed according to a probability distribution over the interval and is limit superior of this delay. The factor accounts for the probability of surviving infected cells during the time period of delay, where is constant. In (2), the probability distribution functions , , are assumed to satisfy , and The function is assumed to be continuously differentiable in the interior of and satisfies the following hypotheses:(H₁), for all ,(H₂), for all and ,(H₃), for all and .(H₄), for all .

The biological meaning of hypothesis (H₁) to (H₄) is given in [10].

Note the following.

The incidence rate given in (2) generalizes many common forms such as [5, 9, 13] (see Section 6).

The distributed delay is more general than the discrete one and it is more adapted to biological phenomena.

or can be infinity.

The present paper is organized as follows. In Section 2, we establish the nonnegativity and boundedness of solutions and we derived the basic reproduction ratios for viral infection and humoral immune response and , respectively. In Section 3, the existence of a possible three positive equilibria, an infection-free equilibrium , an infected equilibrium without cells response , and an infected equilibrium with cells response , is established. In Sections 4 and 5, we show that the global asymptotic stability of these equilibria depend only on the basic reproduction numbers under some hypotheses on the incidence function. In Section 6, some examples are given. A brief discussion is given in the last section to conclude this paper.

2. Preliminary Results

The initial conditions of (2) are given as where , here , with ; denotes the Banach space of continuous functions mapping the interval into .

Theorem 1. Under the initial conditions (4), all solutions of system (2) are nonnegative on and bounded.

Proof. Let us put system (2) in a vector form by setting and where and . It is easy to check that , . Due to [14, Lemma  2], any solution of (2) with , say , is such that for all . Next we show that the solutions are also bounded. It follows from the first equation of (2) that . This implies , so is bounded.
Let Then where and thus . This implies that is bounded and so is . Thus, there exists a such that . It follows from the third equations in (2) that and consequently is bounded. On the other hand, let Then, We have , where ; this implies that is bounded so also for . Finally, all the solutions of system (2) are bounded. This completes the proof.

To simplify the notations we note that Global behaviour of system (2) may depend on the basic reproduction numbers and given by where and with, . Here, and are the basic reproduction ratios for viral infection and humoral immune response of system (2), respectively. Based on the hypotheses (H₂) and (H₃) it is clear that .

3. The Existence of Positive Equilibria

In this section we prove the existence of positive equilibrium. The system (2) always has an infection-free equilibrium . For other possible equilibriums, we have the following theorem.

Theorem 2. Suppose that the conditions (H₁)–(H₃) are satisfied.(1)If , then system (2) has an infected equilibrium without cells response of the form with .(2)If , then system (2) has an infected equilibrium with cells response of the form with .

Proof. The steady states of system (2) satisfy the following equations: From the last equation of (14), we have Equations (15) has two possible solutions, or .
If , (14)₃ yields .
By substituting this into (14)₂, we obtain that which gives or .
If , we obtain the infection-free equilibrium .
If , (14)₁ and (14)₂ yields By substituting this into (14)₃, we obtain Since and , this implies that .
Now, from (H₁), (H₂), and (H₃), the following functional satisfies Hence, we obtain the infected equilibrium without cells response where is the unique zero in of and and are given by (17) and (18).
If , from (15), we obtain and from the first and second equation of (14), we have By substituting this into (14)₃, we obtain which implies that .
Now, from (14)₁ the functional satisfies Hence, we obtain the infected equilibrium with cells response , where is the unique zero of in and and are given by (23) and (24), respectively. This completes the proof.

Remark 3. From (19) we have if . So, as is increasing in the interval , we deduces that and consequently .

4. Global Stability of the Infection-Free Equilibrium

In this section, we study the global stability of the infection-free equilibrium of system (2).

Theorem 4. Suppose that the conditions (H₁)–(H₃) are satisfied. Then the infection-free equilibrium of system (2) is globally asymptotically stable if .

Proof. Define a Lyapunov functional: where and are given in (11).
It is obvious that is defined and continuously differentiable for all , and at . The time derivative of along the solutions of system (2) is given by with , , , , , , and .
At , using , we obtain From (H₂) and (H₃) we have, respectively, Then, ensures that , for all , holds only for , , and from (2)₂ we obtain . It follows that is the largest invariant set in . It follows from LaSalle invariance principle [15] that the infection-free equilibrium is globally asymptotically stable.

5. Global Stability of the Infected Equilibria

In this section, we study the global stability of the infected equilibrium without cells response and the infected equilibrium with cells response of system (2) by the Lyapunov direct method.

We set It is clear that for any , and has the global minimum , with .

Theorem 5. Suppose that the conditions (H₁)–(H₄) are satisfied. Then the equilibrium is globally asymptotically stable if .

Proof. Define a Lyapunov functional where where and are given in (11).
The function verifies
From (H₂), we have for , for , and , so . Consequently is nonnegative defined with respect to the endemic equilibrium , which is a global minimum.
We now prove that the time derivative of is nonpositive. Calculating the time derivative of along the positive solutions of (2), we obtain At , by using and and , we have Calculating the time derivative of , we obtain Combining (36) and (37) and by using , we obtain From (H₂), we have and from (H₃) and (H₄) we have and as is positive, we have From Remark 3 we have for all . It is easy to verify that from (38), the largest invariant set in is the singleton . Using LaSalle invariance principle [15], if , then the equilibrium is globally asymptotically stable. This completes the proof.

Theorem 6. Suppose that the conditions (H₁)–(H₄) are satisfied. Then the equilibrium is globally asymptotically stable if .

Proof. Define a Lyapunov functional where where and are given in (11).
The function verifies From (H₂), we have for , for , and , so . Consequently is nonnegative defined with respect to the endemic equilibrium , which is a global minimum.
We now prove that the time derivative of is nonpositive. Calculating the time derivative of along the positive solutions of (2), we obtain At , by using , , and , we have Calculating the time derivative of , we obtain Combining (47) and (48) and by using , we obtain From (H₂), we have and from (H₃) and (H₄) we have and as is positive, we have Thus, the equilibrium is stable. In this case, note that if and only if , , and and using the third equation of (2), we obtain . Therefore, it follows from LaSalle’s invariance principal [15] that the infected equilibrium with cells response is globally asymptotically stable. This completes the proof.

6. Application

In this section, we give some particular examples. In (2), if we obtain the following model: The global dynamics of model (53) is studied by Elaiw et al. [5]. So the work presented in [5] is a particular case of (2) because the function satisfies the hypothesises (H₁)–(H₄).

Another particular case of (2), if and , we obtain the following model which is presented by Yang et al. [13]: The global asymptotic stability of possible equilibrium of (54) is established in [13].

A last example, in (2), if and , where is the Dirac delta function, we obtain the results presented in [6].

7. Conclusion

In the current paper, we have studied an HIV-1 infection model with humoral immune response and intracellular distributed delays and general incidence rate. The model has two distributed time delays describing time needed for infection of cell and virus replication. The global stability of our model is studied by employing the method of Lyapunov functionals which are motivated by McCluskey [16] for delayed epidemic models. This general incidence represents a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models. We establish that the global dynamics are determined by two threshold parameters, the basic reproduction ratios for viral infection and humoral immune response and , respectively, which depend on the incidence function and the delay. We have proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratios viral infection . In this case, the virus is cleared up. The hypotheses on the general incidence function are used to assure the existence of infected equilibrium without cells response and infected equilibrium with cells response . We prove that if , the infected equilibrium without cells response is globally asymptotically stable and if , the infected equilibrium with cells response is globally asymptotically stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous referees for very helpful suggestions and comments.