Global Analysis of a Virus Dynamics Model with General Incidence Function and Cure Rate

Abstract

A virus dynamics model with logistic function, general incidence function, and cure rate is considered. By carrying out mathematical analysis, we show that the infection-free equilibrium is globally asymptotically stable if the basic reproduction number . If , then the infection equilibrium is globally asymptotically stable under some assumptions. Furthermore, we also obtain the conditions for which the model exists an orbitally asymptotically stable periodic solution. Examples are provided to support our analytical conclusions.

1. Introduction

Mathematical models have proven valuable in understanding the virus dynamics. The basic viral infection model was proposed by Nowak et al. [1, 2] in the following form: where , , and denote uninfected cells, infected cells, and free virus particles, respectively. The uninfected cells are produced at a constant and die at rate . The infected cells are produced from uninfected cells and free virus at rate and die at rate . Free virus is produced from infected cells at rate and declines at rate .

The incidence function in model (1) is based on the law of mass action. However, many researchers suggested that the bilinear incidence function is not sufficient to describe the infection process in detail, and some nonlinear incidence functions were proposed. For example, Li and Ma [3] considered a HIV-1 model with Holling type II function. Min et al. [4] considered a HBV model with standard incidence function. Elaiw [5] considered a virus dynamics model with a more general nonlinear incidence function, which satisfies some conditions.

When HBV infects a cell, infected cells may also revert to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus [6]. Then HBV infection has been modelled using the model including cytokine-mediated “cure” of infected cells [7–9]. Wang et al. [10] considered an improved HBV model with standard incidence function and cure rate. According to the virological basis found in [11], when a HIV enters a resting T-cell, the viral RNA may not be completely reverse transcribed into DNA. If the cell is activated shortly following infection, reverse transcription can proceed to completion. However, the unintegrated virus harbored in resting cells may decay with time and partial DNA transcripts are labile and degrade quickly [12]. Hence a proportion of resting infected cells can revert to the uninfected state [13]. Recently, Zhou et al. [14] considered a model of HIV infection of T-cells with bilinear incidence function and cure rate. Hattaf et al. [15] considered a virus dynamics model with general incidence function and cure rate. However, Tian and Liu [16] have pointed out that the proof for the main results in [15] is not corrected. They introduced a more general nonlinear incidence function including the form in [15].

The population dynamics of target cells is not well understood. Many models with logistic uninfected cell proliferation terms have been introduced. For example, Culshaw and Ruan [17] considered a delay-differential equation model of HIV infection of T-cells with logistic function term. Ji et al. [18] considered a viral infection model of HBV infection with logistic function. Li and Shu [19] considered an in-host model with a logistic mitosis term for the uninfected target cells and a finite intracellular delay in the incidence term.

In this paper, we aim to study the following model with logistic function, general incidence function, and cure rate: with initial condition . Here ,,,, and   defined as earlier. is the maximum proliferation rate of uninfected cells and is the maximum capacity of host's organ cells. is the incidence of new infections. is the rate of cure, that is, noncytolytic loss of infected cells. The function satisfies the following conditions: (i) for all and ;(ii), for all , , and ;(iii) and , for all , , and ;(iv), for all , , and .

Several models for viral dynamics fit model (2). For example, Song and Neumann [20] considered a viral model with and , where is the infection rate constant and is a positive constant. Ji et al. [18] considered a viral infection model of HBV infection with and . Zhou et al. [14] considered a model of HIV infection of T-cells with .

The paper is organized as follows. In Section 2, we carry out mathematical analysis of the model. In Section 3, the local stability of equilibria is proved. In Section 4, the global stability of the infection-free equilibrium or the infection equilibrium is established, respectively. The conditions for the existence of an orbitally asymptotically stable periodic solution are obtained. In Section 5, two examples are provided to illustrate our theorems. The conclusion is given in Section 6.

2. Mathematical Analysis

First, we show that solution of system (2) is bounded.

Theorem 1. The solution of system (2) is positive and bounded.

Proof. Let Computing the derivation of along the solution of system (2), we have which implies that where and . Obviously, and are bounded. From the third equation of system (2) and the boundedness of , it is easy to see that is also bounded. This completes the proof.

It can then be verified that the bounded set is positively invariant with respect to system (2) and is convex.

Following the computation of the basic reproduction number in [21], we have the basic reproduction number

For mathematical convenience, let and is the positive root of . It is clear that and system (2) has an infection-free equilibrium when .

Theorem 2. If and for arbitrary , then there exists a unique infection equilibrium .

Proof. In order to find the infection equilibrium, set which yields Substituting the expression of and by , we have It is obvious that and This implies that there exists such that . Suppose, to the contrary, there exists another infection equilibrium . Without loss of generality, we assume that . Since , we have . This yields and . Hence, we get . On the other hand, we have that . This is a contradiction. Therefore, is the unique infection equilibrium.

3. Local Stability of Equilibria

In this section, we discuss the local stability of the infection-free equilibrium and the infection equilibrium of system (2), respectively.

Theorem 3. If , then the infection-free equilibrium is locally asymptotically stable and becomes unstable when .

Proof. The Jacobian matrix of system (2) at is One eigenvalue of is . The remaining two eigenvalues are solutions of the quadratic equation By the Routh-Hurwitz theorem, is locally asymptotically stable when . When , has a positive eigenvalue and is unstable.

Theorem 4. If and for arbitrary , then the unique endemic equilibrium is locally asymptotically stable.

Proof. The Jacobian matrix of system (2) at is The characteristic equation of can be written as with Since , we have . Therefore, By direct calculation, we have . Then the Routh-Hurwitz theorem implies that the infection equilibrium is locally asymptotically stable.

4. Global Stability of Equilibria

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

Theorem 5. If , then the infection-free equilibrium is globally asymptotically stable.

Proof. Consider a Lyapunov function Calculating the time derivative of along solutions of system (2), we obtain If , from Corollary 5.2 of Kuang [22], is globally asymptotically stable. Also, for , implies that . It is easy to show that the largest invariant set where is the singleton . By the LaSalle invariance principle, is globally asymptotically stable.

Next, we prove that the infection equilibrium is globally asymptotically stable. We need the following theorem.

Theorem 6. If , then system (2) is uniformly persistent.

Proof. The result follows from an application of Theorem 4.6 in [23], with and . We only need to prove that is a weak repeller for .
Suppose that there exists a solution such that . When is sufficiently large, we have where is an arbitrarily small positive constant satisfying . Then,
Consider the following auxiliary system: System (23) always has a trivial equilibrium . Since and continuously differentiable of the function , we have for some sufficiently small. By calculation, system (23) has a unique positive equilibrium , where satisfies the root equation The Jacobian matrix of system (23) at is The eigenvalues and of satisfy We get and . Hence, is locally asymptotically stable when .
Denote the right-hand sides of system (23) by and , respectively. We have Therefore, is globally asymptotically stable by the Bendixson criterion for two-dimensional ordinary differential equations. By the comparison theorem, we have that as for system (23). This is a contradiction to . Hence, is a weak repeller for .

By looking at the Jacobian matrix of system (2) and choosing the matrix as system (2) is competitive in , with respect to the partial order defined by the orthant .

Theorem 7. Suppose and for arbitrary ; then the infection equilibrium of system (2) is globally asymptotically stable.

Proof. Let be a periodic solution whose orbit is contained in . The second compound equation is the following periodic linear system: where and is the second additive compound matrix of the Jacobian matrix of system (2).
The Jacobian matrix of system (2) is and its second additive compound matrix is where
For the solution , system (30) becomes Now, define the function which is a Lyapunov function for system (30). Then, we have From system (36), we have where and . Therefore, where From the second and third equations of system (2), we have Hence, Therefore, We have which implies that as . This means that as , so the linear system (34) is asymptotically stable and the periodic solution is asymptotically orbitally stable.
According to Theorem 4.1 in [24], system (2) satisfies the Poincare-Bendixson property. Using Theorem 1, Theorems 4 and 6, we have that all conditions of Theorem 2.2 in [25] are satisfied for system (2). This completes the proof.

If the condition in Theorem 4 could not be satisfied, there would exist an orbitally asymptotically stable periodic solution. We have the following theorem.

Theorem 8. Suppose and ; then system (2) has an orbitally asymptotically stable periodic solution.

Proof. The nonlinearities in system (2) are analytic in . We obtain that the conclusion follows from Theorem 1.2 in [26]. Take the domain for system (2) to be the interior of the positive orthant, in which the only steady state is . If and , then is unstable. The dissipativity hypothesis of Theorem 1.2 in [26] follows from Theorems 1 and 6. System (2) is competitive in and . Hence, all conditions of Theorem 1.2 in [26] are satisfied. This completes the proof.

5. Examples

In this section, we give two examples to show the application of our theorems.

Example 1. Consider the following system: which is a special case of system (2) by letting and . This model has been investigated by Ji et al. [18]. Applying Theorem 5, Theorems 7 and 8, we have the following result.

Theorem 9. (i)If , then the infection-free equilibrium of system (44) is globally asymptotically stable.(ii)If and , then the unique infection equilibrium of system (44) is globally asymptotically stable.(iii)If and , then system (44) exists an orbitally asymptotically stable periodic solution.

Example 2. Consider the following system: which is a special case of system (2) by letting . This model has been studied by Zhou et al. [14]. By Theorem 5, Theorem 7, and Theorem 8, we get the following result.

Theorem 10. (i)If , then the infection-free equilibrium of system (45) is globally asymptotically stable.(ii)If and for arbitrary , then the unique infection equilibrium of system (45) is globally asymptotically stable.(iii)If and , then system (45) exists an orbitally asymptotically stable periodic solution.

6. Conclusion

In this paper, we have considered a virus dynamics model with logistic function, general incidence function, and cure rate. The basic reproduction number is obtained and it determines the global dynamics of this model. If , then the infection-free equilibrium is globally asymptotically stable. If , then the virus persists in the host, and solutions approach either an infection equilibrium or a periodic orbit. Our model is a generalization of several models that appeared in the literature as its special cases.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Special Fund for the High-Level Project Cultivation in Zhejiang International Studies University (090500442012).