Abstract
This paper investigates the dynamic behavior of a viral infection model with general contact rate between susceptible host cells and free virus particles. If the basic reproduction number of the virus is less than unity, by LaSalle’s invariance principle, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number of the virus is greater than unity, then the virus persists in the host and the endemic equilibrium is locally asymptotically stable.
1. Introduction
Viral infection within-host, such as hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV) infections, is a complicated kinetic process, and mathematical model is always important, which can give a hand to understand the complexity between the responses of the body and variant conditions [1–6].
The basic viral infection model contains three variables, susceptible host cells (), infected host cells (), and free virus particles (), which can be formulated by the following differential equations [7, 8]: in which susceptible host cells are produced at a constant rate, , die at the rate of , and become infected with the rate of . Infected host cells are produced at the rate of and die at the rate of . Free virus particles are released from infected host cells at the rate of and die at the rate of . It is assumed that parameters , , , , , and are all positive constants.
Note that there is an assumption that the infection term is based on the mass-action principle, which means that there is a constant contact rate () between susceptible host cells and virus particles in (1). However, many experiments of microparasitic infections suggest the infection rate may be a nonlinear relationship [3, 9–11], such as dose-dependent infection rate. Thus, to meet more biological practice, we replace the constant contact rate () with a general contact rate () between susceptible cells and virus particles and obtain the following modified viral infection model: where the contact rate function satisfy the following assumption (H1): (H1), continuous and differentiable, , and .
The primary goal of this paper is to carry out a mathematical analysis of system (2) and predict whether the infection disappears or survives. The organization of this paper is as follows. In the next section, some preliminary results are given, including the dissipativity of system (2), the definition of basic reproduction number of the virus, and the existence of the disease-free equilibrium and endemic equilibrium. In Section 3, by analyzing the corresponding characteristic equations, we study the local stability of the equilibria. In Section 4, by using suitable Lyapunov function and LaSalle’s invariance principle [12], we first prove that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Then using Theorem 4.6 in [13], we obtain the uniform persistence of (2) if the basic reproduction number is greater than unity. A brief discussion is given in Section 5 to conclude this work.
2. Preliminary Results
In this section, we first show that all solutions of system (2) are positive and ultimately bounded. Then the existence of feasible equilibria is given under the condition of basic reproduction number of the virus.
Because of the biological meaning of the components , we focus on the model in the first octant of and consider system (2) with initial conditions
The following result shows that system (2) is dissipative.
Theorem 1. Under the initial conditions (3), all solutions of system (2) are positive for and there exists a constant , such that all solutions satisfy , , and for all sufficiently large .
Proof. Note that , and . This implies that for all , provided that . Suppose that is not always positive. Let be the first time such that . By the first equation of (2) we have , which implies for for sufficiently small , a contradiction. Thus, is positive for all . In addition, by the second and third equations of (2), we have
for all . Therefore, it is easy to see that and are positive with initial conditions (3).
Next, we sketch the arguments for ultimate boundedness of solution of (2). Let , , , and . Since all solutions of (2) are positive, we have
Therefore, and for all sufficiently large , and hence, , , and are ultimately bounded by some positive constant .
Note that a free virus particle has an average lifetime of and parameter is the burst size, which means the total number of virions produced by an infected cell during its life span. Thus, at the beginning of the infectious process, the average number of newly virus particles generated from one virus particle, which is the basic reproduction number of virus by [14, 15], can be defined as
Now, we begin to find the equilibria of model (2) by the following algebraic system
Solving the third algebraic equation of (7), we can obtain . By combining this equality with the second equation of (7), we have or . When , it is easy to have and by the third and first equations of (7); that is, system (2) always has a disease-free equilibrium state, denoted as . If , substituting in the first equation of (7), we have Note that Thus, if , that is, , there is a unique positive root for (8).
We summarize the above analyses in the following result.
Proposition 2. For system (2), the disease-free equilibrium always exists. Furthermore, the unique endemic equilibrium exists only if ; here , , and is the unique positive root of (8).
3. Local Stability
In this section, we study the local stability of each of feasible equilibria of system (2) by analyzing the corresponding characteristic equations, respectively.
The Jacobian matrix of (2) at is At disease-free equilibrium , Clearly, the determinant of the lower right-hand matrix is positive and its trace is negative only if , so its eigenvalues have negative real parts in this case. Thus, is locally asymptotically stable if and only if .
When , the endemic equilibrium exists, and the Jacobian matrix at is The characteristic equation of (12) is given by in which Here, we used and the assumption (H1); that is, .
Because and are both positive, by Routh-Hurwitz criterion, is locally asymptotically stable if and only if . After a simple algebraic calculation, we have that is positive because . Thus, is locally asymptotically stable if and only if .
We summarize the above results and Proposition 2 in the following theorem.
Theorem 3. If , then only disease-free equilibrium exists and is locally asymptotically stable. When , is unstable and the endemic equilibrium appears and is locally asymptotically stable.
4. Global Stability and Disease Persistence
For the global stability of the equilibria, we first have the following.
Theorem 4. The disease-free equilibrium is globally asymptotically stable if only exists; that is, .
Proof. Define a Lyapunov function
Along the trajectories of system (2), we have
Based on Theorem 1, we know that all solutions of system (2) are positive for . Taking , we have , ; that is, is a monotone increasing function. Thus, is always valid if . Consequently, all terms of the right hand side of (17) are nonpositive when , which implies that and if and only if and . As a result, the maximal invariant set in is the singleton . According to the results in Theorem 3 and LaSalle’s invariance principle [12], we have that is globally asymptotically stable if .
Next, we investigate the uniform persistence of (2) and have the following result.
Theorem 5. If , then system (2) is uniformly persistent; that is, there exists (independent of initial conditions), such that , , and for all solutions of (2) with initial conditions (3).
Proof. The result follows from an application of Theorem 4.6 in [13], with and . Since the proof is similar to that of Lemma 3.5 in [16], here we only sketch the modifications that is a weak repeller for .
Since , that is, , together with the continuity of the function , there exists a sufficiently small constant such that is valid. Suppose that there exists a solution such that . Thus, when is sufficiently large, we have
By the second equation of (2), we have
Take an auxiliary system of (2) as
Clearly, is the unique equilibrium of (20) and the Jacobian matrix of (20) is given by
After a simple calculation, we have that the determinant of matrix (21)
is valid for some sufficiently small constant if . Thus, is unstable in this case. This is a contradiction to that . As a result, is a weak repeller for .
5. Discussion
Considering the biological practice during viral or microparasitic infection [3, 9–11], we proposed a viral infection model with general contact rate between susceptible cells and virus particles, which is a generalization of the basic viral infection model [7, 8]. The biological meaning of the assumption (H1) is that the accumulation of free virus particles can affect the contact rate between susceptible cells and virus particles, and the contact function is gradually weaker along with the increasing of free virus particles.
Though the rigorous analysis of stability of equilibria is obtained in [17] for the basic model, it is usually very complicated [18] and we cannot obtain the global stability of the endemic equilibrium . However, we have the conditions of globally asymptotic stability of the disease-free equilibrium and persistence of virus. In addition, the phase diagram of system (2) indicates that all solutions tend to the unique disease steady state under different initial conditions (Figure 1). Thus, we conjecture that is globally asymptotically stable only if it exists even though the rigorous mathematical proof remains open.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are very grateful to the anonymous reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Fund of China (nos. 11271369 and 11201434).