Research Article | Open Access
Global Stability of a HIV-1 Model with General Nonlinear Incidence and Delays
A HIV-1 model with two distributed intracellular delays and general incidence function is studied. Conditions are given under which the system exhibits the threshold behavior: the disease-free equilibrium is globally asymptotically stable if ; if , then the unique endemic equilibrium is globally asymptotically stable. Finally, it is shown that the given conditions are satisfied by several common forms of the incidence functions.
The global stability is analyzed for a general mathematical model of HIV-1 pathogenesis proposed by Nelson and Perelson . The general model includes two distributed intracellular delays and a combination therapy with a reverse transcriptase inhibitor and a protease inhibitor. All incidence functions in those papers are the bilinear functions. However, there are some pieces of evidence showing that a bilinear infection rate might not be an effective assumption when the number of target cells is large enough (see [2–4] for the review of evidence and models). The aim of this paper is to establish global stability for a delay integrodifferential equation with a general incidence term , the conditions given here are similar to those given in  for the ODE case.
In this paper, we consider the following HIV-1 model with a side class of nonlinear incidence rates and distributed delays: where , , and are the concentrations of uninfected target cells (T cells), productively infected cells, and infectious virus, respectively.
The form of the incidence function is of fundamental importance. In this paper, we want to work with a function as general as possible but still possesses the properties necessary for conclusions to be made through mathematical analysis. Because of this, we will introduce conditions on with which it may appear technical. However, as shown in Section 5, many commonly used incidence functions satisfy these conditions. For now, we assume only the following. (A1) is a nonnegative differentiable function on the nonnegative quadrant. Furthermore, is positive if and only if both arguments are positive. The partial derivatives of on and are denoted by and , respectively. In Sections 3 and 4, it will be shown how the extra conditions on imply in the local and global stability of an endemic equilibrium.
For the purpose of convenience, we rewrite (1) as where , , , , , , , , , .
The paper is organized as follows. In Section 2, the basic reproduction number is determined and the equilibrium is found. The local stability of the equilibrium is studied in Section 3. The global dynamics are resolved in Section 4. In Section 5, examples are given of incidence functions which satisfy the assumptions that are used throughout the paper.
2. Equilibria and
Assume the kernel functions and satisfy For any given initial condition, system (2) satisfies the hypotheses that are sufficient to ensure the existence, uniqueness, and continuity of solutions [5, Theorems 2.1–2.3]; for the notation of , see [6, page 46]. Recall that and ; thus, . Then, the basic reproduction number defined in  can be rewritten as and for any value of the parameters, the disease-free equilibrium of (2) is given by It is straightforward to show that if , then (2) has only one nonnegative equilibrium, which is called the infection-free equilibrium . The presence and number of endemic equilibrium depend on the form of the nonlinearity , as well as the values of the parameters. In searching for equilibrium, we note that the equilibria of (2) are the same as the equilibria of the corresponding ordinary differential equation system. Here we give the following result.
Theorem 1. Let be the unique solution to system (2) with , , and . Then , and are positive for all . Moreover, the solution is bounded and thus exists globally.
Proof. Using the variation-of-constants formula, we obtain the positivity of and .
For , we claim that if , then for all . Otherwise, there exists such that . Let be the first one that satisfies ; then and which means that is increasing at and it is a contradiction.
Next we will show that every solution is also bounded. It follows from the first equation of (2) that . This implies that .
Let ; then Note that is ultimately bounded; then there exist positive constants and such that
This yields that is eventually bounded and so is . By a similar argument, we can show the boundedness of . Therefore, the system (2) is point dissipative (see ) and hence the solution of (2) exists globally.
Theorem 2. If , then there exists an endemic equilibrium .
Proof. We look for solution of the equations , and . We first note that implies and , so , . Let . Then, whenever . Thus, any zero of in the interval corresponds to an equilibrium with , that is, an endemic equilibrium.
Since , it follows that and . The function is continuous and so a sufficient condition for to have a zero in is that is increasing at . Thus, there is an endemic equilibrium if
Since for all , it follows that and so (10) is equivalent to .
3. Local Stability of the Equilibria
Theorem 3. If , then is locally asymptotically stable.
The proof is normal, so we omitted it. We now give conditions on that are used here and after to show the locally and globally asymptotically stability of the endemic equilibrium. As a precondition, we assume that to guarantee the existence of endemic equilibrium . Consider the following.(A2) for all .(A3) For all , is in the closed interval with endpoints at 1 and .(A4) Either or .
Theorem 4. If , and (A2), (A3), and (A4) hold, then any endemic equilibrium which exists is locally asymptotically stable.
Proof. The characteristic equation of the linearization of (2) at endemic equilibrium is We demonstrate that all zeros of it have negative real part. Since (A2) and (A3) hold, we get that and . Suppose that has nonnegative real part, we deduce and so the solutions with nonnegative real part if and only if all of the inequalities in (12) are in fact equal. The final inequality is strict unless (and ). The second last inequality is strict unless . Assumption (A4) implies at least one is strict. Thus, the endemic equilibrium is locally asymptotically stable.
4. Global Stability of the Equilibria
Theorem 5. If , then is globally stable; that is,
Proof . Let with
where , .
It is clear that and if and only if , . The derivative of along the solution of (2) is Noticing that , , and , and using integration by parts, we calculate the derivatives of and Thus, Recalling that is nonnegative and is positive if and only if both arguments are positive, we must have that is the largest invariant subset of ; then the global stability of follows from the classical Lyapunov-LaSalle invariance principle (see , Theorem ).
Theorem 6. If , and (A2) and (A3) hold, then is globally stable in .
Proof. Let .
With where , , and ; it is easy to see that has the strict global minimum for , for and any positive constant . By Theorem 1, all solutions are positive and bounded. Thus, is well defined and , in which the equality holds if and only if and for almost all . For clarity, the derivatives of , , and will be calculated separately and then combined to obtain .
First, , where Since . That is,
Next, we calculate . Consider
Using the fact that , , and , we get The third quality holds since , . We conclude that where Since the function is monotone on each side of and is minimized at , (A3) implies that . Therefor, noting that and (A2), we have . So the solutions must tend to , the largest invariant subset of .
To have equal zero, it is necessary to have for almost every , which holds if and only if ; and for almost every .
Since is invariant with respect to (2), we get So this determines to be a constant. Using , we obtain and this yields that . Thus, each element of satisfies , , and for all . This shows that This completes the proof.
We now give some examples of incidence functions for which the hypotheses are satisfied.
Example 1 (mass action). Let . The hypotheses (A1)–(A4) are satisfied and so global dynamics are determined by the magnitude of . The global behavior of this model was previously studied in [1, 8] and was fully resolved in .
Example 2 (saturating incidence). Let for some constant , . The hypotheses (A1)–(A4) are satisfied and so global dynamics are determined by the magnitude of . The global properties of this model were studied in .
Example 3 (Holling type II incidence). Let for some constant . The hypotheses (A1)–(A4) are satisfied and so global dynamics are determined by the magnitude of . The global properties of this model were studied in .
Example 4 (Beddington-DeAngelis incidence). Let for some constants and . The hypotheses (A1)–(A4) are satisfied and so global dynamics are determined by the magnitude of . The global properties were studied in [12, 13].
This work was supported by the Technology Development Foundation of Higher Education of Tianjin (20081003).
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Copyright © 2013 Yaping Wang and Fuqin Sun. 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.