Global Exponential Stability of a Delayed HIV Infection Model with a Nonlinear Incidence Rate

Under the assumption that there is a time delay between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, we investigate the exponential stability of the noninfected equilibrium for a delayed HIV infection model with a nonlinear incidence rate. Compared with the global asymptotic stability analysis based on basic reproduction number, exponential stability analysis reveals the change range of various cells in different time periods.


Introduction
As is well-known, acquired immune deficiency syndrome (AIDS) has received widespread attention since its discovery. Initial models used to gain an insight into HIV immunology relied on systems of ordinary differential equations (see, e.g., [1][2][3][4][5][6]). In general, an underlying assumption in such an ODE model for HIV infection is that infection of cells by virions is instantaneous. In fact, in the real world, there may be intracellular delays in the viral infection and replication, and immune response processes. Furthermore, delay-differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could affect the stability of the systems and may lead to some complex dynamic behaviors such as oscillation, chaos, and instability [7,8]. Hence, time delays have been incorporated into HIV infection models by some authors (see [9][10][11][12][13][14][15][16] and the references cited therein).
On the other hand, compared with bilinear incidence rate, nonlinear incidence rate usually has more complex properties. When the number of viruses and susceptible cells is large, the number of susceptible cells contacted by viruses per unit time is limited, so the nonlinear incidence rate should be adopted in this case [17]. Recently, under the assumption that there is a lag between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, Yuan et al. [18] proposed the following delay-differential system for HIV infection models with a nonlinear incidence rate: where x(t), y(t), v(t), and z(t) denote C D4 + cells that are susceptible to infection, productively infected cells, virus, and the effector population of CTLs (cytotoxic T lymphocytes), respectively, at time t; μ denotes the newly added susceptible cell, k represents its death rate constant, α is the infection rate constant, c signifies the infected cell death rate constant, and β represents the killing rate constant of productively infected cell by CTLs; p denotes the rate constant of virus production by infected cell, and d determines the clearance rate constant of virus; effectors are generated in the presence of infected cells at rate δy and die at rate constant q per cell; τ > 0 represents lag between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, and e − mτ is assumed to describe the surviving rate constant of each target cell to get infected. e functions f(x) and h(x) represent the force of infection by the infective at density x and the force of CTLs to kill infected cells at density x, respectively. Furthermore, the functions f(x) and h(x) are locally Lipschitz on [ 0, ∞ ) and satisfy the following: Due to their biological relevance, all parameters in model (1) are positive. In fact, the first equation of (1) has more general form x ′ (t) � n(x(t)) − g(x(t), v(t)), where n(x) is a general function that accounts for both production and turnover of healthy target cells [19,20]. Generally speaking, there are two main forms of this function. In addition to n(x(t)) � λ − dx(t) [21,22], another typical function appearing in the literature is where λ, d, r, K are positive real numbers [23,24]. In particular, if we allow r � 0, the second form of n(x(t)) will become the first one. In order to include the aforementioned two forms, we consider the following delay-differential system: Here, 0 ≤ ρ < k, ρ is the maximum proliferation rate of uninfected cells, and M is the maximum level of uninfected cell concentration in the body. e other parameters are positive and have similar meanings to those in system (1).
For the sake of convenience, we denote by R n the set of all n-dimensional real vectors.
However, eorem 1 does not give us any information about the convergence rate, which is vitally important to the disease prevention and control in real-world applications of theoretical results on epidemic models. In fact, the known convergence rate means that the range of the population is predictable; that is to say, we can estimate the range of changes in the population within a given time range. In particular, since the exponential convergence rate reveals the variation range of population in different time periods, there have been extensive results on the problem of the exponential stability of epidemic models in the literature studies [25][26][27][28]. Now, a question naturally arises: under what conditions is the noninfected equilibrium (x * , y * , v * , z * ) � (x * (ρ), 0, 0, 0) of system (2) with initial conditions (3) are exponentially stable? Here, It is easy to see that lim ρ⟶0 + x * (ρ) � x * (0) � (μ/k). For 0 ≤ ρ < k, we denote x * (ρ) by x * for simplicity of notation.
Motivated by the above discussions, the main purpose of this paper is to establish sufficient conditions for the exponential stability of the noninfected equilibrium (x * , y * , v * , z * ) � (x * , 0, 0, 0) of system (2). To the best of our knowledge, it is the first time to focus on the problem of the exponential stability for (2). In particular, a numerical example is provided to illustrate our theoretical results.
Similar to the reference [29], we give the definition of the exponential stability as follows.
) be the solution of (2) with initial value conditions (3). If there exists a positive constant λ > 0 such that as t ⟶ + ∞, then the noninfected equilibrium is said to be globally exponentially stable.
In the rest of the paper, we give our main result in Section 2. e theoretical result is illustrated with examples as well as numerical simulations in Section 3. Finally, conclusions are made in Section 4.

Main Results
By the fundamental theory of functional differential equations [30], we have that there is a unique solution (x(t), y(t), v(t), z(t)) satisfying system (2) with initial conditions (3). Furthermore, by a similar argument as that in Zhu et al. [31], it is not hard to show that every solution of (2) with initial value conditions (3) is nonnegative and bounded on [ 0, ∞ ). From (2), we have which implies for all t ∈ (0, ∞).
In addition, from (A1), we can get e following theorem is our main result.

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Consequently, In the following, we will show If not, one of the following four cases must occur.
If Case III holds, together with (14) and (16), (23) yields 6 Discrete Dynamics in Nature and Society which contradicts the first equation in (23). us, (23) could not hold.
If Case IV holds, together with (14) and (16), (24) yields which contradicts with the first equation in (24). us, (24) could not hold. erefore, we obtain from the above discussions that holds. Letting ε ⟶ 0 + , we have that which proves (11), where K � K φ ‖X‖ ξ e λt ξ . is completes the proof of eorem 2. According to the above discussion, it is easy to see that our result is also true when ρ � 0. Furthermore, compared with eorem 1, we find that the significantly stronger conclusion of eorem 2 is obtained with only slightly stricter conditions. □

A Numerical Example
In this section, we will show the existence and global exponential stability of the noninfected equilibrium of system (2) by a numerical example.

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Since the results in [3,4,19,32,33] give no opinions about the global exponential stability of the HIV infection models, it is clear that all the results in the above references cannot be applicable to prove the global exponential stability of system (31).

Conclusion
We have proved the global exponential stability of the noninfected equilibrium for a delayed HIV infection model with a nonlinear incidence rate. It is worth pointing out that the required conditions are simple and easy to verify. It is natural to ask whether our methods in this paper are available to study the global exponential stability of the infected equilibrium of the delayed HIV infection models. It is an issue worth our further study. In addition, as pointed out in the literature [34,35], the further extension of the model is to consider the case with reaction-diffusion term, which is also the focus of our further research.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

State variables x(t), y(t), v(t), z(t)
x(t)  Discrete Dynamics in Nature and Society