Abstract
In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate and immune-activated reproduction rate are deduced. When , the system has the unique positive equilibrium . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.
1. Introduction
AIDS is a very dangerous infectious disease caused by HIV-1 virus which attacks the human immune system. At present, it has been proved that there are two different mechanisms for the spread of HIV-1 virus in the host: virus-cell transmission and cell-cell transmission [1]. A great number of studies have considered the above two contagion mechanisms [1–5]. Wang and Zou [4] considered an HIV-I model with humoral immunity, which is a typical virus-to-cell transmissions. In 2017, Lin et al. [5] proposed a cell-to-cell transmission model and described global threshold dynamics of the model.
In recent years, delay differential equation has attracted extensive attention worldwide. It has important applications in many fields such as physics, information, economy, and biomathematics. Depending on the different circumstances, many differential equation models with single delay or multiple delays have been proposed and studied deeply [4, 6–10]. Dong et al. [6] studied the dynamics of the tumor immune system interaction model and investigated the existence of Hopf bifurcation with two time delays as bifurcation parameters. In [7], the authors discussed the influence of awareness coverage and time delays on infectious diseases and found that the endemic equilibrium existed a Hopf bifurcation in both delayed and nondelayed system. A two-delay model with Holling II functional response and stage structure is considered in [8]. The authors have investigated a predator-prey model with a class of Beddington–DeAngelis functional response and two delays in [9]. Recently, in [10], a two-delay HIV-1 virus model with virus-to-cell and cell-to-cell transmission is considered as follows:where , and represent the susceptible cells, infected cells, virus, and B cells, respectively. is the number of carrying target cells. and denote infection rates of virus-to-cell transmission and cell-to-cell transmission. and are the growth and death rate of susceptible cells. and stand for the mortality rates of infected cells and virus, respectively. is the number of free virus particles produced by per infected cell. is the surviving probability the time period from to . describes the virus killed by B cells, and represents the new B cells produced when stimulated by antigen. The time delay stands for the time between virus entering into a cell and producing new virus or the time between infected cells spreading virus into uninfected cells and producing new virus. And is the time that the HIV-1 virus stimulates the production of B cells.
In [10], the coefficients of equations are considered to be constants independent of the time delay. The authors investigated the stability of positive equilibrium and the existence of Hopf bifurcation. The explicit formula for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions was derived. However, in practical situations, virus-cell transmission or cell-cell transmission is not instantaneous but takes some time defined by the average evaluation period . Moreover, the growth of virus at is decided by the number of infected cells at and still alive at . On this occasion, it is inappropriate to regard survival rate as a constant. Let be death rate of infected cells, then the survival rate is from to . Adding time delays to parameters is helpful to further explore the complex dynamic behavior of delay differential equations and their generation mechanism. Based on Sun and Li’work, considering that the two contagion processes are not instantaneous and the probability of cell survival is proportional to the number of cells that can survive to time t at , we propose the following delay model:where is the probability of surviving the time period from to in case is the death rate of infected but not yet virus-producing cells.
The initial conditions for system (2) are given as
A number of scholars investigate the delay differential equations with delay-dependent parameters [11, 12]. Song et al. [11] studied a delayed viral infection model with lytic immune response. The characteristic equations of the system are like . They discussed the influence of delay on the stability of the equilibrium and the local and global asymptotic stability of the disease-free equilibrium. Li and Ma introduced a method for determining the stability of the characteristic equations with delay-dependent parameters in [13]. Reference [14] presented a practical geometric method to study the stability switching properties of the characteristic equation which may result from a stability analysis of a model with two time delays and delay-dependent parameters that depend only on one of the time delay. Jiang and Guo [15] studied a model with double delays and a delay-dependent parameter considering the interaction between nutrients and plankton. The authors took the time delay as the parameter to carry out the dynamic analysis of the system, including the equilibrium stability and Hopf bifurcation existence by using the method in [14].
This paper’s goal is to delve into the local stability and Hopf bifurcation analysis of a two-delay HIV-1 virus model with delay-dependent parameters. The luminescent spots are as follows: (1) in this paper, two different spreads of HIV-1 virus are studied, and we will introduce time delay into the coefficients to be more realistic; (2) the conditions of the local stability of positive equilibrium and the existence of Hopf bifurcation are discussed with the time delay as the bifurcation parameters; and (3) the characteristic equation is when , and the stability is determined by the geometric stability switch criterion [13, 14]; and (4) the influences of different delay values on system stability are investigated.
The rest of this paper is organized as follows. In Section 2, the existence of equilibrium is given. By analyzing the characteristic equation, we discuss the stability of the positive equilibrium and existence of Hopf bifurcation in four different cases. In Section 3, we consider as a parameter and determine the direction and stability of Hopf bifurcation by using the center manifold theorem and the normal form theory. In Section 4, we perform numerical simulations to illustrate our results. Finally, the conclusions of the paper are given in Section 5.
2. Stability of the Positive Equilibrium and Existence of Hopf Bifurcation
Similar to [10], we can derive immune-inactivated reproduction rate and immune-activated reproduction rate , and then the equilibria of system (2) are as follows:(i), then system (2) only has an infection-free equilibrium .(ii), then system (2) has a no-immune response equilibrium except for , where(iii), then system (2) has an immunity-activated equilibrium except for and , where and is a positive, real root of the following quadratic equation:
The characteristic equation of system (2) at iswherethat is,where
Then, the local stability of the positive equilibrium and the existence of local Hopf bifurcation are discussed in the following four cases.
Case 1. .
Equation (9) at the equilibrium reduces toLetFrom the Routh–Hurwitz criterion, we have the following theorem.
Theorem 1. If () and () hold, the positive equilibrium is locally asymptotically stable for .
Case 2. .
The characteristic equation of system (2) isDenotethen equation (13) is reduced towhere .
Denote and . According to the geometrical criterion established by Beretta and Kuang [16], we can easily verify the following conditions for :(a).(b).(c).(d). We can get easily has finite roots.(f)Each positive root of is continuous and differentiable.Let be a root of equation (15) and separate the real and imaginary parts, then we getThen, we can obtainFrom (16), it follows thatwhich is equivalent to for (15). Denote , then we have . Therefore, equation (13) has a pair of pure imaginary roots when if is the positive root of . Then, we denote and . Hence, is a pure imaginary root of equation (13) if and only if is a zero of . We introduce the following theorem [16].
Theorem 2. Assume that have some positive roots for some . Then, a pair of simple pure imaginary roots exists which crosses the imaginary axis from left to right if and crosses the imaginary axis from right to left if , where
Since , equation (19) is equivalent to
Next, we can easily get and is a monotonic increasing function for all . If has no positive root in , then also does not have positive root in . And if has a positive root for some , then there is at least one positive root which satisfies . We introduce the Hopf bifurcation theorem:
Theorem 3. A single parameter system of form where has continuous first and second derivatives for . Define as , where is the derivation of with respect to at , and define . There are two hypotheses:(i)The linear differential equation () has a pure imaginary characteristic root and none of the other roots are multiples of (ii)The hypotheses (i) and (ii) imply that there are nonconstant periodic solutions, and a Hopf bifurcation occurs.
From all above analysis, we have the following theorem.
Theorem 4. Assume that holds, then(i)If has no positive root in , is locally asymptotically stable for all (ii)If has at least one positive root in , there exists such that is locally asymptotically stable for all (iii)If (ii) holds and , a Hopf bifurcation occurs at for
Case 3. .
Denote , then equation (9) reduces toObviously, , is a polynomial with , and its coefficients depend on . Next, we discuss the existence of pure imaginary roots of (21) by using the method introduced in [17].
Let , we need to verify the following conditions:(a)(b)(c)If , then (d)(e) for each has at most a finite number of real zeros where(f)Each positive root of is continuous and differentiable in whenever it existsNext, we verify the above six conditions.(a)It is obviously satisfied.(b) according to , .(c).(d).(e). Substituting into , then we get this condition holds.(f)The implicit function theorem shows that this condition is true.Assuming that is a root of equation (13), then we haveSeparating the real and imaginary parts, we haveTherefore, needs to satisfy the following equations:whereEquation (25) is equivalent to the condition that is the root of .
Let , then we denote , where . . Hence, is the pure imaginary root of (21) and is a necessary and sufficient condition for being the root of . We need to introduce the following theorem [18].
Theorem 5. Assume that has some positive roots for some . Then, a pair of simple pure imaginary roots exists when which crosses the imaginary axis from left to right if and crosses the imaginary axis from right to left if , where
From the above discussion, the following theorem can be obtained.
Theorem 6. Assume that holds, then(i)If or , has no positive root and is locally asymptotically stable for all (ii)If , has at least one positive root and , is locally asymptotically stable when , and a Hopf bifurcation occurs at for
Case 4. .
In this case, we consider as a parameter and . Let be a root of (9), then we getFrom (29), we havewhereWe can also getwhereIf , then and . We obtain that (31) has at least one positive real root. Suppose that there are eight positive real roots of (31) which are . According to (29), we havewhere . DefineTaking the derivative of (9) with respect to , we can getwhereFrom (35), it follows thatwhereIf holds, we can obtain the existence of a Hopf bifurcation as follows.
Theorem 7. For , if holds and , then is locally asymptotically stable when . System (2) undergoes Hopf bifurcations at for .
3. Direction and Stability of the Hopf Bifurcation
In this section, we consider as a parameter and discuss the direction and stability of the Hopf bifurcation of (2) when by using the normal form and the center manifold theorem. Denote as . We can obtain the existence of a Hopf bifurcation at . Let . System (2) can be written as an FDE in as the following form:where , and are shown as follows:whereand .
By the Riesz representation theorem, there exists a function of bounded variation for , such that
Next, we choosewhere is the Dirac delta function. Definewhere . When , equation (39) becomes , where . Definewhere . and are adjoint operators. Let , we know that are eigenvalues of both and . Assume that is the eigenvector of corresponding to . Then, we havewhich yields
Similarly, it can be verified that is the eigenvector of corresponding to , where
According to equation (46), we obtain that
Thus, we havesuch that .
Then, we compute the coordinates that describe the center manifold at . Let be the solution of equation (39) at and define
On the center manifold , we have
where and are local coordinates for center manifold in the direction of and .
When , we havewhere
Then,
From (52), we have . Comparing the coefficients with (54), we getwhere
Next, we can calculate the following values:
Then, for system (2), when , the direction and the stability of periodic solution of Hopf bifurcation are determined by (59).
4. Numerical Simulation
In this section, we consider the following system:and the numerical simulation is carried out under the following four cases: (i). Let , , and , then we have , and is satisfied. Meanwhile, , , , , and holds. Then, is locally asymptotically stable from Theorem 1 (see Figure 1). Let , , and ; by simple calculation, we get , and is satisfied. And , , , and , we know that holds. Then, is locally asymptotically stable from Theorem 1 (see Figure 2).(ii). Choosing , , and , is locally asymptotically stable when (see Figure 3), and Hopf bifurcation occurs when (see Figure 4). It supports the results of Theorem 4.(iii). Considering , , and , we obtain is locally asymptotically stable when (see Figure 5), and Hopf bifurcation occurs when (see Figure 6). The results of Theorem 6 are verified.(iv) Considering , , and , by simple calculation, we get and we choose as a parameter and let . When (see Figure 7) and (see Figure 8), is locally asymptotically stable. And when , Hopf bifurcation occurs (see Figure 9).
5. Conclusion
On the basis of [10], considering that is more practical that stands for the probability of surviving the time period from to , this paper establishes and investigates an HIV-1 virus model with two delay and delay-dependent parameters to describe both virus-to-cell and cell-to-cell transmissions. We find out basic reproductive rate , and system (2) has the unique positive equilibrium when . Without time delays, is asymptotically stable (see Figures 1 and 2). In other words, system (6) does not have any excitable nature. When , there exists a threshold limit beyond which the stability exchange takes place and system shows periodic orbits around (Figures 3 and 4). It shows that the number of cells and virus can fluctuate in time. Similarly, we find out that as and vary, loses its stability and Hopf bifurcations occur when and (see Figures 5–9). Moreover, we obtain the direction and stability of the Hopf bifurcation when using the normal form theory and center manifold theorem. Finally, numerical simulations help us illustrate the main results of model (2). Clearly, we can show that delay value is responsible for changing the stable system to system with periodic cycles. In the discussion of the characteristic equations, when neither nor is zero, the characteristic equation of system (2) is . We only discuss about the case when by applying the geometric stability switch. When , it is very difficult to discuss the root of such characteristic equation. We leave the analysis of the more complicated bifurcations as the future work.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11761040).