Complexity
Volume 2020 (2020), Article ID 2313102, 12 pages
https://doi.org/10.1155/2020/2313102

Research Article

Dynamical Analysis of a Delayed HIV Virus Dynamic Model with Cell-to-Cell Transmission and Apoptosis of Bystander Cells

Tongqian Zhang,1,2 Junling Wang,¹ Yi Song,¹ and Zhichao Jiang³

¹College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
²State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China
³School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang 065000, China

Correspondence should be addressed to Yi Song; songyi_sy@sina.com

Received 20 November 2019; Revised 25 April 2020; Accepted 20 July 2020; Published 17 August 2020

Academic Editor: Zhile Yang

Copyright © 2020 Tongqian Zhang et al. 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.

Abstract.

In this paper, a delayed viral dynamical model that considers two different transmission methods of the virus and apoptosis of bystander cells is proposed and investigated. The basic reproductive number of the model is derived. Based on the basic reproductive number, we prove that the disease-free equilibrium is globally asymptotically stable for by constructing suitable Lyapunov functional. For , by regarding the time delay as bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of endemic equilibrium and cause periodic oscillations. Finally, we give some numerical simulations to illustrate the theoretical findings.

1. Introduction and Model Formulation

AIDS is a serious immune-mediated disorder caused by human immunodeficiency virus (HIV) infection. Continuous high-level HIV replication leads to a chronic fatal infection. By destroying important cells in the human immune system, such as helper T cells (especially CDT cells), macrophages, and dendritic cells, HIV destroys or damages the functions of various immune cells in the immune system, resulting in a wide range of immune abnormalities [1]. With the development of infection, the human immune system begins to weaken, which leads to the extreme decline of the body’s resistance, leading to a variety of opportunistic infections, such as herpes zoster, tuberculosis, Pneumocystis carinii pneumonia, toxoplasmosis, microsporidia, and so on [2, 3].

Since its first discovery in the United States in 1981, AIDS has spread very rapidly. By 2018, AIDS has been found in almost every country in the world. According to the latest UNAIDS report, approximately 37.9 million people worldwide have been infected with HIV by 2018; approximately 23.3 million people are receiving antiretroviral treatment; nearly 770,000 people have died of AIDS-related diseases [4]. AIDS has become one of the most intractable infectious diseases in today’s society, posing a huge threat to the survival and further development of mankind. Especially with the globalization of human activities, the increasing frequency of people’s communication and the continuous evolution of HIV virus pose new challenges to the prevention and treatment of AIDS. Due to several unique characteristics of HIV replication and transmission, such as two transmission mechanisms of the virus and the lack of proofreading mechanisms, effective vaccines and specific medicines that can completely cure AIDS have not been developed until now [5]. Only prophylactic blockers have been developed that can be used before and after the onset of high-risk behavior, and this antiretroviral drug can only delay the progression of the disease. Therefore, a more in-depth study of the HIV virus is very important.

Mathematical modeling plays an important role in the study of biological systems. Mathematical models can provide a more typical, more refined, and more quantitative description of complex systems [6–18]. Due to the uncertainty of clinical measurement, the low quality of samples, and the high risk of repeated trials, the use of mathematical models to describe the infection process qualitatively and quantitatively, to study the mechanism of AIDS infection and the treatment of AIDS, is of great significance to the control of HIV infection [19–21].

In 1996, Nowak et al. [22] proposed a virus infection model of ordinary differential equation describing the relationship among infected cells, uninfected cells, and free viruses as follows:

where are the concentrations of uninfected CDT cells, infected CDT cells, and the free virus particles, respectively. is the rate at which new uninfected CDT cells are produced, and is the natural mortality of uninfected CDT cells. is the mortality of infected CDT cells infected with virus or immune system, is the probability of uninfected CDT cells infected with the free virus, is the rate of CDT cells producing free virus, and is the rate at which the body’s immune system clears the virus. The authors derived the basic reproductive number and proved that when , the infection will not spread and the system will return to the uninfected state, and when , the number of infected cells will increase, while the number of uninfected cells will decrease, and the system will converge to the disease-free equilibrium. In the model, the term is the rate that infected cells are produced by the contact of uninfected cells and free virus; usually this transmission method is called cell-free virus spread. Based on model (1), large mathematical models considering cell-free virus transmission and different functional responses have been proposed and analyzed, for example, in [23], in [24, 25], in [26], in [27], and general nonlinear functional response [28–30].

Perelson and Nelson [31], Wang and Li [32], and Hu et al. [33] extended model (1) by introducing logistic growth for susceptible T cells. However, many research studies have shown that virus can be transmitted directly from cell to cell by virological synapses, i.e., cell-to-cell transmission [34–41]. Then recently, Lai and Zou [42] proposed a viral dynamical model that combines both cell-free virus transmission and cell-to-cell transmission as follows:

where is the target cell growth rate limited by the carrying capacity of the target cell and the constant indicates the restriction of infected cells that are usually applied to the growth of cells of target cells, . They found that if , then the infection will persist and the Hopf bifurcation will occur within a certain range of parameters, where is the basic reproductive number.

What is interesting is that there are some differences between the above models about the decrease of uninfected cells. The decrease of uninfected cells in model (1) is due to the natural death or the infection by free virus, while in model (2), the decrease of is caused by the density-dependent mortality or the infection by free virus or the infection by infected cells. However, recent research studies have shown that apoptosis is an important factor of the decrease of uninfected cells [43], and HIV can produce apoptosis signals that induce apoptosis in uninfected CDT cells [44, 45]. Then, by taking into account the time delay in HIV virus from HIV infection to produce new viral particles [34] and apoptosis of bystander cells, Cheng et al. [46] proposed a virus model for HIV infection with discrete delay as follows:

where is the apoptotic rate of uninfected cells induced by infected CDT cells and represents the probability of survival of infected cells during the incubation period from virus particles entering cells to releasing new virus particles. They proved that when the basic reproductive number , the infection-free equilibrium is globally asymptotically stable, and when , the infected equilibrium is locally asymptotically stable and the system is uniformly persistent. Guo and Ma [47] extended model (3) by using a general functional response and obtained the global properties of the model. Ji et al. [48] expanded model (3) by considering a full logistic growth of uninfected cells.

Then, based on model (3) and motivated by [32, 47, 48], considering the time delay in HIV virus from HIV infection to produce new viral particles and the time delay in the cell-to-cell transmission and the apoptosis effect of uninfected CDT cells, we propose a delayed viral model as follows:

where are the concentrations of uninfected cells, infected cells, and free virus. is the rate that new uninfected cells are produced, is the intrinsic growth rate, and is the maximum level of CDT cells. are the natural mortalities of uninfected cells, infected cells, and free virus, respectively. is the probability of healthy cells being infected by infected cells, represents the survival rate of infected cells before effective infection, is the apoptotic rate of uninfected cells induced by infected CDT cells, and is the survival rate of healthy cells before effective infection by infected cells.

The remainder of the paper is organized as follows. In Section 2, we will give the initial conditions of model (4) and some preliminary properties. In Section 3, we will discuss the stability of the equilibria. In Section 4, we illustrate the main results by some numerical simulations. Finally, we give a brief conclusion and topics for further research in the future.

2. Basic Reproductive Number and the Equilibria

Let be the Banach space of continuous functions mapping from interval to equipped with the sup-norm. The initial condition of the model (4) is given as follows:

where such that . About the boundedness and non-negativity of the solutions of model (4), by using the method in [49], we can prove the following lemma.

Lemma 1. The solution of model (4) with initial condition (5) is existent, unique, and non-negative on and also ultimately bounded. Moreover,

where and .

The basic reproductive number of model (4) is . Then, model (4) always has a virus-free equilibrium , where , and if , there exists a unique epidemic equilibrium , where

3. Stability for and

In this section, we will discuss the stability for and with the increase of . We start with .

Theorem 1. For model (4), if , is globally asymptotically stable for any time delay .

Proof. We consider the linearization system of model (4) at as follows:

The characteristic equation for is given as

It is easy to see that (9) has a root:

Then, we only need to discuss the distribution of the roots of

Obviously, equation (11) can be written as

When , ; therefore, is not a root of (12). If (12) has a pair of pure imaginary root for any , substituting it into (12), we can get that

and then separating its real and imaginary parts, one gets

Simplifying the above formula and letting , we have

On the one hand, it is easy to get that

On the other hand,

where is used.
Clearly, it indicates that , which is in conflict with . This shows that all the roots of (8) have negative real parts for any time delay . Therefore, the disease-free equilibrium is locally asymptotically stable for any time delay .
Next, let us prove that if , is globally attractive for any time delay . Define

It is easy to show that G attracts all solutions of model (4) and is also positively invariant with respect to model (4). If , let us choose a functional on as follows:

where , and is a constant to be chosen later. Since , we have . It can be found that is continuous on the subset in . From the invariance of , for any , the solution of model (4) satisfies for any . It follows from (4) that

Since , it is possible that we can choose the parameter such that . Thus, it has , for any . All these manifest that is a Lyapunov functional on the subset . By using Lyapunov–LaSalle invariance principle, the disease-free equilibrium of model (4) is globally asymptotically stable.
Next, let us analyze the stability of the epidemic equilibrium . And the linearized system of model (4) at is

The characteristic equation for is given as

On the other hand, (22) is equivalent to

where

with

When , (12) reduces to

Notice that

Therefore, if and hold, by the Routh–Hurwitz criterion, the epidemic equilibrium is locally asymptotically stable for any time delay .
Next, let us investigate the stability of when . Since , is not the root of (23). For , we assume that (23) has the pure imaginary root , and substituting it into (23), we can get that

and then separating its real and imaginary parts, we get

Clearly, it is equivalent to

Let ; (22) can be written as

Let ; (33) is equivalent to

Define

and thus

We consider that

and it has two real roots, given as and , where . Next, we get the following lemma from [50].

Lemma 2. For polynomial (34), the following conclusions are given:

(i)If , (34) has at least one positive root.(ii)If and , (34) has no real root.(iii)If and and if and only if , (34) has real roots.

Assume that has positive real roots; therefore, we can suppose that (34) has positive real roots, denoted as , and . Moreover, (33) has positive real roots . From (31), we can get that

In addition, we get the corresponding such that (33) has pure imaginary , where

in which

Define

Differentiating the two sides of (15) with respect to , it follows that

and therefore,

Then,

From (31), we have

Hence,

Since , this is enough to demonstrate that and have the same sign. Combining Lemma 2 with (45), we have the following conclusions.

Theorem 2. If , the following results hold:

(i)If and , then the epidemic equilibrium is locally asymptotically stable.(ii)If or and , then the epidemic equilibrium is asymptotically stable when and unstable when .(iii)If the conditions of (ii) are all satisfied and , then model (1) undergoes a Hopf bifurcation at when .

4. Numerical Simulations

In this section, we give some numerical simulations to validate our main results in Section 3. The basic parameters are set as follows [51]: . Direct calculations with Maple 14 show that and . Figure 1 shows that virus-free equilibrium of the system is stable.

(a)

(b)

(c)

(d)

Figure 1: Illustration of basic behavior of solutions of system (4) with initial condition , where : (a) time series of x (t), (b) time series of y (t), (c) time series of (t), and (d) phase trajectories of x (t), y (t), and (t).

In the following, we change to 100, and direct calculations show system (4) has two equilibria, i.e., and (). We easily get

Then, equation (31) has one positive root , equation (28) has one positive root , and the Hopf bifurcation value is . First, let and select three different initial values , , and , respectively. Numerical simulations show that the solutions of system converge to the same equilibrium, respectively (see Figures 2, 3, and 4). In the biological sense, tiny presence of infected cells or virus can lead to the disease transmission. Next, let ; Figure 5 shows that periodic oscillations occur for . Then, system (4) undergoes a Hopf bifurcation at when .

(a)

(b)

(c)

(d)

Figure 2: Illustration of basic behavior of solutions of system (4) with , where (the initial condition is (1, 0.01, 1700)): (a) time series of x (t), (b) time series of y (t), (c) time series of (t), and (d) phase trajectories of x (t), y (t), and (t).

(a)

(b)

(c)

(d)

Figure 3: Illustration of basic behavior of solutions of system (4) with , where (the initial condition is (1, 80, 0.01)): (a) time series of x (t), (b) time series of y (t), (c) time series of (t), and (d) phase trajectories of x (t), y (t), and (t).

(a)

(b)

(c)

(d)

Figure 4: Illustration of basic behavior of solutions of system (4) with , where (the initial condition is (1, 80, 1700)): (a) time series of x (t), (b) time series of y (t), (c) time series of (t), and (d) phase trajectories of x (t), y (t), and (t).

(a)

(b)

(c)

(d)

Figure 5: Illustration of basic behavior of solutions of system (4) with , where (the initial condition is (1, 80, 1700)): (a) time series of x (t), (b) time series of y (t), (c) time series of (t), and (d) phase trajectories of x (t), y (t), and (t).

5. Conclusion

In this paper, we improved a delayed HIV virus dynamic model by introducing cell-to-cell transmission and apoptosis of bystander cells. We derived the basic reproductive number , and based on the basic reproductive number, the stability of the equilibria of the system was investigated. The mathematical analysis showed that if , then the disease-free equilibrium of system (4) is globally asymptotically stable, and for , the system can produce Hopf bifurcation with the delay changing. In biology, HIV cannot successfully invade healthy cells and will be cleared by the immune system finally if , and if , uninfected cells, infected cells, and free virus can coexist as periodic oscillations under certain conditions.

However, it is very necessary to consider antiretroviral therapy in model (4), since it can make the model more realistic, and many scholars have noticed this topic and achieved excellent results, for example, Wang et al. [52–54] have studied the virus dynamics model with antiretroviral therapy. Their research shows that antiretroviral therapy can increase the number of uninfected T cells and decrease the number of infectious virions and is very effective for the treatment of AIDS patients. In present model, we do not consider this factor, and we will leave this for future research.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Tongqian Zhang was supported by the Shandong Provincial Natural Science Foundation of China (no. ZR2019MA003) and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents. Zhichao Jiang was supported by the National Natural Science Foundation of China (no. 11801014), Natural Science Foundation of Hebei Province (no. A2018409004), Hebei Province University Discipline Top Talent Selection and Training Program (SLRC2019020), and 2020 Talent Training Project of Hebei Province.

References

R. Weiss, “How does HIV cause AIDS?” Science, vol. 260, no. 5112, pp. 1273–1279, 1993.

D. A. Jabs, “Ocular manifestations of HIV infection,” Transactions of the American Ophthalmological Society, vol. 93, pp. 623–683, 1995.

A. Sandhu and A. K. Samra, “Opportunistic infections and disease implications in HIV/AIDS,” International Journal of Pharmaceutical Science Invention, vol. 2, pp. 47–54, 2013.

UNAIDS, “The joint programme on AIDS,” Tech. Rep., UNAIDS, Geneva, Switzerland, 2018, Technical Report, https://www.unaids.org/en.

N. L. Letvin, “Progress and obstacles in the development of an AIDS vaccine,” Nature Reviews Immunology, vol. 6, no. 12, pp. 930–939, 2006.

Y. Wang, M. Lu, and J. Liu, “Global stability of a delayed virus model with latent infection and Beddington-DeAngelis infection function,” Applied Mathematics Letters, vol. 107, Article ID 106463, 2020.

M. Shen, Y. Xiao, and L. Rong, “Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics,” Mathematical Biosciences, vol. 263, pp. 37–50, 2015.

S. Iwami, S. Nakaoka, and Y. Takeuchi, “Mathematical analysis of a HIV model with frequency dependence and viral diversity,” Mathematical Biosciences and Engineering : MBE, vol. 5, no. 3, pp. 457–476, 2008.

T. Guo, Z. Qiu, and L. Rong, “A within-host drug resistance model with continuous state-dependent viral strains,” Applied Mathematics Letters, vol. 104, Article ID 106223, 2020.

W. Wang, W. Ma, and Z. Feng, “Dynamics of reaction-diffusion equations for modeling CD4+ T cells decline with general infection mechanism and distinct dispersal rates,” Nonlinear Analysis: Real World Applications, vol. 51, Article ID 102976, 2020.

T. Loudon, S. Pankavich, and S. Pankavich, “Mathematical analysis and dynamic active subspaces for a long term model of HIV,” Mathematical Biosciences and Engineering, vol. 14, no. 3, pp. 709–733, 2017.

O. M. Otunuga, “Global stability for a 2n + 1 dimensional HIV/AIDS epidemic model with treatments,” Mathematical Biosciences, vol. 299, pp. 138–152, 2018.

Y. Wang, T. Zhao, T. Zhao, and J. Liu, “Viral dynamics of an HIV stochastic model with cell-to-cell infection, CTL immune response and distributed delays,” Mathematical Biosciences and Engineering, vol. 16, no. 6, pp. 7126–7154, 2019.

T. Feng, Z. Qiu, X. Meng, and L. Rong, “Analysis of a stochastic HIV-1 infection model with degenerate diffusion,” Applied Mathematics and Computation, vol. 348, pp. 437–455, 2019.

X. Wang, Q. Ge, and Y. Chen, “Threshold dynamics of an HIV infection model with two distinct cell subsets,” Applied Mathematics Letters, vol. 103, p. 106242, 2020.

Y. Liu and X. Zou, “Mathematical modeling of HIV-like particle assembly in vitro,” Mathematical Biosciences, vol. 288, pp. 46–51, 2017.

N. Bobko, J. P. Zubelli, and J. Zubelli, “A singularly perturbed HIV model with treatment and antigenic variation,” Mathematical Biosciences and Engineering, vol. 12, no. 1, pp. 1–21, 2015.

T. Guo, Z. Qiu, and L. Rong, “Modeling the role of macrophages in HIV persistence during antiretroviral therapy,” Journal of Mathematical Biology, vol. 81, no. 1, pp. 369–402, 2020.

H. C. Tuckwell and F. Y. M. Wan, “First passage time to detection in stochastic population dynamical models for HIV-1,” Applied Mathematics Letters, vol. 13, no. 5, pp. 79–83, 2000.

W. Wang, W. Ma, and Z. Feng, “Complex dynamics of a time periodic nonlocal and time-delayed model of reaction-diffusion equations for modeling CD4+ T cells decline,” Journal of Computational and Applied Mathematics, vol. 367, Article ID 112430, 2020.

W. Wang, X. Wang, and Z. Feng, “Time periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patients,” Nonlinear Analysis: Real World Applications, vol. 57, Article ID 103184, 2021.

M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74–79, 1996.

L. Min, Y. Su, and Y. Kuang, “Mathematical analysis of a basic virus infection model with application to HBV infection,” Rocky Mountain Journal of Mathematics, vol. 38, no. 5, pp. 1573–1585, 2008.

D. Li and W. Ma, “Asymptotic properties of a HIV-1 infection model with time delay,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 683–691, 2007.

X. Song and A. U. Neumann, “Global stability and periodic solution of the viral dynamics,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 281–297, 2007.

G. Huang, W. Ma, and Y. Takeuchi, “Global properties for virus dynamics model with Beddington-DeAngelis functional response,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1690–1693, 2009.

X. Zhou and J. Cui, “Global stability of the viral dynamics with Crowley-Martin functional response,” Bulletin of the Korean Mathematical Society, vol. 48, no. 3, pp. 555–574, 2011.

K. Hattaf and N. Yousfi, “Global stability of a virus dynamics model with cure rate and absorption,” Journal of the Egyptian Mathematical Society, vol. 22, no. 3, pp. 386–389, 2014.

Y. Tian and X. Liu, “Global dynamics of a virus dynamical model with general incidence rate and cure rate,” Nonlinear Analysis: Real World Applications, vol. 16, pp. 17–26, 2014.

F. Li, S. Zhang, and X. Meng, “Dynamics analysis and numerical simulations of a delayed stochastic epidemic model subject to a general response function,” Computational and Applied Mathematics, vol. 38, p. 95, 2019.

A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999.

L. Wang and M. Y. Li, “Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells,” Mathematical Biosciences, vol. 200, no. 1, pp. 44–57, 2006.

Z. Hu, J. Zhang, H. Wang, W. Ma, and F. Liao, “Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment,” Applied Mathematical Modelling, vol. 38, no. 2, pp. 524–534, 2014.

R. V. Culshaw, S. Ruan, and G. Webb, “A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay,” Journal of Mathematical Biology, vol. 46, no. 5, pp. 425–444, 2003.

D. Mcdonald and T. J. Hope, “Recruitment of HIV and its receptors to dendritic cell-T cell junctions,” Science, vol. 300, no. 5623, pp. 1295–1297, 2003.

Q. Sattentau, “Avoiding the void: cell-to-cell spread of human viruses,” Nature Reviews Microbiology, vol. 6, no. 11, pp. 815–826, 2008.

D. M. Phillips, “The role of cell-to-cell transmission in HIV infection,” Aids, vol. 8, no. 6, pp. 719–732, 1994.

C. Jolly, “Cell-to-cell transmission of retroviruses: innate immunity and interferon-induced restriction factors,” Virology, vol. 411, no. 2, pp. 251–259, 2011.

C. R. M. Bangham, “The immune control and cell-to-cell spread of human T-lymphotropic virus type 1,” Journal of General Virology, vol. 84, no. 12, pp. 3177–3189, 2003.

S. Krantic, C. Gimenez, and C. Rabourdin-Combe, “Cell-to-cell contact via measles virus haemagglutinin-CD46 interaction triggers CD46 downregulation,” Journal of General Virology, vol. 76, no. 11, pp. 2793–2800, 1995.

T. Zhang, J. Wang, Y. Li, Z. Jiang, and X. Han, “Dynamics analysis of a delayed virus model with two different transmission methods and treatments,” Advances in Difference Equations, vol. 2020, Article ID 1, 2020.

X. Lai and X. Zou, “Modeling cell-to-cell spread of HIV-1 with logistic target cell growth,” Journal of Mathematical Analysis and Applications, vol. 426, no. 1, pp. 563–584, 2015.

M. F. Cotton, D. N. Ikle, E. L. Rapaport et al., “Apoptosis of CD4+ and CD8+ T cells isolated immediately ex vivo correlates with disease severity in human immunodeficiency virus type 1 infection,” Pediatric Research, vol. 42, no. 5, pp. 656–664, 1997.

T. H. Finkel, G. Tudor-Williams, N. K. Banda et al., “Apoptosis occurs predominantly in bystander cells and not in productively infected cells of HIV- and SIV-infected lymph nodes,” Nature Medicine, vol. 1, no. 2, pp. 129–134, 1995.

N. Selliah and T. H. Finkel, “Biochemical mechanisms of HIV induced T cell apoptosis,” Cell Death & Differentiation, vol. 8, no. 2, pp. 127–136, 2001.

W. Cheng, W. Ma, and S. Guo, “A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells and its stability analysis,” Communications on Pure and Applied Analysis, vol. 15, no. 3, pp. 795–806, 2016.

S. Guo and W. Ma, “Global behavior of delay differential equations model of HIV infection with apoptosis,” Discrete & Continuous Dynamical Systems–B, vol. 21, no. 1, pp. 103–119, 2016.

Y. Ji, W. Ma, and K. Song, “Modeling inhibitory effect on the growth of uninfected T cells caused by infected T cells: stability and hopf bifurcation,” Computational and Mathematical Methods in Medicine, vol. 2018, Article ID 317689, 10 pages, 2018.

S. Hews, S. Eikenberry, J. D. Nagy, and Y. Kuang, “Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth,” Journal of Mathematical Biology, vol. 60, no. 4, pp. 573–590, 2010.

S. Ruan and J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” Mathematical Medicine and Biology, vol. 18, no. 1, pp. 41–52, 2001.

K. Zhuang and H. Zhu, “Stability and bifurcation analysis for an improved HIV model with time delay and cure rate,” Wseas Transactions on Mathematics, vol. 12, pp. 860–869, 2013.

Y. Wang, Y. Zhou, J. Wu, and J. Heffernan, “Oscillatory viral dynamics in a delayed hiv pathogenesis model,” Mathematical Biosciences, vol. 219, no. 2, pp. 104–112, 2009.

Y. Wang, Y. Zhou, F. Brauer, and J. M. Heffernan, “Viral dynamics model with CTL immune response incorporating antiretroviral therapy,” Journal of Mathematical Biology, vol. 67, no. 4, pp. 901–934, 2013.

Y. Wang, J. Liu, and L. Liu, “Viral dynamics of an HIV model with latent infection incorporating antiretroviral therapy,” Advances in Difference Equations, vol. 2016, no. 1, p. 225, 2016.