Computational and Mathematical Methods in Medicine
Volume 2018 (2018), Article ID 3176893, 10 pages
https://doi.org/10.1155/2018/3176893

Research Article

Modeling Inhibitory Effect on the Growth of Uninfected T Cells Caused by Infected T Cells: Stability and Hopf Bifurcation

Yahui Ji, Wanbiao Ma, and Keying Song

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, 100083, Beijing, China

Correspondence should be addressed to Wanbiao Ma; wanbiao_ma@ustb.edu.cn

Received 1 February 2018; Accepted 30 April 2018; Published 12 August 2018

Academic Editor: Ming-shi Yang

Copyright © 2018 Yahui Ji 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.

We consider a class of viral infection dynamic models with inhibitory effect on the growth of uninfected T cells caused by infected T cells and logistic target cell growth. The basic reproduction number is derived. It is shown that the uninfected equilibrium is globally asymptotically stable if . Sufficient conditions for the existence of Hopf bifurcation at the infected equilibrium are investigated by analyzing the distribution of eigenvalues. Furthermore, the properties of Hopf bifurcation are determined by the normal form theory and the center manifold. Numerical simulations are carried out to support the theoretical analysis.

1. Introduction

The human immunodeficiency virus (HIV) is a lentivirus, which replicates by infecting and destroying primarily CD4 T cells. The end stage of HIV viral progression is acquired immune deficiency syndrome (AIDS) (see, for example, [1]), identified when the count of individual’s CD4 cells count falls below 200. Since AIDS was found in America in 1981, it spread worldwide and became the public health and social problem which causes serious damage to human survival and development. In 2016, there exist about 38 million people living with human immunodeficiency virus (HIV) (see, for example, [2]). Thus, it is a challenge to study and control the virus.

It is widely known that mathematical models have made considerable contributions to understanding the HIV infection dynamics. Nowak et al. have proposed a class of classic mathematical model to describe HIV infection dynamics (see, for example, [3–6]),

where , , and denote the concentrations of uninfected cells, infected cells, and free virus at time , respectively. Uninfected cells are produced at the rate   , die at the rate   , and become infected at the rate   . The constant    is the death rate of the infected cells due either to virus or to the immune system. The constant    is the rate of production of virus by infected cells and the constant    is the rate at which the virus is cleared.

Incorporating the life cycle of the virus in the cells, some researchers have considered that the HIV virus from HIV infection to produce new virus takes time. To make a better understanding for this phenomenon in mathematics, HIV models including time delay have been proposed (see, for example, [4, 7–9]). Several researchers have considered that when T cells stimulate by antigen or mitogen, this will differentiate and increase in the number. The HIV model with a full logistic mitosis term has been investigated (see, for example, [6, 10, 11]). Taking into account the growth of uninfected cells, they made a further investigation to add a full logistic term (see, for example, [12, 13]).

In the above model, there are two factors that accelerate the reduction of uninfected cells: one is the natural death of uninfected cells and the other is that uninfected cells become infected cells. HIV gene expression products can be toxic and directly or indirectly induce apoptosis in uninfected cells. Some data show that viral proteins interact with uninfected cells and produce an apoptotic signals that accelerate the death of uninfected cells. Recently, Wang and Zhang proposed a spatial mathematical model to describe the predominance for driving T cells death, which is called caspase-1-mediated pyroptosis (see, for example, [14]).

Based on model (1), Guo and Ma have proposed a class of delay differential equations model of HIV infection dynamics with nonlinear transmissions and apoptosis induced by infected cells (see, for example, [15]). And then, Cheng et al. [16] have considered the following infection model with inhibitory effect on the growth of uninfected cells by infected cells:

where the constant    represents the rate of apoptosis at which infected cells induce uninfected cells.    denotes the surviving rate of infected cells before they become productively infected. The biological meanings of the other parameters in the model (2) are similar to that in the model (1).

Motivated by the above models, in this paper, we will study a delay differential equation model of HIV infection with a full logistic term of uninfected cells,

In this model, the logistic growth of the healthy T cells is described by . The total concentration of T cells is , where denotes the concentration of uninfected cells, is the concentration of infected cells, and is the maximum level of T cells.    is the infection rate of infected cells. The biological meanings of the other parameters in the model (3) are similar to that in the model (2).

The main purpose of this paper is to carry out a pretty theoretical analysis on the stability of the equilibria of the model (3) and to analyze the Hopf bifurcation by related theories of the differential equations. The organization of this paper is as follows. In Section 2, we investigate the existence and the ultimate boundedness of the solutions of the model (3). Then we consider the global stability of the uninfected equilibrium and the Hopf bifurcation at the infected equilibrium. In Section 3, some properties of Hopf bifurcation such as direction, stability, and period are determined. In Section 4, the brief conclusions are given and sets of numerical simulations are provided to illustrate the main results.

2. Local and Global Stability of the Equilibria

According to biological meanings, we assume that the initial condition of the model (3) is given as follows:

where such that . Here, denotes the Banach space of continuous functions mapping from the interval to equipped with the supnorm.

The existence and uniqueness, nonnegativity, and boundedness of the solutions of the model (3) with the initial condition (4) can be given as follows.

Theorem 1. The solution of the model (3) with the initial condition (4) is existent, unique, and nonnegative on and also has

where and .

In fact, by using standard theorems for existence and uniqueness of functional differential equations (see, for example, [17–19]), we can show that the solution of the model (3) with the initial condition (4) is existent, unique and nonnegative on , easily. And the proving of ultimately bounded of the solution is similar to [12, 16].

We can denote the basic reproduction number of the HIV virus for the model (3) as (see, for example, [3]). For the existence of nonnegative equilibria of the model (3), we can obtain the following classifications:

(i) The model (3) always has the uninfected equilibrium .

(ii) If , the model (3) has unique infected equilibrium , where

Theorem 2. If , the uninfected equilibrium of the model (3) is globally asymptotically stable.

Proof. We consider linear system of the model (3) at , we have

The corresponding characteristic equation is given by

Clearly, one of the roots is , so the local stability depends on the other two roots generated by

When , . Therefore, is not root of (9). If (9) has pure imaginary root for some , substituting it into (9) and separating the real and imaginary parts, it has

It follows that

where . Since , , we have , which contradicts . This suggests that all the roots of (8) have negative real parts for any time delay . Therefore, the uninfected equilibrium of the model (3) is locally asymptotically stable.
Define

It is easy to show that attracts all solutions of the model (3) and is also positively invariant with respect to the model (3).
Motivated by the methods in [20, 21], we choose the following Liapunov functional:

for any . The time derivative of along the solutions of the model (3) is

where . By using Liapunov-LaSalle invariance principle [18], the uninfected equilibrium of the model (3) is globally asymptotically stable.
Next, let us study the stability of the infected equilibrium . The linearized system of the model (3) at is

Denote

The corresponding characteristic equation is

Define

where and .
Therefore, (17) becomes

When , (19) becomes . Notice that . Thus, if and hold, by Routh-Hurwitz criterion, the infected equilibrium is locally asymptotically stable when .
Now, let us investigate the stability of when . Rewriting (19) as

where

Since , is not the root of (19). Assume that (19) has pure imaginary    for some ; substituting it into (19), it has , and separating the real and imaginary parts, we have

Therefore, it has

where . Denote ; (23) becomes

Define

hence . Considering

It has two real roots, given as and , where .
Now, we will illustrate the following conclusions, and it has been proved in [22].

Lemma 3. For the polynomial (24), the following conclusions are given:

(i)If , (24) has at least one positive root.(ii)If and , (24) has no real root.(iii)If and , if and only if and , (24) has real roots.

Assume that has positive real roots. Generally, we may suppose that (24) has positive real roots, denoted as , , and . Then, (23) has positive real roots . From (22), we attain

Then, we get the corresponding such that (19) has pure imaginary , where

Define

Differentiating the two sides of (19) with respect to , it follows that

Thus, we get

Then

From (22), we obtain . Therefore,

Since , we get and have the same sign. Combining Lemma 3 with the above , we have the following conclusions.

Theorem 4. and are defined by (28) and (29). If , the following results hold:

(i)If and , then infected equilibrium is locally asymptotically stable.(ii)If or and , then infected equilibrium is locally asymptotically stable when and unstable when .(iii)If the conditions of (ii) are all satisfied and , then model (3) undergoes a Hopf bifurcation at when

3. Properties of Hopf Bifurcation

In the above section, we have given the sufficient condition where the model (3) undergoes a Hopf bifurcation at . In this section, we will use the normal form method and the center manifold theory provided in [23, 24] to analysis direction, stability, and the period of the bifurcating periodic solution. By setting , then is a Hopf bifurcation value of the model (3). Let , , , and

Then, the model (3) is equivalent to the functional differential equations , defined in , where

For , define Here,

Using the Riesz representation theorem, there is a bounded variation matrix function , which exists for , such that holds for any . We can choose , where

For , define

Then, the system is equivalent to the following operator equation:

Let , and adjoint operator of is defined by

Define the bilinear inner product of and as

where .

Since and are adjoint operator and is the eigenvalue of , therefore also is the eigenvalue of . Suppose that the eigenvector of with respect to the eigenvalue is ; the eigenvector of with respect to the eigenvalue is , and they all satisfy .

We choose , and Since , , we get

From and the similar arguments as in [20–22], we attain the following formula:

Following the algorithms given in [23] (see, also [13, 24–26]), it then follows that

where

Then we can obtain the following quantities:

These quantities determine the properties of bifurcating periodic solutions. From the previous discussions, we have the following conclusions.

Theorem 5. Suppose that the conditions in (iii) of Theorem 4 hold, then the infected equilibrium undergoes a Hopf bifurcation at , and determine the direction, stability, and period of the Hopf bifurcation, respectively,

(i)If , a bifurcating periodic solution exists in the sufficiently small neighbourhood.(ii)If (), the bifurcating periodic solution is stable (unstable) when ().(iii)If (), the period of the bifurcating periodic solution decreases (increases).

4. Simulations and Conclusions

For the main results in Sections 2 and 3, we now give some numerical simulations.

Based on the numerical simulations in [16, 27–29], take the following data:

We can get and by direct calculations. The uninfected equilibrium is globally asymptotically stable by Theorem 2. Figure 1 gives the curves and orbits of the model (3) with appropriate initial condition.

(a)

(b)

Figure 1: (a) The solution curves of the model (3) with . (b) The orbits of the model (3) when .

Furthermore, we also simulate the occurrence of Hopf bifurcations as the time delay increases. Take the following data:

By direct calculations, we get that (24) has a positive root , , and . And by simple computations, we attain , , and . From Theorem 4, the infected equilibrium is locally asymptotically stable when and unstable when . Figure 2 gives the stable phase trajectories and orbits of the model (3) when . Figure 3 gives the phase trajectories and orbits of model (3) when and it suggests that Hopf bifurcations occur. From (45), we obtain for . Therefore, both bifurcating periodic solutions are stable.

(a)

(b)

(c)

(d)

Figure 2: (a), (b), and (c) The solution curves of the model (3) with , . (d) The orbits of the model (3) when , .

(a)

(b)

(c)

(d)

Figure 3: (a), (b), and (c) The solution curves of the model (3) with , . (d) The orbits of the model (3) when , .

In this paper, we have proposed a delay HIV infection model (3) with a full logistic term . Then, using the basic reproduction number , we discuss the existence of the uninfected equilibrium and the infected equilibrium . By Routh-Hurwitz criterion, Liapunov-LaSalle invariance principle, and Hopf bifurcation method, we prove the following results.

If , the uninfected equilibrium is globally asymptotically stable when . That is to say, any solution trends to . Biologically, this means that the virus cannot successfully invade uninfected cells and will soon be cleared of the immune system. And as the time increases, the virus will disappear. This suggests that we can control the disease by controlling the .

If , there exists a unique infected equilibrium . The result of Theorem 4 implies that the time delay can destabilize the stability of the infected equilibrium and leads to the occurrence of Hopf bifurcations. And if , the infected equilibrium is locally asymptotically stable. Biologically, this means that the HIV infection may become chronic. The infected equilibrium will be unstable and Hopf bifurcation occurs under some conditions when the time delay exceeds . In biology, this implies that the concentrations of uninfected cells, infected cells, and free virus will first tend to be constants and then oscillate as the time delay increases. In the immune response, this situation is very important for choosing the adequate drug treatment programs.

It can be found that the basic reproduction number for the model (3) is not the same as that for model (2). It is independent on the constant , which represents inhibitory effect on the growth of uninfected cells by infected cells. But depends on the coefficient of the full logistic term . Furthermore, the value of is independent on the coefficient . And the values of and are the increasing functions with respect to . And this paper shows that the time delay can produce richer dynamic behavior. As the time delay increases, the stability changes and periodic oscillations occur.

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

This work is partly supported by the National Natural Science Foundation of China for W. Ma (no. 11471034).

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