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J. Math. Anal. Appl. 434 (2016) 837–857

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Bifurcations of a mathematical model for HIV dynamics

✩

Jianfeng Luo, Wendi Wang

∗

, Hongyan Chen, Rui F u

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

a r t i c l e i n f o a b s t r a c t

Article history:

Received 3 November 2014

Available online 28 September 2015

Submitted by Y. Du

Keywords:

Backward bifurcation

Bistable infections

Stability

Immune response

HIV

This paper investigates bifurcations and stability of an HIV model that incorporates

the immune responses. The conditions for the global stability of infection-free

equilibrium and infection equilibrium are respectively established by the Lyapunov

method and the geometric approach. The backward bifurcation from the infection-

free

equilibrium is examined by analytical analysis. More interestingly, with the

aid of mathematical analysis, we ﬁnd a new type of bifurcations from an infection

equilibrium, where a backward bifurcation curve emerges and can be continued to

the place where the basic reproduction number is less than unity. By numerical

simulations, we ﬁnd a variety of dynamical behaviors of the model, which reveal the

importance and complexity of immune responses in ﬁghting HIV replication.

1. Introduction

Since the emergence of human immunodeﬁciency virus (HIV) in the early 1980s, the disease of HIV

infection has been pandemic around the world and has caused 34 million infected and 25 million deaths so

far [36]. Due to its wide range of infectiousness, strong mortality and incurability, HIV has become a major

problem for human health. Biological studies indicate that HIV infects CD4

T helper cells. The phase of

primary infection is characterized by a strong viral replication, which is rapidly followed by a strong immune

response [3,15]. In the second phase of HIV infection, infected individual displays no symptoms, but has

persistent viral replications, which eventually results in the development of AIDS [15].

Mathematical

models have been employed to study the dynamics of HIV infection in vivo [4,7,8,18,29,

39,44,37,50].

The basic model of three state variables was proposed by [27,30]:

✩

The research is supported by the NSF of China (11171276).

* Corresponding author.

E-mail

address: wendi@swu.edu.cn (W. Wang).

http://dx.doi.org/10.1016/j.jmaa.2015.09.048

838 J. Luo et al. / J. Math. Anal. Appl. 434 (2016) 837–857

= λ − δ

T − βV T,

= βV T − δ

= NI − cV,

(1.1)

where T is the concentration of uninfected target cells, I is the concentration of infected cells, V is the

concentration of virus, λ is the production rate of new target cells, δ

and δ

are the death rates of unin-

fected

cells and infected cells respectively, β is the infection coeﬃcient, N is the burst coeﬃcient of virus

when infected cells die and c is the clearance rate of virus. Model (1.1) has made signiﬁcant contributions to

understand the dynamics of HIV infection and therapy to design speciﬁc treatment regimes [29,46]. How-

ever,

as noted in [29], since immune responses are not explicitly included, model (1.1) is suitable for the

asymptomatic phase of the disease during which immune responses are stable. Indeed, in the primary phase

of HIV infection, the cellular immune response to HIV is strong, grows with the rise in HIV in the blood

and then falls when that response reaches a peak [11,26].

Immune responses have been incorporated in the basic model of HIV infection (see, for example, [1,3,14,

20,21,28,31,32,34,35,43,45,47] and

the references cited therein) to understand how immune systems regulate

the dynamics of HIV infection. Most of these researches study eﬀects of cytotoxic T lymphocyte (CTL)

in ﬁghting HIV replication, and only a few consider the eﬀect of proliferation of CD4

T cells under the

stimulation of pathogens. Indeed, biological studies indicate that CD4

T cell activation plays a fundamental

role in guiding CTL responses [11,26]. In papers [2,33,48], an HIV model is proposed that incorporates CTL

responses to intracellular pathogens, CD4

T cell proliferation due to stimulations of pathogens, and B

cell responses to virus particles. The model admits only three state variables, which makes mathematical

analysis tractable. More importantly, the model captures the main features of the complex interactions of

HIV and immune responses. The dynamics of the variables are described by

⎧

⎪

⎨

⎪

⎩

=(λ + γ

IT)(1 − T/K) − δ

T − βV T,

= βV T − (δ

+ γ

T )I,

= NI − (c + γ

T )V −βV T.

(1.2)

Here, T is the concentration of helper T cells, I is the concentration of infected helper T cells, V is the

concentration of free virus. Parameter γ

represents the clonal ampliﬁcation of helper T cells after stimulation

by infected T cells. Further, the recruitment rate and the proliferation rate of helper T cells are limited

by the carrying capacity K. The model assumes that B cell and CTL responses are proportional to the

concentration of helper T cells. As a consequence, term γ

TI gives the removal rate of infected cells by CTL

responses and term γ

TV denotes the clearance rate of virus particles by antibodies. The last term −βV T

describes the loss rate of virus because of entry into target cells. All the other parameters in (1.2) have the

same meaning as those in (1.1).

Numerical simulations of model (1.2) in [2] found the case of asymptotic coexistence of immune system

and virus, and found the case of virus eradication. The stability of infection-free equilibrium of (1.2) was

examined in [48]. Model (1.2) was also extended by [48] to include stochastic cellular reproduction and

death, stochastic infection process, stochastic immune system activation, and stochastic viral reproduction.

In this paper, we explore the global behaviors of model (1.2) to ﬁnd possible evolutionary outcomes of

HIV infection under the control of immune responses. We establish conditions for the global stability of

infection-free equilibrium and infection equilibrium, which gives the conditions of infection eradication or

J. Luo et al. / J. Math. Anal. Appl. 434 (2016) 837–857 839

moving to the chronic infection. We show that model (1.2) exhibits the classical backward bifurcation from

the infection-free equilibrium (see, for example, [34,35]). More interestingly, we show that the activation of

CD4

T cell stimulated by pathogen can induce a new type of bifurcations at an infection equilibrium where a

backward bifurcation curve emerges and can be continued to the place where the basic reproduction number

is less than unity. This phenomenon reveals more diﬃculties to design treatment regimes. Moreover, by

numerical simulations, we ﬁnd a variety of dynamical behaviors of the model, which indicates the importance

and complexity of immune responses in ﬁghting HIV replication.

The

organization of this paper is as follows. In the next section, we study the existence of equilibria

of model (1.2). In Section 3, we present the local stability analysis of these equilibriums. Then we apply

the Li–Muldowney global-stability criterion [24] to derive suﬃcient conditions for the global stability of an

infection equilibrium. In Section 4, we study the bifurcations of model (1.2). Brief summary and discussions

are shown in the last section.

2. Existence of equilibria

Before going into details, we make the following scaling:

t = δ

T =

I =

V =

Kβγ

λ =

Kδ

,δ=

β =

Kγ

,γ=

Kγ

N =

NβK

,C=

,α=

K(β + γ

)

Dropping the bars, model (1.2) becomes

⎧

⎪

⎨

⎪

⎩

=(λ + IT)(1 − T ) − δT −βV T,

= VT − I − γTI,

= NI − CV −αV T,

(2.1)

where all the parameters are positive. Now, we analyze the equilibria of (2.1) and study the bifurcations on

the basis of distributions of these equilibria. The infection-free equilibrium (IFE) is

=(T

)



λ + δ

, 0, 0). (2.2)

By [40,48], the basic reproduction number of virus in (2.1) is



(1 + γT

)(C + αT

)

An infection equilibrium (

V )satisﬁes

⎧

⎪

⎨

⎪

⎩

(λ +

I)(1 −

T ) − δ

T − β

V =0,

V − (1 + γ

T )

I =0,

I − C

V − α

V =0.

(2.3)

Direct computations lead to

840 J. Luo et al. / J. Math. Anal. Appl. 434 (2016) 837–857

I =

T − λ(1 −

T )

T (1 −

T ) − (β + βγ

T )

V =

C + α

T − λ(1 −

T )

T (1 −

T ) − (β + βγ

T )

, (2.4)

where

T satisﬁes

the equation:

T )=αγ

+ B

T + C =0, (2.5)

in which,

B = α + Cγ −N.

Note

V =

IN/(C + α

T ). To ensure the existence of a positive equilibrium, we let B<0, and require

I>0, which is equivalent to

(

T − T

)g(

T ) > 0, (2.6)

where T

is given in (2.2) and

T )=−

+(1− βγ)

T − β. (2.7)

Set

Δ=B

− 4αγ C, Δ

=(1− βγ)

− 4β.

The positive equilibria of (2.1) are classiﬁed by the signs of Δ

and 1 − βγ. Let us consider two cases.

(C1)

< 0, or Δ

≥ 0 and βγ ≥ 1.

Note

that

:= [(1 + γT

)(C + αT

)]

> 0,

:= −2T

(1 + γT

)(C + αT

)(α + Cγ) < 0,

:= T

(α − Cγ)

> 0. (2.8)

Note also that

:= m

− 4m

=16αγCT

> 0.

We deﬁne

−m



− 4m

, (2.9)

which is clearly positive.

Theorem 2.1. Let B<0 and (C1) hold. Then we have the following conclusions.

(1) When R

> 1, system (2.1) admits a unique chronic-infection equilibrium

(2) Suppose −B/(2αγ) <T

. Then system (2.1) has no chronic-infection equilibrium if R



, has a

unique chronic-infection equilibrium

E if R



, and has two chronic-infection equilibriums

E and

E if



< 1.

(3) Suppose −B/(2αγ) = T

. Then system (2.1) has no chronic-infection equilibrium if R

≤



, has a

unique chronic-infection equilibrium

E if



< 1.

(4) System (2.1) has no chronic-infection equilibrium if −B/(2αγ) >T

and R

< 1.

J. Luo et al. / J. Math. Anal. Appl. 434 (2016) 837–857 841

Proof. Clearly, the assumptions in (C1) imply g(

T ) < 0. It follows from (2.6) that

T must lie in (0, T

). By

direct calculations we see that

f(T

)=(1+γT

)(C + αT

)(1 − R

Thus, f(T

) < 0if R

> 1, and f(T

) > 0if R

< 1. Since f(0) > 0, it follows that (2.5) admits a unique

positive root

T ∈ (0,

)if R

> 1.

Let us consider the case where R

< 1. If −B/(2αγ) >T

, since f(T

) > 0and f(0) > 0, it follows that

(2.5) has no positive root in (0, T

). Similarly, when −B/(2αγ) <T

and Δ > 0, (2.5) admits two positive

roots in (0, T

); when −B/(2αγ) <T

and Δ =0, (2.5) has a positive root in (0, T

); when Δ < 0, (2.5) has

no positive root in (0, T

); when −B/(2αγ) = T

and Δ =0, (2.5) has no positive root in (0, T

); when

−B/(2αγ) = T

and Δ > 0, (2.5) has exactly a positive root in (0, T

To determine the signs of Δin terms of the basic reproduction number, we denote R

by R

so that

N =

(1 + γT

)(C + αT

)

. (2.10)

It is easy to obtain

Δ=m

+ m

where m

are given in (2.8). Since B<0, we get

(α + γC)T

(1 + γT

)(C + αT

)

:= R

It is easy to examine R

< 1. If h(x) = m

+ m

x + m

, we have

h(R

)=m

+ m

= −4T

αγC < 0,

h(1) = m

+ m

=(αγT

− C)

> 0.

It follows that the equation of h(x) =0admits a root R

, which is described by (2.9), in interval (R

, 1).

Clearly, we have Δ =0when R



, Δ > 0when



, and Δ < 0when

√



Consequently, conclusions (2) and (3) of the theorem follow from the discussions of the last paragraph. 2

Let us now consider the second case.

(C2)

> 0 and βγ < 1.

The conditions in Assumption (C2) imply that g(T ) =0has two positive roots:

1 − βγ −

√

1 − βγ +

√

When B<0and Δ > 0, f (T ) =0has two positive roots:

−B −

√

2αγ

−B +

√

2αγ

When B<0and Δ =0, f(T ) =0has only one positive root:

−B

2αγ

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文件中提供了一种实用的方法来模拟传染病的传播，让我能够更好地理解疫情的发展规律。
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