Download An HIV infection model based on a vectored immunoprophylaxis

Journal of Theoretical Biology 313 (2012) 127–135
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Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/yjtbi
An HIV infection model based on a vectored
immunoprophylaxis experiment$
Xiunan Wang, Wendi Wang n
Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China
H I G H L I G H T S
c
c
c
The introduction of immunoprophylaxis can induce a backward bifurcation.
The ignorance of antibodies’ involvement causes the loss of a backward bifurcation.
Burst size of virus induces five types of behaviors including bistability.
a r t i c l e i n f o
abstract
Article history:
Received 21 June 2012
Received in revised form
18 August 2012
Accepted 20 August 2012
Available online 28 August 2012
A medical experiment published in Nature has shown that humanized mice receiving the vectored
immunoprophylaxis can be fully protected from HIV infection. In this paper, a mathematical model is
proposed to investigate the viral dynamics under the effect of antibodies in the experiment. It is shown
that the introduction of vectored immunoprophylaxis can induce the backward bifurcation and the
ignorance of antibodies’ loss due to their involvement with virus may result in the loss of backward
bifurcation. By numerical simulations, it is found that the model also exhibits some other complicated
dynamical behaviors. A subcritical Hopf bifurcation, a fold bifurcation of equilibria and a limit point
bifurcation of limit cycles are detected, which induce five typical patterns of dynamical behaviors
including the bistable phenomenon.
& 2012 Elsevier Ltd. All rights reserved.
Keywords:
Viral dynamics
Global stability
Backward bifurcation
Hopf bifurcation
Bistability
1. Introduction
Since the first case of AIDS was diagnosed on December 1st in
1981 this deadly disease has been pandemic around the world
and has caused about 34 million infected and 25 million deaths so
far. Although the advent of highly active antiretroviral therapy
has dramatically reduced HIV-related morbidity and mortality
(Simon and Ho, 2003), the clinical benefits of combination
therapy are often challenged with the increasing emergence of
drug resistance driven by within-host selective pressure of antiretroviral drugs (Clavel and Hance, 2004). Meanwhile, a latentinfection state can be established in resting memory CD4 þ T cells
(Chun et al., 1995, 1997a). These latently infected cells are
capable of abrogating CTL recognition because they express very
low levels of HIV-1 messenger RNA and proteins (Lambotte et al.,
2004). In the presence of antiretroviral drug therapy memory
CD4 þ T cells can remain in the resting state for a long time,
$
Research supported by the National Science Foundation of PR China
(11171276) and by the Ministry of Education of PR China (20100182110003).
n
Corresponding author. Tel.: þ86 23 68367563; fax: þ 86 23 68252397.
E-mail address: [email protected] (W. Wang).
0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2012.08.023
providing a hiding place for virus (Wong et al., 1997; Finzi et al.,
1997; Chun et al., 1997b). When activated they will start to
produce new virus particles (Chun et al., 1998). Thus, a viral level
rebound seems to be inevitable once the therapy is stopped and
this has become a primary obstacle to the long-term control of
HIV in infected individuals. In addition, viral escape from immune
responses is indeed an impediment for HIV vaccination (Bailey
et al., 2004; Davenport et al., 2008). All these facts have made the
eradication of AIDS an elusive goal.
Recently, however, a vectored immunoprophylaxis experiment
carried out by Balazs et al. (2012) brought new hope to eradicate HIV
infection. In the experiment, vector-mediated gene transfer was used
to engineer secretion of the existing broadly neutralizing antibodies
into the circulation. Vectored immunoprophylaxis (VIP) in mice
induced the lifelong expression of these monoclonal antibodies
at high concentrations from a single intramuscular injection. They
showed that the humanized mice receiving VIP appear to be fully
protected from HIV infection, even when challenged intravenously
with very high doses of replication-competent virus.
Mathematical models have made great contributions to getting
insights into HIV infection and dynamics, as well as on how an
infection can be managed, reduced or even eradicated. Since the
128
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
basic three-dimensional viral dynamical model was proposed
(Perelson et al., 1993, 1996; Bonhoeffer et al., 1997; Perelson and
Nelson, 1999; Nowak and May, 2000) considerable progress has
been made in mathematical modeling of dynamics of viral infections. Some researchers have taken into account the effect of
immune responses including CTL responses or humoral immunity
or both (Wodarz, 2003; Pang et al., 2005; Wang et al., 2006a,b;
Iwami et al., 2009). Some other researchers have incorporated the
effect of intracellular delays into the model (Herz et al., 1996;
Mittler et al., 1999; Nelson et al., 2000; Culshaw and Ruan, 2000;
Nelson and Perelson, 2002; Dixit and Perelson, 2004; Li and Shu,
2010; Yan and Wang, 2012). A number of models use time-varying
drug concentrations to determine the efficacy of antiviral treatment (Wahl and Nowak, 2000; Huang et al., 2003; Dixit and
Perelson, 2004; Wu et al., 2005) and several other models have
been proposed to investigate the effects of nonperfect adherence to
drug regimens (Wahl and Nowak, 2000; Phillips et al., 2001; Huang
et al., 2003; Ferguson et al., 2005; Smith, 2006; Wu et al., 2006;
Rong et al., 2007a). Some more complex models have been
employed to study the question of drug resistance or viral evolution (Kirschner and Webb, 1996; Nelson et al., 2004; Gilchrist et al.,
2004; Iwasa et al., 2005; Iwami et al., 2006; Rong et al., 2007a,b).
In previous papers, most of the models exhibit only forward
bifurcation. However, during the last two decades, backward
bifurcations have been observed in a fraction of epidemic models
(Liu et al., 1986; Hadeler and van den Driessche, 1997; Dushoff
et al., 1998; van den Driessche and Watmough, 2000; Arino et al.,
2003; Martcheva and Thieme, 2003; Inaba and Sekine, 2004; Wang,
2006; Blayneh et al., 2010; Buonomo and Lacitignola, 2011). In
recent years, backward bifurcations are also found in viral dynamical models (Shu and Wang, 2012; Qesmi et al., 2010). When a
backward bifurcation occurs it is insufficient to reduce the basic
reproduction number below 1 to eradicate the infection. In this
case, the basic reproduction number must be reduced further to a
threshold value which is less than 1 in order to guarantee the
eradication. Therefore, identifying the occurrence of a backward
bifurcation has important implications in studying viral dynamics.
The purpose of the present paper is to develop a mathematical
model to depict the viral dynamics under the effects of antibodies
in the experiment (Balazs et al., 2012). We provide theoretical
analysis and implement numerical simulations to investigate the
antibody-based protection against HIV infection by vectored
immunoprophylaxis. We find that the recruitment of antibody
can lead to the occurrence of backward bifurcation and patterns
of bistability, which may be helpful for the selection of treatment
strategies in the future.
The paper is organized as follows. In the next section, we state the
model in consideration and the hypotheses on which it is formulated.
The existence and stability of equilibria and the backward bifurcation
are analyzed in Section 3. Some new and complicated dynamical
behaviors are given by detailed numerical simulations in Section 4
and a general discussion concludes the paper in Section 5.
2. Model formulation
We start from the following system which was proposed by
Bonhoeffer et al. (1997) and Nowak and May (2000), and is the
basis for mathematical studies of HIV dynamics
dT
¼ ldTbTV,
dt
n
dT
¼ bTVdT n ,
dt
dV
¼ N dT n cV,
dt
ð1Þ
where the state variables TðtÞ,T n ðtÞ and V(t) represent the concentrations of healthy CD4 þ T cells, infected cells and free
immunodeficiency virus (HIV) at time t, respectively, l is the
recruitment rate of uninfected CD4 þ T cells, and d represents the
natural death rate of CD4 þ T cells. The term bTV describes the
incidence of HIV infection of uninfected CD4 þ T cells, where b is
the infection coefficient. Infected cells die at a rate d. Furthermore, N represents the total number of free virus particles
released by each productively infected cell over its lifespan, and
c is the clearance rate of virus particles.
The humoral immunity, which is mediated by B lymphocytes
expressing antigen specific receptors, is a very important kind of
adaptive immune responses in HIV infection process. Kajiwara
and Sasaki (2004) and Murase et al. (2005) introduced the effect
of humoral immunity to system (1) to obtain the following
system of differential equations:
dT
¼ ldTbTV,
dt
n
dT
¼ bTVdT n ,
dt
dV
¼ NdT n cVpAV,
dt
dA
¼ aAVbA,
dt
ð2Þ
where A represents the concentration of antibodies in humoral
immune responses. The term pAV depicts the loss rate of virus
under attack of antibodies and p is the killing rate. Since the
humoral immunity is stimulated by virus in HIV infection, the
production rate of antibodies is supposed to be aAV. Moreover,
the natural clearance rate of antibodies is bA.
Considering that the vectored immunoprophylaxis (VIP) in mice
can induce the lifelong expression of monoclonal antibodies at high
concentrations from a single intramuscular injection, we suppose
that neutralizing antibodies are produced at a constant rate m after
the injection. Observe that graph (d) of Fig. 1 in Balazs et al. (2012)
depicts the comparison of the experiments in which, under HIV
challenge, most mice expressing luciferase (i.e., those mice without
receiving VIP) show the dramatic loss of CD4 cells whereas mice
expressing b12 antibody (i.e., those mice receiving VIP) show no
CD4 cell depletion. This indicates that without vectored immunoprophylaxis the antibody response induced by HIV in mice is so
weak that the level of CD4 cells drops dramatically so that it cannot
protect mice from infection at all, and the full protection from HIV
infection in mice results from the effect of immunity induced by
the vectored immunoprophylaxis. Thus, it is reasonable to consider
the case where the immunity by vectored immunoprophylaxis is
much stronger than the humoral immunity induced by HIV, and so
we can neglect the term aAV in (2). Furthermore, since we care
mainly about the effects of antibodies induced by vectored immunolprophylaxis, we ignore also the immune responses from the CTL
immunity induced by infected cells. Note that antibodies, which
are involved in the immune response to virus, can be removed
because antibodies bind to specific antigen, forming precipitation
or cell groups and then swallowed by phagocytes. Thus, we
incorporate the loss rate qAV of antibody from the effect of
antibodies’ involvement with virus. Taking all of the above into
considerations, we obtain the following model:
dT
¼ ldTbTV,
dt
n
dT
¼ bTVdT n ,
dt
dV
¼ NdT n cVpAV,
dt
dA
¼ mbAqAV:
dt
ð3Þ
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
The basic reproduction number for system (3) is
R0 ¼
b
LaSalle–Lyapunov theorem, every nonnegative solution of (3)
approaches E0 as t-1. &
l
cþp m d N:
Now let us consider positive equilibria of (3). An infection
equilibrium E1 ðT 1 ,T n1 ,V 1 ,A1 Þ of system (3) satisfies
8
ldT 1 bT 1 V 1 ¼ 0,
>
>
>
>
< bT 1 V 1 dT n ¼ 0,
1
b
Set
Rn0 ¼
blN
cd
,
Rn0A ¼ a
Nblcd
:
cbb
n
129
n
Then R0 is the basic reproduction number of (2) and R0A is the
antibody immune reproductive number of (2) (see Yan and Wang,
2012). It is shown in Yan and Wang (2012) that the dynamical
behaviors of system (2) are completely determined by Rn0 and Rn0A .
Specifically, the infection-free equilibrium is globally asymptotically stable if Rn0 r1; an immune-free equilibrium is globally
stable if Rn0 41 and Rn0A r1, and a coexistence equilibrium of
healthy CD4 þ T cells, infected cells, virus and antibodies is
globally stable if Rn0 41 and Rn0A 41. However, we will show that
the incorporation of vectored immunoprophylaxis and the loss of
antibodies due to involvement with virus induces rich dynamical
behaviors in (3), including the backward bifurcation, the subcritical Hopf bifurcation, and the bistable phenomena. These
findings may be helpful for designing the strategies of vectored
immunoprophylaxis.
N dT n1 cV 1 pA1 V 1 ¼ 0,
>
>
>
>
: mbA1 qA V 1 ¼ 0:
1
It follows that
T1 ¼
l
d þ bV 1
,
T n1 ¼
blV 1
m
, A1 ¼
,
dðd þ bV 1 Þ
bþ qV 1
where V1 is a positive solution of the following equation:
m2 V 21 þ m1 V 1 þ m0 ¼ 0,
ð4Þ
in which
m2 ¼ cqb,
m1 ¼ cdq þcbb þ pmb
dqðcb þ pmÞ
R0 ,
b
m0 ¼ dðbc þ pmÞð1R0 Þ:
3. Equilibria and backward bifurcation
There is always an infection-free equilibrium E0 ¼ ðl=d,0,0, m=bÞ
in (3). Evaluating the Jacobian matrix of system (3) at E0 gives
2
3
0
d 0
bl
d
6
7
bl
6 0 d
0 7
6
7
d
JðE0 Þ ¼ 6
7:
6 0 Nd c pm 0 7
b
4
5
b
0
0
qbm
Its characteristic equation is
pm
dðbc þpmÞ
ð1R0 Þ ¼ 0:
ðo þ dÞðo þ bÞ o2 þ d þc þ
oþ
b
b
If R0 o1, then all the solutions of the above characteristic
equation have negative real parts. If R0 41, the above characteristic equation has one positive root. Thus, by the Routh–Hurwitz
criterion, we have the following theorem:
Theorem 1. The infection-free equilibrium E0 is locally asymptotically stable if R0 o 1 and is unstable if R0 41.
Next we analyze the global stability of the infection-free
equilibrium E0 by constructing a Lyapunov function.
Theorem 2. The infection-free equilibrium E0 is globally asymptotically stable if Rn0 o 1, i.e., R0 obc=ðbc þ pmÞ.
Proof. We show below that E0 attracts the nonnegative solutions
of (3). From the first equation of system (3), we have dT=dt
r ldT: It follows that for any given E 40, there is some t0 such
that TðtÞ r l=d þ E,t r t 0 .
Define a Lyapunov function:
L ¼ NT n þ V:
For positive equilibria to exist, the solutions of Eq. (4) must be
positive. Note that m2 4 0 and m0 has the same sign as 1R0 . It
follows that (4) admits a unique positive solution if R0 4 1, which
means that (3) admits a unique positive equilibrium whenever
R0 41.
To consider the case of R0 r 1, we set D ¼ m21 4m2 m0 . When
R0 o1, it is easy to see that (4) admits two positive solutions if
m1 o 0
and
D 4 0,
ð5Þ
admits a unique positive solution if
m1 o 0
and
D ¼ 0,
ð6Þ
and there is no positive solution in (4) if
m1 o 0
and
D o 0:
ð7Þ
Note that m1 o 0 is equivalent to
R0 4
ðpmb þ cdq þ cbbÞb
:¼ R01 :
dqðcb þ pmÞ
ð8Þ
Since we aim to find positive equilibria of (3) when R0 o 1, we
need to ensure R01 o1 according to (8). R01 o 1 equals to the
following inequality:
ðpmb þ cdq þcbbÞb
o 1:
dqðcb þ pmÞ
ð9Þ
Solving b from (9) we get the following condition:
bo
dqpm
:¼ b1 :
bðcb þpmÞ
ð10Þ
By direct calculations we obtain
2
b D ¼ a2 R20 þ a1 R0 þa0 ,
ð11Þ
where
2
a2 ¼ d q2 ðcb þ pmÞ2 ,
Since Rn0 ¼ Nbl=ðcdÞ o 1, we can select E small enough such that
Nbðl=d þ EÞc o 0. Thus,
dL
l
¼ N bTVcV pAV r Nb
þ E c V r 0,
dt
d
a1 ¼ 2bdðcb þ pmÞqðcbbcdqpmbÞ,
and dL=dt ¼ 0 if and only if V ¼0. The largest compact invariant set
in fðT,T n ,V,AÞ : dL=dt ¼ 0g is the singleton fE0 g. Therefore, by the
If D0 ¼ a21 4a2 a0 , by direct calculations we see that D0 o 0 if
b 4 dq=b. This case is not interesting because b1 o dq=b. When
2
a0 ¼ b ðcbb þ pmbcdqÞ2 :
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X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
b o b1 , we have D0 4 0. Note that (10) implies
2
2
2
b D9R0 ¼ R01 ¼ 4cbb ðpmbb þ cbb dqpmÞ o0,
ð12Þ
and
2
2
b D9R0 ¼ 1 ¼ ðcbb þ pmbbdqpmÞ2 4 0:
ð13Þ
Since a2 4 0, it follows from (12) and (13) that the equation of
D ¼ 0 in R0 admits a root Rc0 in the interval ðR01 ,1Þ where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1 þ a21 4a2 a0
Rc0 ¼
ð14Þ
2a2
dynamical behaviors of system (3). By fixing q ¼ 104 , which is
higher than q ¼ 106 for the case of forward bifurcation in Fig. 1,
and choosing R0 as a bifurcation parameter, we observe a backward bifurcation, which occurs through the branch point ðBPÞ at
R0 ¼ 1 (see Fig. 2). In addition, we find a Hopf bifurcation point ðHÞ
at R0 ¼ 0.8243 and a fold bifurcation point ðLPÞ at R0 ¼ 0.6111 (see
Fig. 2). At the Hopf point ðHÞ, the values of the four variables T, Tn,
11000
10000
9000
such that D ¼ 0 when R0 ¼ Rc0 , D 40 when Rc0 oR0 o1, and D o 0
when R0 oRc0 . Consequently, we can state the following results for
positive equilibria of (3).
Theorem 3 indicates that system (3) exhibits multiple infection equilibria via a backward bifurcation when (10) holds. This
has two biological implications. First, the introduction of the
vectored immunoprophylaxis induces the backward bifurcations.
In contrast, model (2) with only humoral immunity exhibits the
global stability of an equilibrium (Yan and Wang, 2012). Second,
since (10) cannot be satisfied when q¼ 0, it follows that the
ignorance of antibodies’ loss due to the involvement with virus
may result in the loss of backward bifurcation, and thus give
wrong predictions of HIV evolutions.
7000
Virus
Theorem 3. System (3) has a unique infection equilibrium whenever
R0 4 1, and has two infection equilibria when Rc0 o R0 o1 and (10)
holds. Moreover, there exists a unique infection equilibrium of
multiplicity 2 in (3) when (10) holds and R0 ¼ Rc0 o 1.
8000
6000
5000
4000
3000
2000
1000
0
0.6
0.7
0.8
0.9
1
1.1
1.2
2.2
x 104
2
1.8
1.6
1.4
V
In this section, we implement numerical simulations to testify
the theoretical results above and explore more patterns of
dynamical behaviors of model (3). The parameter values are
chosen from literatures (Zack et al., 1990; Perelson et al., 1996;
Pawelek et al., 2012; Rong et al., 2007b; Stafford et al., 2000) by
fixing l ¼ 50 cells ml day 1, d ¼0.008 day 1, b ¼ 5 107 ml
1
virion day 1, d ¼ 0:8 day , c¼3 day 1. The estimate of burst
size, N, varies from 100 to a few thousands (Haase et al., 1996;
Hlavacek et al., 1999) and possibly could be significantly larger
(Chen et al., 2007). The value of m is mainly determined by the
dosage of vectored immunoprophylaxis and the values of the
other three parameters p, b and q may vary with various kinds
of antibodies. We set m ¼ 12 cells ml day 1, p ¼ 5 103 ml
virion day 1 and b¼0.02 day 1 in all the following simulations
(see also Table 1).
A forward bifurcation is shown in Fig. 1 where q ¼ 106 and R0
is a bifurcation parameter. With the help of the MatCont package
(Dhooge et al., 2003, 2008) we obtain some more complicated
1.2
1
H
0.8
0.6
0.4
0.2
0
0.4
LP
BP
0.5
0.6
0.7
0.8
R0
0.9
1
Parameter
Value
Description
Reference
l
50 cells ml day 1
0.008 day 1
5 10 7 ml virion day 1
0.8 day 1
Varied
3 day 1
Varied
Varied
Varied
Varied
Recruitment rate of uninfected cells
Death rate of uninfected cells
Infection rate of target cells by virus
Death rate of infected cells
Burst size of virus
Clearance rate of free virus
Killing rate of antibody
Recruitment rate of antibody
Clearance rate of antibody
Involvement rate of antibody
Stafford et al. (2000)
Stafford et al. (2000)
Stafford et al. (2000)
Zack et al. (1990)
See text
Perelson et al. (1996)
See text
See text
See text
See text
N
c
p
m
b
q
1.1
1.2
Fig. 2. Graph of V component in equilibria versus R0 by fixing
l ¼ 50,d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,q ¼ 104 ,
where H denotes a Hopf bifurcation point, LP is a fold bifurcation point and BP
represents a branch point.
Table 1
Parameter definitions and values used in numerical simulations.
b
d
1.4
Fig. 1. The diagram of a forward bifurcation. The parameters are
l ¼ 50,d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,q ¼ 106 .
4. Numerical simulations
d
1.3
R0
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
V and A are 3866:5166,23:8348,9863:0727 and 11.9248, respectively. At the fold bifurcation point ðLPÞ, the values of T,T n ,V and A
are 5689:0464,5:6095,1577:6382 and 67.5053, respectively. At
the branch point ðBPÞ, the values of these four variables are
6250:0000,0:0000,0:0000 and 600.0000, respectively. The first
Lyapunov coefficient at the Hopf point is l1 ¼ 3:2165 104 ,
which means that the Hopf bifurcation is subcritical and the
periodic orbits are born unstable. The fold bifurcation and the
backward bifurcation lead to the existence of multiple infection
equilibria for 0:6111oR0 o1. Note that the qualitative difference
between Figs. 1 and 2 results from the fact that the value of q in
Fig. 1 is smaller than it in Fig. 2. This indicates that a stronger loss
1600
1400
1200
Period
1000
800
600
400
200
LPC
0
0.6
0.8
1
1.2
1.4
1.6
1.8
2
R0
Fig. 3. Bifurcation curve of period of cycles versus R0 by fixing
l ¼ 50,d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,q ¼ 104 ,
where LPC represents the limit point bifurcation of cycles.
131
of antibodies’ involvement with virus can lead to a backward
bifurcation, which supports the result of Theorem 3.
Using the detected Hopf point in Fig. 2 to start a limit cycle
continuation we find that the periodic orbits are initially unstable.
A limit point of cycles (LPC) is detected at R0 ¼1.8214 and stability
is gained at this moment. At this limit point (LPC), the period of
the cycle is 125.6932. As the value of R0 decreases, the stability is
preserved but the period appears to be infinity as the value of R0
tends to 1.0000 from right (see Fig. 3), which suggests that the
stable limit cycle turns to a homoclinic orbit as R0 approaches
1.0000.
From Figs. 2 and 3 we conclude that if we increase R0 from
0 continuously, we can obtain the five typical patterns of
dynamical behaviors. These typical patterns of dynamical behaviors are listed briefly in Table 2. As an example, we choose a
value of N within each interval corresponding to each pattern
to illustrate the five dynamical behaviors of system (3). The
other parameters are fixed as the same as above: l ¼ 50,
d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,
q ¼ 104 .
Firstly, if 0 oR0 r 0:6111, i.e., 0 oN r 1173, that is, the reproduction number (or equivalently, the burst size) is relatively
small, then the infection-free equilibrium is the unique steady
state and is globally asymptotically stable. Fig. 4 illustrates this
pattern where the burst size is chosen as N ¼1000 (R0 ¼0.5208).
Secondly, if 0:6111 oR0 r 0:8243, i.e., 1173 oN r 1582, that is,
the burst size becomes a little larger, then the stable infectionfree equilibrium coexists with two unstable positive equilibria.
For this pattern, we choose N ¼1400 (R0 ¼0.7292) to obtain Fig. 5,
in which a numerical solution of system (3) tends also to the
infection-free equilibrium.
Thirdly, if 0:8243o R0 o1:0000, i.e., 1582 o N o1920, then
there is a stable infection-free equilibrium, an unstable positive
Table 2
Five typical patterns of dynamical behaviors of system (3).
Pattern
Range of R0
(Range of N)
Steady states of system (3)
1
0 o R0 r 0:6111
ð0 o N r 1173Þ
0:6111 o R0 r 0:8243
ð1173 o N r 1582Þ
0:8243 o R0 o 1:0000
ð1582 o N o 1920Þ
1:0000 o R0 r 1:8214
ð1920 o N r 3497Þ
R0 41:8214
ðN 4 3497Þ
A stable infection-free equilibrium
2
3
4
5
4
A stable infection-free equilibrium and two unstable positive equilibria
A stable infection-free equilibrium, an unstable positive equilibrium, a stable positive equilibrium and an unstable limit cycle
An unstable infection-free equilibrium, a stable positive equilibrium, an unstable limit cycle and a stable limit cycle
An unstable infection-free equilibrium and a stable positive equilibrium
x 104
700
3.5
600
3
500
Antibody
Virus
2.5
2
1.5
400
300
1
200
0.5
100
0
0
0
20
40
60
80 100 120 140 160 180 200
Time
0
100 200 300 400 500 600 700 800 900 1000
Time
Fig. 4. A numerical solution of system (3) tends to the infection-free equilibrium as time tends to infinity, where N ¼ 1000,R0 ¼ 0:5208. The initial values are
Tð0Þ ¼ 8000,T n ð0Þ ¼ 0,V ð0Þ ¼ 2000,Að0Þ ¼ 0. (a) Time series of V. (b) Time series of A.
132
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
2.5
x 104
700
600
2
Antibody
Virus
500
1.5
1
400
300
200
0.5
100
0
0
0
0
100 200 300 400 500 600 700 800
100 200 300 400 500 600 700 800 900 1000
Time
Time
Fig. 5. A numerical solution of system (3) tends to the infection-free equilibrium as time tends to infinity, where N ¼ 1400,R0 ¼ 0:7292. The initial values are
Tð0Þ ¼ 4400,T n ð0Þ ¼ 9,Vð0Þ ¼ 7000,Að0Þ ¼ 0. (a) Time series of V. (b) Time series of A.
4.5
x 104
700
4
600
3.5
500
Antibody
Virus
3
2.5
2
1.5
400
300
200
1
100
0.5
0
0
0
50 100 150 200 250 300 350 400 450 500
0
100
200
Time
300
400
500
600
700
800
Time
Fig. 6. A numerical solution of system (3) tends to the infection-free equilibrium as time tends to infinity, where N ¼ 1700,R0 ¼ 0:8854. The initial values are
Tð0Þ ¼ 4080,T n ð0Þ ¼ 30,Vð0Þ ¼ 11,800,Að0Þ ¼ 0. (a) Time series of V. (b) Time series of A.
1.24
x 104
1.23
15
1.22
Antibody
Virus
1.21
1.2
1.19
1.18
1.17
5
0
27.5
1.16
1.15
1.14
10
0
500
1000 1500 2000 2500 3000 3500 4000
Time
27
26.5
Inf
ect
ed
Tc
ells
26
25.5
3575
3580
3585
3590
ted T
c
Uninfe
3595
3600
cells
Fig. 7. A numerical solution of system (3) tends to the positive equilibrium as time tends to infinity, where N ¼ 1700,R0 ¼ 0:8854. The initial values are
Tð0Þ ¼ 3580,T n ð0Þ ¼ 27,V ð0Þ ¼ 11,870,Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space.
equilibrium, a stable positive equilibrium and an unstable limit
cycle which is born from the subcritical Hopf bifurcation. The
limit cycle encloses the stable positive equilibrium and it contracts as R0 increases in the interval. Hence, the bistability occurs,
where a stable disease-free equilibrium coexists with a stable
positive equilibrium. For this pattern, the burst size is chosen
to be N ¼ 1700 (R0 ¼0.8854). As depicted in Figs. 6 and 7, the solution with the initial condition Tð0Þ ¼ 4080,T n ð0Þ ¼ 30,Vð0Þ ¼
11 800,Að0Þ ¼ 0 tends to the infection-free equilibrium while the
solution with the initial condition Tð0Þ ¼ 3580,T n ð0Þ ¼ 27,Vð0Þ ¼
11870,Að0Þ ¼ 0 tends to the stable positive equilibrium inside the
unstable limit cycle.
Fourthly, if 1:0000 o R0 r1:8214, i.e., 1920o N r3497, that is,
the burst size is larger than that in the third case, then another
type of bistability occurs, where a stable positive equilibrium
coexists with a stable limit cycle. Specifically, the infection-free
equilibrium becomes unstable and it coexists with a stable
positive equilibrium and two limit cycles, where the smaller limit
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
cycle is unstable and the larger one is stable. The smaller limit
cycle encloses the positive equilibrium and firstly contracts and
then enlarges as R0 increases in the interval. The larger limit
cycle encloses the smaller limit cycle and contracts as R0
increases in the interval. These two limit cycles collide when
R0 reaches 1.8214. In this case, the ultimate infection state
in vivo depends on the initial infection state. If the initial state
lies in the basin of attraction for the stable positive equilibrium,
then the infection state will tend to this positive equilibrium at
last. If the initial state locates in the basin of attraction for the
stable limit cycle, then stable periodic oscillations will occur as
time tends to infinity. For this pattern, we choose N ¼3400
(R0 ¼1.7708). In Fig. 8, the solution with the initial condition
Tð0Þ ¼ 1770, T n ð0Þ ¼ 120,Vð0Þ ¼ 40,392,Að0Þ ¼ 0 tends to the positive equilibrium. In Fig. 9, the solution with the initial condition
Tð0Þ ¼ 6000,T n ð0Þ ¼ 0,Vð0Þ ¼ 106 ,Að0Þ ¼ 0 tends to the stable
limit cycle. The phenomenon of bistability is depicted in
Fig. 10. The blue and the green orbits, whose initial values lie
in the basin of attraction for the stable limit cycle, both oscillate
to the stable limit cycle. The magenta orbit, whose initial value
locates in the basin of attraction for the stable positive equilibrium, tends to the stable positive equilibrium also in an
oscillatory form.
Fifthly, if R0 4 1:8214, i.e., N 43497, that is, the burst size
is large enough, then the unstable infection-free equilibrium coexists
10
16
133
x 105
14
12
Virus
10
8
6
4
2
0
0
1000
2000
3000
4000
Uninfected T cells
5000
6000
Fig. 10. Graph of bistability (a stable positive equilibrium and a stable limit cycle)
on the TV-plane. A blue orbit, with initial values Tð0Þ ¼ 6000,T n ð0Þ ¼ 0,
Vð0Þ ¼ 106 ,Að0Þ ¼ 0, tends to the stable limit cycle from outside. A green orbit,
with initial values Tð0Þ ¼ 3770,T n ð0Þ ¼ 120,V ð0Þ ¼ 40,392,Að0Þ ¼ 0, tends to the
stable limit cycle from inside. A magenta orbit, with initial values Tð0Þ ¼ 2770,
T n ð0Þ ¼ 120,Vð0Þ ¼ 40,392,Að0Þ ¼ 0, tends to the positive equilibrium inside the
cycle, where N ¼ 3400,R0 ¼ 1:7708. (For interpretation of the references to color in
this figure caption, the reader is referred to the web version of this article.)
x 104
9
8
Antibody
7
Virus
6
5
4
10
5
0
150
2
1
2000
1900
100
Inf
ec
ted
Tc
3
1800
ells
0
100
200
300
400
500
600
700
1700
50
ed T
1600
1500
0
800
ect
ninf
s
cell
U
Time
Fig. 8. A numerical solution of system (3) tends to the positive equilibrium as time tends to infinity, where N ¼ 3400, R0 ¼1.7708. The initial values are
Tð0Þ ¼ 1770,T n ð0Þ ¼ 120,V ð0Þ ¼ 40,392,Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space.
16
x 105
14
600
Antibody
12
10
8
400
200
6
0
2000
4
2
0
0
500
1000
Time
1500
1500
1000
Infe
cted
500
T ce
lls
0
0
4000
2000
6000
8000
ells
dTc
cte
ninfe
U
Fig. 9. A numerical solution of system (3) tends to a stable limit cycle as time tends to infinity. N ¼ 3400, R0 ¼1.7708. The initial values are
Tð0Þ ¼ 6000,T n ð0Þ ¼ 0,V ð0Þ ¼ 106 ,Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space.
134
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
8
x 105
400
6
300
Antibody
7
Virus
5
4
3
2
1
0
50 100 150 200 250 300 350 400 450 500 550 600
200
100
0
2500
2000
1500
Infe
1000
cte
500
dT
cell
s
0
0
2000
4000
ected
Uninf
8000
6000
T cell
s
Time
Fig. 11. A numerical solution of system (3) tends to the positive equilibrium as time tends to infinity, where N ¼ 5000, R0 ¼ 2.6042. The initial values are
Tð0Þ ¼ 6000,T n ð0Þ ¼ 0, Vð0Þ ¼ 106 , Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space.
with a stable positive equilibrium, but there is no limit cycle. All
positive orbits tend to the positive equilibrium. For this pattern, the
burst size is chosen as N¼5000 (R0 ¼2.6042). As depicted in Fig. 11,
the solution tends to the positive equilibrium in this case.
Acknowledgments
We are very grateful to anonymous reviewers for their valuable comments and suggestions.
5. Concluding remarks
In this paper, we have developed a mathematical model based on
a vectored immunoprophylaxis experiment published in Nature. By
combining theoretical analysis and numerical simulations we have
found that the model can exhibit some new and complicated
dynamical behaviors. We have shown that the infection-free equilibrium is globally asymptotically stable when the basic reproduction
number is smaller than a threshold. In this case, the mouse can be
completely protected from the infection. We have also proved that
the introduction of the vectored immunoprophylaxis can induce the
backward bifurcation and the ignorance of antibodies’ loss due to
their involvement with virus may result in the loss of backward
bifurcation. This means that driving the basic reproduction number
below 1 is not enough to eradicate the infection.
Another interesting result is that the model can exhibit five
different patterns of dynamical behaviors if we choose the burst
size N as a bifurcation parameter. When the burst size is relatively
small, the infection-free equilibrium is globally asymptotically
stable. When the burst size is a little larger, the stable infectionfree equilibrium coexists with two unstable positive equilibria. In
these two cases, the infection could be thoroughly eradicated. But
when the burst size is larger than a critical value, there is a stable
infection-free equilibrium, an unstable positive equilibrium, a
stable positive equilibrium and an unstable limit cycle. In this
case, whether the infection can be eradicated depends on the
initial state. When the burst size is larger than the second critical
value, the unstable infection-free equilibrium coexists with a
stable positive equilibrium, a small unstable limit cycle and a
large stable limit cycle. Thus, the bistability occurs in this case. If
the burst size is large enough, the model admits a unique stable
positive equilibrium. In the last two cases, the infection persists
in vivo.
The mathematical analysis and numerical simulations of the
model proposed in this paper provide insights into the underlying
viral infection mechanisms under vectored immunoprophylaxis.
It is interesting to consider the synthetical effects of immunoprophylaxis and cell immunity. We leave this as a future work.
References
Arino, J., McCluskey, C.C., van den Driessche, P., 2003. Global results for an
epidemic model with vaccination that exhibits backward bifurcation. SIAM J.
Appl. Math. 64, 260–276.
Bailey, J., Blankson, J.N., Wind-Rotolo, M., Siliciano, R.F., 2004. Mechanisms of HIV1 escape from immune responses and antiretroviral drugs. Curr. Opin.
Immunol. 16, 470–476.
Balazs, A.B., Chen, J., Hong, C.M., Rao, D.S., Yang, L., Baltimore, D., 2012. Antibodybased protection against HIV infection by vectored immunoprophylaxis.
Nature 481, 81–84.
Blayneh, K.W., Gumel, A.B., Lenhart, S., Clayton, T., 2010. Backward bifurcation and
optimal control in transmission dynamics of West Nile Virus. Bull. Math. Biol.
72, 1006–1028.
Bonhoeffer, S., May, R.M., Shaw, G.M., Nowak, M.A., 1997. Virus dynamics and drug
therapy. Proc. Natl. Acad. Sci. U.S.A. 94, 6971–6976.
Buonomo, B., Lacitignola, D., 2011. On the backward bifurcation of a vaccination
model with nonlinear incidence. Nonlinear Anal. Model. 16, 30–46.
Chen, H.Y., Mascio, M.D., Perelson, A.S., Ho, D.D., Zhang, L., 2007. Determination of
virus burst size in vivo using a single-cycle SIV in rhesus macaques. Proc. Natl.
Acad. Sci. U.S.A. 104, 19079–19084.
Chun, T.W., Carruth, L., Finzi, D., Shen, X., DiGiuseppe, J.A., Taylor, H., Hermankova,
M., Chadwick, K., Margolick, J., Quinn, T.C., Kuo, Y.H., Brookmeyer, R., Zeiger,
M.A., Barditch-Crovo, P., Siliciano, R.F., 1997a. Quantification of latent tissue
reservoirs and total body viral load in HIV-1 infection. Nat. Rev. Microbiol. 387,
183–188.
Chun, T.W., Engel, D., Mizell, S.B., Ehler, L.A., Fauci, A.S., 1998. Induction of HIV-1
replication in latently infected CD4 þ T cells using a combination of cytokines.
J. Exp. Med 188, 83–91.
Chun, T.W., Finzi, D., Margolick, J., Chadwick, K., Schwartz, D., Siliciano, R.F., 1995.
In vivo fate of HIV-1-infected T cells: quantitative analysis of the transition to
stable latency. Nat. Med. 1, 1284–1290.
Chun, T.W., Stuyver, L., Mizell, S.B., Ehler, L.A., Mican, J.A., Baseler, M., Lloyd, A.L.,
Nowak, M.A., Fauci, A.S., 1997b. Presence of an inducible HIV-1 latent reservoir
during highly active antiretroviral therapy. Proc. Natl. Acad. Sci. U.S.A. 94,
13193–13197.
Clavel, F., Hance, A.J., 2004. HIV drug resistance. N. Engl. J. Med. 350, 1023–1035.
Culshaw, R.V., Ruan, S.G., 2000. A delay-differential equation model of HIV
infection of CD4þ T cells. Math. Biosci. 165, 27–39.
Davenport, M.P., Loh, L., Petravic, J., Kent, S.J., 2008. Rates of HIV immune escape
and reversion: implications for vaccination. Trends Microbiol. 16, 561–566.
Dhooge, A., Govaerts, W., Kuznetsov, Y.A., 2003. Matcont: A Matlab package for
numerical bifurcation analysis of odes. ACM Trans. Math. Software 29,
141–164.
Dhooge, A., Govaerts, W., Kuznetsov, Y.A., Meijer, H.G.E., Sautois, B., 2008. New
features of the software matcont for bifurcation analysis of dynamical
systems. Math. Comput. Model. Dyn. Syst. 14, 145–175.
X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135
Dixit, N.M., Perelson, A.S., 2004. Complex patterns of viral load decay under
antiretroviral therapy: influence of pharmacokinetics and intracellular delay. J.
Theor. Biol. 226, 95–109.
van den Driessche, P., Watmough, J., 2000. A simple SIS epidemic model with a
backward bifurcation. J. Math. Biol. 40, 525–540.
Dushoff, J., Huang, W., Castillo-Chavez, C., 1998. Backward bifurcations and
catastrophe in simple models of fatal diseases. J. Math. Biol. 36, 227–248.
Ferguson, N.M., Donnelly, C.A., Hooper, J., Ghani, A.C., Fraser, C., Bartley, L.M., Rode,
R.A., Vernazza, P., Lapins, D., Mayer, S.L., Anderson, R.M., 2005. Adherence to
antiretroviral therapy and its impact on clinical outcome in HIV-infected
patients. J. R. Soc. Interface 2, 349–363.
Finzi, D., Hermankova, M., Pierson, T., Carruth, L.M., Buck, C., Chaisson, R.E., Quinn,
T.C., Chadwick, K., Margolick, J., Brookmeyer, R., Gallant, J., Maarkowitz, M., Ho,
D.D., Richman, D.D., Siliciano, R.F., 1997. Identification of a reservoir for HIV-1
in patients on highly active antiretroviral therapy. Science 278, 1295–1300.
Gilchrist, M.A., Coombs, D., Perelson, A.S., 2004. Optimizing within-host viral
fitness: infected cell lifespan and virion production rate. J. Theor. Biol. 229,
281–288.
Haase, A.T., Henry, K., Zupancic, M., Sedgewick, G., Faust, R.A., Melroe, H., Cavert,
W., Gebhard, K., Staskus, K., Zhang, Z.Q., Dailey, P.J., Balfour, H.H.J., Erice, A.,
Perelson, A.S., 1996. Quantitative image analysis of HIV-1 infection in
lymphoid tissue. Science 274, 985–989.
Hadeler, K.P., van den Driessche, P., 1997. Backward bifurcation in epidemic
control. Math. Biosci. 146, 15–35.
Herz, V., Bonhoeffer, S., Anderson, R., May, R., Nowak, M., 1996. Viral dynamics
in vivo: limitations on estimates of intracellular delay and virus decay. Proc.
Natl. Acad. Sci. U.S.A. 93, 7247–7251.
Hlavacek, W.S., Wofsy, C., Perelson, A.S., 1999. Dissociation of HIV-1 from follicular
dendritic cells during HAART: mathematical analysis. Proc. Natl. Acad. Sci.
U.S.A. 96, 14681–14686.
Huang, Y., Rosenkranz, S.L., Wu, H., 2003. Modeling HIV dynamics and antiviral
response with consideration of time-varying drug exposures, adherence and
phenotypic sensitivity. Math. Biosci. 184, 165–186.
Inaba, H., Sekine, H., 2004. A mathematical model for chagas disease with
infection-age-dependent infectivity. Math. Biosci. 190, 39–69.
Iwami, S., Miura, T., Nakaoka, S., Takeuchi, Y., 2009. Immune impairment in HIV
infection: existence of risky and immunodeficiency thresholds. J. Theor. Biol
260, 490–501.
Iwami, S., Nakaoka, S., Takeuchi, Y., 2006. Frequency dependence and viral
diversity imply chaos in an HIV model. Phys D: Nonlinear Phenom. 223,
222–228.
Iwasa, Y., Franziska, M., Nowak, M.A., 2005. Virus evolution with patients increases
pathogenicity. J. Theor. Biol. 232, 17–26.
Kajiwara, T., Sasaki, T., 2004. A note on the stability analysis of pathogen-immune
interaction dynamics. Discrete Cont. Dyn. B 4, 615–622.
Kirschner, D.E., Webb, G.F., 1996. A model for treatment strategy in the
chemotherapy of AIDS. Bull. Math. Biol. 58, 367–390.
Lambotte, O., Chaix, M.L., Gubler, B., Nasreddine, N., Wallon, C., Goujard, C.,
Rouzioux, C., Taoufik, Y., Delfraissy, J.F., 2004. The lymphocyte HIV reservoir
in patients on long-term HAART is a memory of virus evolution. AIDS 18,
1147–1158.
Li, M.Y., Shu, H., 2010. Global dynamics of an in-host viral model with intracellular
delay. Bull. Math. Biol. 72, 1492–1505.
Liu, W.M., Levin, S.A., Iwasa, Y., 1986. Influence of nonlinear incidence rates upon
the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204.
Martcheva, M., Thieme, H.R., 2003. Progression age enhanced backward bifurcation in an epidemic model with superinfection. J. Math. Biol. 46, 385–424.
Mittler, J.E., Markowitz, M., Ho, D.D., Perelson, A.S., 1999. Refined estimates for
HIV-1 clearance rate and intracellular delay. AIDS 13, 1415–1417.
Murase, A., Sasaki, T., Kajiwara, T., 2005. Stability analysis of pathogen–immune
interaction dynamics. J. Math. Biol. 51, 247–267.
Nelson, P.W., Gilchrist, M.A., Coombs, D., Hyman, J.M., Perelson, A.S., 2004. An agestructured model of HIV infection that allows for variations in the production
rate of viral particles and the death rate of productively infected cells. Math.
Biosci. Eng. 1, 267–288.
135
Nelson, P.W., Murray, J.D., Perelson, A.S., 2000. A model of HIV-1 pathogenesis that
includes an intracellular delay. Math. Biosci. 163, 201–215.
Nelson, P.W., Perelson, A.S., 2002. Mathematical analysis of delay differential
equation models of HIV-1 infection. Math. Biosci. 179, 73–94.
Nowak, M.A., May, R.M., 2000. Virus Dynamics: Mathematical Principles of
Immunology and Virology. Oxford University Press, Oxford.
Pang, H., Wang, W., Wang, K., 2005. Global properties of virus dynamics model
with immune response. J. Southwest China Norm. Univ. (Nat. Sci.) 30,
796–799.
Pawelek, K.A., Liu, S., Pahlevani, F., Rong, L., 2012. A model of HIV-1 infection with
two time delays: mathematical analysis and comparison with patient data.
Math. Biosci. 235, 98–109.
Perelson, A.S., Kirschner, D.E., Boer, R.D., 1993. Dynamics of HIV infection of CD4 þ
T cells. Math. Biosci. 114, 81–125.
Perelson, A.S., Nelson, P.W., 1999. Mathematical analysis of HIV-1 dynamics
in vivo. SIAM Rev. 41, 3–44.
Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D., 1996. HIV-1
dynamics in vivo: virion clearance rate, infected cell life-span, and viral
generation time. Science 271, 1582–1586.
Phillips, A.N., Youle, M., Johnson, M., Loveday, C., 2001. Use of a stochastic model to
develop understanding of the impact of different patterns of antiretroviral
drug use on resistance development. AIDS 15, 2211–2220.
Qesmi, R., Wu, J., Wu, J., Heffernan, J.M., 2010. Influence of backward bifurcation in
a model of hepatitis B and C viruses. Math. Biosci. 224, 118–125.
Rong, L., Feng, Z., Perelson, A.S., 2007a. Emergence of HIV-1 drug resistance during
antiretroviral treatment. Bull. Math. Biol. 69, 2027–2060.
Rong, L., Gilchrist, M.A., Feng, Z., Perelson, A.S., 2007b. Modeling within-host HIV-1
dynamics and the evolution of drug resistance: trade-offs between viral
enzyme function and drug susceptibility. J. Theor. Biol. 247, 804–818.
Shu, H., Wang, L., 2012. Role of CD4 þ T-cell proliferation in HIV infection under
antiretroviral therapy. J. Math. Anal. Appl. 394, 529–544.
Simon, V., Ho, D.D., 2003. HIV-1 dynamics in vivo: implications for therapy. Nat.
Rev. Microbiol. 1, 181–190.
Smith, R.J., 2006. Adherence to antiretroviral HIV drugs: how many doses can you
miss before resistance emerges?. Proc. R. Soc. B 273, 617–624.
Stafford, M.A., Corey, L., Cao, Y., Daar, E.S., Ho, D.D., Perelson, A.S., 2000. Modeling
plasma virus concentration during primary HIV infection. J. Theor. Biol. 203,
285–301.
Wahl, L.M., Nowak, M.A., 2000. Adherence and drug resistance: predictions for
therapy outcome. Proc. R. Soc. Lond. B 267, 835–843.
Wang, K., Wang, W., Liu, X., 2006a. Global stability in a viral infection model with
lytic and nonlytic immune responses. Comput. Math. Appl. 51, 1593–1610.
Wang, K., Wang, W., Liu, X., 2006b. Viral infection model with periodic lytic
immune response. Chaos Solutions Fractals 28, 90–99.
Wang, W., 2006. Backward bifurcation of an epidemic model with treatment.
Math. Biosci. 201, 58–71.
Wodarz, D., 2003. Hepatitis C virus dynamics and pathology: the role of CTL and
antibody responses. J. Gen. Virol. 84, 1743–1750.
Wong, J.K., Hezareh, M., Gunthard, H.F., Havlir, D.V., Ignacio, C.C., Spina, C.A.,
Richman, D.D., 1997. Recovery of replication-competent HIV despite prolonged suppression of plasma viremia. Science 278, 1291–1295.
Wu, H., Huang, Y., Acosta, E.P., Park, J.G., Yu, S., Rosenkranz, S.L., Kuritzkes, D.R.,
Eron, J.J., Perelson, A.S., Gerber, J.G., 2006. Pharmacodynamics of antiretroviral
agents in HIV-1 infected patients: using viral dynamic models that incorporate
drug susceptibility and adherence. J. Pharmacokinet. Pharmacodyn. 33,
399–419.
Wu, H., Huang, Y., Acosta, E.P., Rosenkranz, S.L., Kuritzkes, D.R., Eron, J.J., Perelson,
A.S., Gerber, J.G., 2005. Modeling long-term HIV dynamics and antiretroviral
response: effects of drug potency, pharmacokinetics, adherence, and drug
resistance. J. Acquir. Immune Defic. Syndr. 39, 272–283.
Yan, Y., Wang, W., 2012. Global stability of a five-dimensional model with immune
responses and delay. Discrete Cont. Dyn. B 17, 401–416.
Zack, J.A., Arrigo, S.J., Weitsman, S.R., Go, A.S., Haislip, A., Chen, I.S., 1990. HIV-1
entry into quiescent primary lymphocytes: molecular analysis reveals a labile
latent viral structure. Cell 61, 213–222.