Mathematical Biosciences 235 (2012) 98–109
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Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
A model of HIV-1 infection with two time delays: Mathematical analysis
and comparison with patient data
Kasia A. Pawelek a,1, Shengqiang Liu b,1, Faranak Pahlevani c, Libin Rong a,⇑
a
Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA
The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, Harbin 150080, China
c
Division of Science and Engineering, Penn State University, Abington College, Abington, PA 19001, USA
b
a r t i c l e
i n f o
Article history:
Received 27 June 2011
Received in revised form 31 October 2011
Accepted 4 November 2011
Available online 13 November 2011
Keywords:
Mathematical model
Virus dynamics
Stability analysis
Delays
Data fitting
a b s t r a c t
Mathematical models have made considerable contributions to our understanding of HIV dynamics. Introducing time delays to HIV models usually brings challenges to both mathematical analysis of the models
and comparison of model predictions with patient data. In this paper, we incorporate two delays, one the
time needed for infected cells to produce virions after viral entry and the other the time needed for the
adaptive immune response to emerge to control viral replication, into an HIV-1 model. We begin model
analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free
and infected steady states, and uniform persistence of the system. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of the steady states. We also fit the model including two
delays to viral load data from 10 patients during primary HIV-1 infection and estimate parameter values.
Although the delay model provides better fits to patient data (achieving a smaller error between data and
modeling prediction) than the one without delays, we could not determine which one is better from the
statistical standpoint. This highlights the need of more data sets for model verification and selection when
we incorporate time delays into mathematical models to study virus dynamics.
2011 Elsevier Inc. All rights reserved.
1. Introduction
Human immunodeficiency virus (HIV) infection is characterized
by three different phases, namely the primary infection, clinically
asymptomatic stage (chronic infection), and acquired immunodeficiency syndrome (AIDS) or drug therapy. During primary infection,
viral load in the peripheral blood experiences a substantial increase to the peak level, followed by decline to the steady state,
which is referred to as the viral set point [13,45]. Extremely high
viral load during primary infection leads to the activation of
CD8+ T cells, which are recognized as cytotoxic T cells (CTL) capable
of suppressing viral replication. Viral decline from the peak is due
to the control by these immune cells and/or limited target cell
availability [14]. The viral set point has been shown to be predictive for the pace of disease development [33]. The higher the viral
set point, the more quickly disease progresses to full-blown AIDS. A
better understanding of the virus dynamics will provide more insights into the viral control during primary infection. Knowledge
about the host immune response to HIV infection could be crucial
for the development of HIV vaccines and treatment of the infection
by enhancing immune response [20].
⇑ Corresponding author. Tel.: +1 248 370 3446; fax: +1 248 370 4184.
1
E-mail address: [email protected] (L. Rong).
These authors contributed equally to this study.
0025-5564/$ - see front matter 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2011.11.002
Mathematical models have made considerable contributions to
our understanding of HIV infection, immune responses, and antiretroviral treatment [5,6,14,28,37,39,41,43,44,47,48,55,56,59] (see
reviews in [38,40,42,51,63]). Time delays have also been incorporated into mathematical models to study virus dynamics. To characterize the time between the initial viral entry into a target cell
and subsequent viral production, an intracellular delay was first
introduced by Herz et al. [21] to analyze the clinical data. Assuming that the level of target cells is constant and that the protease
inhibitor is 100% effective, they obtained the expression of the viral
load and explored the effect of the intracellular delay on viral load
change [21]. Nelson et al. [36] analyzed the delay model with
imperfect drug treatment. They provided an analytical expression
of the dominant eigenvalue that determines the rate of viral decay.
Their result explains why there was no change in the estimate of
the infected cell death using the delay model in [21]. The delay effect canceled out due to the assumption of 100% effectiveness of
the protease inhibitor. Combining the intracellular delay with less
than perfect antiretroviral treatment resulted in a significant increase in the estimated value for the infected cell death compared
with the case of perfect drug therapy [36]. Because the conversion
of a newly infected cells into a productively infected one is a multistep process, all infected cells would not finish all these processes
in the same time. In [35,39], a gamma distribution was introduced
to describe a continuous time delay between viral infection and
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
production. By fitting to patient data, they found the estimate of
the viral clearance rate using the model without delay was underestimate. However, no change was found in the estimate of the infected cell death due to the assumption of a perfect drug [39]. More
within-host HIV models including time delays can be found in
[2,3,9,11,12,16,26,61,66].
Although time delay representing either the time needed for infected cells to produce virions after viral entry or the time needed
for the CD8+ T cell immune response to emerge to control viral replication has been included in HIV models, very few, if any, models
have considered both delays which are biologically reasonable during HIV infection. In general, including more than one time delay
will bring challenges to both mathematical analysis of the models
and comparison of model predictions with experimental data. In
this paper, we incorporated the two delays into an HIV-1 model.
We began model analysis with proving the positivity and boundedness of the solutions, local stability of the steady states, and uniform persistence of the system. By developing a few different
Lyapunov functionals, we obtained conditions ensuring global stability of the infection-free and infected steady states. We also fit
the model to viral load data from 10 patients during primary HIV
infection and estimated parameter values. We used an F-test to
compare the data fits using both the two-delay model and the
model without delays from a statistical viewpoint. This work presents mathematical properties of the solutions of the two-delay
model, and provides more information for model verification and
selection when delays are included to study virus dynamics.
99
parameters. Here, we assumed the same generation rate for effector cells as in [9]. The model including the two delays is given by
d
TðtÞ ¼ s dT kVT
dt
d H
T ðtÞ ¼ k1 Vðt s1 ÞTðt s1 Þ dT H dx ET H
dt
d
VðtÞ ¼ NdT H cV
dt
ð1Þ
d
EðtÞ ¼ pT H ðt s2 Þ dE E
dt
Model (1) has two steady states: the infection-free steady state
e ; 0; 0; 0 with T
e ¼ s=d and the infected steady state
E0 ¼ T
E1 ¼ T; T H ; V; E where
2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi3
!2
u
2 2
2
u kNd2 d
kNd
d
kk
N
d
d
6
E
E
1
E7
T¼
dþt
d þ 4s
4
5
cdx p
c2 dx p
2kk1 N2 d2 dE cdx p
c2 dx p
!
dE k1 NdT
d
T ¼
c
dx p
H
Nd H
T
c
p
E ¼ TH
dE
V¼
ð2Þ
2. Model description
Ciupe et al. [9] incorporated one time delay into an HIV-1 model
to account for the time needed to activate the CD8+ T cell response,
i.e., the immune cells at time t were activated by infected cells at
time t s2, where s2 is a constant. Here we include this immune
delay as well as an intracellular delay, s1, between viral entry
and viral production (this phase is referred to as the eclipse phase).
The model is described by a system of differential Eqs. (1). Schematic diagram at the model is given in Fig. 1. It includes four variables: uninfected target cells T(t), productively infected cells Tw(t),
free virus V(t), and effector cells E(t). The parameter s represents
the rate at which target cells are created, d is the death rate of target cells, k is the infection rate, and d is the death rate of producas
tively infected cells. As described in [36], we assume k1 ¼ ke 1 ,
where a (d < a < d) is the death rate of infected cells before viral
production commences. Thus, eas1 is the probability that an infected cell survives the eclipse phase to produce virions. The constant dx represents the killing rate of infected cells by effector
cells. N is the number of virions produced by an infected cell during
its lifespan, and c is the viral clearance rate constant. Effector cells
are assumed to be generated at a rate proportional to the level of
productively infected cells, and die at a rate dE. Note that the generation of effector cells was described using a mass action term
pETw in other studies [31,67]. However, it generates a steady state
of infected cells, Tw = dE/p, which is independent of any viral
H
From T > 0, we have that the infected steady state exists if and
only if k1 NT=c > 1, which is equivalent to k1Ns/(dc) > 1, i.e.,
e =c > 1. Note that R0 ¼ k1 Ns=ðdcÞ is the basic reproductive rak1 N T
tio of the basic model (without the immune response) [48]. Beas
cause k1 ¼ ke 1 ; R0 is intracellular delay-dependent.
We will study the mathematical properties of the solutions of
model (1). We will also fit both the two-delay model and the model
without delays to the data from 10 patients during primary HIV
infection [59], and compare the data fits using a statistical test.
3. Analysis of the delay model (1)
3.1. Initial conditions
We denote by X ¼ C ½s; 0; R4þ the Banach space of continuous
functions mapping the interval [s, 0] into R4þ equipped with the
sup-norm, where s = max{s1, s2}. By the standard theory of functional differential equations (see [19]) we know that for any
/ 2 C ½s; 0; R4þ there exists a unique solution
Yðt; /Þ ¼ ðTðt; /Þ; T H ðt; /Þ; Vðt; /Þ; Eðt; /ÞÞ
of the system (1), which satisfies Y0 = /. The initial conditions are
given by
TðhÞ ¼ /1 ðhÞ; T H ðhÞ ¼ /2 ðhÞ; VðhÞ ¼ /3 ðhÞ; EðhÞ ¼ /4 ðhÞ; h 2 ½s; 0ð3Þ
where / ¼ ð/1 ; . . . ; /4 Þ 2 R4þ with /i(h) P 0 (h 2 [s, 0], i = 1, . . . , 4)
and /2(0), /3(0), /4(0) > 0.
3.2. Positiveness and boundedness of solutions
Proposition 1. Let Y(t, /) be the solution of the delayed system (1)
with the initial condition (3). T(t), Tw(t), V(t), E(t) > 0, ("t P 0) are
ultimately bounded. Moreover, there exists an > 0 such that
lim inft?1T(t) P .
Fig. 1. Schematic representation of model (1).
Proof. From (1) we obtain
100
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
TðtÞ ¼ Tð0Þe
Rt
0
ðdþkVðfÞÞdf
þ
Z
t
se
Rt
c
ðdþkVðfÞÞdf
dc
0
T H ðtÞ ¼ T H ð0Þe
Rt
0
ðdþdx EðfÞÞdf
þ
Z
t
k1 Tðc s1 ÞVðc s1 Þe
Rt
c
ðdþdx EðfÞÞdf
dc
0
VðtÞ ¼ Vð0Þect þ
Z
t
NdT H ðcÞecðtcÞ dc
0
EðtÞ ¼ Eð0ÞedE t þ
Z
t
pT H ðc s2 ÞedE ðtcÞ dc
ð4Þ
0
Using (3) we know that the solution of model (1) is positive for all
t P 0.
Next, we show that the solution is ultimately bounded. From
the T(t) equation in (1), we have
dT
6 s dT
dt
s
d
Thus, lim supt!1 TðtÞ 6 and T(t) is ultimately bounded. We define a
Lyapunov functional
while the modulus of the right-hand side of (6) satisfies jcdeks1 R0 j
6 cdR0 < cd. This leads to a contradiction. Thus, all the eigenvalues
have negative real parts, and hence the infection-free steady state is
locally asymptotically stable when R0 < 1.
When R0 > 1, we define a function f ðkÞ ¼ ðk þ cÞðk þ dÞ
cdeks1 R0 . It is clear that f(0) < 0 and f(k) ? 1 when k ? 1. By
the continuity we know there exits at least one positive root. Thus,
the infection-free steady state is unstable if R0 > 1. h
At the infected steady state, the characteristic Eq. (5) can be
simplified to
ðk þ d þ kVÞðk þ cÞ½ðk þ R0 dÞðk þ dE Þ þ ðR0 1ÞdE deks2 ¼ ðk þ dÞðk þ dE ÞR0 cdeks1
ð7Þ
For a special case of s2 = 0, we have the following theorem for the
stability of the infected steady state.
Theorem 2. The infected steady state of model (1) is locally asymptotically stable when R0 > 1 in the case of s2 = 0.
Proof. In the case of s2 = 0, the characteristic equation is
k
UðtÞ ¼ TðtÞ þ T H ðt þ s1 Þ
k1
ðk þ d þ kVÞðk þ cÞ½ðk þ R0 dÞðk þ dE Þ þ ðR0 1ÞdE d
Thus, U(t) P 0 for t P 0. Differentiating U(t) along the solution of
system (1), we obtain
dUðtÞ
dk
6 s dTðtÞ T H ðt þ s1 Þ ¼ s dTðtÞ þ dTðtÞ dUðtÞ
dt
k1
6 s þ dTðtÞ dUðtÞ 6 C 1 dUðtÞ
where C 1 ¼ s þ dsd > 0. Thus, lim supt!1 UðtÞ 6 Cd1 and Tw(t) is ultimately bounded. It follows from the third and fourth equation of
(4) that V(t) and E(t) are also ultimately bounded. From the first
equation of (1), one can show that
_
TðtÞ
P s T ðd þ kV u Þ;
for a large t
where Vu is the upper bound of V(t). This shows that T(t) is uniformly bounded away from zero. h
3.3. Local stability of the steady states
To study the local stability of the steady states of model (1), we
linearized the system and obtained the characteristic equation, given by the following determinant:
d kV k
0
kT
k Veks1
ks1
d
d
E
k
k
1
x
1 Te
0
Nd
c k
0
peks2
0
0
dx T H
0
dE k
¼0
ð5Þ
Obviously, Eq. (8) does not have a nonnegative real solution.
Now we prove that (8) does not have any complex root k with a
nonnegative real part. Suppose, by contradiction, that k = x + iy
with x P 0 is a root of (8). Because its complex conjugate k = x iy
is also a root of (8), we can assume that y > 0.
When R0 ! 1, we have V ! 0. Thus, Eq. (8) reduces to
ðk þ dÞðk þ cÞ ¼ cdeks1 . Using the same arguments as above, we
can show that it does not have any root with a nonnegative real
part.
By the continuous dependence of roots of the characteristic
equation on R0 , we know that the curve of the roots must cross
the imaginary axis as R0 decreases sufficiently close to 1. That is,
the characteristic Eq. (8) has a pure imaginary root, say, iy0, where
y0 > 0. From (8), we have
ðd þ kV þ iy0 Þðc þ iy0 Þ½ðR0 d þ iy0 ÞðdE þ iy0 Þ þ ðR0 1ÞdE d
¼ ðd þ iy0 ÞðdE þ iy0 ÞR0 cdeis1 y0
ð9Þ
jðR0 d þ iy0 ÞðdE þ iy0 Þ þ ðR0 1ÞdE dj > R0 djdE þ iy0 j:
ð10Þ
In fact, after straightforward computations, we have
Theorem 1. The infection-free steady state of model (1) is locally
asymptotically stable when R0 < 1 and unstable when R0 > 1.
Proof. We first prove the local stability when R0 < 1. At the infection-free steady state, the characteristic equation becomes
ð6Þ
If k has a nonnegative real part, then the modulus of the left-hand
side of (6) satisfies
jðk þ cÞðk þ dÞj P cd
ð8Þ
We claim that the following inequality holds:
where k is an eigenvalue. We have the following result for the infection-free steady state.
ðk þ cÞðk þ dÞ ¼ cdeks1 R0
¼ ðk þ dÞðk þ dE ÞR0 cdeks1
jðR0 d þ iy0 ÞðdE þ iy0 Þ þ ðR0 1ÞdE dj2 ðR0 dÞ2 jdE þ iy0 j2
2
¼ y20 ðR0 1ÞdE d þ 2R0 ðR0 1ÞðdE dÞ2 þ ðdE y0 Þ2 > 0
Thus, (10) holds. It follows from jd þ kV þ iy0 j P jd þ iy0 j; jc þ iy0 j >
c, and the inequality (10) that the modulus of the left-hand side of
(9) is greater than the modulus of the right-hand side. This leads to
the contradiction. Therefore, we conclude that the characteristic Eq.
(8) does not have any root with a nonnegative real part. Thus, the
infected steady state is locally asymptotically stable when R0 > 1
in the case of s2 = 0. h
In the case of s2 > 0, the analysis of the characteristic Eq. (7) is
challenging. When s1 = 0 and s2 > 0, Ciupe et al. [9] used the
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K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
Routh–Hurwitz criteria to obtain a sufficient condition for the local
stability of the infected steady state. We will show below that a positive immune delay s2 is able to destabilize the infected steady
state even for a simplified model of (1). To reduce the dimension
of the system (i.e., the order of the characteristic equation), we
simplified model (1) by assuming the level of target cells is a constant, T0. We will show that for this simplified model, the infected
steady state is locally stable for s2 < s2 and bifurcation occurs
when s2 ¼ s2 in the case of s1 = 0, where s2 > 0 is a threshold of
the immune delay.
The simplified model is
d H
T ðtÞ ¼ k1 Vðt s1 ÞT 0 dT H dx ET H
dt
d
VðtÞ ¼ NdT H cV
dt
ð11Þ
V¼
NdT H
;
c
E¼
pT H
dE
0
Here, R0 ¼ k1 NT
is the basic reproductive ratio of model (11). The inc
fected steady state exists if and only if R0 > 1.
The characteristic equation for the linearized system is
d dx E k k1 T 0 eks1
Nd
c d
peks2
0
dx T H ¼0
0
d k ð12Þ
E
where k is an eigenvalue.
At the infection-free steady state, the characteristic equation
becomes
ðk þ dE Þ ðk þ cÞðk þ dÞ R0 cdeks1 ¼ 0
Using the similar arguments in Theorem 1, we know the infectionfree steady state is locally asymptotically stable when R0 < 1 and
unstable when R0 > 1.
At the infected steady state, the characteristic equation is
ðk þ cÞ ðk þ R0 dÞðk þ dE Þ þ ðR0 1ÞdE deks2 ¼ ðk þ dE ÞR0 cdeks1
ð13Þ
Similar to Theorem 2, we can show that the infected steady state of
model (11) is locally asymptotically stable when R0 > 1 in the case
of s2 = 0. We show in the following theorem that bifurcation occurs
when s2 > 0.
Theorem 3. In the case of s1 = 0 and s2 > 0, the infected steady state
is
locally
asymptotically
stable
when
s2 < s2 , where
n
o
1;j
2;j
1;j
2;j
s2 ¼ minj2N s2 ; s2 with s2 and s2 defined below by (22) and
(23), respectively. Moreover, a Hopf bifurcation occurs at the infected
steady state when s2 ¼ s2 .
Proof. In the case of s1 = 0 and s2 > 0, the characteristic Eq. (13) is
kðk þ dE Þðk þ c þ R0 dÞ þ ðk þ cÞðR0 1ÞddE eks2 ¼ 0
ð14Þ
2
ks2
k þ a1 k þ a2 k þ a3 ðk þ cÞe
¼0
xðx2 a2 Þ ¼ a3 ½x cosðxs2 Þ c sinðxs2 Þ
ð17Þ
Taking squares and adding the two equations, we have
a21 x4 þ x2 ðx2 a2 Þ2 ¼ a23 ðc2 þ x2 Þ
ð18Þ
y3 þ y2 a21 2a2 þ y a22 a23 a23 c2 ¼ 0
ð15Þ
where a1 ¼ dE þ c þ R0 d; a2 ¼ dE ðc þ R0 dÞ; a3 ¼ ðR0 1ÞdE d. Note
that these coefficients are independent of the immune delay.
ð19Þ
We define a function F(y) as the left-hand side of the above
equation.
2
It is easy to verify that a21 2a2 ¼ dE þ ðc þ R0 dÞ2 > 0, and that
2
2
2
a2 a3 ¼ dE ðc þ dÞðc þ 2R0 d dÞ > 0. By Descartes’ rule of signs,
there is one positive root of Eq. (19). We denote the positive root
pffiffiffiffiffi
by y⁄. Thus, the corresponding value of x is x ¼ y .
Next, we solve for sin(xs2) and cos(xs2) from (16) and (17),
and obtain
cosðxs2 Þ ¼
a1 cx2 þ x2 ðx2 a2 Þ
a3 ðc2 þ x2 Þ
ð20Þ
sinðxs2 Þ ¼
a1 x3 þ cxða2 x2 Þ
a3 ðc2 þ x2 Þ
ð21Þ
The unique solution h = xs2 2 [0, 2p] of (20) and (21) is h = arccos
(a1cx 2 + x2(x2 a2)/(a3(c2 + x2))) if sin (h) > 0, i.e., if a1x2 +
a2c cx2 > 0, and h = 2p arccos (a1cx2 + x2(x2 a2)/(a3(c2 +
x2))) if sin (h) 6 0, i.e., if a1x2 + a2c cx2 6 0. Therefore, for the
n o
and
imaginary root k = ix of (15) we have two sequences s1;j
2
n o
2;j
s2 for j 2 N
s1;j
2 ¼
1
s2;j
2 ¼
1
arccos
x
and
x
a1 cx2 þ x2 ðx2 a2 Þ
þ 2jp
a3 ðc2 þ x2 Þ
2p arccos
ð22Þ
a1 cx2 þ x2 ðx2 a2 Þ
þ 2jp :
a3 ðc2 þ x2 Þ
ð23Þ
o
2;j
, i.e., s2 is the minimum value
Assuming s2 ¼ minj2N s1;j
2 ; s2
associated with the imaginary solution ix⁄ of the characteristic
n
o
, where
Eq. (15) we found above, we determine sign dReðkÞ
j
ds2 s2 ¼s
n
2
sign is the sign function and Re(k) is the real part of k. We assume that k(s2) = m(s2) + ix(s2) is a solution of (15). Thus,
mðs2 Þ ¼ 0 and xðs2 Þ ¼ x . Taking derivative of (15) with respect
to s2, we have
2
dk
3k þ 2a1 k þ a2 þ a3 ð1 s2 ðk þ cÞÞeks2
¼ a3 kðk þ cÞeks2
ds2
From (15), we have
eks2 ¼ a3 ðk þ cÞ
k þ a1 k2 þ a2 k
3
Thus,
For the ease of notation, we rewrite (14) as
3
ð16Þ
Letting y = x , the above equation can be simplified to
The simplified model has two steady states: the
stea infection-free
dy state (0, 0, 0) and the infected steady state T H ; V; E , where
dE dðR0 1Þ
;
dx p
a1 x2 ¼ a3 ½c cosðxs2 Þ þ x sinðxs2 Þ
2
d
EðtÞ ¼ pT H ðt s2 Þ dE E
dt
TH ¼
We will determine if the solution curve of the characteristic Eq.
(15) crosses the imaginary axis. Suppose that ix (x > 0) is a root of
(15). Substituting k = ix into (15) and separating the real and imaginary parts, we have
dk
ds2
1
¼
Evaluating
we have
3k2 þ 2a1 k þ a2
2
2
k ðk þ a1 k þ a2 Þ
dk
ds2
1
at
þ
1
s2
kðk þ cÞ k
s2 ¼ s2 (i.e., k = ix⁄) and taking the real part,
102
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
2
1 dk
Re4
ds2
3
2
2
2
2 1
5 ¼ ða2 3x Þða2 x Þ þ 2a1 x 2
2
2
2
2 þ x2
2
½ða x Þ þ a x c
x
2
1
s2 ¼s2
Differentiating M(t) along the solution of system (1), we obtain
H
dM
dM 1 dM2 dM 3 dT Te dT k dT
þ
¼
þ
þ
¼
dt ð1Þ
dt ð1Þ
dt ð1Þ
dt ð1Þ dt T dt k1 dt
dV
dE
þ b2
þ kðVT Vðt s1 ÞTðt s1 ÞÞ
dt
dt
H
H
þ b2 pðT T ðt s2 ÞÞ ¼ ðs dT kVT s þ d Te Þ
Te
k
ðs dT kVTÞ þ ½k1 Vðt s1 ÞTðt s1 Þ dT H
T
k1
dx ET H þ b1 ðNdT H cVÞ þ b2 ½pT H ðt s2 Þ dE E
þ b1
From (18), we have
2
2
2
2
x ½ða2 x Þ2 þ a21 x ¼ a23 ðc2 þ x Þ
Thus,
2
dk
Re4
ds2
1 3
2
2
2
2a21 x a23
5 ¼ ða2 3x Þða2 x Þ þ
2
2 2
a3 ðc þ x Þ
þ kVT kVðt s1 ÞTðt s1 Þ þ b2 pT H b2 pT H ðt s2 Þ
1
kdx H
kd
¼ d ðT Te Þ2 ET b2 dE E þ b2 pT H þ kV Te T H
T
k1
k1
1
kdx H
2
H
e
þ b1 NdT b1 cV ¼ d ðT T Þ ET b2 dE E
T
! k1 e
kdðc k1 N T Þ kNd e kd
ks
þ V k Te þ TH
T
þ
ck1
c
k1
d
2 kd
1
x
T Te ET H b2 dE E 6 0
¼ d
T
k1
s2 ¼ s2
From the definition of FðyÞ ¼ y3 þ y2 ða21 2a2 Þ þ yða22 a23 Þ
a23 c2 , we find the numerator of the right-hand side of the above
2
equation is exactly F0 (y) evaluated at y ¼ x . Thus,
2
1 dk
4
Re
ds2
3
5¼
s2 ¼s2
2
F 0 ðx Þ
a23 ðc2 þ x2 Þ
Therefore,
8 2
(
)
1 <
dReðkÞ
dk
4
sign
¼ sign Re
:
ds2 s2 ¼s
ds2
39
=
n
o
5 ¼ sign F 0 ðx2 Þ
;
s2 ¼s2
2
Because
h
i
2
4
2
2
2
F 0 ðx Þ ¼ 3x þ 2x dE þ ðc þ R0 dÞ2 þ dE ðc þ dÞðc þ 2R0 d dÞ > 0
we know
dReðkÞ
ds2
is positive at
s2 ¼ s2 . Thus, the solution curve of the
characteristic Eq. (15) crosses the imaginary axis. This shows that a
Hopf bifurcation occurs at s2 ¼ s2 > 0. When s2 < s2 , the infected
steady state is locally asymptotically stable by continuity. h
3.4. Global stability of the steady states of model (1)
3.4.1. Infection-free steady state
Theorem 4. If the basic reproductive number R0 ¼ k1dcNs < 1, then the
infection-free equilibrium E0 of model (1) is globally asymptotically
stable.
Proof. By Theorem 1, it suffices for us to prove the global attractiveness of E0. Inspired by the work of McCluskey [32], we define
a Lyapunov functional
MðtÞ ¼ M1 ðtÞ þ M 2 ðtÞ þ M 3 ðtÞ
with
T
T
k
ln 1 þ T H þ b1 V þ b2 E
M 1 ¼ Te
k1
Te
Te
M2 ¼ k
Z
VðtÞTðtÞds
t
3.4.2. Permanence as R0 > 1
We show the uniform persistence of system (1), for which we
apply the persistence theory by Smith and Zhao [57] for infinite
dimensional systems (also see [65]). The methods and techniques
we are using have been recently employed in [29, Theorem 2],
[54, Theorem 6.1], [62, Theorem 3.1] for distributed and infinite
delay systems and in [30] for a discrete delay system.
To proceed, we introduce the following notation and terminology. Denote by P(t), t P 0 the family of solution operators corresponding to (1). The x-limit set x(x) of x consists of y 2 X such
that there exists a sequence tn ? 1 as n ? 1 with P(tn)x ? y as
n ? 1.
Theorem 5. Assuming R0 > 1 and the initial conditions (3), system
(1) is uniformly persistent; that is, there exists g0 > 0 such that lim
inft?1T(t) P g0, lim inft?1Tw(t) P g0, lim inft?1E(t) P g0, and lim
inft?1V(t) P g0.
Proof. Let
X 0 ¼ f/ 2 X : /2 ð0Þ > 0; /3 ð0Þ > 0g
Now we prove X0 is positively invariant for P(t). By the second
and third equations of (1) we have
T H ðtÞds
ts2
e ¼ s ; b1 ¼ ks ¼ k T
e , and b2 ¼ kdðck1 NeT Þ ¼ kdð1R0 Þ > 0 since
where T
cpk1
pk1
d
cd
c
R0 < 1. We know that
dt ð1Þ
for delay systems in [19, Theorem 5.3.1], the infection-free steady
state E0 is globally attracting. Further, it was showed in Theorem
1 that E0 is locally asymptotically stable. Therefore, E0 is globally
asymptotically stable. h
which is relatively closed in X.
ts1
M 3 ¼ b2 p
V(t) = 0. Using E(t) = V(t) = 0 in the second equation of (1), we have
Tw(t) = 0. Therefore, the maximal compact invariant set in
n o
dM
¼ 0 is the singleton E0. By the LaSalle invariance principle
@X ¼ X n X 0 ¼ f/ 2 X : /2 ð0Þ ¼ 0 or /3 ð0Þ ¼ 0g
t
Z
It follows that M(t) is bounded and non-increasing. Thus limt?1M(t)
e and E(t) = 0. Substitutexists. Note that dM
¼ 0 if and only if TðtÞ ¼ T
dt
e
ing TðtÞ ¼ T into the first equation of (1), one can directly obtain
T
T
ln 1 P 0 for all T(t) > 0 (the equality
eT
e ). From the definition of M(t) and Propholds if and only if TðtÞ ¼ T
osition 1, we know M(t) is well-defined and M(t) P 0. The equality
e ; T H ðtÞ ¼ VðtÞ ¼ EðtÞ ¼ 0.
holds if and only if TðtÞ ¼ T
eT
H
dT ðtÞ
P dT H ðtÞ;
dt
dVðtÞ
P cVðtÞ;
dt
8t P 0
ð24Þ
Since Tw(0, /) = /2(0) > 0, we have V(0, /) = /3(0) > 0. It follows from
(24) that
T H ðt; /Þ P /2 ð0Þ edt > 0;
Vðt; /Þ P /3 ð0Þ ect > 0;
8t P 0
103
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
Thus X0 is positively invariant for P(t).
We set
lim inf t!1 ðT H ðtÞ; VðtÞÞ P ðg1 ; g1 Þ
M @ ¼ f/ 2 X : YðtÞ/ satisfies ð1Þ and YðtÞ/ 2 @X; 8t P 0g
We claim that
M @ ¼ fðT; 0; 0; EÞg
ð25Þ
Assuming Y(t) 2 M@ , "t P 0, it suffices to show that
Tw(t) = V(t) = 0, "t P 0. If it is not true, then there exists t0 > 0 such
that either (a) Tw(t0) > 0,V(t0) = 0; or (b) Tw(t0) = 0, V(t0) > 0.
For case (a), from the third equation of (1) we have
dV ¼ NdT H ðt 0 Þ > 0
dt t¼t0
Hence there is an e0 > 0 such that V(t) > 0, "t 2 (t0, t0 + e0). On the
other hand, from Tw(t0) > 0 there exists an e1 (0 < e1 < e0) such that
Tw(t) > 0, "t 2 (t0, t0 + e1). Thus, we have Tw(t) > 0, V(t) > 0,
"t 2 (t0, t0 + e1), which contradicts the assumption that
(T(t), Tw(t), V(t), E(t)) 2 M@ , "t P 0. Similarly, we can obtain a contradiction for case (b). This proves the claim (25).
Let A ¼ \x2Ab xðxÞ, where Ab is the global attractor of P(t) restricted to @X. We show that A ¼ fE0 g. In fact, from A # M @ and
the second and first equations of (1), we have limt?1E(t) = 0 and
e . Thus, {E0} is the isolated invariant set in X.
limt!1 TðtÞ ¼ T
T
Next we show that Ws(E0) X0 = ;. If this is not true, then there
H
exists a solution T t ; T t ; V; Et 2 X 0 such that
s
lim TðtÞ ¼ Te ¼ ;
t!1
d
lim T H ðtÞ ¼ 0;
t!1
lim VðtÞ ¼ 0;
t!1
EðtÞ < e for all t P T 0
dT H ðtÞ
P k1 ð Te ÞVðt 1 Þ dT H
dt
dVðtÞ
¼ NdT H cV; t P T 0 þ
dt
e
s
dx eT H
s
w
If T (t),V(t) ? 0, as t ? 1, then by a standard comparison argument and the nonnegativity, the solution ðnT H ðtÞ; nV ðtÞÞ of the following monotone system
( dn
ðtÞ
¼ k1 ð Te dt
dnV ðtÞ
¼ NdnT H ðtÞ
dt
eÞ nV ðt s1 Þ dnT H ðtÞ dx enT H ðtÞ
cnV ðtÞ; t P T 0 þ s
TH
ð26Þ
with the initial condition nT H ðtÞ ¼ T H ðtÞ; nV ðtÞ ¼ VðtÞ; 8t 2
f ðtÞ ¼ 0, where
½T 0 ; T 0 þ s converges to (0, 0) as well. Thus limt!1 W
f ðtÞ > 0 is defined by
W
Z t
e
f ðtÞ ¼ n H ðtÞ þ k1 ð T eÞ nV ðtÞ þ k1 ð Te eÞ
W
nV ðnÞdn
T
c
ts1
f ðtÞ with respect to time gives
Differentiating W
f ðtÞ
dW
dt ¼
ð26Þ
1
Ndk1 ð Te eÞ d dx e nT H ðtÞ
c
Because R0 > 1, we have
1
c
Theorem 6. If the basic reproductive number satisfies 1 < R0 6 1þ
dx g0 =ð2dÞ where g0 was defined in Theorem 5, then the infected
equilibrium E1 of model (1) is globally attracting.
Proof. We define a Lyapunov functional
WðtÞ ¼ W 1 ðtÞ þ W 2 ðtÞ þ W 3 ðtÞ
with
W 1 ðtÞ ¼ T H
W 2 ðtÞ ¼ kVT
e eÞ d dx e > 0 for a suffi Ndk1 ð T
f ðtÞ goes to either infinity or a positive
ciently small e. Therefore W
number as t ? 1, which leads to a contradiction with
f ðtÞ ¼ 0. Thus we have W s ðE0 Þ T X 0 ¼ ;.
limt!1 W
Define p : X ! Rþ by
pð/Þ ¼ minf/2 ð0Þ; /3 ð0Þg; 8/ 2 X
It is clear that X 0 ¼ p1 ð0; 1Þ and @X ¼ p1 ð0Þ. Thus by [57, Theorem
3] we have
T
T
Z
Z
k
TH
þ TH H
k1
TH
t
H
ts1
W 3 ðtÞ ¼ a2 p
For the constant e given above, it follows from the second and third
equations of (1) that
(
3.4.3. Infected steady state
lim EðtÞ ¼ 0
t!1
For any sufficiently small constant e > 0, there exists a positive constant T0 = T0(e) such that
TðtÞ > Te e > 0;
for some constant g1 > 0. Moreover, by the fourth equation of (1), we
have lim inft?1E(t) P g2 for some constant g2 > 0. Let g0 = min{g1,
g2, }, where is from Proposition 1 such that lim inft?1T(t) P > 0.
We showed that lim inft?1T(t) P g0, lim inft?1Tw(t) P g0, lim
inft?1E(t) P g0, and lim inft?1V(t) P g0. This finishes the proof of
Theorem 5. h
!
þ a1 V H
V
V
þ a2 E H
E
E
VðsÞTðsÞ
ds
VT
t
T H ðsÞds
ts2
where HðxÞ ¼ x 1 ln x; a1 ¼ kc T, and a2 ¼ kkd1 px E. Proposition 1 implies that W(t) is well-defined and that W(t) P 0, "t P 0. The
equality holds if and only if TðtÞ ¼ T; T H ðtÞ ¼ T H ; VðtÞ ¼ V and
EðtÞ ¼ E. We calculate the derivatives of W1, W2, and W3 separately
as follows
!
!
!
H
dW 1 T dT k
T H dT
V dV
þ
¼ 1
1 H
þ a1 1 T dt k1
V dt
dt ð1Þ
dt
T
!
!
E dE
T
ðs dT kVT s þ dT þ kV TÞ
þ a2 1 ¼ 1
E dt
T
!
k
TH
1 H ðk1 Vðt s1 ÞTðt s1 Þ dT H dx ET H Þ
þ
k1
T
!
!
V
E
H
ðNdT cVÞ þ a2 1 ðpT H ðt s2 Þ dE EÞ
þ a1 1 V
E
1
¼ d ðT TÞ2 þ ðkV T þ a1 cVÞ þ ðkVT a1 cVÞ kVT
T
kV T 2 kd H
kdx H
ET
T þ kVðt s1 ÞTðt s1 Þ k1
T
k1
H
T Vðt s1 ÞTðt s1 Þ kd H
VT H
k
þ T þ a1 NdT H a1 Nd
H
k1
V
T
kdx H
ET H ðt s2 Þ
H
þ a2 pT ðt s2 Þ þ
T E a2 dE E a2 p
E
k1
1
kV T 2
þ a2 dE E ¼ d ðT TÞ2 þ 2kV T kVT T
T
kd H a2 p H
þ kVðt s1 ÞTðt s1 Þ T ET
k1
E
H
T Vðt s1 ÞTðt s1 Þ kd H
VT H
k
þ T þ a1 NdT H a1 Nd
H
k1
V
T
H
ET
ðt
s
Þ
2
þ a2 pT H ¼ D0 þ kV T D1
þ a2 pT H ðt s2 Þ a2 p
E
kd
kd
þ T H a2 pT H
þ a2 pT H D2 þ T H a1 Nd k1
k1
104
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
k k1 Nd
kNd
kd
ET H
a2 p
T d Tþ
k1
c
c
k1
E
!
H
b þ 2kdT H TN 1 a2 p ET
¼X
c
k1
E
where
þ TH
1
D0 ¼ d ðT TÞ2
T
VT
D1 ¼ 2 VT
ET H
D2 ¼ 2 E
TH
T Vðt s1 ÞTðt s1 Þ T H V T H Vðt s1 ÞTðt s1 Þ
þ
T
THV T
VT
THV
þ
T H ðt s2 Þ
TH
ET H
dW 2 VT Vðt s1 ÞTðt s1 Þ
Vðt s1 ÞTðt s1 Þ
þ ln
þ kV T D1 ¼ kV T D1 þ kV T
dt ð1Þ
VT
VT
"
#
VT T Vðt s1 ÞTðt s1 Þ T H V T H Vðt s1 ÞTðt s1 Þ
¼ kVT 2 þ
VT T
VT
THV
THV T
VT Vðt s1 ÞTðt s1 Þ
Vðt s1 ÞTðt s1 Þ
þ ln
þ kVT
VT
VT
"
#
T T H V T H Vðt s1 ÞTðt s1 Þ
Vðt s1 ÞTðt s1 Þ
þ
ln
¼ kVT 2 T THV
VT
THV T
¼ kVT ðX0 þ X1 þ X2 Þ kV T
where
T H Vðt s1 ÞTðt s1 Þ
X0 ¼ 1 H
T VT
T
T
X1 ¼ 1 þ ln
X2 ¼ 1 T V
THV
þ ln
T H Vðt s1 ÞTðt s1 Þ
T HV T
T V
dW 3 þ a2 pT H D2 ¼ a2 pðT H T H ðt s2 ÞÞ
dt ð1Þ
þ a2 pT H 2 ¼ a2 pT H a2 p
!
ET H T H ðt s2 Þ ET H ðt s2 Þ
þ
E TH
TH
ET H
ET H
ET H ðt s2 Þ
þ 2a2 pT H a2 p
E
E
b ¼ D0 þ kVT ðX0 þ X1 þ X2 Þ and combining the derivaAssuming X
tives of W1, W2, and W3, we have
dW dW 1 dW 2 dW 3 ¼
þ
þ
¼
dt ð1Þ
dt ð1Þ
dt ð1Þ
dt ð1Þ
kd
D0 þ kV T ðX0 þ X1 þ X2 Þ kV T þ T H a1 Nd k1
kd H
ET H
T a2 pT H þ a2 pT H a2 p
þ 2a2 pT H
k1
E
ET H ðt s2 Þ b
6 X þ a2 pT H þ a2 pT H kV T
E
kd
kd
ET H
b
þ T H a2 p
¼X
þ T H a1 Nd k1
k1
E
kd
kdx
kd
þ
þ T H a2 p þ a1 Nd E T H kV T þ T H
k1
k1
k1
!
H
ET
b þ T H 2kTNd 2kd
¼X
a2 p
c
k1
E
a2 p
dW b þ k T H ð2dðR0 1Þ dx g Þ 6 k T H ð2dðR0 1Þ dx g Þ
6X
0
0
dt ð1Þ
k1
k1
6 0;
8t > t M :
b 6 0; dW ¼ 0 implies that
6X
dt ð1Þ
Because
dW dt ð1Þ
T ¼ T;
T H V ¼ T H V:
Substituting TðtÞ ¼ T into the first equation of (1), one can get
VðtÞ ¼ V. From T H V ¼ T H V, we have T H ¼ T H . Substituting the above
equalities into the second equation of (1), we
have EðtÞ ¼ E. There ¼ 0g is the infected
fore, the largest compact invariant set in fdW
dt ð1Þ
steady state E1.
Using the similar arguments in Theorem 4, we prove that
limðTðtÞ; T H ðtÞ; VðtÞ; EðtÞÞ ¼ E1 : T HV
Because 1 + lnx x 6 0 for all x > 0 we have X0, X1, X2 6 0.
The derivative of W3 gives
þ
From Theorem 5 we know that there exists tM > 0 such that
E(t) P g0 for all t > tM. Thus,
Theorem 7. If s2 = 0 in system (1) and R0 > 1, then the infected
steady state E1 is globally attracting.
H
þ ln
H
t!1
T
T
H
b þ 2kd T H ðR0 1Þ a2 p ET
¼X
k1
E
k
H
b
¼ X þ T ð2dðR0 1Þ dx EÞ
k1
ET H ðt s2 Þ
Next, we calculate the derivative of W2.
Proof. We replace the term a2 p ET
H
a2 p ETE , and get
H
ðts2 Þ
E
in
dW of
dt ð1Þ
Theorem 6 with
dW kd
kd
þ T H a2 pT H
¼ D0 þ T H a1 Nd dt ð1Þ
k1
k1
þ kV T ðX0 þ X1 þ X2 Þ kV T þ a2 pT H a2 p
ET H
E
ET H
kd
H
H
þ 2a2 pT a2 p
6 a2 pT þ a2 pT kV T þ T H a1 Nd k1
E
kd H
ET H
ET H
kd
H
þ T a2 p
a2 p
¼ T a1 Nd þ a2 p
k1
k1
E
E
!
E
E
kd
þ a2 pT H kV T þ T H
a2 pT H þ
k1
E E
kd
6 T H a1 Nd þ a2 p 2a2 pT H
k1
k1 Nd
kNd
kd
H k
þT
T d Tþ
k1
c
c
k1
kd
H
¼ T a1 Nd a2 p
k1
H
Considering a1 ¼ kc T and a2 ¼ kkd1 px E, we have a1 Nd kkd1 a2 p ¼ 0.
6 0. Again using the similar arguments in
Thus, for s2 = 0, dW
dt ð1Þ
Theorem 4, we show that E1 is globally attracting. h
4. Comparison with patient data
We compared modeling predictions with/without time delays
to the plasma viral load data obtained from 10 patients during primary HIV-1 infection [59]. For most of the patients, the time between initial infection and the time of the first data point is
105
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
7
7
−1
6
log10 HIV−1 RNA ml
log10 HIV−1 RNA ml−1
Patient 1
6.5
6
5.5
5
4.5
4
5
4
3
2
Patient 2
3.5
0
100
200
300
Days
4
3
0
20
40
60
80
100
200
300
Days
6
−1
5
0
5
4.5
4
3.5
3
100
0
20
40
−1
5
100
200
300
Days
Patient 6
5
4.5
4
3.5
3
0
100
200
300
Days
6.5
400
500
Patient 8
−1
log10 HIV−1 RNA ml
log10 HIV−1 RNA ml −1
6
5.5
5
0
50
100
150
Days
200
6
5.5
5
4.5
250
0
50
100
−1
Patient 9
log10 HIV−1 RNA ml
5
4.5
4
0
100
200
300
Days
150
Days
5.5
3.5
100
5.5
500
Patient 7
6
−1
400
6.5
4.5
80
6
log10 HIV−1 RNA ml
−1
log10 HIV−1 RNA ml
5.5
7
log10 HIV−1 RNA ml
6.5
Patient 5
0
60
Days
6
4.5
500
Patient 4
Days
6.5
400
5.5
log10 HIV−1 RNA ml
−1
1
500
Patient 3
6
log10 HIV−1 RNA ml
400
400
500
Patient 10
7
6
5
4
0
100
200
300
Days
400
500
Fig. 2. Data fits using the two-delay model (solid curves) and the non-delay model (dashed curves).
unknown and needs to be estimated. Here, we used the same time
shift as in [59]. Because CD4+ T cell data are not available for these
patients, we fixed the parameters, such as s, d, and k, at the same
values as in [59] as well as in a later study [9] for each patient.
Although the current estimate of the viral clearance rate c is higher
[46], we chose an earlier estimate from [43], c = 3 day1, for comparisons because the same value was used in the data fitting using
the model without a delay in [59].
The precise values of the intracellular time delay s1 and the immune delay s2 are unknown. Previous estimates of s1 suggested
values of 61.5 days [34–36,43]. The CTL immune response to HIV
infection is observed in the first few weeks of infection, coincident
with the initial decline in the plasma viral load [22]. The best-fit
values for s2 range from 19 to 32 days in [9]. Here, we assumed
that s1 is 61.5 days and s2 is 65 weeks, and estimated both of
them by data fitting. The ratio k1/k represents the proportion of
new infections that progress to the productive state. When performing data fitting, we found that by setting k1 equal to k the
resulting error was very similar to the error obtained with k1 fitted.
Thus, we assumed k1 is equal to k during data fitting, reducing the
106
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
Table 1
Best-fit parameter values using the non-delay model. The values of d, k, and s were chosen from Stafford et al. [59].
Patient
dx 104 ml cells-day1
p day1
dE day1
N viron cells1
d day1
k 107 ml viron-day1
s cells ml-day1
d day1
1
2
3
4
5
6
7
8
9
10
Mean
SD
2.2
10
5.4
6.8
1.0
7.2
1.0
1.0
1.0
9.7
3.8
3.7
0.07
2
0.01
0.01
0.6
2
1
0.01
0.01
0.01
0.04
0.8
0.01
0.55
0.02
4.07
1.13
2.13
5.00
0.97
2.87
0.30
1.05
1.76
5101
2966
5617
668
3843
1341
4493
6689
1415
18621
4168
5158
0.013
0.02
0.0065
0.0046
0.017
0.012
0.017
0.0085
0.006
0.0043
0.010
0.006
0.46
3.6
6.4
48
6.3
7.5
8
6.6
25
1.9
6.5
14
130
200
65
46
170
120
170
85
60
43
103
57.3
0.75
0.80
0.10
0.13
0.22
0.59
0.32
0.10
0.10
0.50
0.27
0.28
Table 2
Best-fit parameter values using the two delay model. Parameters values of d, k, and s were chosen from Stafford et al. [59].
Patient
dx 104 ml cells-day1
p day1
dE day1
s1 days
s2 days
N viron cells1
d day1
k 107ml viron-day1
s cells ml-day1
d day1
1
2
3
4
5
6
7
8
9
10
Mean
SD
3.9
8.8
3.7
9.4
1.0
9.9
1.0
2.7
1.3
3.7
3.7
3.5
0.02
0.4
0.3
0.02
0.09
0.01
2
0.02
0.01
0.08
0.05
0.6
0.01
0.41
1.81
0.81
0.45
0.01
1.65
5.00
0.04
0.01
0.43
1.55
0.1
0.4
0.1
0.5
0.1
0.1
0.2
0.4
0.6
0.1
0.2
0.2
11.2
8.5
9.2
16.1
7.0
35.0
20.5
8.7
13.8
29.9
12.5
9.7
5505
6167
2374
1261
3360
2244
5308
7381
2175
23528
4334
6508
0.013
0.02
0.0065
0.0046
0.017
0.012
0.017
0.0085
0.006
0.0043
0.010
0.006
0.46
3.6
6.4
48
6.3
7.5
8
6.6
25
1.9
6.5
14
130
200
65
46
170
120
170
85
60
43
103
57.3
0.26
0.53
0.30
0.10
0.30
0.35
0.44
0.14
0.10
0.56
0.30
0.17
number of fitted parameters by one. In fact, k is a good approximation of k1 because both a and s1 are small.
The initial target cell, infected cell, and viral concentrations
6
were set to be T0 = 10,000 cells/ml, T H
0 ¼ 0 cells/ml, and V0 = 10
RNA copies/ml, respectively [59]. We also assumed that no virusspecific immune cells exist before virus infection, i.e., E0 = 0 cells/
ml. The rest of the parameters were estimated by fitting the predictions of the two-delay model to the viral load data from 10 patients
during primary HIV infection [59]. The data fitting was performed
using a commercial software package Berkeley Madonna. The program generates parameter values that give the best fit of the model
to the data set.
Data fits using the two delay model are shown in Fig. 2. For
comparison, we also included the fits using the model without delays to the same viral load data in the same figure. The delay model
(green2 solid) seems to fit the data better than the model without
delays (red dotted) for some patients. For example, for patient 2
and patient 4, the delay model captured most of the data points,
although the prediction using the delay model exhibited frequent
oscillations for patient 2. However, for some other patients such as
patients 3 and 5, the two fits using models with and without delays
generated very similar fits. The parameter values corresponding to
the best fits using the model without and with delays are given in
Tables 1 and 2, respectively.
For each patient, we also calculated the RMS (root mean square)
error, shown in Table 3, which represents the deviation between
patient data and the best fit. We found that the delay model
achieved a smaller error than the model without time delays for
each patient. However, this does not mean that the delay model
is better in fitting to the data because it has two more parameters
than the model without delays. To further compare the best fits
2
For interpretation of colour in Fig. 2, the reader is referred to the web version of
this article.
using the two models, we performed an F-test, which is able to
determine which one of the two nested models provides a better
fit to the same data set from a statistical standpoint. Notice that
the model without delays is a special case of the delay model. Thus,
they are nested models with 7 parameters for the delay model and
5 for the model without delays.
We define the F-test statistic
F¼
½RSSnon RSStwo =½dfnon dftwo RSStwo =dftwo
where RSS is the residual sum of squares, df is the degree of freedom
(i.e., the number of data points minus the number of fitted parameters), and the subscripts non and two represent the model without
delays and with two delays, respectively. It is clear that
RSS = n RMS2 where n is the number of data points. To calculate
the p-value for the F test, we compute the F value with the numerator degree of freedom dfnon dftwo and the denominator degree of
freedom dftwo. Results were shown in Table 3. We found that the
two-delay model provides significantly better fits for patients 7
and 10 (with the p-value < 0.05), although the p-value is close to
0.05 for these patients. For the other patients, there is no significant
difference in the fits, although the delay model generates a smaller
error.
In Figs. 3 and 4, we showed how the variations of time delays
and other parameter values change the virus dynamics in some patients. Similar changes with varying parameters were observed in
other patients. In Fig. 3A–D, we fixed the intracellular delay s1 to
be the best fit for each patient and reduced the immune delay s2
from the best fit value to 0. We found that the immune delay
was able to generate a periodic solution (patient 2). This further
confirms our theoretical prediction in Theorem 3. We also observed that the immune delay did not affect the magnitude of
the viral peak and the time to reach the peak. This is due to the
small values of the killing rate (dx) and generation rate (p) of
107
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
the best fit and changed the intracellular delay. We found as the
intracellular delay increased, the time to reach the peak was delayed. The level of the viral peak was lower for a larger intracellular
delay. We also studied the effects of different killing rate (dx) by
immune cells and the viral burst size on the viral load change in
Fig. 4. As expected, when the killing rate increases or the viral burst
size decreases, the viral set point (or the average viral load in the
case of a periodic solution) decreases.
Table 3
Comparison of the fits using models with and without delays.
Patient
Number of
data points
Non-delay
model RMS
Two delay
model RMS
p-value for
the F-test
1
2
3
4
5
6
7
8
9
10
11
10
9
8
9
15
11
16
10
15
0.324
0.622
0.254
0.229
0.236
0.647
0.210
0.172
0.259
0.516
0.218
0.291
0.222
0.048
0.193
0.506
0.097
0.155
0.205
0.354
0.205
0.102
0.764
0.210
0.669
0.140
0.046
0.392
0.496
0.049
5. Summary and discussion
immune cells. Thus, there is only a minor difference in the killing of
infected cells by immune cells with and without the immune delay
before viral peak. Around viral peak, a high level of infected cells
activated a large number of immune cells, which led to an effective
killing of infected cells. In Fig. 3E–H, we fixed the immune delay at
7
−1
B
Patient 1
τ =11.18
4
2
τ2=0
0
100
200
300
Days
400
log10 HIV−1 RNA ml
−1
C
5
4
3
Patient 6
τ2=35
1
5
4
3
2
Patient 2
τ =8.47
1
τ =0
2
2
6
2
6
0
500
τ =0
0
100
200
300
Days
400
2
500
D
7
log10 HIV−1 RNA ml −1
log10 HIV−1 RNA ml
5
7
−1
7
A
6
3
log10 HIV−1 RNA ml
Mathematical models, in conjunction with experimental data,
have provided important insights into virus infection, antiviral
therapy, emergence of drug resistance, and immune responses.
Time delays are inevitable in many engineering and biological systems [4,23,58], and have also been incorporated into viral dynamic
models. In this paper, we included two delays in a model to study
HIV-1 dynamics. One represents the time needed for infected cells
to produce virions after viral entry (intracellular delay), and the
6
5
4
3
Patient 10
τ2=30
2
τ =0
2
0
0
100
200
300
Days
400
500
0
6.5
100
200
300
Days
400
7
F
log10 HIV−1 RNA ml
−1
5.5
5
4.5
Patient 1
τ1=0.1
4
τ =1.5
log10 HIV−1 RNA ml
−1
E
6
6
5
4
3
Patient 2
τ1=0.41
2
1
τ =1.5
1
3.5
0
100
200
300
Days
400
1
0
500
0
100
200
300
Days
400
7
5
4
3
Patient 6
τ =0.14
1
1
τ1=1.5
0
log10 HIV−1 RNA ml −1
log10 HIV−1 RNA ml
−1
G
6
2
500
500
H
7
6
5
Patient 10
τ =0.1
4
1
τ1=1.5
3
0
100
200
300
Days
400
500
0
100
200
300
Days
400
500
Fig. 3. The effect of varying immune delay (A-D) or intracellular delay (E-H) on virus dynamics.
108
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
6
log10 HIV−1 RNA ml
5.5
5
4.5
Patient 1
dx=3.9×10−6
4
3.5
3
dx=3.9×10−4
0
100
200
300
Days
400
log10 HIV−1 RNA ml −1
7
6
−1
6.5
5
4
3
1
0
500
Patient 2
dx=8.8×10−6
2
dx=8.8×10−4
0
100
200
300
Days
400
500
−1
6
5
4
3
Patient 1
N=1504
N=5504
N=9504
2
1
0
0
100
200
300
Days
400
500
log10 HIV−1 RNA ml
log10 HIV−1 RNA ml
−1
7
6
4
Patient 2
N=2166
N=6166
N=10166
2
0
0
100
200
300
Days
400
500
Fig. 4. The effect of varying killing rate due to immune cells (top panels) or the viral burst size (bottom panels) on virus dynamics.
other denotes the time needed for the adaptive immune response
to emerge to control viral replication (immune delay). We studied
how these delays impact the dynamics. We defined a basic reproductive ratio R0 and showed that this ratio plays an important role
in determining the stability of the steady states of the delay model.
More specifically, we showed that the infection-free steady state is
locally asymptotically stable and globally attracting when R0 < 1.
In the case of s2 = 0 (no immune delay), the infected steady state
is locally asymptotically stable and globally attracting when
R0 > 1. In the case of s1 = 0 (no intracellular delay), we showed
that a Hopf bifurcation takes place at a critical threshold of the im
mune delay s2 > 0 even for a simplified model assuming the target cell concentration is constant. When s2 < s2 , the infected
steady state is still locally asymptotically stable. We also derived
conditions under which the infected steady state is globally
attracting when both delays are positive. These results suggest that
introducing the intracellular delay does not change the stability results (note that the basic reproductive ratio is intracellular delaydependent in this case). Incorporating the immune response delay
into the model generates rich dynamics.
Determining the stability switching regions for a model with
two positive delays is challenging. A few papers have studied the
mathematical properties of the steady states of a model with two
delays [1,10,25,24,52,53]. For a general linear scalar system with
two delays, Gu et al. [18] showed the stability crossing set can be
expressed by a few inequality constraints. Moreover, the crossing
curves fall into a few categories of curves, namely, closed curves,
open ended curves, and spiral-like curves oriented horizontally,
vertically, or diagonally. Identifying the local stability regions
when both the intracellular delay and the immune delay vary
within their biologically plausible ranges remains a potential topic
for future investigation.
We used a simple linear term, s dT, to model the generation
and death of target cells. Some studies included a logistic term
qT(t)[1 T(t)/Tmax] to represent the proliferation of target cells
[15,42]. They showed that periodic oscillations can occur through
Hopf bifurcation. Another logistic term, q T(t)[1 (T(t) + Tw(t))/
Tmax], was included to describe the proliferation of target cells in
a coupled way [11,60]. They showed that for an open set of parameter values the infected steady state can be unstable and periodic
solutions may exist. Recently, Li and Shu [27] studied the effects
of both the intracellular delay and target cell proliferation on virus
dynamics.
The model predicts that the infection-free steady state is stable
when R0 < 1, which indicates that the infection can be cleared if
antiretroviral therapy is sufficiently efficient in reducing the basic
reproductive ratio to below 1. However, current treatment regimens cannot eradicate the virus. The establishment of HIV-1 infection in a few cell populations, such as latently infected CD4+ T cells
[8,17,64] and hematopoietic progenitor cells [7], represents a major obstacle to viral elimination. Mathematical models have been
developed to describe low viral load persistence, slow decay of
the latent reservoir, and emergence of intermittent viral blips
above the detection limit in the setting of effective combination
therapy [49,50] (see a recent review in [51]). In this paper, we
did not include the latent reservoir. The major goal of this study
is to examine the effects of both the intracellular and immune delays on virus dynamics.
Model comparison with experimental data showed that including a time delay can affect the estimate of model parameters
[9,21,36]. We also fit the model with two delays to the viral load
data from 10 patients during primary HIV-1 infection. Although
we obtained a smaller error between data and fit for each patient,
a statistical test suggested that the improvement is not significant
for most of the patients. This highlights that more data are needed
for model verification and selection when we incorporate time delays into mathematical models to study virus dynamics.
Acknowledgments
Portions of this work were performed during the first three
authors’ visit to the Fields Institute (Summer 2010 Thematic Program on the Mathematics of Drug Resistance in Infectious Diseases). They would like to acknowledge the hospitality received
there. The work is supported in part by the NSF Grant DMS1122290 and NIH P30-EB011339 (LR), the NNSF of China (No.
61075037), the Fundamental Research Funds for the Central Universities (No. HIT.NSRIF.2010052) and Program of Excellent Team
in Harbin Institute of Technology (SL). The authors also thank the
referees for the comments that improved this manuscript.
K.A. Pawelek et al. / Mathematical Biosciences 235 (2012) 98–109
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