Post on 12-Nov-2023
transcript
Seediscussions,stats,andauthorprofilesforthispublicationat:https://www.researchgate.net/publication/26298298
RichdynamicsofahepatitisBviralinfectionmodelwithlogistichepatocytegrowth
ARTICLEinJOURNALOFMATHEMATICALBIOLOGY·JUNE2009
ImpactFactor:1.85·DOI:10.1007/s00285-009-0278-3·Source:PubMed
CITATIONS
35
READS
48
4AUTHORS,INCLUDING:
YangKuang
ArizonaStateUniversity
136PUBLICATIONS5,312CITATIONS
SEEPROFILE
Availablefrom:YangKuang
Retrievedon:04February2016
unco
rrec
ted p
roof
J. Math. Biol.
DOI 10.1007/s00285-009-0278-3 Mathematical Biology
Rich dynamics of a hepatitis B viral infection model
with logistic hepatocyte growth
Sarah Hews · Steffen Eikenberry ·
John D. Nagy · Yang Kuang
Received: 20 August 2008 / Revised: 18 May 2009
© Springer-Verlag 2009
Abstract Chronic hepatitis B virus (HBV) infection is a major cause of human1
suffering, and a number of mathematical models have examined within-host dynamics2
of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes3
regenerate at a constant rate from a source outside the liver; and/or (b) the infection4
takes place via a mass action process. Assumption (a) contradicts experimental data5
showing that healthy hepatocytes proliferate at a rate that depends on current liver size6
relative to some equilibrium mass, while assumption (b) produces a problematic basic7
reproduction number. Here we replace the constant infusion of healthy hepatocytes8
with a logistic growth term and the mass action infection term by a standard incidence9
function; these modifications enrich the dynamics of a well-studied model of HBV10
pathogenesis. In particular, in addition to disease free and endemic steady states, the11
system also allows a stable periodic orbit and a steady state at the origin. Since the12
system is not differentiable at the origin, we use a ratio-dependent transformation to13
show that there is a region in parameter space where the origin is globally stable.14
When the basic reproduction number, R0, is less than 1, the disease free steady state15
is stable. When R0 > 1 the system can either converge to the chronic steady state,16
experience sustained oscillations, or approach the origin. We characterize parameter17
regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and18
show how they depend on the basic reproduction number and the intrinsic growth rate19
of hepatocytes.20
S. Hews (B) · S. Eikenberry · Y. Kuang
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
e-mail: sarah.hews@asu.edu
Y. Kuang
e-mail: kuang@asu.edu
J. D. Nagy
Department of Biology, Scottsdale Community College, Scottsdale, AZ 85256, USA
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
Keywords HBV · Ratio-dependent transformation · Logistic hepatocyte growth ·21
Origin stability · Hopf bifurcation · Homoclinic bifurcation22
1 Introduction23
Hepatitis B virus (HBV) causes severe disease characterized by liver inflammation.24
Although a vaccine has been available since 1982 and distributed in over 116 countries,25
8–10% of the developing world is currently infected with HBV. The virus is contracted26
through contact with blood or other bodily fluids and is 50–100 times more infectious27
than HIV. Of those who contract HBV, 17.5% will develop chronic infections and are28
likely to die from cirrhosis of the liver or liver cancer. Children, especially infants,29
infected with HBV run the highest risk of chronic infection. Those with acute disease30
still experience severe symptoms for up to a year, including jaundice, extreme fatigue,31
nausea, vomiting and abdominal pain (Arguin et al. 2007; World Health Organization32
2000).33
Although HBV can be treated using interferon or lamivudine therapy, these treat-34
ments are expensive and therefore largely unavailable in the world’s poorest areas35
where disease burden is highest. So, a better understanding of HBV and its dynamics36
are crucial to developing cheaper vaccines and treatments for the developing countries37
predominantly affected by HBV.38
HBV primarily infects humans, but it has been known to infect some non-human39
primates (Grethe et al. 2000). Thus, there have been few animal and tissue models40
of human HBV, although this problem has been ameliorated to some degree by the41
discoveries of a number of related hepadnaviruses in wild animals. The woodchuck42
hepatitis virus (WHV) displays a very similar life-cycle and causes chronic disease43
similar to that seen in chronic HBV. Other related viruses infect ground squirrels44
and there are several avian hepadnaviruses; the duck HBV in particular has been45
a useful model (Tennant and Gerin 2001). Moreover, HBV transgenic mice have46
recently been used to study the effect of HBV on liver regeneration (Dong et al.47
2007). Mathematical models have the potential to complement such animal models48
and play a significant role in improving understanding of the in vivo dynamics of the49
disease.50
Most HBV models were not developed specifically to describe HBV dynamics, but51
rather were adaptations of HIV models to HBV. One of the earliest of these models,52
used by Nowak et al. (1996) and Nowak and May (2000) and commonly referred to53
as the basic virus infection model (BVIM), focuses on the dynamics of the number or54
mass of healthy cells (x)—in the current context, these are hepatocytes, HBV infected55
hepatocytes (y) and free virions (v). In particular,56
dx
dt= r − dx(t) − βv(t)x(t),57
dy
dt= βv(t)x(t) − ay(t), (1)58
dv
dt= γ y(t) − µv(t).59
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
In this model, healthy hepatocytes enter the liver at a constant rate r and die at per60
capita rate d. Infection of hepatocytes occurs through a simple mass action process61
at rate βv(t)x(t), where β is the mass action constant. Infected hepatocytes then die62
at per capita rate a. Each infected hepatocyte produces virions at per capita rate γ ,63
which die at per capita rate µ. Many subsequent models have adapted the structure of64
(1) to include immune system dynamics (Ciupe et al. 2007a,b) and various treatments65
(Long et al. 2008).66
A fundamental problem with model (1) is that it predicts a biologically unlikely67
relationship between liver size and susceptibility to HBV infection. Specifically, R068
depends on rd
, the homeostatic liver size. This relationship was shown by Gourley69
et al. (2008) to be an artifact of the mass action formulation of infection. To correct the70
problem, and to place the model on more sound biological grounds, Min et al. (2008)71
and Gourley et al. (2008) replace the mass action process with a standard incidence72
function.73
A further problem with model (1) is the assumption that healthy hepatocytes are74
replenished through “reseeding” from an outside source, perhaps the bone marrow, for75
example. However, it is well established that liver recovery following injury is facil-76
itated by widespread hepatocyte proliferation (Michalopoulos 2007). Several models77
have corrected this problem by introducing a logistic function for healthy hepatocyte78
growth (Ciupe et al. 2007a,b; Eikenberry et al. 2009). The addition of the logistic func-79
tion can result in an additional steady state at the origin. This combination of logistic80
growth and standard incidence functions produces the mathematical complication of81
a singularity at the origin. Since the origin is also a steady state, mathematical analysis82
of its stability properties becomes more complicated.83
Ciupe et al. (2007a,b) developed several models that included additional variables84
to explicitly model the immune system response. Mathematical analysis is difficult85
due to the complexity of their models, so the effect of the logistic hepatocyte growth86
term on the dynamics is difficult to determine.87
Eikenberry et al. (2009) extended Gourley et al.’s (2008) model to include logistic88
hepatocyte growth. This model also considered the latency period from cell infection89
to active virion production with an explicit time delay. In addition to the dynamics seen90
in Gourley et al. (2008), Eikenberry et al. observed the emergence of a stable periodic91
orbit with a period that depends on both the infection’s virulence and the hepatocyte92
regeneration rate. In addition, as the parameters in Eikenberry et al. move toward val-93
ues representing a more virulent disease state, the chronic equilibrium switches from94
stable to unstable, and a stable periodic orbit arises. Since the onset of large amplitude95
oscillations is quite sudden, Eikenberry et al. predicted that such a switch in stability96
could lead to, and therefore predict, the onset of acute liver failure (ALF). To investi-97
gate the cause and nature of this periodic orbit, here we study a simplified version of98
the Eikenberry et al. model.99
To analyze the dynamics near the origin, we use a ratio-dependent transformation.100
The dynamics seen in Eikenberry et al. (2009) are preserved in the model studied101
here, but we also find a homoclinic bifurcation and a region where the origin is glob-102
ally stable. We present a thorough description of the model dynamics and discuss the103
biological relevance.104
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
2 Full model and basic properties105
Our model uses the structure in (1) with few but significant changes. Like Gourley106
et al. (2008), Min et al. (2008), and Eikenberry et al. (2009), we replace the mass107
action process with a standard incidence function, and we use the logistic function108
for hepatocyte growth justified in Ciupe et al. (2007b,a) and Eikenberry et al. (2009).109
This leads to the following system:110
dx
dt= r x(t)
(
1 −T (t)
K
)
−βv(t)x(t)
T (t), (2)111
dy
dt=
βv(t)x(t)
T (t)− ay(t), (3)112
dv
dt= γ y(t) − µv(t), (4)113
with114
T (t) = x(t) + y(t),115
where x(t) is the mass of healthy hepatocytes, y(t) is the mass of infected hepa-116
tocytes, and v(t) is the mass of free virions. Healthy hepatocytes grow at a rate that117
depends on the homeostatic liver size, K , at a maximum per capita proliferation rate r .118
Hepatocytes become infected at maximum rate β and die at rate a. Free virions are119
created by infected hepatocytes at per-capita rate γ , and virions either disintegrate or120
are destroyed by the immune system at rate µ. All parameters in (2)–(4) are strictly121
positive.122
The basic reproduction number of model (2)–(4) is123
R0 =βγ
aµ.124
Since we are interested in HBV pathogenesis and not initial processes of infection,125
we assume that the initial data for the system (2)–(4) has the form126
x(0) = x0 > 0, y(0) = y0 > 0, v(0) = v0 > 0,127
K ≥ T (0) = x(0) + y(0). (5)128
We show in the first proposition that solutions of system (2)–(4) behave in a biologi-129
cally reasonable manner.130
Proposition 1 Each component of the solution of system (2)–(4), subject to (5) remains131
bounded and non-negative for all t > 0.132
Proof Notice that system (2)–(4) is locally Lipschitz at t = 0. Hence, a solution of133
system (2)–(4) subject to (5) exists and is unique on [0, b) for some b > 0.134
Assume first that there is a t1 such that b > t1 > 0, x(t1) = 0, and x(t) > 0, y(t) >135
0, v(t) > 0 for t ∈ (0, t1). Observe that136
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
dT
dt= r x(t)
(
1 −T (t)
K
)
− ay(t).137
It is easy to show that 0 < T (t) ≤ K for t ∈ [0, t1]. In fact, we can see that138
dTdt
≥ −aT (t) for t ∈ [0, t1], which yields139
T (t) ≥ T (0)e−at1 .140
Clearly y(t) ≤ K for t ∈ [0, t1], which implies that dv(t)dt
≤ γ K − µv(t). Therefore,141
v(t) ≤γ Kµ
+v(0)e−µt , which implies that v(t) ≤ V ≡ max{v(0),γ Kµ
} for t ∈ [0, t1].142
These observations imply that for t ∈ [0, t1], we have143
dx(t)
dt≥ −
βV e(a)t1
T (0)x(t).144
Hence a contradiction is obtained as145
x(t1) ≥ x(0)e−
βV e(a)t1
T (0)t1 > 0.146
Assume now that there is a t1 with b > t1 > 0 such that y(t1) = 0 and x(t) >147
0, y(t) > 0, v(t) > 0 for t ∈ (0, t1). Equation (2.3) implies that y′(t) ≥ −ay(t) for148
t ∈ [0, t1] which yields y(t1) > y(0)e−t1 > 0, also a contradiction.149
Finally, we assume again that b > t1 > 0 such that v(t1) = 0 and x(t)>0, y(t)>0,150
v(t) > 0 for t ∈ (0, t1). Clearly, this case is similar to the case of y(t1) = 0, and a151
contradiction can be obtained.152
The above contradictions together show that components of the solution of system153
(2)–(4) subject to (5) are non-negative for all t ∈ [0, b). This together with the uniform154
boundedness of T (t) and v(t) on [0, b) imply that b = ∞. This completes the proof155
of the proposition.156
System (2)–(4) has steady states E0 = (0, 0, 0), E f = (K , 0, 0), and E∗ = (x∗ > 0,157
y∗ > 0, v∗ > 0). E0 symbolizes complete liver failure, E f is a healthy, disease free,158
mature liver, and E∗ represents persistent, chronic HBV infection. E∗ is given by159
x∗ =K a
r
(
R∗
R0− 1
)
, (6)160
y∗ =K a
r
(
R∗
R0− 1
)
(R0 − 1), (7)161
v∗ =Kγ a
rµ
(
R∗
R0− 1
)
(R0 − 1), (8)162
where163
R∗ =r + a
a.164
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
E∗ exists only when165
1 ≤ R0 ≤ R∗.166
Since R∗ depends on the hepatocyte proliferation and death rate, we call R∗ the cel-167
lular vitality index. It is helpful to think of the basic reproduction number, R0, as the168
parameter grouping that explains the virus dynamics and the cellular vitality index, R∗,169
as the parameter grouping that explains the hepatocyte dynamics. One of our primary170
conclusions is that the behavior of (2)–(4) is determined not only by the values of R0171
and R∗, but their relationship to each other.172
3 Hepatocyte infection173
One possible criticism of (2)–(4) is that the infected hepatocytes do not proliferate. The174
effect of HBV infection on hepatocyte proliferation is controversial, with conflicting175
data showing both induction and inhibition of proliferation (Kwun and Jang 2004),176
and both pro- and anti-apoptotic effects (Wu et al. 2006). HBV X protein has severely177
impaired liver regeneration in some mouse models (Tralhao et al. 2002; Wu et al.178
2006; Dong et al. 2007), but had little effect in others (Hodgson et al. 2008). Natural179
variation in the HBV virus itself may explain these conflicting results (Kwun and Jang180
2004), and our model represents the limiting case where infection completely blocks181
hepatocyte proliferation.182
One important biological implication of (2)–(4) is that even without infected hepa-183
tocyte proliferation, the majority of hepatocytes still become infected during a chronic184
HBV infection. For example, Fig. 1 shows a time-series of an infection yielding a185
chronic steady state, along with the percentage of total hepatocytes infected as a func-186
tion of time. At the peak of infection nearly all of the hepatocytes are infected; this187
fraction drops somewhat when the chronic state is reached, but it is still quite high.188
Specifically, the fraction of infected hepatocytes during a chronic steady state is given189
by190
y∗
x∗ + y∗=
(R0 − 1)x∗
x∗(1 + (R0 − 1))=
R0 − 1
R0= 1 −
1
R0.191
Surprisingly, the fraction of infected hepatocytes in the chronic state is determined192
only by R0, which represents virus virulence, and not by the maximum healthy hepa-193
tocyte proliferation rate, r . For realistic values of R0, approximately 4 < R0 < 10, at194
least 75% of hepatocytes are infected (see Eikenberry et al. (2009) for model param-195
etrization).196
The picture is similar for infections that result in convergence to a periodic orbit.197
The fraction of hepatocytes infected peaks in early infection, and afterwards the frac-198
tion of infected hepatocytes changes cyclically, with nearly 100% of the hepatocytes199
infected at the peak. An example of such an infection is shown in Fig. 2. Figure 3 shows200
the percentage of hepatocytes infected at the peak of infection and at the chronic steady201
state as function of R0, for infections resulting in chronic disease.202
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
Fig. 1 The upper panel shows the time series solution of a chronic infection under the following parameter
values, r = 1, K = 2e11, β = 0.0014, γ = 280, a = 0.0693, and µ = 0.693. The lower panel shows the
percentage of hepatocytes that are infected over the course of infection
Fig. 2 Convergence to periodic solution under infection represented by the following parameter values,
r = 1, K = 2e11, β = 0.0014, γ = 320, a = 0.0693, and µ = 0.693. The upper panel shows the
time-series solution, and the lower panel shows the percentage of hepatocytes infected as a function of time
4 Stability for E f and E∗203
To gain a thorough understanding of (2)–(4), we study the local and global stabilities204
of all steady states. Since (2)–(4) is not differentiable at E0, the stability of the origin205
cannot be studied using a standard linearization approach. This issue will be addressed206
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
Fig. 3 The percentage of hepatocytes that are infected in a chronic steady state as a function of R0 . The frac-
tion in the chronic state is given byy∗
x∗+y∗ = 1− 1R0
, while the peak fraction is determined computationally
later in the paper. Due to the difficulties in the stability of the origin and to create a207
cohesive understanding of (2)–(4), we explore the dynamics as we increase R0. We208
start with R0 < 1. As with other infection models, we expect the disease free state, E f ,209
to be locally and globally asymptotically stable, and this is confirmed in Propositions 2210
and 3. When R0 > 1, we expect E f to become unstable, as is shown in Proposition 4.211
Proposition 2 If R0 < 1, then E f is locally asymptotically stable.212
Proof The Jacobian matrix of the vector field corresponding to (2)–(4) at E f is213
J (x, y, v)|E∗ =
⎛
⎝
−r −r −β
0 −a β
0 γ −µ
⎞
⎠ .214
The eigenvalues of the matrix are given by215
λ1 = −r , (9)216
λ2,3 = −1
2(a + µ) ±
1
2
√
(a + µ)2 − 4aµ(1 − R0). (10)217
λ2,3 are negative and thus E f is locally asymptotically stable when R0 < 1.218
Proposition 3 If R0 < 1, then E f is globally stable.219
Proof It is sufficient to show that (y(t), v(t)) → (0, 0). From there, it is clear that220
x(t) → K . From positivity of solutions, y and v satisfy the differential inequality:221
dy(t)
dt≤ βv(t) − ay(t) =
dY (t)
dt, (11)222
dv(t)
dt≤ γ y(t) − µv(t) =
dV (t)
dt. (12)223
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
Fig. 4 Bifurcation diagram for various values of R0 with r = 0.9961, K = 1.9922e11, β = 0.0014,
a = 0.0693, and µ = 0.693. For R0 < 1, E f is globally stable. When δ > σ , E∗ is locally stable
and then as R0 increases, (2)–(4) approaches a stable periodic orbit. There is a homoclinic bifurcation at
R0 =
(
1 − aµ
)
R∗ where E0 is stable even though E∗ is still positive, and then E∗ becomes negative with
E0 retaining stability
Since R0 < 1 and Y , V are linear, (Y (t), V (t)) → (0, 0) as t → ∞. Since y(t) ≤ Y (t)224
and v(t) ≤ V (t), (y(t), v(t)) → (0, 0) as t → ∞ by a simple comparison argument.225
Thus E f is globally stable.226
Proposition 4 If R0 > 1, then E f is unstable.227
Proof The eigenvalues of the Jacobian evaluated at E f are given by (9)–(10). At228
least one eigenvalue becomes positive when R0 > 1, so the steady state is229
unstable.230
The disease free steady state, E f , is locally and globally stable when R0 < 1 and231
unstable when R0 > 1 (Fig. 4). Note that stability of E f depends only on infection232
rate, production rate of free virions, death rate of infected cells, and death rate of free233
virions. In particular, the maximum per capita proliferation rate of healthy cells and234
homeostatic liver size do not affect the stability.235
As the reproduction number crosses the bifurcation point of R0 = 1, the sta-236
bility of E f is transferred to E∗ as it crosses into the positive quadrant. Recall237
that E∗ only exists in the positive quadrant when 1 < R0 < R∗. For the con-238
dition R0 < R∗ to hold, the proliferation rate has to be sufficiently large; specif-239
ically, r >βγ−aµ
µ. There is a region where E∗ is locally asymptotically stable240
before crossing a Hopf bifurcation point and entering a region with a stable limit241
cycle. The region of stability for E∗ and the Hopf bifurcation point are presented in242
Theorem 1.243
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
Theorem 1 Let δ = a2(
R∗
R0− 1
)
(R0−1)+ a2
R20
(R0−1)− arR0
(R0−1)+aµ
(
R∗
R0− 1
)
,244
and σ =−(µa2 R0+a3 R∗)(R∗−R0)(R0−1)
R0(µR0+a R∗). If δ > σ , then E∗ is locally asymptotically sta-245
ble and δ = σ is the Hopf bifurcation point.246
Proof
J (x, y, v)|E f=
⎛
⎜
⎜
⎜
⎝
r(1 −2x∗+y∗
K) −
βv∗ y∗
(x∗+y∗)2 − r x∗
K+
βv∗x∗
(x∗+y∗)2 −βx∗
x∗+y∗
βv∗ y∗
(x∗+y∗)2 −βv∗x∗
(x∗+y∗)2 − aβx∗
x∗+y∗
0 γ −µ
⎞
⎟
⎟
⎟
⎠
,247
The eigenvalues of J satisfy248
λ3 + a2λ2 + a1λ + a0 = 0,249
where250
a2 = µ + aR∗
R0,251
a1 = 2a2
(
R∗
R0− 1
)
(R0 − 1) +a2
R20
(R0 − 1)2 −ar
R0(R0 − 1) + aµ
(
R∗
R0− 1
)
,252
a0 = a2µ(R0 − 1)
(
R∗
R0− 1
)
.253
Clearly a2 > 0 and a0 > 0 when E∗ exists. Let,254
δ = a2
(
R∗
R0−1
)
(R0−1)+a2
R20
(R0−1)−ar
R0(R0−1)+aµ
(
R∗
R0−1
)
,255
a2a1 − a0 = µa2 + aR∗
R0a2 − a2µ(R0 − 1)
(
R∗
R0− 1
)
256
= µδ + aR∗
R0δ +
(
µa2 + a3 R∗
R0
)(
R∗
R0− 1
)
(R0 − 1) > 0.257
Therefore, a2a1 > a0 when258
δ >−(µa2 R0 + a3 R∗)(R∗ − R0)(R0 − 1)
R0(µR0 + a R∗)= σ.259
By Routh-Hurwitz criteria, we determine a condition for E∗ to be locally asymptoti-260
cally stable. Since the Routh-Hurwitz criteria are necessary and sufficient for stability,261
there is a Hopf bifurcation point at δ = σ .262
After increasing R0 past the Hopf bifurcation point, numerical solutions show that263
there is an attracting limit cycle in this region as shown in Fig. 4. Unlike in Eikenberry264
et al. (2009), these periodic solutions are not sustained as R0 increases indefinitely.265
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
Numerical investigation also suggests that there is a homoclinic bifurcation where the266
periodic solutions cease and E0 is stable (Fig. 4). Biologically, this represents ALF267
resulting from chronic HBV infection. The stability of the origin is therefore an inte-268
gral part of the system dynamics with an important biological meaning. These results269
are confirmed analytically in the following section.270
5 Ratio-dependent transformation and results for E0271
As mentioned previously, the stability of E0 cannot be studied using standard lineari-272
zation techniques. To overcome this difficulty, we use ratio-dependent transformations273
used in Hwang and Kuang (2003), Hsu et al. (2001), and Berezovsky et al. (2005).274
The first transformation gives a global stability result for E0. The second and third275
transformations yield a complete local stability result and an explicit form for the276
homoclinic bifurcation.277
The first transformation is a change of variable (x, y, v) → (x, z, w) where z =yx
278
and w = vx
. This transforms (2)–(4) to the following system:279
dx
dt= r x(t)
(
1 −x(t)(1 + z(t))
K
)
−βw(t)x(t)
1 + z(t), (13)280
dz
dt= βw(t) − az(t) − r z(t)
(
1 −x(t)(1 + z(t))
K
)
, (14)281
dw
dt= γ z(t) − µw(t) − rw(t)
(
1 −x(t)(1 + z(t))
K
)
+βw(t)2
1 + z(t). (15)282
The steady states of the transformed system are283
U0 = (0, 0, 0), Un = (0, zn, wn), U f = (K , 0, 0), U∗ =
(
x∗,y∗
x∗,v∗
x∗
)
,284
where285
zn =R∗(1 + r
µ) − R0
R∗(
aµ
− 1)
+ R0
, wn =a + r
βzn,286
and x∗, y∗ and v∗ are given by (6)–(8). Un is nonnegative when − aµ
< R0R∗ − 1 < r
µ,287
and U∗ is nonnegative when R∗ < R0 < 1. The steady states are preserved in that288
E f = U f and E∗ = U∗, while E0 has been blown up into two steady states: U0289
and Un . It is important to realize that the transformed system is not bounded. The290
following results will prove that there is a region where E0 is globally stable.291
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
Lemma 1 U0 is always unstable.292
Proof The variational matrix of the system (13)–(15) evaluated at U0 is293
J (x, z, w)|E0 =
⎛
⎝
r 0 0
0 −a − r β
0 γ −µ − r
⎞
⎠ ,294
where295
λ1 = r,296
λ2 = −1
2(a + µ) − r +
1
2
√
(a − µ)2 + 4kβ,297
λ3 = −1
2(a + µ) − r −
1
2
√
(a − µ)2 + 4kβ.298
Since λ1 = r > 0, U0 is always unstable.299
When R0 >
(
rµ
+ 1)
R∗, U0 is the only steady state of (13)–(15). Since it is always300
unstable, there are no stable steady states in (13)–(15) that map back to the origin.301
Lemma 2 and Theorem 2 show that if R0 >
(
rµ
+ 1)
R∗ and µ > a, E0 is globally302
stable.303
Lemma 2 If R0 >
(
rµ
+ 1)
R∗, then z, w → ∞ as t → ∞.304
Proof
dz
dt= βw − az − r z
(
1 −x(1 + z)
K
)
> βw − (a + r)z,305
dw
dt= γ z − µw − rw
(
1 −x(1 + z)
K
)
+βw2
1 + z> γ z − (µ + r)w.306
Let307
d Z
dt= βW − (a + r)Z , (16)308
dW
dt= γ Z − (µ + r)W . (17)309
(0, 0) is the only steady state of (16)–(17) and is unstable when R0 >
(
rµ
+ 1)
R∗.310
Since there are no other steady states, and Z , W are unbounded, Z , W → ∞ as311
t → ∞. Since dzdt
> d Zdt
, dwdt
> dWdt
, z, w → ∞ as t → ∞, and R0 >
(
rµ
+ 1)
R∗.312
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
Theorem 2 If R0 >
(
rµ
+ 1)
R∗ and µ > a, then x → 0 as t → ∞.313
Proof Since314
dx
dt= r x(t)
(
1 −x(t)(1 + z(t))
K
)
−βw(t)x(t)
1 + z(t)<
(
r −βw(t)
1 + z(t)
)
x(t),315
it is sufficient to show316
limt→∞
βw(t)
1 + z(t)> r. (18)317
Let θ(t) =βw(t)1+z(t)
. Then318
dθ
dt=
β dwdt
1 + z(t)−
βw(t) dzdt
(1 + z(t))2319
= [βγ + (a + r)θ(t)]z(t)
1 + z(t)− (µ + r)θ(t) +
r x(t)z(t)θ(t)
K320
≥ [βγ + (a + r)θ(t)]z(t)
1 + z(t)− (µ + r)θ(t). (19)321
By Lemma (2), for all ǫ > 0, ∃ t∗ s.t. ∀ t ≥ t∗,322
z(t)
1 + z(t)> 1 − ǫ. (20)323
Combining (19) and (20) for t ≥ t∗,324
dθ
dt> βγ (1 − ǫ) + ((a + r)(1 − ǫ) − µ − r)θ(t).325
Letting Ŵ(ǫ) = (a + r)(1 − ǫ) − µ − r and solving for θ(t) yields326
θ(t) >βk(1 − ǫ)
−Ŵ(ǫ)+ θ(t∗)eŴ(ǫ)(t−t∗) = (t).327
Since µ > a, Ŵ(ǫ) < 0. Therefore, limt→∞
(t) =βγ (1−ǫ)−Ŵ(ǫ)
. Since R0 >
(
rµ
+ 1)
R∗,328
∃ ǫ∗ > 0 s.t. ∀ ǫ ∈ (0, ǫ∗],βγ (1−ǫ)
µ+r> a + r . So,329
limt→∞
βw(t)
1 + z(t)≥
βγ (1 − ǫ)
−Ŵ(ǫ)>
βγ (1 − ǫ)
µ + r> a + r > r.330
Thus, (18) is satisfied.331
Since y = xz, v = xw, and (2)–(4) is bounded, y, z → 0 when x → 0 and332
z, w → ∞.Therefore, Theorem 2 and Lemma 2 prove that E0 in (2)–(4) is globally333
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
stable when R0 >
(
rµ
+ 1)
R∗ and µ > a. Biologically, this implies that a sufficiently334
virulent infection will result in complete liver failure.335
The second transformation, that yields the homoclinic bifurcation point and a larger336
region for stability of E0, is (x, y, v) → (g, h, v) where g = xv
and h =yv
. This337
yields the system,338
dg
dt= rg(t)
(
1 −v(t)(g(t) + h(t))
K
)
−βg(t)
g(t) + h(t)− γ g(t)h(t) + µg(t), (21)339
dh
dt=
βg(t)
g(t) + h(t)− ah(t) − γ h(t)2 + µh(t), (22)340
dv
dt= γ h(t)v(t) − µv(t). (23)341
In addition to there still being a singularity at (0, 0, 0), (21)–(23) exhibits three342
steady states.343
U † =
(
0,µ − a
γ, 0
)
, Un =
(
1
wn
,zn
wn
, 0
)
, U∗ =
(
x∗
v∗,
y∗
v∗, v∗
)
,344
where zn and wn are given by (16) and x∗, y∗ and v∗ are given by (6)–(8). U † is the345
only steady state that is not present in the original system or the other transforma-346
tions. It is only positive when µ > a and provides the condition for the homoclinic347
bifurcation.348
Lemma 3 For the system (21)–(23), if µ > a, the following results hold:349
(a) If(
1 − aµ
)
R∗ > R0, then U † is a saddle point;350
(b) If(
1 − aµ
)
R∗ < R0, then U † is locally asymptotically stable.351
Proof The Jacobian of system (21)–(23) evaluated at U † is352
J (g, h, v)|U † =
⎛
⎜
⎜
⎜
⎝
r −βγ
µ−a0 0
βγµ−a
a − µ 0
0 0 −a
⎞
⎟
⎟
⎟
⎠
.353
Since the Jacobian is a triangular matrix, the eigenvalues are given by354
λ1 = r −βγ
µ − a,355
λ2 = a − µ,356
λ3 = −a.357
Since µ > a, λ2,3 are always negative. λ1 can be rewritten as aµ−a
((µ − a)358
R∗ − µR0) , which is positive when(
1 − aµ
)
R∗ > R0 and negative when(
1 − aµ
)
359
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
R∗ < R0. Therefore, when µ > a and(
1 − aµ
)
R∗ > R0, U † is a saddle point, and360
when µ > a and(
1 − aµ
)
R∗ < R0, U † is locally asymptotically stable.361
Since U † maps back to E0, we have the following:362
Theorem 3 For the system (2)–(4), if µ > a, the following results hold:363
(a) If(
1 − aµ
)
R∗ > R0, then E0 is a saddle point;364
(b) If(
1 − aµ
)
R∗ < R0, then E0 is locally asymptotically stable.365
Computationally, we have confirmed that at(
1 − aµ
)
R∗ = R0, the periodic solu-366
tions collide with the saddle point, thus creating a homoclinic bifurcation point.367
Furthermore,(
1 − aµ
)
R∗ < R0 includes the entire parameter regime where Un is368
positive, so it is not necessary to analyze the stability of Un as long as µ > a. Mathe-369
matically, the homoclinic bifurcation point corresponds to the onset of complete liver370
failure. Biologically, liver failure might occur prior to this point as metabolic demands371
overwhelm hepatocyte proliferation when liver mass is very low (Rozga 2002).372
The final transformation is (x, y, v) → (m, y, n) where m = xy
and n = vy. It373
yields all of the same steady states and results as (21)–(23) so it is omitted here.374
6 Discussion375
In (1), Nowak et al. (1996) modeled the infection of healthy hepatocytes by free virions376
as a mass action process. This makes the viral basic reproduction number dependent377
on the homeostatic liver size, rd
, implying that individuals with smaller livers are378
more susceptible to HBV infection. Gourley et al. (2008), Min et al. (2008), and379
Eikenberry et al. (2009), and the current model all replace this mass action process380
with a standard incidence function, eliminating this artifact. Here, we also keep the381
logistic proliferation term used in Eikenberry et al. (2009), and similar to that in Ciupe382
et al. (2007a,b), since hepatocytes are produced in the liver and their numbers are383
homeostatically regulated. Together, these improvements significantly increase the384
richness of the predicted dynamics.385
Here we introduce the concept of R∗, the cellular vitality index, which is a combi-386
nation of parameters that describes the hepatocyte behavior and includes their prolif-387
eration and death rates. One of our main conclusions is that the relationship between388
R0 and R∗ generates significant insight into both the mathematical behavior and bio-389
logical interpretation of model (2)–(4). The difference between them ultimately deter-390
mines whether the infection will be chronic, undergo oscillations, or induce ALF. This391
result suggests that treatments could focus not just on reducing R0, which represents392
virus virulence, but also on increasing R∗, which represents the hepatocytes ability to393
regenerate.394
In addition to disease free (E f ), and chronic (E∗) equilibria, we also see the emer-395
gence of a stable periodic orbit and a steady state at the origin (E0). Similar to previous396
models, E f is globally asymptotically stable when R0 < 1 and unstable when R0 > 1.397
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
For R0 > 1, we prove the existence of a region where E∗ is stable and find the closed398
form for the Hopf bifurcation point. We also computationally determine the region399
where a stable periodic orbit exists (Fig. 4).400
In contrast to system (2)–(4), the origin does not appear to be stable anywhere in401
the Eikenberry et al. (2009) system. Apparently the delay term in Eikenberry et al.’s402
model obliterates stability at the origin; however, in our system, the origin is stable403
for sufficiently large values of R0. Specifically, we implement a change of variable404
technique as shown in Hwang and Kuang (2003), Hsu et al. (2001), and Berezovsky405
et al. (2005) to prove that E0 is globally stable when R0 > (1 + rµ)R∗ and that there406
is a homoclinic bifurcation point at R0 =
(
1 − aµ
)
R∗. This technique is called a407
ratio-dependent or a blow-up transformation.408
Changes in the dynamics within different parameter regimes are dominated by R0409
and R∗. When R0 < 1, a perturbed (i.e. infected) liver will always return to a healthy,410
disease free state. At R0 = 1, there is a transcritical bifurcation as E f = E∗. As R0411
becomes larger than 1, the stability of E f is transferred to the chronic state, E∗, which412
crosses into the positive quadrant. As R0 increases further, the system crosses a super-413
critical Hopf bifurcation point and all solutions approach an attracting periodic orbit.414
Increasing R0 further causes the system to cross a homoclinic bifurcation point where415
E0 becomes stable. Beyond this point, the liver has failed completely. The changes in416
dynamics are shown in Fig. 4.417
Despite our analysis of (2)–(4), there are still a few open questions. For example,418
we have yet to prove a region where the chronic steady state is globally stable. We419
anticipate that a Lyapunov function can be employed to complete the proof. We also420
have yet to completely characterize the region where there is an attracting periodic421
orbit and prove its existence.422
When the delay term in Eikenberry et al. (2009) approaches 0, we would expect the423
dynamics to converge to those in (2)–(4). However, in computational investigations,424
this is not the case. For increasingly virulent infections, sustained oscillations are425
observed in the delay model, but the origin is never stable. While the period of these426
oscillations increases marginally as the delay becomes very small, there is apparently427
no convergence to the origin, in contrast to the behavior seen in our model. Although428
the essential biological prediction of Eikenberry et al. (2009), that ALF in HBV infec-429
tion can be induced by a switch in stability and may be preceded by oscillations in430
viral load, is preserved, origin stability for a sufficiently virulent infection is seen only431
in our model, implying a fundamental change in the system dynamics when the delay432
is omitted.433
In our model, we have assumed that infected hepatocytes do not proliferate. Infec-434
tion by HBV can clearly affect both hepatocyte proliferation and apoptosis, and these435
changes are linked to virus-induced hepatocellular carcinoma (Wu et al. 2006), yet436
how infection affects these processes remains controversial. HBV X (HBx) protein437
has been widely studied in this context. HBx affects cell-cycle progression, and it can438
induce entry into the cell cycle, DNA synthesis, and proliferation (Benn and Schneider439
1995). HBx has variously been reported to have either pro- or anti-apoptotic effects440
(Wu et al. 2006). While much work has demonstrated that HBx can induce prolifera-441
tion, it may also block or prolong the G1 → S transition by acting upon the cell cycle442
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
Rich dynamics of a hepatitis B viral infection model
inhibitor p21 (Park et al. 2000; Kwun and Jang 2004). Natural variants in the HBx443
protein may affect p21 differently, and thus affect cell proliferation differently (Kwun444
and Jang 2004).445
Eikenberry et al. (2009) justified the assumption of logistic hepatocyte growth for446
healthy cells on the basis of the pattern of healthy liver regeneration seen in 2/3 partial447
hepatectomies (PHx). Several studies in HBx transgenic mice have shown that this448
protein severely inhibits liver regeneration and hepatocyte proliferation following PHx449
(Wu et al. 2006; Tralhao et al. 2002). Wu et al. (2006) found that the G1 → S transition450
was blocked in such transgenic mice, but these mice also had steatotic livers which can451
independently impede regeneration. Several studies support a paracrine role for HBV452
viral proteins in inhibiting regeneration. Tralhao et al. (2002) found that transplanta-453
tion of HBx expressing hepatocytes into a healthy liver could impede regeneration, and454
Dong et al. (2007) found that natural killer T cells inhibited regeneration in HBV-tg455
transgenic livers, largely through cytokine (interferon-γ ) inhibition of proliferation.456
However, Hodgson et al. (2008) found that HBx induced early entry into the cell cycle457
in regenerating hepatocytes and did not impair liver regeneration.458
Hepatocyte proliferation also affects HBV virus expression; Ozer et al. (1996)459
found that arrest in either G1 or G2 increased virus expression while passage through460
S and DNA synthesis inhibited viral mRNA. While increased proliferation in infected461
hepatocytes may be expected to enhance infection, rapid hepatocyte turnover aided462
infection clearance in a duck model of chronic duck HBV (Fourel et al. 1994). Hepa-463
tocyte proliferation has been observed to inhibit production of or possibly destroy464
viral nucleocapsids (Guidotti et al. 1994), and dilution of HBV cccDNA caused by465
proliferation may cause spontaneous recovery (Fourel et al. 1994; Guidotti et al. 1994).466
Thus, HBV infection may either block or induce hepatocyte proliferation, and467
proliferation itself can affect virus replication and expression. Therefore, patterns of468
hepatocyte proliferation may be central to the dynamics of virus infection and are469
one area where modeling has the potential to generate significant insight. Ciupe et al.470
incorporated infected hepatocyte proliferation in several mathematical models of acute471
HBV infection (Ciupe et al. 2007a,b) and also considered the possibility of spontaneous472
recovery through cccDNA dilution (Ciupe et al. 2007b). Our model represents the lim-473
iting case where infection completely blocks hepatocyte proliferation. We have found474
that, under this assumption, the majority of hepatocytes become infected in the chronic475
disease state, and that biologically plausible dynamics are observed overall. The impor-476
tance of infected hepatocyte proliferation and the effect of proliferation on viral repli-477
cation in our model of chronic disease are two problems that we are currently studying.478
Acknowledgments This research is partially supported by the NSF grant DMS-0436341 and the grant479
DMS/NIGMS-0342388 jointly funded by NIH and NSF. We would like to thank the anonymous reviewers480
for their valuable comments and suggestions.481
References482
Arguin PM, Kozarsky PE, Reed C (eds) (2007) CDC health information for international travel 2008.483
Elsevier, Philadelphia484
Benn J, Schneider RJ (1995) Hepatitis B virus HBx protein deregulates cell cycle checkpoint controls. Proc485
Natl Acad Sci USA 92:11215–11219486
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of
unco
rrec
ted p
roof
S. Hews et al.
Berezovsky F, Karev G, Song B, Castillo-Chavez C (2005) A simple epidemic model with surprising487
dynamics. Math Biol Eng 2:133–152488
Ciupe SM, Ribeiro RM, Nelson PW, Dusheiko G, Perelson AS (2007a) The role of cells refractory to489
productive infection in acute hepatitis B viral dynamics. Proc Natl Acad Sci USA 104:5050–5055490
Ciupe SM, Ribeiro RM, Nelson PW, Perelson AS (2007b) Modeling the mechanisms of acute hepatitis B491
virus infection. J Theor Biol 247:23–35492
Dong Z, Zhang J, Sun R, Wei H, Tian Z (2007) Impairment of liver regeneration correlates with activated493
hepatic NKT cells in HBV transgenic mice. Hepatology 45:1400–1412494
Eikenberry S, Hews S, Nagy JD, Kuang Y (2009) The dynamics of a delay model of hepatitis B virus495
infection with logistic hepatocyte growth. Math Biol Eng 6(2):283–299496
Fourel I, Cullen JM, Saputelli J, Aldrich CE, Schaffer P, Averett DR, Pugh J, Mason WS (1994) Evidence497
that hepatocyte turnover is required for rapid clearance of duck hepatitis B virus during antiviral498
therapy of chronically infected ducks. J Virol 68:8321–8330499
Ganem D, Prince A (2004) Hepatitis B virus infection – natural history and clinical consequences. N Engl500
J Med 350:1118–1129501
Gourley SA, Kuang Y, Nagy JD (2008) Dynamics of a delay differential model of hepatitis B virus. J Biol502
Dyn 2:140–153503
Grethe S, Heckel JO, Rietschel W, Hufert FT (2000) Molecular epidemiology of hepatitis B virus variants504
in nonhuman primates. J Virol 74:5377–5381505
Guidotti LG, Martinez V, Loh YT, Rogler CE, Chisari FV (1994) Hepatitis B virus nucleocapsid particles506
do not cross the hepatocyte nuclear membrane in transgenic mice. J Virol 68:5469–5475507
Hodgson AJ, Keasler VV, Slagle BL (2008) Premature cell cycle entry induced by hepatitis B virus regu-508
latory HBx protein during compensatory liver regeneration. Cancer Res 68:10341–10348509
Hsu SB, Hwang TW, Kuang Y (2001) Global analysis of the Michaelis-Menten-type ratio-dependent pred-510
ator-prey system. J Math Biol 42:489–506511
Hwang TW, Kuang Y (2003) Deterministic extinction effect of parasites on host populations. J Math Biol512
46:17–30513
Kwun HJ, Jang KL (2004) Natural variants of hepatitis B virus X protein have differential effects on the514
expression of cyclin-dependent kinase inhibitor p21 gene. Nucleic Acids Res 32:2202–2213515
Long C, Qi H, Huang SH (2008) Mathematical modeling of cytotoxic lymphocyte-mediated immune516
responses to hepatitis B virus infection. J Biomed Biotechnol 2008:1–9517
Michalopoulos GK (2007) Liver regeneration. J Cell Physiol 213:286–300518
Min L, Su Y, Kuang Y (2008) Mathematical analysis of a basic virus infection model with application to519
HBV infection. Rocky Mount J Math 38:1573–1585520
Nowak MA, May RM (2000) Virus dynamics. Oxford University Press, Oxford521
Nowak MA, Bonhoeffer S, Hill AM, Boehme R, Thomas HC, McDade H (1996) Viral dynamics in hepatitis522
B virus infection. Proc Natl Acad Sci USA 93:4398–4402523
Ozer A, Khaoustov VI, Mearns M, Lewis DE, Genta RM, Darlington GJ, Yoffe B (1996) Effect of hepa-524
tocyte proliferation and cellular DNA synthesis on hepatitis B virus replication. Gastroenterology525
110:1519–1528526
Park US, Park SK, Lee YI, Park JG, Lee YI (2000) Hepatitis B virus-X protein upregulates the expression527
of p21waf1/cip1 and prolongs G1–>S transition via a p53-independent pathway in human hepatoma528
cells. Oncogene 19:3384–3394529
Rozga J (2002) Hepatocyte proliferation in health and in liver failure. Med Sci Monit 8:RA32–RA38530
Tennant BC, Gerin JL (2001) The woodchuck model of hepatitis B virus infection. ILAR J 42:89–102531
Tralhao JG, Roudier J, Morosan S, Giannini C, Tu H, Goulenok C, Carnot F, Zavala F, Joulin V,532
Kremsdorf D, Brchot C (2002) Paracrine in hepatitis B virus X protein (HBx) on liver cell proliferation:533
an alternative mechanism of HBx-related pathogenesis. Proc Natl Acad Sci USA 99:6991–6996534
World Health Organization (2000) Hepatitis B fact sheet No. 204. WHO website535
Wu BK, Li CC, Chen HJ, Chang JL, Jeng KS, Chou CK, Hsu MT, Tsai TF (2006) Blocking of G1/S tran-536
sition and cell death in the regenerating liver of Hepatitis B virus X protein transgenic mice. Biochem537
Biophys Res Commun 340:916–928538
123
Journal: 285 Article No.: 0278 MS Code: JMB327.1 TYPESET DISK LE CP Disp.:2009/6/3 Pages: 18 Layout: Small-X
Au
tho
r P
ro
of