Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

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

https://www.researchgate.net/publication/26298298_Rich_dynamics_of_a_hepatitis_B_viral_infection_model_with_logistic_hepatocyte_growth?enrichId=rgreq-4894a7a6-378d-4588-bbb6-48afed8244bc&enrichSource=Y292ZXJQYWdlOzI2Mjk4Mjk4O0FTOjk4NTg0Mjg4NTYzMjE4QDE0MDA1MTU3MzQ5Mjc%3D&el=1_x_2

https://www.researchgate.net/publication/26298298_Rich_dynamics_of_a_hepatitis_B_viral_infection_model_with_logistic_hepatocyte_growth?enrichId=rgreq-4894a7a6-378d-4588-bbb6-48afed8244bc&enrichSource=Y292ZXJQYWdlOzI2Mjk4Mjk4O0FTOjk4NTg0Mjg4NTYzMjE4QDE0MDA1MTU3MzQ5Mjc%3D&el=1_x_3

https://www.researchgate.net/?enrichId=rgreq-4894a7a6-378d-4588-bbb6-48afed8244bc&enrichSource=Y292ZXJQYWdlOzI2Mjk4Mjk4O0FTOjk4NTg0Mjg4NTYzMjE4QDE0MDA1MTU3MzQ5Mjc%3D&el=1_x_1

https://www.researchgate.net/profile/Yang_Kuang?enrichId=rgreq-4894a7a6-378d-4588-bbb6-48afed8244bc&enrichSource=Y292ZXJQYWdlOzI2Mjk4Mjk4O0FTOjk4NTg0Mjg4NTYzMjE4QDE0MDA1MTU3MzQ5Mjc%3D&el=1_x_4


https://www.researchgate.net/institution/Arizona_State_University?enrichId=rgreq-4894a7a6-378d-4588-bbb6-48afed8244bc&enrichSource=Y292ZXJQYWdlOzI2Mjk4Mjk4O0FTOjk4NTg0Mjg4NTYzMjE4QDE0MDA1MTU3MzQ5Mjc%3D&el=1_x_6


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: [email protected]

Y. Kuang

e-mail: [email protected]

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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

unco

rrec

ted p

roof


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


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


Au

tho

r P

ro

of

Date post:	12-Nov-2023
Category:	Documents
Upload:	asu
View:	0 times
Download:	0 times

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

Documents