J. Math. Biol. (2010) 60:573–590DOI 10.1007/s00285-009-0278-3 Mathematical Biology

Rich dynamics of a hepatitis B viral infection modelwith logistic hepatocyte growth

Sarah Hews · Steffen Eikenberry ·John D. Nagy · Yang Kuang

Received: 20 August 2008 / Revised: 18 May 2009 / Published online: 17 June 2009© Springer-Verlag 2009

Abstract Chronic hepatitis B virus (HBV) infection is a major cause of humansuffering, and a number of mathematical models have examined within-host dynamicsof the disease. Most previous HBV infection models have assumed that: (a) hepatocytesregenerate at a constant rate from a source outside the liver; and/or (b) the infectiontakes place via a mass action process. Assumption (a) contradicts experimental datashowing that healthy hepatocytes proliferate at a rate that depends on current liver sizerelative to some equilibrium mass, while assumption (b) produces a problematic basicreproduction number. Here we replace the constant infusion of healthy hepatocyteswith a logistic growth term and the mass action infection term by a standard incidencefunction; these modifications enrich the dynamics of a well-studied model of HBVpathogenesis. In particular, in addition to disease free and endemic steady states, thesystem also allows a stable periodic orbit and a steady state at the origin. Since thesystem is not differentiable at the origin, we use a ratio-dependent transformation toshow that there is a region in parameter space where the origin is globally stable.When the basic reproduction number, R0, is less than 1, the disease free steady stateis stable. When R0 > 1 the system can either converge to the chronic steady state,experience sustained oscillations, or approach the origin. We characterize parameterregions for all three situations, identify a Hopf and a homoclinic bifurcation point, andshow how they depend on the basic reproduction number and the intrinsic growth rateof hepatocytes.

S. Hews (B) · S. Eikenberry · Y. KuangDepartment of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USAe-mail: sarah.hews@asu.edu

Y. Kuange-mail: kuang@asu.edu

J. D. NagyDepartment of Biology, Scottsdale Community College, Scottsdale, AZ 85256, USA

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Keywords HBV · Ratio-dependent transformation · Logistic hepatocyte growth ·Origin stability · Hopf bifurcation · Homoclinic bifurcation

Mathematics Subject Classification (2000) 34C23 · 34C25 · 92C50

1 Introduction

Hepatitis B virus (HBV) causes severe disease characterized by liver inflammation.Although a vaccine has been available since 1982 and distributed in over 116 countries,8–10% of the developing world is currently infected with HBV. The virus is contractedthrough contact with blood or other bodily fluids and is 50–100 times more infectiousthan HIV. Of those who contract HBV, 17.5% will develop chronic infections and arelikely to die from cirrhosis of the liver or liver cancer. Children, especially infants,infected with HBV run the highest risk of chronic infection. Those with acute diseasestill experience severe symptoms for up to a year, including jaundice, extreme fatigue,nausea, vomiting and abdominal pain (Arguin et al. 2007; World Health Organization2000).

Although HBV can be treated using interferon or lamivudine therapy, these treat-ments are expensive and therefore largely unavailable in the world’s poorest areaswhere disease burden is highest. So, a better understanding of HBV and its dynamicsare crucial to developing cheaper vaccines and treatments for the developing countriespredominantly affected by HBV.

HBV primarily infects humans, but it has been known to infect some non-humanprimates (Grethe et al. 2000). Thus, there have been few animal and tissue modelsof human HBV, although this problem has been ameliorated to some degree by thediscoveries of a number of related hepadnaviruses in wild animals. The woodchuckhepatitis virus (WHV) displays a very similar life-cycle and causes chronic diseasesimilar to that seen in chronic HBV. Other related viruses infect ground squirrels andthere are several avian hepadnaviruses; the duck HBV in particular has been a usefulmodel (Tennant and Gerin 2001). Moreover, HBV transgenic mice have recently beenused to study the effect of HBV on liver regeneration (Dong et al. 2007). Mathematicalmodels have the potential to complement such animal models and play a significantrole in improving understanding of the in vivo dynamics of the disease.

Most HBV models were not developed specifically to describe HBV dynamics, butrather were adaptations of HIV models to HBV. One of the earliest of these models,used by Nowak et al. (1996) and Nowak and May (2000) and commonly referred toas the basic virus infection model (BVIM), focuses on the dynamics of the number ormass of healthy cells (x)—in the current context, these are hepatocytes, HBV infectedhepatocytes (y) and free virions (v). In particular,

dx

dt= r − dx(t) − βv(t)x(t),

dy

dt= βv(t)x(t) − ay(t), (1)

dv

dt= γ y(t) − µv(t).

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Rich dynamics of a hepatitis B viral infection model 575

In this model, healthy hepatocytes enter the liver at a constant rate r and die at percapita rate d. Infection of hepatocytes occurs through a simple mass action processat rate βv(t)x(t), where β is the mass action constant. Infected hepatocytes then dieat per capita rate a. Each infected hepatocyte produces virions at per capita rate γ ,which die at per capita rate µ. Many subsequent models have adapted the structure of(1) to include immune system dynamics (Ciupe et al. 2007a,b) and various treatments(Long et al. 2008).

A fundamental problem with model (1) is that it predicts a biologically unlikelyrelationship between liver size and susceptibility to HBV infection. Specifically, R0depends on r

d , the homeostatic liver size. This relationship was shown by Gourleyet al. (2008) to be an artifact of the mass action formulation of infection. To correct theproblem, and to place the model on more sound biological grounds, Min et al. (2008)and Gourley et al. (2008) replace the mass action process with a standard incidencefunction.

A further problem with model (1) is the assumption that healthy hepatocytes arereplenished through “reseeding” from an outside source, perhaps the bone marrow, forexample. However, it is well established that liver recovery following injury is facil-itated by widespread hepatocyte proliferation (Michalopoulos 2007). Several modelshave corrected this problem by introducing a logistic function for healthy hepatocytegrowth (Ciupe et al. 2007a,b; Eikenberry et al. 2009). The addition of the logistic func-tion can result in an additional steady state at the origin. This combination of logisticgrowth and standard incidence functions produces the mathematical complication ofa singularity at the origin. Since the origin is also a steady state, mathematical analysisof its stability properties becomes more complicated.

Ciupe et al. (2007a,b) developed several models that included additional variablesto explicitly model the immune system response. Mathematical analysis is difficultdue to the complexity of their models, so the effect of the logistic hepatocyte growthterm on the dynamics is difficult to determine.

Eikenberry et al. (2009) extended Gourley et al.’s (2008) model to include logistichepatocyte growth. This model also considered the latency period from cell infectionto active virion production with an explicit time delay. In addition to the dynamics seenin Gourley et al. (2008), Eikenberry et al. observed the emergence of a stable periodicorbit with a period that depends on both the infection’s virulence and the hepatocyteregeneration rate. In addition, as the parameters in Eikenberry et al. move toward val-ues representing a more virulent disease state, the chronic equilibrium switches fromstable to unstable, and a stable periodic orbit arises. Since the onset of large amplitudeoscillations is quite sudden, Eikenberry et al. predicted that such a switch in stabilitycould lead to, and therefore predict, the onset of acute liver failure (ALF). To investi-gate the cause and nature of this periodic orbit, here we study a simplified version ofthe Eikenberry et al. model.

To analyze the dynamics near the origin, we use a ratio-dependent transformation.The dynamics seen in Eikenberry et al. (2009) are preserved in the model studiedhere, but we also find a homoclinic bifurcation and a region where the origin is glob-ally stable. We present a thorough description of the model dynamics and discuss thebiological relevance.

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2 Full model and basic properties

Our model uses the structure in (1) with few but significant changes. Like Gourleyet al. (2008), Min et al. (2008), and Eikenberry et al. (2009), we replace the massaction process with a standard incidence function, and we use the logistic functionfor hepatocyte growth justified in Ciupe et al. (2007b,a) and Eikenberry et al. (2009).This leads to the following system:

dx

dt= r x(t)

(1 − T (t)

K

)− βv(t)x(t)

T (t), (2)

dy

dt= βv(t)x(t)

T (t)− ay(t), (3)

dv

dt= γ y(t) − µv(t), (4)

with

T (t) = x(t) + y(t),

where x(t) is the mass of healthy hepatocytes, y(t) is the mass of infected hepa-tocytes, and v(t) is the mass of free virions. Healthy hepatocytes grow at a rate thatdepends on the homeostatic liver size, K , at a maximum per capita proliferation rate r .Hepatocytes become infected at maximum rate β and die at rate a. Free virions arecreated by infected hepatocytes at per-capita rate γ , and virions either disintegrate orare destroyed by the immune system at rate µ. All parameters in (2)–(4) are strictlypositive.

The basic reproduction number of model (2)–(4) is

R0 = βγ

aµ.

Since we are interested in HBV pathogenesis and not initial processes of infection,we assume that the initial data for the system (2)–(4) has the form

x(0) = x0 > 0, y(0) = y0 > 0, v(0) = v0 > 0,

K ≥ T (0) = x(0) + y(0). (5)

We show in the first proposition that solutions of system (2)–(4) behave in a biologi-cally reasonable manner.

Proposition 1 Each component of the solution of system (2)–(4), subject to (5) remainsbounded and non-negative for all t > 0.

Proof Notice that system (2)–(4) is locally Lipschitz at t = 0. Hence, a solution ofsystem (2)–(4) subject to (5) exists and is unique on [0, b) for some b > 0.Assume first that there is a t1 such that b > t1 > 0, x(t1) = 0, and x(t) > 0, y(t) >

0, v(t) > 0 for t ∈ (0, t1). Observe that

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Rich dynamics of a hepatitis B viral infection model 577

dT

dt= r x(t)

(1 − T (t)

K

)− ay(t).

It is easy to show that 0 < T (t) ≤ K for t ∈ [0, t1]. In fact, we can see thatdTdt ≥ −aT (t) for t ∈ [0, t1], which yields

T (t) ≥ T (0)e−at1 .

Clearly y(t) ≤ K for t ∈ [0, t1], which implies that dv(t)dt ≤ γ K − µv(t). Therefore,

v(t) ≤ γ Kµ

+v(0)e−µt , which implies that v(t) ≤ V ≡ max{v(0),γ Kµ

} for t ∈ [0, t1].These observations imply that for t ∈ [0, t1], we have

dx(t)

dt≥ −βV e(a)t1

T (0)x(t).

Hence a contradiction is obtained as

x(t1) ≥ x(0)e− βV e(a)t1T (0)

t1 > 0.

Assume now that there is a t1 with b > t1 > 0 such that y(t1) = 0 and x(t) >

0, y(t) > 0, v(t) > 0 for t ∈ (0, t1). Equation (2.3) implies that y′(t) ≥ −ay(t) fort ∈ [0, t1] which yields y(t1) > y(0)e−t1 > 0, also a contradiction.

Finally, we assume again that b > t1 > 0 such that v(t1) = 0 and x(t)>0, y(t)>0,

v(t) > 0 for t ∈ (0, t1). Clearly, this case is similar to the case of y(t1) = 0, and acontradiction can be obtained.

The above contradictions together show that components of the solution of system(2)–(4) subject to (5) are non-negative for all t ∈ [0, b). This together with the uniformboundedness of T (t) and v(t) on [0, b) imply that b = ∞. This completes the proofof the proposition.

System (2)–(4) has steady states E0 = (0, 0, 0), E f = (K , 0, 0), and E∗ = (x∗ > 0,

y∗ > 0, v∗ > 0). E0 symbolizes complete liver failure, E f is a healthy, disease free,mature liver, and E∗ represents persistent, chronic HBV infection. E∗ is given by

x∗ = K a

r

(R∗

R0− 1

), (6)

y∗ = K a

r

(R∗

R0− 1

)(R0 − 1), (7)

v∗ = Kγ a

rµ

(R∗

R0− 1

)(R0 − 1), (8)

where

R∗ = r + a

a.

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578 S. Hews et al.

E∗ exists only when

1 ≤ R0 ≤ R∗.

Since R∗ depends on the hepatocyte proliferation and death rate, we call R∗ the cel-lular vitality index. It is helpful to think of the basic reproduction number, R0, as theparameter grouping that explains the virus dynamics and the cellular vitality index, R∗,as the parameter grouping that explains the hepatocyte dynamics. One of our primaryconclusions is that the behavior of (2)–(4) is determined not only by the values of R0and R∗, but their relationship to each other.

3 Hepatocyte infection

One possible criticism of (2)–(4) is that the infected hepatocytes do not proliferate. Theeffect of HBV infection on hepatocyte proliferation is controversial, with conflictingdata showing both induction and inhibition of proliferation (Kwun and Jang 2004),and both pro- and anti-apoptotic effects (Wu et al. 2006). HBV X protein has severelyimpaired liver regeneration in some mouse models (Tralhao et al. 2002; Wu et al.2006; Dong et al. 2007), but had little effect in others (Hodgson et al. 2008). Naturalvariation in the HBV virus itself may explain these conflicting results (Kwun and Jang2004), and our model represents the limiting case where infection completely blockshepatocyte proliferation.

One important biological implication of (2)–(4) is that even without infected hepa-tocyte proliferation, the majority of hepatocytes still become infected during a chronicHBV infection. For example, Fig. 1 shows a time-series of an infection yielding achronic steady state, along with the percentage of total hepatocytes infected as a func-tion of time. At the peak of infection nearly all of the hepatocytes are infected; thisfraction drops somewhat when the chronic state is reached, but it is still quite high.Specifically, the fraction of infected hepatocytes during a chronic steady state is givenby

y∗

x∗ + y∗ = (R0 − 1)x∗

x∗(1 + (R0 − 1))= R0 − 1

R0= 1 − 1

R0.

Surprisingly, the fraction of infected hepatocytes in the chronic state is determinedonly by R0, which represents virus virulence, and not by the maximum healthy hepa-tocyte proliferation rate, r . For realistic values of R0, approximately 4 < R0 < 10, atleast 75% of hepatocytes are infected (see Eikenberry et al. (2009) for model param-etrization).

The picture is similar for infections that result in convergence to a periodic orbit.The fraction of hepatocytes infected peaks in early infection, and afterwards the frac-tion of infected hepatocytes changes cyclically, with nearly 100% of the hepatocytesinfected at the peak. An example of such an infection is shown in Fig. 2. Figure 3 showsthe percentage of hepatocytes infected at the peak of infection and at the chronic steadystate as function of R0, for infections resulting in chronic disease.

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Rich dynamics of a hepatitis B viral infection model 579

Fig. 1 The upper panel shows the time series solution of a chronic infection under the following parametervalues, r = 1, K = 2e11, β = 0.0014, γ = 280, a = 0.0693, and µ = 0.693. The lower panel shows thepercentage 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 thetime-series solution, and the lower panel shows the percentage of hepatocytes infected as a function of time

4 Stability for E f and E∗

To gain a thorough understanding of (2)–(4), we study the local and global stabilitiesof all steady states. Since (2)–(4) is not differentiable at E0, the stability of the origincannot be studied using a standard linearization approach. This issue will be addressed

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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 by y∗x∗+y∗ = 1− 1

R0, while the peak fraction is determined computationally

later in the paper. Due to the difficulties in the stability of the origin and to create acohesive understanding of (2)–(4), we explore the dynamics as we increase R0. Westart with R0 < 1. As with other infection models, we expect the disease free state, E f ,to be locally and globally asymptotically stable, and this is confirmed in Propositions 2and 3. When R0 > 1, we expect E f to become unstable, as is shown in Proposition 4.

Proposition 2 If R0 < 1, then E f is locally asymptotically stable.

Proof The Jacobian matrix of the vector field corresponding to (2)–(4) at E f is

J (x, y, v)|E∗ =⎛⎝−r −r −β

0 −a β

0 γ −µ

⎞⎠ .

The eigenvalues of the matrix are given by

λ1 = −r , (9)

λ2,3 = −1

2(a + µ) ± 1

2

√(a + µ)2 − 4aµ(1 − R0). (10)

λ2,3 are negative and thus E f is locally asymptotically stable when R0 < 1.

Proposition 3 If R0 < 1, then E f is globally stable.

Proof It is sufficient to show that (y(t), v(t)) → (0, 0). From there, it is clear thatx(t) → K . From positivity of solutions, y and v satisfy the differential inequality:

dy(t)

dt≤ βv(t) − ay(t) = dY (t)

dt, (11)

dv(t)

dt≤ γ y(t) − µv(t) = dV (t)

dt. (12)

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Rich dynamics of a hepatitis B viral infection model 581

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 stableand 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)and v(t) ≤ V (t), (y(t), v(t)) → (0, 0) as t → ∞ by a simple comparison argument.Thus E f is globally stable.

Proposition 4 If R0 > 1, then E f is unstable.

Proof The eigenvalues of the Jacobian evaluated at E f are given by (9)–(10). Atleast one eigenvalue becomes positive when R0 > 1, so the steady state isunstable.

The disease free steady state, E f , is locally and globally stable when R0 < 1 andunstable when R0 > 1 (Fig. 4). Note that stability of E f depends only on infectionrate, production rate of free virions, death rate of infected cells, and death rate of freevirions. In particular, the maximum per capita proliferation rate of healthy cells andhomeostatic liver size do not affect the stability.

As the reproduction number crosses the bifurcation point of R0 = 1, the sta-bility of E f is transferred to E∗ as it crosses into the positive quadrant. Recallthat E∗ only exists in the positive quadrant when 1 < R0 < R∗. For the con-dition R0 < R∗ to hold, the proliferation rate has to be sufficiently large; specif-ically, r >

βγ−aµµ

. There is a region where E∗ is locally asymptotically stablebefore crossing a Hopf bifurcation point and entering a region with a stable limitcycle. The region of stability for E∗ and the Hopf bifurcation point are presented inTheorem 1.

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Theorem 1 Let δ = a2(

R∗R0

− 1)

(R0−1)+ a2

R20(R0−1)− ar

R0(R0−1)+aµ

(R∗R0

− 1)

,

and σ = −(µa2 R0+a3 R∗)(R∗−R0)(R0−1)R0(µR0+a R∗) . If δ > σ , then E∗ is locally asymptotically sta-

ble and δ = σ is the Hopf bifurcation point.

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 − a βx∗x∗+y∗

0 γ −µ

⎞⎟⎟⎟⎠ ,

The eigenvalues of J satisfy

λ3 + a2λ2 + a1λ + a0 = 0,

where

a2 = µ + aR∗

R0,

a1 = 2a2(

R∗

R0− 1

)(R0 − 1) + a2

R20

(R0 − 1)2 − ar

R0(R0 − 1) + aµ

(R∗

R0− 1

),

a0 = a2µ(R0 − 1)

(R∗

R0− 1

).

Clearly a2 > 0 and a0 > 0 when E∗ exists. Let,

δ = a2(

R∗

R0−1

)(R0−1)+ a2

R20

(R0−1)− ar

R0(R0−1)+aµ

(R∗

R0−1

),

a2a1 − a0 = µa2 + aR∗

R0a2 − a2µ(R0 − 1)

(R∗

R0− 1

)

= µδ + aR∗

R0δ +

(µa2 + a3 R∗

R0

)(R∗

R0− 1

)(R0 − 1) > 0.

Therefore, a2a1 > a0 when

δ >−(µa2 R0 + a3 R∗)(R∗ − R0)(R0 − 1)

R0(µR0 + a R∗)= σ.

By Routh-Hurwitz criteria, we determine a condition for E∗ to be locally asymptoti-cally stable. Since the Routh-Hurwitz criteria are necessary and sufficient for stability,there is a Hopf bifurcation point at δ = σ .

After increasing R0 past the Hopf bifurcation point, numerical solutions show thatthere is an attracting limit cycle in this region as shown in Fig. 4. Unlike in Eikenberryet al. (2009), these periodic solutions are not sustained as R0 increases indefinitely.

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Rich dynamics of a hepatitis B viral infection model 583

Numerical investigation also suggests that there is a homoclinic bifurcation where theperiodic solutions cease and E0 is stable (Fig. 4). Biologically, this represents ALFresulting from chronic HBV infection. The stability of the origin is therefore an inte-gral part of the system dynamics with an important biological meaning. These resultsare confirmed analytically in the following section.

5 Ratio-dependent transformation and results for E0

As mentioned previously, the stability of E0 cannot be studied using standard lineari-zation techniques. To overcome this difficulty, we use ratio-dependent transformationsused in Hwang and Kuang (2003), Hsu et al. (2001), and Berezovsky et al. (2005).The first transformation gives a global stability result for E0. The second and thirdtransformations yield a complete local stability result and an explicit form for thehomoclinic bifurcation.

The first transformation is a change of variable (x, y, v) → (x, z, w) where z = yx

and w = vx . This transforms (2)–(4) to the following system:

dx

dt= r x(t)

(1 − x(t)(1 + z(t))

K

)− βw(t)x(t)

1 + z(t), (13)

dz

dt= βw(t) − az(t) − r z(t)

(1 − x(t)(1 + z(t))

K

), (14)

dw

dt= γ z(t) − µw(t) − rw(t)

(1 − x(t)(1 + z(t))

K

)+ βw(t)2

1 + z(t). (15)

The steady states of the transformed system are

U0 = (0, 0, 0), Un = (0, zn, wn), U f = (K , 0, 0), U∗ =(

x∗, y∗

x∗ ,v∗

x∗

),

where

zn = R∗(1 + rµ) − R0

R∗(

aµ

− 1)

+ R0

, wn = a + r

βzn,

and x∗, y∗ and v∗ are given by (6)–(8). Un is nonnegative when − aµ

< R0R∗ − 1 < r

µ,

and U∗ is nonnegative when R∗ < R0 < 1. The steady states are preserved in thatE f = U f and E∗ = U∗, while E0 has been blown up into two steady states: U0and Un . It is important to realize that the transformed system is not bounded. Thefollowing results will prove that there is a region where E0 is globally stable.

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Lemma 1 U0 is always unstable.

Proof The variational matrix of the system (13)–(15) evaluated at U0 is

J (x, z, w)|E0 =⎛⎝ r 0 0

0 −a − r β

0 γ −µ − r

⎞⎠ ,

where

λ1 = r,

λ2 = −1

2(a + µ) − r + 1

2

√(a − µ)2 + 4kβ,

λ3 = −1

2(a + µ) − r − 1

2

√(a − µ)2 + 4kβ.

Since λ1 = r > 0, U0 is always unstable.

When R0 >(

rµ

+ 1)

R∗, U0 is the only steady state of (13)–(15). Since it is always

unstable, there are no stable steady states in (13)–(15) that map back to the origin.

Lemma 2 and Theorem 2 show that if R0 >(

rµ

+ 1)

R∗ and µ > a, E0 is globally

stable.

Lemma 2 If R0 >(

rµ

+ 1)

R∗, then z, w → ∞ as t → ∞.

Proof

dz

dt= βw − az − r z

(1 − x(1 + z)

K

)> βw − (a + r)z,

dw

dt= γ z − µw − rw

(1 − x(1 + z)

K

)+ βw2

1 + z> γ z − (µ + r)w.

Let

d Z

dt= βW − (a + r)Z , (16)

dW

dt= γ Z − (µ + r)W . (17)

(0, 0) is the only steady state of (16)–(17) and is unstable when R0 >(

rµ

+ 1)

R∗.

Since there are no other steady states, and Z , W are unbounded, Z , W → ∞ as

t → ∞. Since dzdt > d Z

dt , dwdt > dW

dt , z, w → ∞ as t → ∞, and R0 >(

rµ

+ 1)

R∗.

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Rich dynamics of a hepatitis B viral infection model 585

Theorem 2 If R0 >(

rµ

+ 1)

R∗ and µ > a, then x → 0 as t → ∞.

Proof Since

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),

it is sufficient to show

limt→∞

βw(t)

1 + z(t)> r. (18)

Let θ(t) = βw(t)1+z(t) . Then

dθ

dt= β dw

dt

1 + z(t)− βw(t) dz

dt

(1 + z(t))2

= [βγ + (a + r)θ(t)] z(t)

1 + z(t)− (µ + r)θ(t) + r x(t)z(t)θ(t)

K

≥ [βγ + (a + r)θ(t)] z(t)

1 + z(t)− (µ + r)θ(t). (19)

By Lemma (2), for all ε > 0, ∃ t∗ s.t. ∀ t ≥ t∗,

z(t)

1 + z(t)> 1 − ε. (20)

Combining (19) and (20) for t ≥ t∗,

dθ

dt> βγ (1 − ε) + ((a + r)(1 − ε) − µ − r)θ(t).

Letting �(ε) = (a + r)(1 − ε) − µ − r and solving for θ(t) yields

θ(t) >βk(1 − ε)

−�(ε)+ θ(t∗)e�(ε)(t−t∗) = (t).

Since µ > a, �(ε) < 0. Therefore, limt→∞ (t) = βγ (1−ε)

−�(ε). Since R0 >

(rµ

+ 1)

R∗,

∃ ε∗ > 0 s.t. ∀ ε ∈ (0, ε∗], βγ (1−ε)µ+r > a + r . So,

limt→∞

βw(t)

1 + z(t)≥ βγ (1 − ε)

−�(ε)>

βγ (1 − ε)

µ + r> a + r > r.

Thus, (18) is satisfied.

Since y = xz, v = xw, and (2)–(4) is bounded, y, z → 0 when x → 0 andz, w → ∞.Therefore, Theorem 2 and Lemma 2 prove that E0 in (2)–(4) is globally

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586 S. Hews et al.

stable when R0 >(

rµ

+ 1)

R∗ and µ > a. Biologically, this implies that a sufficiently

virulent infection will result in complete liver failure.The second transformation, that yields the homoclinic bifurcation point and a larger

region for stability of E0, is (x, y, v) → (g, h, v) where g = xv

and h = yv

. Thisyields the system,

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)

dh

dt= βg(t)

g(t) + h(t)− ah(t) − γ h(t)2 + µh(t), (22)

dv

dt= γ h(t)v(t) − µv(t). (23)

In addition to there still being a singularity at (0, 0, 0), (21)–(23) exhibits threesteady states.

U † =(

0,µ − a

γ, 0

), Un =

(1

wn,

zn

wn, 0

), U∗ =

(x∗

v∗ ,y∗

v∗ , v∗)

,

where zn and wn are given by (16) and x∗, y∗ and v∗ are given by (6)–(8). U † is theonly steady state that is not present in the original system or the other transforma-tions. It is only positive when µ > a and provides the condition for the homoclinicbifurcation.

Lemma 3 For the system (21)–(23), if µ > a, the following results hold:

(a) If(

1 − aµ

)R∗ > R0, then U † is a saddle point;

(b) If(

1 − aµ

)R∗ < R0, then U † is locally asymptotically stable.

Proof The Jacobian of system (21)–(23) evaluated at U † is

J (g, h, v)|U † =

⎛⎜⎜⎜⎝

r − βγµ−a 0 0

βγµ−a a − µ 0

0 0 −a

⎞⎟⎟⎟⎠ .

Since the Jacobian is a triangular matrix, the eigenvalues are given by

λ1 = r − βγ

µ − a,

λ2 = a − µ,

λ3 = −a.

Since µ > a, λ2,3 are always negative. λ1 can be rewritten as aµ−a ((µ − a)

R∗ − µR0) , which is positive when(

1 − aµ

)R∗ > R0 and negative when

(1 − a

µ

)

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Rich dynamics of a hepatitis B viral infection model 587

R∗ < R0. Therefore, when µ > a and(

1 − aµ

)R∗ > R0, U † is a saddle point, and

when µ > a and(

1 − aµ

)R∗ < R0, U † is locally asymptotically stable.

Since U † maps back to E0, we have the following:

Theorem 3 For the system (2)–(4), if µ > a, the following results hold:

(a) If(

1 − aµ

)R∗ > R0, then E0 is a saddle point;

(b) If(

1 − aµ

)R∗ < R0, then E0 is locally asymptotically stable.

Computationally, we have confirmed that at(

1 − aµ

)R∗ = R0, the periodic solu-

tions collide with the saddle point, thus creating a homoclinic bifurcation point.

Furthermore,(

1 − aµ

)R∗ < R0 includes the entire parameter regime where Un is

positive, so it is not necessary to analyze the stability of Un as long as µ > a. Mathe-matically, the homoclinic bifurcation point corresponds to the onset of complete liverfailure. Biologically, liver failure might occur prior to this point as metabolic demandsoverwhelm hepatocyte proliferation when liver mass is very low (Rozga 2002).

The final transformation is (x, y, v) → (m, y, n) where m = xy and n = v

y . Ityields all of the same steady states and results as (21)–(23) so it is omitted here.

6 Discussion

In (1), Nowak et al. (1996) modeled the infection of healthy hepatocytes by free virionsas a mass action process. This makes the viral basic reproduction number dependenton the homeostatic liver size, r

d , implying that individuals with smaller livers aremore susceptible to HBV infection. Gourley et al. (2008), Min et al. (2008), andEikenberry et al. (2009), and the current model all replace this mass action processwith a standard incidence function, eliminating this artifact. Here, we also keep thelogistic proliferation term used in Eikenberry et al. (2009), and similar to that in Ciupeet al. (2007a,b), since hepatocytes are produced in the liver and their numbers arehomeostatically regulated. Together, these improvements significantly increase therichness of the predicted dynamics.

Here we introduce the concept of R∗, the cellular vitality index, which is a combi-nation of parameters that describes the hepatocyte behavior and includes their prolif-eration and death rates. One of our main conclusions is that the relationship betweenR0 and R∗ generates significant insight into both the mathematical behavior and bio-logical interpretation of model (2)–(4). The difference between them ultimately deter-mines whether the infection will be chronic, undergo oscillations, or induce ALF. Thisresult suggests that treatments could focus not just on reducing R0, which representsvirus virulence, but also on increasing R∗, which represents the hepatocytes ability toregenerate.

In addition to disease free (E f ), and chronic (E∗) equilibria, we also see the emer-gence of a stable periodic orbit and a steady state at the origin (E0). Similar to previousmodels, E f is globally asymptotically stable when R0 < 1 and unstable when R0 > 1.

123

588 S. Hews et al.

For R0 > 1, we prove the existence of a region where E∗ is stable and find the closedform for the Hopf bifurcation point. We also computationally determine the regionwhere a stable periodic orbit exists (Fig. 4).

In contrast to system (2)–(4), the origin does not appear to be stable anywhere inthe Eikenberry et al. (2009) system. Apparently the delay term in Eikenberry et al.’smodel obliterates stability at the origin; however, in our system, the origin is stablefor sufficiently large values of R0. Specifically, we implement a change of variabletechnique as shown in Hwang and Kuang (2003), Hsu et al. (2001), and Berezovskyet al. (2005) to prove that E0 is globally stable when R0 > (1 + r

µ)R∗ and that there

is a homoclinic bifurcation point at R0 =(

1 − aµ

)R∗. This technique is called a

ratio-dependent or a blow-up transformation.Changes in the dynamics within different parameter regimes are dominated by R0

and R∗. When R0 < 1, a perturbed (i.e. infected) liver will always return to a healthy,disease free state. At R0 = 1, there is a transcritical bifurcation as E f = E∗. As R0becomes larger than 1, the stability of E f is transferred to the chronic state, E∗, whichcrosses into the positive quadrant. As R0 increases further, the system crosses a super-critical Hopf bifurcation point and all solutions approach an attracting periodic orbit.Increasing R0 further causes the system to cross a homoclinic bifurcation point whereE0 becomes stable. Beyond this point, the liver has failed completely. The changes indynamics are shown in Fig. 4.

Despite our analysis of (2)–(4), there are still a few open questions. For example,we have yet to prove a region where the chronic steady state is globally stable. Weanticipate that a Lyapunov function can be employed to complete the proof. We alsohave yet to completely characterize the region where there is an attracting periodicorbit and prove its existence.

When the delay term in Eikenberry et al. (2009) approaches 0, we would expect thedynamics to converge to those in (2)–(4). However, in computational investigations,this is not the case. For increasingly virulent infections, sustained oscillations areobserved in the delay model, but the origin is never stable. While the period of theseoscillations increases marginally as the delay becomes very small, there is apparentlyno convergence to the origin, in contrast to the behavior seen in our model. Althoughthe essential biological prediction of Eikenberry et al. (2009), that ALF in HBV infec-tion can be induced by a switch in stability and may be preceded by oscillations inviral load, is preserved, origin stability for a sufficiently virulent infection is seen onlyin our model, implying a fundamental change in the system dynamics when the delayis omitted.

In our model, we have assumed that infected hepatocytes do not proliferate. Infec-tion by HBV can clearly affect both hepatocyte proliferation and apoptosis, and thesechanges are linked to virus-induced hepatocellular carcinoma (Wu et al. 2006), yethow infection affects these processes remains controversial. HBV X (HBx) proteinhas been widely studied in this context. HBx affects cell-cycle progression, and it caninduce entry into the cell cycle, DNA synthesis, and proliferation (Benn and Schneider1995). HBx has variously been reported to have either pro- or anti-apoptotic effects(Wu et al. 2006). While much work has demonstrated that HBx can induce prolifera-tion, it may also block or prolong the G1 → S transition by acting upon the cell cycle

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Rich dynamics of a hepatitis B viral infection model 589

inhibitor p21 (Park et al. 2000; Kwun and Jang 2004). Natural variants in the HBxprotein may affect p21 differently, and thus affect cell proliferation differently (Kwunand Jang 2004).

Eikenberry et al. (2009) justified the assumption of logistic hepatocyte growth forhealthy cells on the basis of the pattern of healthy liver regeneration seen in 2/3 partialhepatectomies (PHx). Several studies in HBx transgenic mice have shown that thisprotein severely inhibits liver regeneration and hepatocyte proliferation following PHx(Wu et al. 2006; Tralhao et al. 2002). Wu et al. (2006) found that the G1 → S transitionwas blocked in such transgenic mice, but these mice also had steatotic livers which canindependently impede regeneration. Several studies support a paracrine role for HBVviral proteins in inhibiting regeneration. Tralhao et al. (2002) found that transplanta-tion of HBx expressing hepatocytes into a healthy liver could impede regeneration, andDong et al. (2007) found that natural killer T cells inhibited regeneration in HBV-tgtransgenic livers, largely through cytokine (interferon-γ ) inhibition of proliferation.However, Hodgson et al. (2008) found that HBx induced early entry into the cell cyclein regenerating hepatocytes and did not impair liver regeneration.

Hepatocyte proliferation also affects HBV virus expression; Ozer et al. (1996)found that arrest in either G1 or G2 increased virus expression while passage throughS and DNA synthesis inhibited viral mRNA. While increased proliferation in infectedhepatocytes may be expected to enhance infection, rapid hepatocyte turnover aidedinfection clearance in a duck model of chronic duck HBV (Fourel et al. 1994). Hepa-tocyte proliferation has been observed to inhibit production of or possibly destroyviral nucleocapsids (Guidotti et al. 1994), and dilution of HBV cccDNA caused byproliferation may cause spontaneous recovery (Fourel et al. 1994; Guidotti et al. 1994).

Thus, HBV infection may either block or induce hepatocyte proliferation, andproliferation itself can affect virus replication and expression. Therefore, patterns ofhepatocyte proliferation may be central to the dynamics of virus infection and areone area where modeling has the potential to generate significant insight. Ciupe et al.incorporated infected hepatocyte proliferation in several mathematical models of acuteHBV infection (Ciupe et al. 2007a,b) and also considered the possibility of spontaneousrecovery through cccDNA dilution (Ciupe et al. 2007b). Our model represents the lim-iting case where infection completely blocks hepatocyte proliferation. We have foundthat, under this assumption, the majority of hepatocytes become infected in the chronicdisease state, and that biologically plausible dynamics are observed overall. The impor-tance of infected hepatocyte proliferation and the effect of proliferation on viral repli-cation in our model of chronic disease are two problems that we are currently studying.

Acknowledgments This research is partially supported by the NSF grant DMS-0436341 and the grantDMS/NIGMS-0342388 jointly funded by NIH and NSF. We would like to thank the anonymous reviewersfor their valuable comments and suggestions.

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