AN AGE-STRUCTURED MODEL FOR THE TRANSMISSION DYNAMICS …ruan/MyPapers/ZouRuanZhang... · 2010. 11....

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SIAM J. APPL. MATH. c© 2010 Society for Industrial and Applied MathematicsVol. 70, No. 8, pp. 3121–3139

AN AGE-STRUCTURED MODEL FOR THE TRANSMISSIONDYNAMICS OF HEPATITIS B∗

LAN ZOU† , SHIGUI RUAN‡, AND WEINIAN ZHANG†

Abstract. Hepatitis B virus (HBV) infection is endemic in many parts of the world. One ofthe characteristics of HBV transmission is the age structure of the host population. In this paper,we propose an age-structured model for the transmission dynamics of HBV. The host population isstratified by age and is divided into six subclasses: susceptible, latently infected, acutely infectious,carrier, recovered, and vaccinated individuals. By determining the basic reproduction number, westudy the existence and stability of the disease-free and endemic steady state solutions of the model.Numerical simulations are performed to find optimal strategies for controlling the transmission ofHBV.

Key words. HBV, age structure, steady state, stability, basic reproduction number

AMS subject classifications. 35Q92, 35B35, 92D30

DOI. 10.1137/090777645

1. Introduction. Hepatitis B virus (HBV) infection is widespread in many partsof the world, especially in Africa, Southeast Asia, the Middle East, South and WesternPacific islands, the interior Amazon River basin, and certain parts of the Caribbean(Centers for Disease Control and Prevention (CDC) [5]). By the estimation of theWorld Health Organization (WHO) [30], about 2 billion people have been infectedwith HBV. An estimate of 600,000 persons die each year due to the acute or chronicconsequences of the virus [30]. HBV is the most common serious viral infection anda leading cause of death in China. Around 130 million people in China are carriers ofHBV, almost a third of the people infected with HBV worldwide and about 10% of thegeneral population in the country; among them, 30 million are chronically infected.Every year, 300,000 people die from HBV-related diseases in China, accounting for40–50% of HBV-related deaths worldwide (Jia and Zhuang [17], Liu et al. [19]). Thespecific character of HBV infection in China is that there is a very large number ofHBV carriers.

Hepatitis B is transmitted through body fluids like blood, semen, and vaginalsecretions. One of the most important factors influencing the probability of developingcarriage of HBV is age. Acute HBV infection causes chronic (long-term) infectionin 30–90% of persons infected as infants or young children and in less than 5% ofadolescents and adults (Shepard et al. [26], Goldstein et al. [11]). There is no widelyavailable effective treatment for chronic HBV carriers. Immunization against HBVis now considered a viable option. In 1991, WHO recommended that hepatitis Bvaccination should be included in the national immunization program in countrieswith an HBsAg carrier prevalence of 8% by 1995 and all countries by 1997 [29].

∗Received by the editors November 18, 2009; accepted for publication (in revised form) August 25,2010; published electronically November 4, 2010.

http://www.siam.org/journals/siap/70-8/77764.html†Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu,

Sichuan 610064, People’s Republic of China ([email protected], [email protected]).The first author’s research was partially supported by the China Scholarship Council. The thirdauthor’s research was partially supported by NSFC grant 10825104 and SRFDP grant 200806100002.

‡Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250 ([email protected]). This author’s research was partially supported by NSF grant DMS-1022728.

3121


3122 LAN ZOU, SHIGUI RUAN, AND WEINIAN ZHANG

In the last two decades, many mathematical models have been proposed to in-vestigate the transmission dynamics of HBV in various countries and regions in theworld. Anderson, May, and Nokes [2] described a model for the sexual transmissionof HBV, which includes heterogeneous mixing with respect to age and sexual activity.Edmunds et al. [9] illustrated the relation between the age at infection with HBVand the development of the carrier state. McLean and Blumberg [21] and Edmunds,Medley, and Nokes [7] studied models of HBV transmission in developing countries.Williams et al. [28] presented a model for HBV infection in the UK that takes intoconsideration differences of age and sex. Zhao, Xu, and Lu [31] developed a mathemat-ical model to predict the HBV transmission dynamics and to evaluate the long-termeffectiveness of the vaccination program in China. Medley et al. [23] gave a modelto show that the prevalence of infection is determined by a feedback mechanism thatrelates the rate of transmission, average age at infection, and age-related probabilityof developing carriage following infection.

The age structure of a population has been regarded as an important factorfor the dynamics of disease transmission since the work of Sharpe and Lotka [25]and McKendrick [22]. Great effort has been devoted to determining the thresholdconditions for the disease to become endemic, the stability of steady state solutions,and the global behavior of age-structured epidemic models; we refer to Anderson andMay [1], Busenberg, Cooke, and Iannelli [3], Busenberg, Iannelli, and Thieme [4], Feng,Huang, and Castillo-Chavez [10], Greenhalgh [12, 13], Iannelli [14], Inaba [15, 16], Li,Gupur, and Zhu [18], Martcheva and Crispino-O’Connell [20], Webb [27], and thereferences cited therein. Age-structured models have also been used to model thetransmission dynamics of HBV by some researchers; see Edmunds et al. [9], McLeanand Blumberg [21], Zhao, Xu, and Lu [31], etc.

In a previous study (Zou, Zhang, and Ruan [32]), we developed a mathematicalmodel described by ordinary differential equations to study the transmission dynamicsand control of HBV in mainland China, taking account of the specific character ofthe virus infection in the country. We analyzed the existence and stability of thedisease-free and disease-endemic steady states of the model and used the model tosimulate the HBV data reported by the Ministry of Health of China (MOHC) [24].

Based on our previous work [32], in this article we propose an age-structuredmodel for the transmission of HBV. We qualitatively analyze the model, includingthe existence of positive solutions, and the stability and existence of the disease-freeand disease-endemic steady states in terms of the basic reproduction number. Somenumerical simulations of the model are also given to illustrate the results and to findoptimal strategies in controlling HBV infection.

The paper is organized as follows. In section 2, we present the mathematical modelfor the transmission dynamics of hepatitis B. The existence of positive solutions isconsidered in section 3. Section 4 deals with the existence and stability of the disease-free steady state. In section 5, we give conditions that guarantee the existence andstability of disease-endemic steady states. In section 6, some numerical simulationsof the model are presented. The paper ends with a brief discussion in section 7.

2. The model. We propose an age-structured model to study the transmissiondynamics of HBV. As the model includes age-dependent processes such as the forceof infection and the probability of developing the carrier state, the host population isstratified by age. We divide the total population of size N(a, t) into six subclasses:susceptible S(a, t), latently infected L(a, t), acutely infectious I(a, t), carrier C(a, t),recovered R(a, t), and vaccinated V (a, t), with age distribution at time t. The vari-


AN AGE-STRUCTURED MODEL FOR HEPATITIS B 3123

Fig. 1. Flowchart of HBV transmission in a population.

ables and model structure are described in Figure 1.

The age-structured model for the transmission of HBV is described by the follow-ing system of partial differential equations:

(2.1)

∂S

∂a+∂S

∂t= b(a)ω(N(a, t)− νC(a, t)) + ψV (a, t)− (μ(a) + λ(a, t) + p(a, t))S(a, t),

∂L

∂a+∂L

∂t= λ(a, t)S(a, t)− (μ(a) + σ)L(a, t),

∂I

∂a+∂I

∂t= σL(a, t)− (μ(a) + γ1)I(a, t),

∂C

∂a+∂C

∂t= b(a)ωνC(a, t) + q(a)γ1I(a, t)− (μ(a) + μ1(a) + γ2(a))C(a, t),

∂R

∂a+∂R

∂t= γ2(a)C(a, t) + (1− q(a))γ1I(a, t)− μ(a)R(a, t),

∂V

∂a+∂V

∂t= b(a)(1− ω)N(a, t) + p(a, t)S(a, t)− (ψ + μ(a))V (a, t),

with initial and boundary conditions

(2.2)

S(0, t) =

∫ a2

a1

b(a)ω[S(a, t) + L(a, t) + I(a, t) +R(a, t) + V (a, t) + (1− ν)C(a, t)]da,

C(0, t) =

∫ a2

a1

b(a)ωνC(a, t)da,

V (0, t) =

∫ a2

a1

b(a)(1− ω)[S(a, t) + L(a, t) + I(a, t) + C(a, t) +R(a, t) + V (a, t)]da,

L(0, t) = I(0, t) = R(0, t) = 0,

S(a, 0) = S0(a), L(a, 0) = L0(a), I(a, 0) = I0(a), C(a, 0) = C0(a),

R(a, 0) = R0(a), V (a, 0) = V0(a),



where

λ(a, t) =

∫ a+

0

β(a, a′)I(a′, t) + εC(a′, t)

N(a′, t)da′,

a+ > 0 is the maximum age of an individual, and the interval (a1, a2) represents thefemale fertile period. Then we obtain the following equations for the total populationN(a, t):

(2.3)

∂N(a, t)

∂a+∂N(a, t)

∂t= (b(a)− μ(a))N(a, t)− μ1(a)C(a, t),

N(0, t) =

∫ a2

a1

b(a)N(a, t)da, N(a, 0) = N0(a).

The parameters are described as follows:

b(a) – birth rateμ(a) – natural mortality rateμ1(a) – HBV-related mortality rate

β(a, a′) – probability that an infective individual of age a′ will have contact with andsuccessfully infect a susceptible individual of age a

ε – reduced transmission rate from chronic carriers compared to acute infec-tions

σ – rate moving from latent to acuteγ1 – rate moving from acute to carrier

γ2(a) – rate moving from carrier to vaccinatedp(a, t) – vaccination rate against HBV

(1− ω) – proportion of births with successful vaccinationq(a) – probability an individual fails to clear an acute infection and develops to

carrier stateψ – rate of waning vaccine-induced immunityν – proportion of perinatally infected (from carrier mothers).

Let

s(a, t) =S(a, t)

N(a, t), �(a, t) =

L(a, t)

N(a, t), x(a, t) =

I(a, t)

N(a, t),

c(a, t) =C(a, t)

N(a, t), v(a, t) =

V (a, t)

N(a, t), r(a, t) =

R(a, t)

N(a, t).

Let k(a) be the age-specific (average) probability of becoming infected through con-tact with infectious individuals, and let β(a′) be the age-specific per-capita contact/activity rate. Following the discussion of infection coefficients in Anderson and May[1], we make the following assumption.

Assumption 2.1. Assume that

(2.4) β(a, a′) = k(a)β(a′).

Under the above assumption, system (2.1) can be normalized as the following



system:

(2.5)

∂s

∂a+∂s

∂t= b(a)ω(1− νc(a, t)) + ψv(a, t)− (b(a)− μ1(a)c(a, t))s(a, t)

− (λ(a, t) + p(a, t))s(a, t),

∂�

∂a+∂�

∂t= λ(a, t)s(a, t)− (σ + b(a)− μ1(a)c(a, t))�(a, t),

∂x

∂a+∂x

∂t= σ�(a, t)− (γ1 + b(a)− μ1(a)c(a, t))x(a, t),

∂c

∂a+∂c

∂t= b(a)ωνc(a, t) + q(a)γ1x(a, t)− (μ1(a)(1 − c(a, t)) + b(a) + γ2(a))c(a, t),

∂r

∂a+∂r

∂t= γ2(a)c(a, t) + (1− q(a))γ1x(a, t)− (b(a)− μ1(a)c(a, t))r(a, t),

∂v

∂a+∂v

∂t= b(a)(1− ω) + p(a, t)s(a, t)− (ψ + b(a)− μ1(a)c(a, t))v(a, t),

with boundary conditions

s(0, t) = ω

(1−

∫ a2

a1

νc(a, t)da

), c(0, t) =

∫ a2

a1

ωνc(a, t)da,

v(0, t) = 1− ω, �(0, t) = x(0, t) = r(0, t) = 0,

where

λ(a, t) = k(a)

∫ a+

0

β(a)(x(a, t) + εc(a, t))da.

In the following, we consider system (2.5) with initial conditions

(2.6)s(a, 0) = s0(a), �(a, 0) = �0(a), x(a, 0) = x0(a),

c(a, 0) = c0(a), r(a, 0) = r0(a), v(a, 0) = v0(a).

3. Existence of positive solutions. Since r(a, t) can be obtained by

r(a, t) = 1− s(a, t)− �(a, t)− x(a, t)− c(a, t)− v(a, t),

it suffices to discuss the system in terms of only s(a, t), �(a, t), x(a, t), c(a, t), andv(a, t). Since in a short period of time the change of p is small, we assume thatthe vaccination rate p(a, t) = p(a) depends only on the age a. Thus, we obtain thefollowing system:

(3.1)

∂s

∂a+∂s

∂t= b(a)ω(1− νc(a, t)) + ψv(a, t)− (b(a)− μ1(a)c(a, t))s(a, t)

− (λ(a, t) + p(a))s(a, t),

∂�

∂a+∂�

∂t= λ(a, t)s(a, t)− (σ + b(a)− μ1(a)c(a, t))�(a, t),

∂x

∂a+∂x

∂t= σ�(a, t)− (γ1 + b(a)− μ1(a)c(a, t))x(a, t),

∂c

∂a+∂c

∂t= b(a)ωνc(a, t) + q(a)γ1x(a, t)− (μ1(a)(1 − c(a, t)) + b(a) + γ2(a))c(a, t),

∂v

∂a+∂v

∂t= b(a)(1− ω) + p(a)s(a, t)− (ψ + b(a)− μ1(a)c(a, t))v(a, t),




s(0, t) = ω

(1−

∫ a2

a1

νc(a, t)da

), c(0, t) =

∫ a2

a1

ωνc(a, t)da,

v(0, t) = 1− ω, �(0, t) = x(0, t) = 0.

Consider the Banach space

X = L1(0, a+)× L1(0, a+)× L1(0, a+)× L1(0, a+)× L1(0, a+)

endowed with the norm

‖φ‖ =

5∑i=1

‖φi‖ for φ(a) = (φ1(a), φ2(a), φ3(a), φ4(a), φ5(a))T ∈ X,

where ‖ · ‖ is the norm of L1(0, a+). The state space of system (3.1) is

Ω := {(s, l, x, c, v) ∈ X+|0 ≤ s+ �+ x+ c+ v ≤ 1},whereX+ = L1

+(0, a+)×L1+(0, a+)×L1

+(0, a+)×L1+(0, a+)×L1

+(0, a+), and L1+(0, a+)

denotes the positive cone of L1(0, a+).Let A be a linear operator defined by

(3.2) (Aφ)(a) = (A1, A2, A3, A4, A5)T ,

where

A1 =

(−dφ1da

− (b(a) + p(a))φ1, 0, 0, −b(a)ωνφ4, ψφ5),

A2 =

(0, −dφ2

da− (σ + b(a))φ2, 0, 0, 0

),

A3 =

(0, σφ2, −dφ3

da− (γ1 + b(a))φ3, 0, 0

),

A4 =

(0, 0, q(a)γ1φ3, −dφ4

da− (μ1(a) + γ2(a)− b(a)ων)φ4, 0

),

A5 =

(p(a)φ1, 0, 0, 0, −dφ5

da− (ψ + b(a))φ5

),

φ(a) = (φ1(a), φ2(a), φ3(a), φ4(a), φ5(a))T ∈ D(A),

and the domain D(A) is given as

D(A) =

{φ ∈ X|φi ∈ AC[0, a+),

φ(0) =

(ω

(1−

∫ a2

a1

νφ3(a)da

), 0, ων

∫ a2

a1

φ3(a)da, 0, 1− ω

)T},

where AC[0, a+) denotes the set of absolutely continuous functions on [0, a+).We also define a nonlinear operator F : X → X by

(3.3) (Fφ)(a) =

⎛⎜⎜⎜⎜⎝

b(a)ω + μ1(a)φ4φ1 − ((Qφ3)(a) + ε(Qφ4)(a))φ1((Qφ3)(a) + ε(Qφ4)(a))φ1 + μ1(a)φ4φ2

μ1(a)φ4φ3μ1(a)φ

24

b(a)(1− ω) + μ1(a)φ4φ5

⎞⎟⎟⎟⎟⎠ ,



where Q is a bounded linear operator on L1(0, a+) given by

(Qf)(a) =

∫ a+

0

β(a, a′)f(a′)da′.

Let u(t) = (s(·, t), �(·, t), x(·, t), c(·, t), v(·, t)). Thus, we can rewrite the system as anabstract Cauchy problem

(3.4)du(t)

dt= Au(t) + F (u(t)), u(0) = u0 ∈ X,

where u0(a) = (s0(a), �0(a), x0(a), c0(a), v0(a))T .

For A and F , following Inaba [16] and Webb [27], we have the following results.Lemma 3.1. The operator A generates a C0 semigroup etA and the space Ω is

positively invariant with respect to the semiflow defined by etA.Lemma 3.2. The operator F is continuously Frechet differentiable on X.Theorem 3.3. For each u0 ∈ X+, there are a maximal interval of existence

[0, t0) and a unique continuous mild solution u(t, u0) ∈ X+, t ∈ [0, t0) for (3.4) suchthat

(3.5) u(t) = eAtu0 +

∫ t

0

eA(t−τ)F (u(τ))dτ.

4. The disease-free steady state. A steady state (s(a), �(a), x(a), c(a), v(a))of system (3.1) must satisfy the following time-independent system of ordinary differ-ential equations:

(4.1)

ds

da= b(a)ω(1− νc(a)) + ψv(a)− (b(a)− μ1(a)c(a))s(a) − (λ(a) + p(a))s(a),

d�

da= λ(a)s(a)− (σ + b(a)− μ1(a)c(a))�(a),

dx

da= σ�(a)− (γ1 + b(a)− μ1(a)c(a))x(a),

dc

da= b(a)ωνc(a) + q(a)γ1x(a) − (μ1(a)(1 − c(a)) + b(a) + γ2(a))c(a),

dv

da= b(a)(1− ω) + p(a)s(a) − (ψ + b(a)− μ1(a)c(a))v(a),

with initial value conditions

s(0) = ω

(1−

∫ a2

a1

νc(a)da

), c(0) =

∫ a2

a1

ωνc(a)da, v(0) = 1−ω, �(0) = x(0) = 0,

where

λ(a) = k(a)

∫ a+

0

β(a)(x(a) + εc(a))da.

Therefore, we obtain the disease-free steady state

E0 = (s0(a), �0(a), x0(a), c0(a), v0(a)),

where

s0(a) = ωe−ψa−∫

a0(b(ξ)+p(ξ))dξ +

∫ a

0

e−ψ(a−η)−∫ aη(b(ξ)+p(ξ))dξ(b(η)ω + ψ)dη,

�0(a) = x0(a) = c0(a) = 0, v0(a) = 1− s0(a).



To study the stability of the disease-free steady state E0, we denote the perturba-tions by s(a, t), �(a, t), x(a, t), c(a, t), v(a, t), respectively. The perturbations satisfythe following equations:

(4.2)

∂s

∂a+∂s

∂t= (ψ − b(a)ων)c(a, t)− (b(a) + p(a))s(a, t) + (μ1(a)c(a, t)− λ(a, t))s0(a),

∂�

∂a+∂�

∂t= λ(a, t)s0(a)− (σ + b(a))�(a, t),

∂x

∂a+∂x

∂t= σ�(a, t)− (γ1 + b(a))x(a, t),

∂c

∂a+∂c

∂t= b(a)ωνc(a, t) + q(a)γ1x(a, t)− (μ1(a) + b(a) + γ2(a))c(a, t),

∂v

∂a+∂v

∂t= p(a)s(a, t)− (ψ + b(a))v(a, t) + μ1(a)c(a, t)v

0(a),


s(0, t) = −ω∫ a2

a1

νc(a, t)da, c(0, t) =

∫ a2

a1

ωνc(a, t)da,

�(0, t) = x(0, t) = v(0, t) = 0,

where λ(a, t) = k(a)∫ a+0

β(a)(x(a, t) + εc(a, t))da.

Now, we consider the exponential solutions of system (4.2) of the form

s(a, t) = s(a)e�t, �(a, t) = �(a)e�t, x(a, t) = x(a)e�t,

c(a, t) = c(a)e�t, v(a, t) = v(a)e�t,

where s(a), �(a), x(a), c(a), v(a), and the growth exponent � satisfy the followingequations:

(4.3)

ds

da= −�s(a) + (ψ − b(a)ων)c(a)− (b(a) + p(a))s(a) + (μ1(a)c(a)− λ(a))s0(a),

d�

da= −��(a) + λ(a)s0(a)− (σ + b(a))�(a),

dx

da= −�x(a) + σ�(a)− (γ1 + b(a))x(a),

dc

da= −�c(a) + b(a)ωνc(a) + q(a)γ1x(a)− (μ1(a) + b(a) + γ2(a))c(a),

dv

da= −�v(a) + p(a)s(a)− (ψ + b(a))v(a) + μ1(a)c(a)v

0(a),


s(0) = −ω∫ a2

a1

νc(a)da, c(0) =

∫ a2

a1

ωνc(a)da, �(0) = x(0) = v(0) = 0,

where

λ(a) = k(a)

∫ a+

0

β(a′)(x(a′) + εc(a′))da′.



Let Λ =∫ a+0

β(a)(x(a) + εc(a))da. Then λ(a) = k(a)Λ. From the equations of(4.3), we obtain that

�(a) = Λ

∫ a

0

e−�(a−η)−σ(a−η)−∫aηb(ξ)dξk(η)s0(η)dη,(4.4)

x(a) = σ

∫ a

0

e−∫ aη(�+γ1+b(ξ))dξ l(η)dη,(4.5)

c(a) = c(0)e−∫

a0(�+b(ξ)ων+μ1(ξ)+b(ξ)+γ2(ξ))dξ

+

∫ a

0

e−∫

aη(�+b(ξ)ων+μ1(ξ)+b(ξ)+γ2(ξ))dξq(η)γ1x(η)dη.(4.6)

Since the perinatal infection is about 11% of carrier mothers (Edmunds, Medley,and Nokes [7]), which is much smaller than the proportion of all infected individuals,in the rest of the discussion we make the following assumption.

Assumption 4.1. Assume that ν = 0; i.e., we ignore the proportion of thoseperinatally infected from carrier mothers.

Define

Φ(a) :=

∫ a

0

e−�(a−η)−σ(a−η)−∫ aηb(ξ)dξk(η)s0(η)dη.

Substituting (4.4) into (4.5), we have

(4.7) x(a) = Λσ

∫ a

0

e−∫ aη(�+γ1+b(ξ))dξΦ(η)dη.

Under Assumption 4.1, we substitute (4.7) into (4.6) and obtain that

(4.8) c(a) = Λσγ1

∫ a

0

∫ η

0

e−�(a−ζ)−γ1(η−ζ)−∫

aη(μ1(ξ)+γ2(ξ))dξ−

∫aζb(ξ)dξq(η)Φ(ζ)dζdη.

We further substitute (4.7) and (4.8) into the expression of λ and have that

Λ = Λσ

∫ a+

0

β(a)

∫ a

0

e−∫

aη(�+γ1+b(ξ))dξΦ(η)dηda

+ εΛσγ1

∫ a+

0

β(a)

∫ a

0

∫ η

0

e−�(a−ζ)−γ1(η−ζ)−∫ aη(μ1(ξ)+γ2(ξ))dξ−

∫ aζb(ξ)dξq(η)Φ(ζ)dζdηda,

which yields the characteristic equation

1 = σ

∫ a+

0

β(a)

∫ a

0

[e−

∫aη(�+γ1+b(ξ))dξΦ(η)

+ εγ1

∫ η

0

e−�(a−ζ)−γ1(η−ζ)−∫ aη(μ1(ξ)+γ2(ξ))dξ−

∫ aζb(ξ)dξq(η)Φ(ζ)dζ

]dηda.(4.9)

Denote the right-hand side of (4.9) by F (�). We define the basic reproductionnumber (Diekmann, Heesterbeek, and Metz [6]) as R0 = F (0), or explicitly as

R0 = σ

∫ a+

0

β(a)

∫ a

0

[e−

∫ aη(γ1+b(ξ))dξΦ(η)

+ εγ1

∫ η

0

e−γ1(η−ζ)−∫

aη(μ1(ξ)+γ2(ξ))dξ−

∫aζb(ξ)dξq(η)Φ(ζ)dζ

]dηda,(4.10)



where

Φ(a) =

∫ a

0

e−σ(a−η)−∫

aηb(ξ)dξk(η)s0(η)dη.

We have the following result on the local stability of the disease-free steady state.Theorem 4.2. Under Assumption 4.1, the disease-free steady state E0 is locally

asymptotically stable if R0 < 1 and unstable if R0 > 1.Proof. The function F (�) is a decreasing and continuous function of � which

approaches ∞ when � → −∞ and zero when � → ∞. Therefore, the characteristicequation (4.9) has a unique real solution �∗. If F (0) < 1, or, equivalently, R0 < 1,then the real solution of F (�) = 1 satisfies �∗ < 0 and the unique real eigenvalue isnegative.

Furthermore, we consider other solutions of (4.9). Let � = α+γi. Since Re e� ≤eRe�, from (4.9) we know that Re F (�) = 1, Im F (�) = 0, and 1 ≤ F (Re�).We immediately obtain that Re� ≤ �∗ since F (�) is a decreasing function of �.That is, all complex solutions of (4.9) have real parts smaller than �∗, and thereforeare negative. Hence, the disease-free steady state is locally asymptotically stable ifR0 > 1. On the other hand, if F (0) > 1, or, equivalently, R0 > 1, the unique realsolution of (4.9) is positive, and the disease-free steady state is unstable.

5. Disease-endemic steady states. In this section, we discuss the existenceand stability of the disease-endemic steady states under Assumption 4.1. Thoughmany people die from HBV-related diseases every year, the number of deaths directlyrelated to HBV is small. For example, the reported mortality rate of HBV in Chinawas 0.0464% in January 2009 (see [24]). Based on this observation, we make thefollowing additional assumption.

Assumption 5.1. Assume that μ1(a) = 0; i.e., we ignore deaths directly relatedto HBV.

We have the following result on the existence of the disease-endemic steady state.Theorem 5.2. If Assumptions 4.1 and 5.1 are satisfied and R0 > 1, then there is

a unique disease-endemic steady state E∗ = (s∗(a), l∗(a), x∗(a), c∗(a), v∗(a)) of system(2.5).

Proof. Under Assumptions 4.1 and 5.1, the coordinates of E∗ are given by

s∗(a) = ωe−∫

a0(b(ξ)+k(ξ)Λ∗+p(ξ))dξ

+

∫ a

0

e−∫

aη(b(ξ)+k(ξ)Λ∗+p(ξ))dξ[b(η)ω + ψv∗(η)]dη,(5.1)

�∗(a) =∫ a

0

e−σ(a−η)−∫ aηb(ξ)dξk(η)Λ∗s∗(η)dη,(5.2)

x∗(a) = σ

∫ a

0

e−γ1(a−η)−∫ aηb(ξ)dξ�∗(η)dη,(5.3)

c∗(a) =∫ a

0

e−∫

aη(γ2(ξ)+b(ξ))dξq(η)γ1x

∗(η)dη,(5.4)

v∗(a) = (1− ω)e−∫ a0(ψ+b(ξ))dξ

+

∫ a

0

e−∫ aη(ψ+b(ξ))dξ(b(η)(1 − ω) + p(η)s∗(η))dη,(5.5)

where Λ∗ =∫ a+0

β(a)(x∗(a) + εc∗(a))da.



Substituting (5.5) into (5.1), we obtain

(5.6) s∗(a) = G(a) +

∫ a

0

K(a, ς,Λ∗)s∗(ς)dς,

where

G(a) = ωe−∫ a0(b(ξ)+k(ξ)Λ∗+p(ξ))dξ +

∫ a

0

eb(ξ)+k(ξ)Λ∗+p(ξ)dξ

[b(η)ω

+ ψ(1 − ω)

(e−

∫η0(ψ+b(ξ))dξ +

∫ η

0

e−∫ ης(ψ+b(ξ))dξb(ς)dς

)]dη,

K(a, ς,Λ∗) = ψp(ς)

∫ a

ς

e−∫

aη(b(ξ)+k(ξ)Λ∗+p(ξ))dξ−∫

ης(ψ+b(ξ))dξdη.

Then for every Λ∗ there is a unique solution s∗(a,Λ∗) which depends continuouslyon Λ∗.

Substituting (5.2) into (5.3), we obtain

x∗(a) = Λ∗σ∫ a

0

∫ η

0

e−γ1(a−η)−σ(η−ς)−∫ aςb(ξ)dξk(ς)s∗(ς)dςdη,

which together with (5.4) implies

c∗(a) = γ1σΛ∗∫ a

0

∫ η

0

∫ ς

0

e−∫

aηγ2(ς)dς−γ1(η−ς)−

∫aρb(ξ)dξ−σ(ς−ρ)q(η)k(ρ)s∗(ρ)dρdςdη.

Substituting the expressions of x∗(a) and c∗(a) into the expression of Λ∗ and cancelingΛ∗, we get

1 =

∫ a+

0

σβ(a)

∫ a

0

∫ η

0

{e−γ1(a−η)−σ(η−ς)−

∫aςb(ξ)dξk(ς)s∗(ς)

+ εγ1

∫ ς

0

e−∫

aηγ2(ξ)dξ−γ1(η−ς)−σ(ς−ρ)−

∫aρb(ξ)dξq(η)k(ρ)s∗(ρ)dρ

}dςdηda.(5.7)

Denote the right-hand side of (5.7) by H(Λ∗). Then H(Λ∗) is a continuous decreasingfunction of Λ∗. Moreover, s∗(a, 0) is s0, which is the disease-free steady state. Hence,H(0) = R0. That is, R0 > 1 implies H(0) > 1. On the other hand, for Λ∗ > 0,

(5.8) H(Λ∗) =1

Λ∗

∫ a+

0

β(a)[x∗(a) + εc∗(a)]da <1

Λ∗

∫ a+

0

β(a)da.

The right-hand side of (5.8) approaches zero as Λ∗ approaches infinity. Therefore,H(Λ∗) = 1 has a unique solution in (0,+∞). Furthermore, if Λ∗ ≥ ∫ a+

0β(a)da, then

H(Λ∗) < 1. Hence, (5.7) has a unique solution in (0,∫ a+0

β(a)da) and a disease-endemic steady state exists.

To consider the linear stability of the disease-endemic steady state, let s(a, t),

�(a, t), x(a, t), c(a, t), and v(a, t) be the perturbations of s∗(a), �∗(a), x∗(a), c∗(a),and v∗(a), respectively. Consider the exponential solutions of the system for theperturbations

s(a, t) = e�ts(a), �(a, t) = e�t�(a), x(a, t) = e�tx(a),

c(a, t) = e�tc(a), v(a, t) = e�tv(a).



We have

(5.9)

ds

da= −�s(a) + ψv(a)− b(a)s(a)− k(a)Λs∗(a)− k(a)Λ∗s(a)− p(a)s(a),

d�

da= −��(a) + k(a)Λs∗(a) + k(a)Λ∗s(a)− (σ + b(a))�(a),

dx

da= −�x(a) + σ�(a)− (γ1 + b(a))x(a),

dc

da= −�c(a) + q(a)γ1x(a)− (b(a) + γ2(a))c(a),

dv

da= −�v(a) + p(a)s(a)− (ψ + b(a))v(a),

with the initial conditions

s(0) = �(0) = x(0) = c(0) = v(0) = 0,

where

Λ =

∫ a+

0


Assume that Λ �= 0. Let s = s/Λ, � = �/Λ, x = x/Λ, c = c/Λ, and v = v/Λ.Then system (5.9) becomes

(5.10)

ds

da= −�s(a) + ψv(a)− b(a)s(a)− k(a)s∗(a)− k(a)Λ∗s(a)− p(a)s(a),

d�

da= −��(a) + k(a)s∗(a) + k(a)Λ∗s(a)− (σ + b(a))�(a),

dx

da= −�x(a) + σ�(a)− (γ1 + b(a))x(a),

dc

da= −�c(a) + q(a)γ1x(a)− (b(a) + γ2(a))c(a),

dv

da= −�v(a) + p(a)s(a)− (ψ + b(a))v(a),

with the initial conditions

s(0) = �(0) = x(0) = c(0) = v(0) = 0.

Then the solution of (5.10) satisfies the equation

(5.11) 1 =

∫ a+

0


Denote the right-hand side of (5.11) by H(�). Then it satisfies the followingproposition.

Proposition 5.3. (1) H(�) is a decreasing function of �, which tends to zero as�→ ∞; (2) H(0) < 1.



Proof. From (5.10), we have

s(a) =

∫ a

0

e−∫

aη(�+b(ξ)+k(ξ)Λ∗+p(ξ))dξ(ψv(η) − k(η)s∗(η))dη,(5.12)

�(a) =

∫ a

0

e−∫ aη(�+b(ξ)+σ)dξ(k(η)s∗(η) + k(η)Λ∗s(η))dη,(5.13)

x(a) =

∫ a

0

e−∫ aη(�+b(ξ)+γ1)dξσ�(η)dη,(5.14)

c(a) =

∫ a

0

e−∫

aη(�+b(ξ)+γ2(ξ))dξq(η)γ1x(η)dη,(5.15)

v(a) =

∫ a

0

e−∫ aη(�+ψ+b(ξ))dξp(η)s(η)dη.(5.16)

Substituting (5.14) into (5.15) and using the expression of H(�), we get

H(�) = σ

∫ a+

0

β(a)

∫ a

0

e−�(a−η)−∫

aηb(ξ)dξ

[e−γ1(a−η)�(η)

+ εγ1q(η)e− ∫ a

ηγ2(ξ)dξ

∫ η

0

e−�(η−ς)−γ1(η−ς)−∫ ηςb(ξ)dξ�(ς)dς

]dη.(5.17)

From (5.11) we have x(a) + εc(a) > 0, which implies that �(a) > 0. Therefore, from(5.17) we know that H(�) decreases exponentially in � and H(�) approaches 0 as �tends to infinity.

Substituting (5.13) into (5.17) and taking � = 0, we have

H(0) = σ

∫ a+

0

β(a)

∫ a

0

e−∫

aηb(ξ)dξ

[e−γ1(a−η)−σ(η−ς)k(ς)s∗(ς)

+ εγ1e−γ1(η−ς)−

∫ aηγ2(ξ)dξq(η)

∫ ς

0

e−σ(ς−τ)−∫

ςτb(ξ)dξk(τ)s∗(τ)dτ

]dςdηda

+ Λ∗σ∫ a+

0

β(a)

∫ a

0

∫ η

0

e−∫

aςb(ξ)dξ

[e−γ1(a−η)−σ(η−ς)k(ς)s(ς)

+ εγ1e−γ1(η−ς)−

∫ aηγ2(ξ)dξq(η)

∫ ς

0

e−σ(ς−τ)−∫

ςτb(ξ)dξk(τ)s(τ)dτ

]dςdηda.(5.18)

Equation (5.7) implies that the first integral of (5.18) is equal to one. Furthermore,let y(a) = s(a) + �(a) + x(a) + c(a) + v(a). From (5.10), we have

(5.19)dy

da= −�y(a)− b(a)y(a)− (1− q(a))γ1x(a) − γ2(a)c(a),

which has a solution

(5.20) y(a) = −∫ a

0

e−∫ aη(ω+b(ξ))dξ((1 − q(η))γ1x(η) + γ2(η)c(η))dη.

By x(a) + εc(a) > 0 and (5.15), we know that x(a) > 0 and c(a) > 0. Then (5.20)implies that y(a) < 0. Moreover, we have �(a) > 0 from (5.14). Thus, s(a)+v(a) < 0.On the other hand, (5.16) implies that v(a) has the same sign as s(a). Therefore,s(a) < 0. Finally, from (5.18) we obtain that H(0) < 1.



Consequently, H(�) = 1 has a unique real solution which is negative, and allcomplex solutions have real parts smaller than the unique real solution. Therefore,we obtain the following result on the stability of the disease-endemic steady state.

Theorem 5.4. If Assumptions 4.1 and 5.1 are satisfied and R0 > 1, then thedisease-endemic steady state E∗ of system (3.1) is stable.

6. Numerical simulations. Since hepatitis B infection is most endemic inChina and the model is based on our previous study on HBV transmission dynamicsin China, our numerical simulations are based on some main parameters used or de-rived in Zhao, Xu, and Lu [31] and Zou, Zhang, and Ruan [32] for HBV infection inChina.

We first have the parameter λ(a, 0) given by

λ(a, 0) =

{0.13074116− 0.01362531a+ 0.00046463a2 − 0.00000489a3, 0 ≤ a ≤ 47.5,λ(47.5, 0), a > 47.5.

(6.1)

The probability q(a) that an individual fails to clear an acute infection and developsto carrier state is

(6.2) q(a) = 0.176501 exp(−0.787711a) + 0.02116.

The remaining parameters are given in Table 1.

Table 1

Parameter values used in numerical simulations.

Parameter Interpretation Value Referenceb Birth rate 0.0121 [24]ε Reduced transmission rate 0.16 [7]σ Rate moving from latent to acute 6 per yr [7]γ1 Rate moving from acute to carrier 4 per yr [7]γ2 Rate moving from carrier to immune 0.025 per yr [7]ψ Rate of waning vaccine-induced immunity 0.1 [8]

Using a constant p for p(a), we simulate the properties of disease-endemic steadystates. In Figure 2, the solid curves represent the states with p = 0.4, while thedash curves represent the states with p = 0.3. It is obvious from Figure 2 that themaximum values of acute infections and carriers with p = 0.3 are larger than thosewith p = 0.4.

To control HBV infection, we would like to see what parameters can reduce thebasic reproduction number R0 defined by (4.10). From Figure 3(a) we can see that forfixed p, R0 decreases when 1− ω increases. Similarly, in Figure 3(b) when ω is fixed,R0 decreases if p increases. Consequently, R0 decreases as both 1− ω and p increase(Figure 3(c)). Thus, the optimal control strategy is a combination of immunizationof newborns and retroactive immunization of susceptible adults.

Using equivalent time steps in the age and time directions, we choose 50 stepsper year and solve the system numerically. Figure 4 presents the age distribution ofthe proportion of carriers c(a, t), where the initial value c(a, 0) is constructed basedon the data in Table 3 of Zhao, Xu, and Lu [31].

7. Discussion. The transmission of HBV is characterized by two age-dependentprocesses: the per-capita rate of infection and the risk of becoming a carrier (Ed-munds, Medley, and Nokes [7]). Taking age-dependent heterogeneity into considera-tion, several groups of researchers have proposed age-structured models to study the



(a)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Susceptible s*

Acute

infec

tions

x*

(b)

0 10 20 30 40 500

1

2

3

4

5

6

7

8x 10

−3

Age

Acute

infec

tions

(c)

0 10 20 30 40 500

1

2

3

4

5

6

7x 10

−3

Age

Carri

ers

Fig. 2. (a) The disease-endemic steady states of susceptible and acute infections. (b) Thepredicted proportion of acute infections by age. (c) The predicted proportion of carriers by age.



(a)

0 0.2 0.4 0.6 0.8 10.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1−ω

R 0

(b)

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

p

R 0

(c)

00.2

0.40.6

0.81

0

0.5

10

1

2

3

4

5

6

R 0

1−ωp

Fig. 3. The plots of the basic reproduction number R0 in terms of some parameters. (a) R0 interms of 1− ω. (b) R0 as a function of p. (c) R0 in terms of 1− ω and p.



010

2030

4050

60

0

10

20

300

0.02

0.04

0.06

0.08

0.1

0.12

at

c(t,a

)

Fig. 4. The age distribution of the carriers c(a, t).

transmission dynamics of HBV; see Edmunds et al. [9], McLean and Blumberg [21],Zhao, Xu, and Lu [31], etc. Some of these models have also been used to evaluate theeffectiveness of the vaccination program in newborns in some countries. For example,Zhao, Xu, and Lu [31] used an age-structured model to predict the transmission dy-namics of HBV and to evaluate the long-term effectiveness of the vaccination programin China. Their results suggest that HBV infection in China can be controlled in justone generation and eventually eliminated if all infants are immunized throughout thecountry, especially in poor rural areas. However, achieving such a high vaccinationrate for infants in countries such as China is almost impossible. In fact, despite aneffective national vaccination program for newborn babies since the 1990s, which hasreduced chronic HBV infection in children, the incidence of hepatitis B in China isstill increasing (Zou, Zhang, and Ruan [32]).

Mass vaccination in infants increases the average age of infection in unimmunizedindividuals and shifts the average age at infection to older age groups (Edmunds,Medley, and Nokes [7]). This indicates that mass vaccination in infants might benot enough to control the infection and eradicate the virus. Different immuniza-tion programs can be evaluated by considering the prevalence of carriers after theimplementation of immunization. The specific character of HBV infection in Chinais that there is a large number of HBV carriers. In a previous study (Zou, Zhang,and Ruan [32]), we developed an ordinary differential equations model to study thetransmission dynamics and control of HBV in mainland China, taking into accountthe character of the virus infection in the country. Based on our previous work [32],in this article we proposed an age-structured model for the transmission of HBV.We studied the existence and stability of the disease-free and disease-endemic steadystates in terms of the basic reproduction number. The analytical results and numeri-



cal simulations of the model suggest that the optimal control strategy is a combinationof immunization of newborns and retroactive immunization of susceptible adults.

Acknowledgment. We would like to thank the referees for their helpful com-ments and valuable suggestions.

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