A MATHEMATICAL MODEL FOR HEPATITIS B WITH INFECTION-AGE STRUCTURE

. A model with age of infection is formulated to study the possible eﬀects of variable infectivity on HBV transmission dynamics. The stability of equilibria and persistence of the model are analyzed. The results show that if the basic reproductive number R 0 < 1, then the disease-free equilibrium is globally asymptotically stable. For R 0 > 1, the disease is uniformly persistent, and a Lyapunov function is used to show that the unique endemic equilibrium is globally stable in a special case.


Introduction. Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus(HBV)
, which is a circular genome composed of partially double-stranded DNA and can hardly be cleared after infection because of the formation of cccDNA. It is estimated that approximately 2 billion people have serological evidence of past or present HBV infection. Over 350 million people are chronic carriers of HBV and every year about 600,000 people die from HBV-related liver disease or hepatocellular carcinoma(HCC). Although there is no widely available treatment for chronic HBV carriage, the infection can be safely and effectively prevented by vaccination, which includes passive immunoprophylaxis and after birth active immunization [1].
HBV can cause an acute illness and a chronic liver infection which is characterized by persistent serum level of HBV surface antigen(HBsAg), IgG anti-core antigen(anti-HBc) and HBV DNA [3]. Chronic infection may later develop into cirrhosis of the liver or liver cancer [11]. The incapability to clear infection and the subsequent development of the carrier state is almost certainly due to host factors [4]. It has become increasingly acknowledged that the probability of becoming chronically infected is dependent on age of the host [11,4,12,6]. In general, the average age at which individuals become infected is largely determined by the prevalence of infection in the population. Mathematical model with age structure may be more reasonable to study the important consequences of age for the HBV infection.
In many epidemiological models, it has been assumed that all infected individuals are equally infectious during their infectivity period, which is proved to be reasonable for some diseases such as influenza. Actually, the infectivity of HBV individuals varies at different age of infection, thus it is necessary to formulate age structure models to describe the heterogeneity in infectious individuals, which will lead to a system of partial differential equations. In general, there are two different age structures in disease models, i.e., biological age and infection age. Although their dynamical analysis is particularly challenging, the epidemic models with age structures have been extensively studied recently [9,17,2,5]. Since the progress of the acute and chronic stages of HBV is complicated, depending on the time that has passed since the moment of infection, it is imperative to develop models with the infection age structure to show its influence on transmission dynamics of HBV infection.
Several recent studies have focused on modeling HBV transmission dynamics [13,25,20,16]. These models mainly investigated the influence of prevention and control measures including vaccination and antivirus treatment in some regions and countries, with the form of ordinary equations. Age-structured models have also been developed to study the epidemiology of HBV [4,23,24]. Medley et al observed a feedback mechanism that determined the prevalence of HBV infection, relating the rate of transmission, average age at infection and age-related probability becoming carriers [12]. Based on sero-survey data, Zhao et al constructed an agestructured model to predict the dynamics of HBV and to evaluate the long-term effectiveness of the vaccination programme [23]. Zou et al in [24] proposed a model that incorporated age structure into individuals to study the transmission dynamics of HBV, analysing the existence and stability of the disease-free and endemic steady state solutions. They used numerical simulations to illustrate the optimal strategies for controlling the disease.
In order to study the possible effects of variable infectivity on HBV transmission, we develop a mathematical model with infection age to describe the HBV infection dynamics, which extends the existing one of ordinary equations [21]. In this model, the infectivity is allowed to depend on the age of infection. The parameters, such as the transmission coefficient, the rate moving from acute to chronic stage, and the rate moving from carrier to immunized class, are age-dependent. A detailed analysis of stability and uniform persistence is conducted. The dynamic behaviors of the system are determined by the basic reproductive number R 0 , and a special case is studied by constructing a Lyapunov function to analyse the global stability of endemic equilibrium.
The organization of the remaining part is as follows. In section 2, we formulate the model with infection-age structure for HBV transmission. In Section 3, we study the existence and stability of steady states. Section 4 focuses on the analysis of uniform persistence. A special case is presented in section 5 to illustrate the global stability of endemic equilibrium. Section 6 gives the summary and conclusions.
2. Formulation of the model. The transmission begins when a susceptible subject acquires an acute HBV infection through effective contact with a temporary or a chronic HBV carrier, shifts to the latent period, and then the individual becomes an acute HBV for 3 months on average. If the acute infection does not progress to a chronic one, the host clears HBV, recovers, and becomes immune. A chronic HBV carrier that lasts for many years can also follow acute infection. A few chronic carriers clear HBV and become immune. From the natural history of HBV, it can be seen that the infection age of an infected individual may be important for the transmission of HBV among population.
Two major transmission routes are included in the model, that is, perinatal and horizontal transmission. We assume that all the newborns are vaccinated at the same efficacy, and all the vertical infected infants are the chronic carriers since the risk of becoming chronic carriers is extremely high (90%) for the neonates who acquire HBV infection perinatally. Based on these characteristics and assumptions of HBV transmission, we construct the following age-structured model for HBV Here, S(t), E(t) and R(t) denote population of susceptible, exposed and immunized at time t, respectively. i(a, t) and c(a, t) denote the density of acute infections and chronic HBV carriers with infection age a at time t. The population of acute HBV is I(t) = a1 0 i(a, t)da, and C(t) = ∞ a1 c(a, t)da represents the chronic HBV carriers, where the critical infection age a 1 equals to 3 months, since an acute HBV lasts for 3 months on average, and then the chronic stage begins. The definitions of the parameters in system (S 1 ) are listed in Table 2 proportion of children that is unsuccessfully immunized at birth p vaccination rate of susceptible children and adults α infectiousness of carriers relative to acute infections σ rate of transfer from exposed to acute infection v(a) proportion of children developing to HBV carriers born to carrier mothers of infection age a β(a) age-dependent transmission coefficient γ 1 (a) age-dependent rate moving from acute to chronic or immunized class γ 2 (a) age-dependent rate moving from carrier to immunized q(a) rate leaving acute infection and progressing to carrier with age a θ(a) HBV induced death rate with age-dependence In our analysis, the functions v(a), β(a), γ 1 (a), γ 2 (a), q(a) and θ(a) are assumed to be nonnegative, bounded and integrable in their definition intervals. The function β(a) describes the variable probability of infectiousness as the disease progresses within an infected individual. The other constant parameters are nonnegative, and the initial conditions satisfy S 0 ≥ 0, E 0 ≥ 0, R 0 ≥ 0, i 0 (a) ∈ L 1 + (0, ∞) and c 0 (a) ∈ L 1 + (0, ∞). To simplify expressions, we introduce the following notations π 1 (a) = e − a 0 (µ+γ1(s))ds , a ∈ [0, a 1 ], π 2 (a) = e − a a 1 (µ+γ2(s)+θ(s))ds , a ∈ [a 1 , ∞), a1 v(a)π 2 (a)da, here π 1 (a) is the age-specific survival probability of an acute infected individual, and π 2 (a) is that of a chronic one. With the given boundary and initial conditions, integrating i(a, t) and c(a, t) along the characteristic lines(t − a =constant) yields and Using classical existence and uniqueness results for functional differential equations, it is seen that the integro-differential system (S 1 ) has a unique solution, in which i(a, t) and c(a, t) are substituted with the expression (1) and (2), respectively. By (1) and (2), it is easy to see that i(a, t) and c(a, t) remains nonnegative for any nonnegative initial value. Further, if there exists t * such that S(t * ) = 0 and S(t) > 0 for 0 < t < t * , then from the first equation of (S 1 ) we can get Similarly it can be shown that E(t) ≥ 0 and R(t) ≥ 0 for all t ≥ 0 and for all nonnegative initial values.
For epidemic model, the basic reproductive number is an important threshold parameter, which gives the expected number of secondary cases produced in a completely susceptible population by a typical infective individual. As for the HBV transmission model S 1 , we define the basic reproductive number as W 1 gives the number of individuals infected by one acute infection, and W 2 is that of leaving acute infection and progressing to chronic stage. The number of infections produced by an HBV carrier is expressed by αW 3 . bωW 5 is the quantity of vertical infected infants.
Substituting i * (a) and c * (a) into the boundary conditions about yields c * (a 1 ) = σE * W 2 + bωc * (a 1 )W 5 , so it follows that c * (a 1 ) = σE * W2 1−bωW5 . Solving the second and first equations of system S 1 in terms of S and E gives Notice that the variable R does not appear in equations for other variables, thus the equation of R can be ignored when studying the dynamics of HBV infection. We denote the system of the remaining equations as S 2 , which will be studied in the rest part of the paper.
We consider the stability of disease-free equilibrium P 0 , and have the following result Theorem 3.1. The disease-free equilibrium P 0 is globally stable when R 0 < 1 and it is unstable when R 0 > 1.
Next, we will deal with the global attraction of P 0 .
Substituting (1) and (2) into the boundary condition about c(a 1 , t), we get Observe that when time t → ∞, age a in the integral is fixed at a 1 , i.e., a = a 1 . Denoting f ∞ = lim t→∞ sup f (t), by the above equality, we have Since then by the results in [18], there exists a sequence {t n } as t n → ∞ such that by (4) and (5), we obtain Hence if R 0 < 1, then E ∞ = 0, which implies that lim t→a1 i(a, t) = 0, since i(0, t) = σE(t). By the equation of c(a, t) and its initial condition, it can be easily inferred that lim t→∞ c(a, t) = 0, and lim t→∞ S(t) = bω µ . Theorem 3.2. The endemic equilibrium P * is locally stable when R 0 > 1.
If Rz > 0, where Rz denotes the real part of z, then which is a contradiction, so the characteristic equation (6) has eigenvalues only with negative real part.
4. Uniform persistence. In this section, we establish the uniform persistence for the model when R 0 > 1. We reformulate the system S 2 as an abstract Cauchy problem, and let u = (S, E, i, c) T ∈X = R 2 × L 1 ((0, +∞), R 2 ), which is endowed with the following norm In order to take into account the boundary condition, we extend the state space and setX with Dom(A) = {v ∈ X 0+ | u 3 (.), u 4 (.) ∈ W 1,1 [0, +∞)}, where W 1,1 denotes a Sobolev space. Define nonlinear operator F : X 0 → X as Then we rewrite the system S 2 as the following abstract Cauchy problem with v(0) = v 0 ∈ X 0+ for any t ≥ 0. By Theorem 2 and Theorem 3 in [19], it can be verified that there exists a unique solution semiflow U (t) : X 0+ → X 0+ defined by system (7). Let N (t) denote the total population size, then we have For the system dy dt = b − µy, the equilibrium b µ is globally asymptotically stable. By the comparison principle, it follows that lim t→∞ sup N (t) ≤ b µ , which implies that all solutions of system S 2 , and correspondingly, the solutions of system (7) are ultimately bounded. Moreover, when N (t) > b µ , we have dN (t) dt < 0, which implies that all solutions are uniformly bounded. Therefore, the solution semiflow U (t) : X 0+ → X 0+ is point dissipative. And it holds that the set µ } is positively invariant absorbing under the semiflow U (t) on X 0+ , and U (t) maps any bounded set to a precompact set in X 0+ . Consequently, by Theorem 3.6.1 in [7], U (t) is compact(completely continuous) for any t > 0. Thus, Theorem 1.1.3 in [22] implies that U (t) has a compact global attractor in X 0+ . Consequently, we have the following result Lemma 4.1. The model system (7) has a unique solution semiflow U (t) in X 0+ . Furthermore, there is a compact global attractor A for U (t).

5.
A special case. In this section, we will study the global stability of endemic equilibrium P * in a special case θ(a) = 0.
6. Discussion. Globally, hepatitis B is among the most serious communicable diseases because of its incidence of new cases, the prevalence of carriers and the burden of acute and chronic disease. It is imperative to have better understanding on HBV disease dynamics. Since the infection of HBV is characterized by replicative and non-replicative phases based on virus-host interaction, the infectivity of HBV individuals varies at different age of infection. Generally, the age structure of the host population has been regarded as an important factor for the dynamics of disease transmission. According to the characteristics of HBV, we formulate a mathematical model that incorporates infection-age structure to describe the possible effects of variable infectivity on the transmission dynamics.
Although dynamical analysis for age-related models with partial differential or integro-differential equations is challenging, mathematical properties of our system are studied analytically in this paper. A detailed analysis of stability and uniform persistence is conducted, which shows that the dynamic behaviors of the system are determined by the basic reproductive number R 0 . Specifically, by linearizing the system at the steady states, it is proved that the disease-free equilibrium is stable when R 0 < 1, and the endemic equilibrium is locally stable when R 0 > 1. Furthermore, in the case θ(a) = 0, it can be seen that the endemic equilibrium is globally asymptotically stable by constructing a Lyapunov function. When R 0 > 1, the model system is described by an abstract Cauchy problem and the uniform persistence for the system is established. Maybe the possible effects of infection-age on the transmission dynamics of HBV can be illustrated numerically by using some reasonable parameters relating age of infection in our future research.