A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China

Shoujun Zhaoa,b, Zhiyi Xua and Ying Lub

a Shanghai Medical University, Shanghai, 200032, PR China.
b University of California, San Francisco, 94143–1349, USA.

Reprint requests to: Shoujun Zhao, Osteoporosis and Arthritis Research Group, Department of Radiology, University of California, San Francisco, CA 94143–1349, USA. E-mail: shoujun.zhao{at}oarg.ucsf.edu


    Abstract
 Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion and Conclusion
 References
 
Background Before universal infant immunization against hepatitis B virus (HBV) in 1986 China was a region endemic for HBV infection. The prevalence of HBV infection in the population was about 60% and the proportion of chronic HBV carriers around 10%. These HBV carriers could progress to chronic hepatitis B, cirrhosis, and primary hepatocellular carcinoma. Since 1976, large-scale sero-surveys of HBV infection have been carried out and a lot of data have been collected.

Method This paper describes a mathematical model developed to predict the dynamics of HBV transmission and to evaluate the long-term effectiveness of the vaccination programme. We used a compartment model expressed by a set of partial differential equations based on the characteristics of HBV infection.

Results All parameters, expressed in the model as a non-linear function of age and time since vaccination, were estimated using sero-survey data. The model fits well with both pre-vaccination and post-vaccination sero-surveys. The observed and estimated age-specific prevalence rates of HBV infection and HBV carriage agree with each other. According to our model, if all newborns are vaccinated according to schedule, the rate of HBV carriage will decline sharply over time to 0.2% in 70 years. By then, the ratio of acute hepatitis B will be less than 0.5% and the ratio of chronic hepatitis B will be around 5%.

Conclusions The results suggest that HBV infection in China can be controlled in just one generation, and eventually eliminated. Our model shows that vaccination coverage is the most important indicator for the elimination of HBV transmission. The higher the vaccination coverage, the better the long-term effectiveness of immunization. Thus, the key to controlling and eliminating HBV transmission in China is to find ways to immunize all infants throughout the country, especially in poor, rural areas.

Keywords Hepatitis B, compartment models, differential equation, epidemiology, infectious diseases, hepatitis B vaccination, immunization strategy

Accepted 5 January 2000


    Introduction
 Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion and Conclusion
 References
 
In China and other Asian countries, hepatitis B virus (HBV) infection has been a major public health problem. The prevalence of HBV infection in China was around 60% and the proportion of chronic HBV carriage as high as 10%1,2 before the vaccination programme. These HBV carriers could progress to chronic hepatitis B, cirrhosis, and primary hepatocellular carcinoma. The development of the hepatitis B vaccine was a landmark in the control and elimination of HBV infection in humans.3 In China prior to 1986, most HBV infections occurred in infancy and early childhood.4–6 In 1986, China implemented a large-scale programme to immunize all newborns against HBV. This programme has resulted in enormous social and economic benefits.7–10 In children under 10 years old, the proportion carrying HBV has decreased to 0.53% in Shanghai.7,8,10 In some rural areas HBV carriage has been reduced to 1–2%, as long as all newborns are vaccinated with a low dose programme (10µg x 3) according to the schedule.9,10 However, the long-term effectiveness of the programme in China still needs to be evaluated. It is very important to understand changes in the dynamics of HBV transmission following implementation of the programme, and the ways that best control and eliminate HBV infection in the population throughout China.11 The aim of this study was to establish a mathematical model for HBV transmission dynamics so it can be used for predicting the long-term effectiveness of the immunization programme and help in selecting optimal strategies for nationwide HBV immunization.


    Materials and Methods
 Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion and Conclusion
 References
 
A compartment model
A compartment mathematical model expressed by a set of first-order partial differential equations was developed.11–15 Based on the characteristics of HBV transmission, the population was divided into five compartments: (1) susceptibles S(a,t); (2) latent period (the time interval from infection to development of infectiousness), L(a,t); (3) temporary HBV carriers, T(a,t); (4) chronic HBV carriers C(a,t); and (5) the immune I(a,t). Here ‘a’ represents the age and ‘t’ represents the length of follow-up. Of the five stages, compartments 3 and 4 are infectious. According to the natural history of HBV,11,16–19 a susceptible subject acquires an acute HBV infection through effective contact with a temporary or a chronic HBV carrier, and shifts to the next compartment—the latent period. The average latent period persists 45 days, and then the individual becomes a temporary HBV carrier for 3 months on average.17–19 If the acute infection does not progress to a chronic one, the host clears HBV, recovers, and becomes immune. A chronic HBV carrier state that lasts for many years can also follow acute infection. A few chronic carriers clear HBV and become immune. The age-related annual rate of the viral clearance has been reported to be as low as 1–2% on average.20–22 A proportion of susceptibles move directly to the immune state when they are successfully immunized with hepatitis B vaccine. Clinical manifestations, such as acute and chronic hepatitis B, were beyond the scope of this paper and were not considered. The acute hepatitis B state which harbours HBV transiently was attributed to the temporary HBV carrier and chronic hepatitis B to the chronic HBV carrier state.16,17 In this model, birth rate was considered as a constant; age-specific death rates were collected from death notification systems.23 The immune status was assumed to be lifelong and newborns were assumed to be susceptible. A few infants born to both hepatitis B surface antigen (HBsAg) and hepatitis B e antigen (HBeAg) positive mothers24,25 can be infected by HBV in utero. The rate was reported to be about 3–5%, and the proportion of their mothers in all pregnant women was only 2–3%. Therefore, the probability of intrauterine fetal infection was very low (about 0.0006–0.0015). For simplicity of modelling, intrauterine HBV infection, the short period of newborn maternal antibody protection and sex differences were ignored.

The five compartments and model variables are illustrated in Figure 1Go. More specifically, the model parameters were defined as the following: {lambda}(a,t) is the force of infection; {alpha} is the rate of transition from latent period to temporary HBV viraemia; ß(a) is the risk of transient viraemia progressing to chronic HBV carriage; {epsilon} is the rate of transition from temporary HBV viraemia to immune per time unit; {nu}(a) is the rate of HBV clearance in chronic HBV carriers; {tau}(a) is the mortality rate of HBV related diseases; µ(a) is the age-specific mortality rate of non-HBV related diseases; Vc(a,t) is the effectiveness of hepatitis B immunization. These parameters must satisfy the following partial differential equations (1)Go:



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Figure 1 Flowchart of hepatitis B virus (HBV) transmission in a population

Each box represents a compartment of the process.

Definition of the parameters:

{lambda}(a,t): the force of HBV infection.

{alpha}: the rate of transition from latent period to temporary HBV viraemia.

ß(a): the risk of transient viraemia progressing to chronic HBV carrier state.

{epsilon}: the rate of transition from temporary HBV viraemia to immune per time unit.

{nu}(a): the rate of HBV clearance in chronic HBV carriers.

{tau}(a): the mortality rate of HBV related diseases.

µ(a): the age-specific mortality rate of non-HBV related diseases.

Vc(a,t): the effectiveness of hepatitis B vaccine immunization.


ß(a) = 0.706004exp(–0.787711a) + 0.08464

{nu}(a) = 0.00227005a – 0.00011211a2 + 0.00000149a3

{tau}(a) = 1/[1 + exp(11.80965 – 0.16887177a + 0.0007375a2)]

 

(1)




Epidemiological data sets
Data from the following studies were used in this paper to estimate the model parameters. More detailed descriptions of these studies are given in the Appendix.

  1. A cross-sectional sero-epidemiological survey of HBV markers in 10 484 subjects, aged 0–70 years in four provinces (Hebei, Hunan, Heilongjiang and Henan) of China, in 1985.1,26
  2. A 2-year longitudinal study of sero-conversion to HBV markers and incidence by age of HBV infection in the above population, 1986–1988.27,28
  3. An 8-year follow-up study of HBV sero-reversion from positivity to negativity in 227 chronic HBV carriers in Hebei Province, 1985–1993.22
  4. An epidemiological survey of age-specific mortality rates of HBV related chronic liver diseases in samples of 613 939 subjects in the four provinces of China, 1984–1987.29
  5. Age-specific mortality rates in China 1990.23,30
  6. Ten-year follow-up studies of vaccination effectiveness data in newborn babies were used to evaluate the precision of the model prediction.7–10

All the parameters in the model were estimated by the maximum likelihood method based on the data from the above epidemiological surveys.31,32


    Results
 Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion and Conclusion
 References
 
Estimation of model parameters
Parameter {lambda}(a,t), the force of HBV infection, is the instantaneous per capita rate for susceptibility of acquiring the infection at age a and time t. At the start of vaccination (t = 0), this parameter, {lambda}(a,0), can be derived based on data set 1. The estimating parameter equation of {lambda}(a,0) is:


(2)


(3)

The estimated prevalence of HBV infection coincided well with the field survey data (Table 1Go, {chi}2 = 8.6497, d.f. = 10, P = 0.5656). Figure 2aGo shows the parameter {lambda}(a,0). It peaked in infancy and early childhood, declined rapidly with age, dropped to a low level at age 15 and remained at that level afterwards.1,26 The peak has mainly been associated with maternal-infant transmission and improperly sterilized needles and syringes in China.1,4–6,33


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Table 1 Comparison of values estimated by parameter equation 3Go with observed prevalence of hepatitis B virus (HBV)
 


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Figure 2 Model parameters estimated using the data from sero-epidemiological surveys

 
Parameter {alpha} is designated as the rate of transition from latent period to temporary HBV viraemia. Assuming that the rate per time unit for a shift from latent to temporary viraemia is constant for the entire latent period, the average time of the shift should be 1/{alpha}.17–19 There are two ways of leaving the latent state; either moving to temporary viraemia or dying of other diseases. Thus, this parameter was calculated based on the average latent time, 1/{alpha}, and the mortality rate, µ(a). The estimated average latent time was 1.5 months.

Parameter ß(a), the risk of temporary HBV viraemia progressing to chronic HBV carriage, described the relationship between age of infection and development of chronic HBV carriage. Based on data set 2, we estimated the parameter equation as follows (Table 2Go, {chi}2 = 4.14, d.f. = 5, P = 0.5296):


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Table 2 Comparison of values estimated by parameter equation 4Go with observed data
 

(4)

The risk was a function of age and was very high in infancy but remained at a low level after 5 years of age (Figure 2bGo).27,28,34

There are three ways of leaving the state of temporary HBV viraemia: becoming immune, or a chronic HBV carrier and death from the causes other than HBV related diseases. Let {epsilon} be the rate of transition from temporary HBV viraemia to immune. The rate {epsilon} per time unit for transition from temporary HBV viraemia to immune is assumed to be equal for the entire period of the transient viraemia. Thus, the average time of a transition is 1/{epsilon}. This parameter was calculated based on 1/{epsilon}, ß(a) and µ(a). The estimated average time of transition was 3 months.16–19

Parameter {nu}(a) is designated as the rate of HBV clearance in chronic HBV carriers. A follow-up study (data set 3) had shown that there were significant differences in the rates of reversion from HBV carrier to negativity among different age groups. The rate after age 50 is much higher than before 50. No reversion was observed in people 0–4 years old. A low reversion level was observed in people 5–45 years old. A high rate of reversion in the elderly has not been reported in the literature and can be explained by the sudden decrease in HBV carriers after age 50. It also provides an interpretation for self-limitation of chronic HBV infection proposed by Szmuness (Figure 2cGo).22

Parameter {tau}(a) is the mortality rate of HBV related diseases, such as chronic hepatitis B, cirrhosis, and primary hepatocellular carcinoma, among the chronic HBV carriers. It was estimated using the data of age-specific mortality rates from data set 4 (Figure 2dGo).29

Parameter µ(a) is the age-specific mortality rate of non-HBV related diseases. It was estimated using the age-stratified death notification data (data set 5).23,30

Finally, parameter Vc(a,t) is the effectiveness of hepatitis B vaccine immunization at age a and time t and was estimated based on dataset 6.8–10

Model evaluation
Based on the parameters estimated above, we calculated all the probabilities in the model (1), including S(a,t), L(a,t), T(a,t), C(a,t) and I(a,t), at age a and time t, by the integral of the partial differential equations. These estimates fit the dynamics of HBV transmission in the population during the pre-vaccination period. The observed and estimated values for the age-specific prevalence rates of HBV and for the proportion of HBV carriers in the population were all very close and the model corresponded well with the sero-epidemiological surveys (Figure 3Go) before the vaccination programme. It is interesting to show that the model successfully simulated not only the age-specific HBV carrier rates observed in the 1985 sero-surveys in four provinces, but also those observed in the 1978 sero-surveys among 176 068 subjects in all 29 provinces of China (Figure 3bGo).1,2



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Figure 3 Prevalence of hepatitis B virus (HBV) infection and HBV carriage rate by age

 
We applied the following formula proposed by Anderson13,15 to estimate the force of the HBV infection after vaccination:


(5)

Different WAIFW (Who Acquires Infection From Whom) matrices were created to determine the term ß (a',a), and the best WAIFW matrix which predicted the age-specific proportions of HBV infection and carriers at baseline and after vaccination for 10 years, was selected (Figures 3 and 4GoGo). Because {lambda}(a,t) is concomitant with the mass vaccination year by year, the dynamics of HBV transmission can be simulated under the model 1. The model-predicted proportions of HBV carriers in the population were in agreement with the proportions observed among randomly sampled vaccinated children (Figure 4Go).



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Figure 4 Hepatitis B virus carriage rate in vaccinated children

 
Prediction of long-term effectiveness of hepatitis B vaccination
Since the model can fit HBV transmission dynamics before and after vaccination, we utilized it to predict the long-term effectiveness of hepatitis B immunization and to depict the transmission dynamics of HBV in the population.

The proportion of HBV carriers in a vaccinated cohort will decrease sharply depending on both the coverage of immunization and the doses received by infants.7–10,35,36 If all newborn babies are immunized according to the schedule set in Shanghai, the proportion of HBV carriage in immunized children will decrease to 0.53%.7,8,10,37 In some urban or rural areas with the low dose schedules, the proportion will also decrease to a low level of 1–2%.9,10 Table 3Go shows the predicted proportions of HBV carriers following the immunization programme with 100% coverage in the population and Figure 5Go illustrates the dynamics of HBV carriers. The majority of HBV carriers will shift gradually in age from children to the elderly and fade away in 70 years. After the vaccination programme has been implemented for 70 years, the average HBV carrier rate will decrease to 0.2%. Since the HBV carrier state lasts for many years and the annual rate of HBV clearance in chronic carriers less than 50 years old was as low as 1–2%, the reduction in the HBV carrier rate among those unvaccinated adults will remain until all carriers die. Of course, if a new drug can be developed to cure chronic HBV carriage, the time it takes to control HBV transmission will be shorter than our estimates.


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Table 3 Age-specific proportion of hepatitis B virus (HBV) carriers in the population predicted by model 1 at different intervals from start of immunization programme (coverage = 100%, effectiveness = 90%)
 


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Figure 5 Dynamics of hepatitis B virus (HBV) carriers concomitant with mass vaccination

 
To assess the impact of vaccination on the future incidence of acute and chronic hepatitis B, we defined two incidence ratios for acute and chronic hepatitis B as functions of age and time since vaccination. The incidence of acute disease can be considered as a linear function of proportion of new acute HBV infection. The incidence of chronic hepatitis B is a function of proportion of chronic HBV carriers because almost all attacks of chronic hepatitis B were a reactivation of the chronic carrier status.16,17

The first ratio is the incidence ratio of acute hepatitis B, Ra(a1,a2:t). It is defined as the number of acute cases in the age range from a1 to a2 at time t divided by the corresponding number of acute cases at t = 0, the baseline before vaccination. Mathematically,


(6)

The range in equation (6)Go has been defined as a1 = 10 and a2 = 45, because the peak of the incidence curve for acute hepatitis B was observed in the age interval 10–45 years old. Incidence in other age groups is very low.16,17 The Ra(a1,a2:t) at time t with different vaccination coverage is shown in Figure 6Go. It decreases steeply at the beginning of the hepatitis B vaccination programme. The higher the vaccination coverage level, the steeper the decrease of the ratio. The decrease slows down in a few years after the start of the vaccination programme. At a coverage rate of 100%, the ratio will be reduced from 1 to less than 0.5% 70 years from the start of the vaccination programme.



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Figure 6 Ratios of acute hepatitis B incidence following the immunization programme

 
The incidence ratio of chronic hepatitis B, Rc(a1,a2:t) is defined as the number of chronic HBV carriers in the age range from a1 to a2, at time t divided by the corresponding number of chronic HBV carriers at t = 0, the baseline before vaccination. Mathematically


(7)

In equation (7)Go, a1 = 25 and a2 = 70. Most chronic liver diseases were observed in adults >=25 years. The disease incidence in the younger age group was negligible.16,17 The Rc(a1,a2:t), at time t with different vaccination coverage, is shown in Figure 7Go. It remained almost unchanged at the beginning of the vaccination programme, and dropped rapidly only after 25 years of immunization. Again, the decrease in the ratio is closely related to vaccination coverage. At 100% vaccination coverage, the ratio will be around 5% 70 years from the start of the vaccination programme.



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Figure 7 Ratios of chronic hepatitis B incidence following the immunization programme

 

    Discussion and Conclusion
 Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion and Conclusion
 References
 
Hepatitis B virus is highly prevalent and control of HB is a major public health concern in China. Since the assays of HBV markers were developed, the prevalence and incidence rates as well as the age distribution of HBV infection and HBV carriage have remained very similar across most provinces of China for decades. This stable state, expressed as ‘equilibrium’ between the virus and human population, has provided good opportunities for using the mathematical models to study the disease's dynamics.1,2

Much field data were accumulated and can be used to develop appropriate mathematical models. Results of several large-scale population studies based on sero-surveys as well as special follow-up studies were analysed, which allowed us to successfully estimate the model parameters. The correspondence of these parameters to the observed field data is demonstrated in Figures 2–4GoGoGo.

The model has simulated well, not only HBV transmission dynamics, but also the proportion of age-specific HBV carriers obtained from two large-scale, cross-sectional sero-surveys undertaken 7 years apart (Figure 3Go). It is also a powerful tool to study the impact of parameters on long-term vaccine effectiveness. It demonstrates that the HBV carrier rate, the most important indicator of the vaccine's effectiveness, will fall from 10% to less than 0.2% 70 years after the start of the universal infant vaccination programme (Figure 5Go). Thus, long-term vaccination effectiveness is foreseeable and the disease is eradicable. The model also suggests that vaccination coverage is the most important parameter for vaccine effectiveness (Figures 5, 6, and 7GoGoGo). Compared to different vaccination strategies being applied in China, our model has shown that a low dose strategy with higher vaccination coverage and lower vaccine efficacy provided higher long-term effectiveness than a high dose strategy with lower coverage and higher efficacy.11

One limitation of this model, however, may be underestimation. The formula (2) indicated that the decreased number of infectious subjects in the post-vaccination period would result in a reduction in transmission of HBV infection. Thus, vaccine effectiveness should be higher than the vaccine efficacy. However, the formula did not completely consider the role of herd immunity, established after the universal immunization programme, in reducing infection transmission. This is especially important for a chronic infection. Therefore, this formula awaits further modification and improvement.

A universal infant hepatitis B immunization programme has been underway for more than 10 years in China. A set of administrative systems for the hepatitis B immunization programme has been well established in most developed regions, especially in major cities. Most newborns are vaccinated according to the schedule. For example, in Shanghai, the vaccination coverage rate has remained at more than 95% in recent years, and the proportion of HBV carriage among vaccinated children has decreased to a very low level. In addition, disinfection of the medical instruments and syringes has also contributed to the reduction in HBV transmission. However, the hepatitis B immunization programme is different from those of other viral infectious diseases. Some infants, if they remain unvaccinated and are infected by HBV, will become chronic HBV carriers. They will be the new sources of infection, which will last for many years. These study results show that the goal of eliminating HBV transmission in some developed regions of China will be realized in just one generation.

In summary, this paper established a mathematical model to describe the epidemic dynamics of HBV infections. Based on several large-scale epidemiological surveys and follow-up studies, we estimated model parameters. The model corresponded with observed data and can be used to evaluate the long-term effectiveness of the vaccination programme and help determine the optimal strategy to reduce and eventually eliminate HBV infections.


    Acknowledgments
 
We thank Ms Victoria Vandenberg for her editorial help. The first author also thanks Harry Genant, MD for his support and guidance during the author's visit to the University of California, San Francisco.


    References
 Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion and Conclusion
 References
 
1 Liu CB, Xu ZY, Cao HL et al. Sero-epidemiology of HBV infection in four provinces in China. Chin J Virol 1991;7(Suppl.):8–14.

2 Ju ZY. An epidemiological study on the distribution of HBsAg and Anti-HBs in China. Chin J Microbiol Immunol 1986;11(Suppl.): 20–40.

3 World Health Organization. The Yaounde Declaration on the Elimination of Hepatitis B Infection. Geneva: WHO, 1992.

4 Liu LH, Wang HX, Yin DH et al. Prevalence and incidence of hepatitis B virus infection in 3–5 years old kindergarten children. Chin J Virol 1991;7(Suppl.):25–29.

5 Xi LF, Xu ZY, Sun YD et al. The horizontal and prenatal transmission of hepatitis B virus infection. Chin J Virol 1991;7(Suppl.):21–24.

6 Beasley RP, Hwang LY, Lin CC et al. Incidence of hepatitis B virus infections in preschool children in Taiwan. J Infect Dis 1982;146: 198–204.[ISI][Medline]

7 Wu WS, Shao ZP. A review of hepatitis B immunization. Chin J Vaccine Immunization 1996;2:61–66.

8 Lin XM, Xu ZY, Ouyang PY et al. Eight year survey for hepatitis B vaccine efficacy to newborns after universal immunization. Chin J Exp Clin Virol 1995;9(Suppl.):55–58.

9 Li RC, Yang JY, Wang SS et al. The effect of hepatitis B vaccination on epidemiology of hepatitis B virus. Chin J Vaccine Immunization 1996; 2:56–60.

10 Xu ZY, Cao HL, Liu CB et al. A long-term effectiveness evaluation of the universal infant hepatitis B vaccination program in China. Chin J Exp Clin Virol 1995;9(Suppl.):13–16.

11 Zhao SJ, Xu ZY. Mathematical simulation of hepatitis B transmission and application in immunization police. In: Zhen XW (ed.). Progress in Epidemiology, Beijing. 1995;8:162–81.

12 Anderson RM, May RM. Vaccination against rubella and measles quantitative investigations of different policies. J Hygiene 1983;90: 259–325.[ISI]

13 Anderson RM, Grenfell BT. Quantitative investigations of different vaccination policies for the control of congenital rubella syndrome (CRS) in the United Kingdom. J Hygiene 1986;96:305–33.[ISI]

14 Anderson RM, May RM. Age-related changes in the rate of disease transmission: implications for the design of vaccination programs. J Hygiene 1985;94:365–436.[ISI]

15 Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford, New York: Oxford University Press, 1991.

16 Xu ZY. Impact and control of viral hepatitis in China. In: Hollinger FB (ed.). Viral Hepatitis and Liver Diseases. Baltimore: Williams & Wilkins, 1991, pp.700–06.

17 Szmuness W. Sociodemograghic aspect of the epidemiology of Hepatitis B. In: Vyas GN, Cohen SN, Schmidt R (eds). Viral Hepatitis. Philadelphia, PA: Franklin Institute Press, 1978, pp.296–320.

18 Centers for Disease Control. Protection against viral hepatitis: recommendation of immunization practices advisory committee (ACIP). MMWR 1990;39:5–22.

19 Shi LY. Viral hepatitis. In: Zhihao L (ed.). Epidemiology. Beijing: People's Health Press, 1994, pp.225–43.

20 Alward WLM. The long-term serological course of asymptotic hepatitis B virus carriers and the development of primary hepatocellular carcinoma. J Infect Dis 1985;151:604–09.[ISI][Medline]

21 Hsu HY, Chang MH, Lee CY et al. Spontaneous loss of HBsAg in children with chronic hepatitis B virus infection. Hepatology 1992; 15:382–86.[ISI][Medline]

22 Zhao SJ, Xu ZY, Ma JC et al. A follow up study of spontaneous clearance rates on hepatitis B surface agent persistent carriers. Chin J Prev Med 1994;29:378–79.

23 Census Data Manual of China. Beijing: Center of Chinese Vital Statistical Information, 1985.

24 Tang SX. Study of the mechanisms and influential factors of intrauterine infection of hepatitis B virus. Chin J Epidemiol 1991;12:325–28.

25 Wang SS. Transplacental transmission of hepatitis B virus. Chin J Epidemiol 1991;12:33–35.

26 Zhao SJ, Xu ZY. A study of hepatitis B infection force in China. Chin J Epidemiol 1991;14(Suppl.):70–74.

27 Cao HL, Sun YD, Yan TQ et al. A study on the dynamics of HBsAg carrier status in four study areas. Chin J Virol 1991;7(Suppl.):15–20.

28 Zhao SJ, Xu ZY, Cao HL et al. An estimating model of relationship between the infection age of hepatitis B virus and the age-specific chronic carrier rate. Chin J Exp Clin Virol 1995;9(Suppl.):101–04.

29 Liu CB, Xu ZY, Cao HL et al. A field study of incidence of acute and chronic viral hepatitis and the mortality of hepatitis related diseases and hepatocellula carcinoma. Chin J Virol 1991;7(Suppl.):1–7.

30 Li JL. Age-specific mortality rate model with its application. Chin J Biometrics 1992;1:41–45.

31 Grenfell BT, Anderson RM. The estimation of age-related rates of infection from case notifications and serological data. J Hygiene 1985; 95:419–36.[ISI]

32 Farrington CP. Modeling force of infection for measles, mumps and rubella. Stat Med 1990;9:953–67.[ISI][Medline]

33 Ma JC, Sun YD, Xie YF et al. A study on the relationship between disinfection of syringes, acupuncture needle transfusion sets and HBV infection in children under 2 years. Chin J Virol 1991;7(Suppl.):30–34.

34 McMahon BJ, Alward WLM, Hall DB et al. Acute hepatitis B virus infection: relation of age to the clinical expression of disease and subsequent development of the carrier state. J Infect Dis 1985;151: 599–603.[ISI][Medline]

35 Xu ZY, Liu CB, Wen YM et al. Prevention of perinatal acquisition of hepatitis B virus carriage using vaccine: preliminary report of a randomized, double-blind, placebo-controlled and comparative trial. Pediatrics 1985;90:259–63.[ISI][Medline]

36 Xu ZY, Liu CB, Yan TQ et al. Evaluation of effectiveness of large-scale hepatitis B vaccination in neonates. Chin J Virol 1991;7(Suppl.): 48–51.

37 Zhao SJ, Xu ZY, Lu Y et al. Evaluating and predicting the long-term effectiveness of the hepatitis B vaccination in Shanghai using a mathematical model. 1997 Proceedings of the Epidemiology Section, American Statistical Association, pp.71–74.