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- 1. ARTICLE IN PRESS Journal of Theoretical Biology ] (]]]]) ]]]–]]] Contents lists available at ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy Simon Thornley a,Ã, Chris Bullen b, Mick Roberts c a Auckland Regional Public Health Service, Cornwall Complex, Floor 2, Building 15, Greenlane Clinical Centre, Private Bag 92605, Symonds Street, Auckland 1150, New Zealand b Clinical Trials Research Unit, School of Population Health, University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland 1142, New Zealand c Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand a r t i c l e in fo abstract Article history: Background: Chronic hepatitis B (CHB) is a vaccine preventable disease of global public health Received 31 January 2008 importance. The prevalence of CHB in New Zealand’s Tongan population is over 10%, a level consistent Received in revised form with endemic infection, which contrasts to the low overall New Zealand prevalence (o0.5%). 14 May 2008 Despite the introduction of infant vaccination in 1988, coverage among Tongan children is estimated Accepted 25 June 2008 to be only 53%. Aims: To estimate the population beneﬁt of additional public health control measures besides ‘business Keywords: as usual’ infant vaccination for hepatitis B in high prevalence populations. Hepatitis B Methods: A mathematical model of hepatitis B virus (HBV) transmission was used to predict future CHB Models prevalence in the New Zealand Tongan population under different infection control strategies. Health policy Results: Prevalence of CHB is predicted to plateau at 2% in the New Zealand Tongan population if Vaccination New Zealand coverage remains at current levels, which are therefore insufﬁcient to achieve long-term elimination of HBV. The critical proportion of immunisation coverage for elimination of the virus is estimated to be 73%. The effect of screening for HBV carriage and early disease management was unable to be quantiﬁed, but is likely to reduce the population burden of HBV infection and thus contribute to accelerating elimination. Conclusions and recommendations: Mathematical models are a useful tool to forecast the future burden of CHB under a range of control strategy scenarios in high prevalence populations. Serosurveillance and targeted vaccination has similarly arrested HBV transmission in time-series prevalence studies from Taiwan and Alaska. Such a policy may demonstrate similar efﬁcacy in New Zealand ethnic groups with endemic HBV infection. & 2008 Elsevier Ltd. All rights reserved. 1. Background hepatocellular carcinoma (HCC) and liver cirrhosis compared to European New Zealanders (Blakely et al., 1998). Hepatitis B virus (HBV) infection is a disease of global health The most current and comprehensive estimate of New Zealand importance. An estimated 350 million people worldwide have CHB prevalence comes from the New Zealand Hepatitis B Screen- been infected with HBV (Maddrey, 2000), with 5% developing ing Programme (HBSP), a government-funded initiative that ran chronic hepatitis B (CHB) as a result (Gjorup et al., 2003). Between from 2000 to 2003 and tested adult Maori, Paciﬁc Island and Asian 500,000 and 1.2 million deaths are attributable to CHB each year, New Zealanders for HBV infection status (Robinson et al., 2005). making it the 10th leading cause of death worldwide (Lavanchy, Within the Paciﬁc group, the prevalence of CHB was highest 2004). Despite New Zealand’s low overall population prevalence among Tongan adults (13.4%) (Robinson et al., 2005; Herman (1–3%), CHB is responsible for two-thirds of liver-related et al., 2006). HBV serology taken from family contacts of index deaths and one-third of adult liver transplants (Gane, 2005). CHB cases showed that HBV carriage (2.0%) was high among Maori, Paciﬁc Island and Asian populations in New Zealand, Paciﬁc Island children aged 0–14 years, as was non-immune together comprising 30% of the total population, have an status (42.4%), suggesting a failure of infant vaccination pro- increased prevalence of CHB and far higher rates of HBV-related grammes to reach these populations (Herman, 2006). The HBSP also offered participants and contacts of cases found to be susceptible (non-immune/non-carriers) a free course of Ã Corresponding author. Tel.: +64 9 623 4600. hepatitis B immunisations. Individuals with CHB (‘carriers’) were E-mail address: sithor@woosh.co.nz (S. Thornley). enrolled in lifelong six-monthly surveillance for HCC, and where 0022-5193/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2008.06.022 Please cite this article as: Thornley, S., et al., Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control.... J. Theor. Biol. (2008), doi:10.1016/j.jtbi.2008.06.022
- 2. ARTICLE IN PRESS 2 S. Thornley et al. / Journal of Theoretical Biology ] (]]]]) ]]]–]]] appropriate-offered treatment. Anti-viral therapy for CHB, devel- 5-year-old children (Ministry of Health, 2001). However, there is oped over the last 15 years, offers effective treatment of disease in currently no explicit plan as to how this may be achieved across some individuals (Lok and McMahon, 2003) and may reduce all population groups, apart from increasing infant immunisation infectivity of cases by reducing viral load. In general, CHB is coverage. asymptomatic and most cases go undetected until irreversible Infectious disease models are increasingly used to develop sequelae prompt medical attention. Out of an estimated 67,000 disease control strategies. For example, a New Zealand measles cases nationally, the HBSP identiﬁed 10,176 cases of CHB epidemic in 1997 was predicted and prevented using a mathe- (Gane, 2005). This leaves around 50,000 cases undetected among matical model, leading to an effective catch-up vaccination high-risk populations, a proportion of whom would potentially strategy (Tobias et al., 1997; Roberts and Tobias, 2000). A range beneﬁt from treatment and surveillance for sequelae. However, of models of hepatitis B transmission in endemic populations current government policy does not favour continuing the provides the opportunity for such models to guide policy in New screening programme. Zealand. HBV transmission occurs following contact with infected blood or body ﬂuids. In high prevalence populations, transmission is largely vertical (mother to child during delivery), or horizontal 2. Aims through household contact (e.g., via skin breaches such as open sores or scratches) in the early years of life. In contrast, HBV To apply an established mathematical model of disease transmission in low endemicity populations typically occurs in transmission (Medley et al., 2001) to develop a strategy for adults via parenteral exposures (such as intravenous drug use) or eliminating HBV from high prevalence population groups. through sexual contact. Effective control measures for hepatitis B infection exist: clinical trials of hepatitis B vaccine among high-risk groups have 3. Methods found efﬁcacy rates of 85–95% after three doses (Ministry of Health, 2006). Transmission in a population can be almost entirely Deterministic models of HBV transmission have been reﬁned prevented by high levels of vaccine coverage (Harpaz et al., 2000). over the last decade (Edmunds et al., 1993, 1996, 1999; Medley In New Zealand, the widespread introduction of infant hepatitis B et al., 2001). We used the model described by Medley et al. (2001), vaccination in 1988 led to a dramatic decline in cases of acute HBV combined with data from the HBSP on the prevalence of CHB in infection (Ministry of Health, 2002). Subsequently, Ministry of New Zealand Tongan adults, together with estimates of vaccine Health policy has focused almost exclusively on infant vaccination coverage levels, to estimate population-speciﬁc infection trans- as the sole means of reducing the prevalence of CHB to 2% in mission parameters. Vaccination coverage levels were converted (1- c) Birth rate - susceptible infants Susceptible, x x x Latent, h h h Death Acute infection, y rates y q( ) 1y Carrier, c c 2c Protective immunity, z z c [1-q( )] 1y (1- ) Birth rate - carrier infants Birth rate - vaccinated infants Fig. 1. Diagrammatic, mathematical schema for the ﬂow of individuals in a population through different epidemiological states of HBV infection. s—rate of latent individuals becoming infectious (6 per year); g1—recovery rates of acute infections (4 per year); g2—recovery rates of carriers (CHB) (0.025 per year); n—the proportion of unimmunised children born to carrier mothers that develop carriage. Estimated at 0.11 (perinatal rate of children born to women with acute infection is ignored). (1-o)—is the proportion of births successfully immunised and hence protected from perinatal and postnatal infection. It is also manipulated to provide data on the effect of immunisation coverage. m—per capita death and birth rate, (1/70 for a typical western population (1/average life expectancy)). l—per capita rate at which susceptible individuals become infected such that l ¼ b(y+ac) where a—is the infectiousness of CHB relative to acute infections and b the transmission coefﬁcient, representing the rate of contacts and probability of transmission between individuals within a population. Hence, the average probability that an individual fails to clear an acute infection and develops the carrier state is denoted by q(l). This is further described as q(l) ¼ f+(1Àf)exp[À0.645 lÀ0.455] where f represents the lowest value this probability can take, and therefore represents the probability of carriage development in adults (0.05). Please cite this article as: Thornley, S., et al., Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control.... J. Theor. Biol. (2008), doi:10.1016/j.jtbi.2008.06.022
- 3. ARTICLE IN PRESS S. Thornley et al. / Journal of Theoretical Biology ] (]]]]) ]]]–]]] 3 Table 1 Epidemiological data and source for the Tongan population used to model hepatitis B transmission Epidemiological data (symbol) Source (year) Population from which data was drawn Value (95% CI) Prevalence of CHB infection (%) (c) New Zealand Hepatitis B Screening Tongan adults, aged 15–40 years, resident in 13.4% (12.7–14.1) Programme (2000–2003) the North Island of New Zealand Vaccination coverage (%) (1Ào) North Health’s 1996 Immunisation Coverage Cluster sample of Paciﬁc children aged 2–3 53%a Survey (1996) years in the upper half of the North Island a 95% CI not published. to modelled immunisation estimates by multiplying vaccination 16% coverage by 90% efﬁcacy (Ministry of Health, 2002). The model and parameter values are shown in Fig. 1, and data relating to the Tongan population are presented in Table 1. In the model, 14% the host population is divided into ﬁve different epidemiological Current immunisation classes, expressed as the proportion: susceptible-x, infected but (47%) 12% not infectious (latent)-h, with acute infection-y, with CHB (carriers)-c, with protective immunity-z. The relevant differential Screening and current immunisation Carrier prevalence equations are 10% dx Target immunisation ¼ Àðl þ mÞx þ moð1 À ncÞ dt 8% (85%) dh ¼ lx À ðs þ mÞh 6% dt dy 4% ¼ sh À ðg1 þ mÞy dt dc 2% ¼ qðlÞg1 y À ðg2 þ mÞc þ monc dt 0% dz 0 100 200 300 ¼ ½1 À qðlÞg1 y þ g2 c À mz þ mð1 À oÞ dt Years where the force of infection is l ¼ b(y+ac). Fig. 2. Projected CHB prevalence in the Tongan community with different Only four of these differential equations are necessary to intensities of vaccination coverage and early disease identiﬁcation and manage- describe the system as the host population is assumed to be of ment. constant size, and all the proportions must add to one. Of particular interest is the proportion of the population that has been infected with HBV, which is equal to 1Àx. The critical epidemiological parameter to be determined is the 4. Results basic reproduction number (R0). This is deﬁned as the expected number of secondary cases that would arise from a primary case The model shows that improved levels of infant vaccination in a fully susceptible population. Diekmann and Heesterbeek coverage were required to swiftly arrest disease transmission (2000) provide a mathematical deﬁnition: R0 is the largest within the Tongan population. It predicts the seropositive eigenvalue of the next generation matrix, which is in this case: proportion to be 96%, with an R0 of 3.65. Hence, 72.6% immunisation coverage (pc) of the Tongan population will result 0 1 sb qð0Þg1 a sba in elimination of infection assuming the parameters described B ðs þ mÞðg þ mÞ 1 þ g þ m ðs þ mÞðg2 þ mÞ C above are ﬁxed. B 1 2 C K¼B C In Fig. 2, time-series changes in Tongan prevalence of CHB @ qð0Þg1 mn mn A ðg1 þ mÞðg2 þ mÞ g2 þ m show that sustained high levels of infant vaccination coverage may eliminate HBV infection from the subpopulation. Current The method used to deﬁne a relationship between the levels of coverage (53% vaccination, 47% immunisation) are proportion seropositive (1Àx) and the basic reproduction number contrasted with theoretically feasible scenarios of improved (R0) was to ﬁrst ﬁx all parameters except for b, then assume a coverage—say, 72.6% (pc)—and with government policy target value for l and use this to ﬁnd steady state solutions of the levels (95% vaccination, 85% immunisation). Under business-as- differential equations above. For the two to be consistent, the usual vaccination coverage CHB prevalence takes 250 years to value of b is deﬁned. This was substituted into the next generation decline to 2%, with minimal further reduction in prevalence. matrix and R0 was calculated. This process was repeated with l in Alternatively, if immunisation coverage is increased to 85%, increments from zero until a suitable range of R0 values had been transmission is halted, and HBV infection eliminated within a covered. generation (as CHB cases die and are removed from the Once R0 is determined, the critical proportion required to be population). The beneﬁt of screening for CHB and early disease vaccinated to eliminate infection from a population (pc) may be management was not possible to quantify. However, this model determined by the equation pc ¼ 1 À ð1=R0 Þ. shows that a 50% reduction in infectivity as a surrogate for Please cite this article as: Thornley, S., et al., Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control.... J. Theor. Biol. (2008), doi:10.1016/j.jtbi.2008.06.022
- 4. ARTICLE IN PRESS 4 S. Thornley et al. / Journal of Theoretical Biology ] (]]]]) ]]]–]]] identifying and treating CHB cases (reduction in a from 1 to 0.5 numbers of children included as contacts of CHB cases. The model with the same level of immunisation coverage) produces a assumes a total population prevalence including all ages. dramatic reduction in CHB prevalence similar to that from 95% Second, although several parameters are known—such as vaccination coverage. those that estimate the latent period of infection, and the length To check the sensitivity of this parameter to change, we of acute infection—other parameters speciﬁc to the population speculate that if all carriers are identiﬁed, close contacts were uncertain. Estimates of CHB infection transmission (a), and vaccinated and treatment offered a realistic range of reduced vaccination coverage (1Ào) proved especially difﬁcult to quantify. transmission is estimated to be 20–80%. Taking the worst-case The former is unknown, although we do know that anti-viral scenario of a 20% reduction in a0 and current immunisation therapy will result in reduced viral load, and hence lower coverage remaining the same, the model predicts a new steady probability of transmission. Vaccination coverage is estimated state CHB prevalence of 0.3%. Under a near ideal 95% vaccination by record-based surveys (North Health, 1996); however, the coverage scenario, the virus is eliminated from the population, so model is designed to use prevalence of serological immunity to in this very optimistic scenario, improved coverage is better than a the virus (HBsAg positivity). We also assumed that the Tongan sustained screening programme. The importance of CHB cases population had a stable prevalence of CHB. While this is likely to in maintaining the endemicity of the disease is consequently be accurate in the high prevalence adult population, this may demonstrated. be quite different in younger age groups who have received vaccination and transmission is likely to be reduced as a result. Further work may more accurately predict the likely impact of 5. Discussion targeted disease control strategies. However, the experience of other high prevalence regions is instructive. Time-series serosur- These models of HBV transmission show that high levels of veillance of endemic HBV infection in Alaska and Taiwan shows vaccination coverage alone have the potential to hasten elimina- that high vaccination coverage in infancy can halt infection tion of infection from high prevalence populations in New transmission. Alaska’s indigenous Inuit population has had Zealand, with a signiﬁcant effect produced by herd immunity. endemic hepatitis B infection and its eradication was prioritised. Equally, they suggest that the ongoing application of the now A system of serosurveillance of anti-HBsAg antibodies in children discontinued HBSP to such populations may swiftly arrest the aged o15 years plus catch-up vaccination has almost stopped transmission of infection within a similar timeframe, without any transmission in this group (Harpaz et al., 2000). Taiwan has high change in vaccination coverage. levels of vaccination coverage in which uptake is carefully There are a number of caveats to these statements. First, the monitored by a Hepatitis Information System established by the model did not completely ﬁt the data supplied for the Tongan Department of Health. Vaccination coverage has been reported to population from the HBSP. For example, there was a difference be between 84% and 96%. Prevalence of CHB has fallen among 6 between model predictions and actual serological data. The most year olds from 10.5% to 1.7% between 1989 and 1991 (Chen et al., important discrepancy was an overestimation of the infected class 1996). These results are broadly consistent with those the model in the modelled population. From the HBSP, 20.4% of Tongan presented here for the Tongan population. participants were non-immune, 66.2% were immune and 13.4% In New Zealand, hepatitis B infection control policy is currently had CHB. This departed from model predictions if CHB prevalence not distinguished from that of other vaccine preventable diseases (13.4%) was used as the primary reference and the population (Ministry of Health, 2001). Yet, the very different behaviour and assumed to be at steady state—82% immune (z) and 4% in the sequelae of the virus, compared to other such diseases, with the non-immune (x) classes were forecast. In summary, the model need for long-term sustained action to arrest transmission, argues overestimates the extent of virus exposure when compared to for separate consideration of this unique and important disease. data from the Tongan population. The HBSP data show a low prevalence of immunity amongst To check the sensitivity of the model to changes in the primary Paciﬁc Island child contacts of those with CHB links with data used (for modeling), an alternative scenario is described. If previously identiﬁed evidence of low immunisation coverage the starting point is considered to be a seropositive proportion of amongst Paciﬁc Island children in New Zealand (North Health, 66% (95% CI 65–67), then the proportion of carriers predicted is 1996). Hence, those in most need of HBV vaccination have the 2.5%, a very different value from that forecast originally from the lowest level of coverage (Fiscella and Shin, 2005). New Zealand’s carrier proportion (13.4%). The original data describing the Tongan Paciﬁc Island communities have relatively low levels of access to CHB prevalence has narrow 95% CIs (12.7–14.1%), so such a health care services in New Zealand (Barwick, 2000), given their discrepancy may not be explained by sampling variability. burden of illness and need for preventive services, so additional However, the corresponding R0 under this scenario is reduced to resources to improve uptake of hepatitis B vaccination for this 3.41 (cf. 3.65) with the corresponding pc of 70% (cf. original 76%). community should be a priority. So, although there is a discrepancy in the original data used to One way to promote a high uptake of vaccination in high build the model, the effect on R0 and pc is small. prevalence populations is by serosurveillance. This approach is Possible reasons for the disparity between modelled and attractive due to high prevalence HBV populations being small observed data include limitations in the model design, imperfect and geographically discrete. Rapid, point of assessment, ﬁnger- epidemiological data and imprecise or unknown parameter prick assays yielding near instant determination of immune status estimates. The model provides a caricature of the spread of the could simplify the roll out of the proposed programme. The virus in populations, with assumptions simplifying highly com- Tongan population is particularly suitable for such a strategy, plex mechanisms of disease transmission. For example, it does not being settled in a discrete geographic area (South Auckland), and discriminate between vaccination given to neonates of mothers with a church-oriented community that may be an efﬁcient point known to have CHB and routine infant vaccination. It also assumes of access. Economic analysis would be the next essential step in a constant population size whereas census data show a dramatic building the case for introducing such a programme and would growth in the New Zealand Tongan community (Statistics New include sensitivity analyses and discounting of future beneﬁts. Zealand, 2001). CHB prevalence—used to estimate population- A recent review of the economics of hepatitis B vaccination speciﬁc transmission parameters amongst the Tongan commu- supported the use of deterministic models such as that described nity—was drawn from an adult cohort (aged 15–40), with small here of this sort (Beutels et al., 2002.). Please cite this article as: Thornley, S., et al., Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control.... J. Theor. Biol. (2008), doi:10.1016/j.jtbi.2008.06.022
- 5. ARTICLE IN PRESS S. Thornley et al. / Journal of Theoretical Biology ] (]]]]) ]]]–]]] 5 6. Conclusions Fiscella, K., Shin, P., 2005. The inverse care law: implications for healthcare of vulnerable populations. J. Ambul. Care Manage. 28 (4), 304–312. Gane, E., 2005. Screening for chronic hepatitis B infection in New Zealand: While this attempt to quantify the long-term beneﬁts of unﬁnished business. N. Z. Med. J. 118 (1211), 1–5. targeting a high HBV prevalence population with preventive Gjorup, I.E., Skinhoj, P., et al., 2003. Changing epidemiology of HBV infection in health measures has a number of limitations, it is consistent with Danish children. J. Infect. 47 (3), 231–235. Harpaz, R., McMahon, B.J., et al., 2000. Elimination of new chronic hepatitis B virus international experience of hepatitis B control. Time-series data infections: results of the Alaska immunization program. J. Infect. Dis. 181 (2), from serosurveillance of a high prevalence population would 413–418. allow for improved model assessment and parameter estimation. Herman, J., 2006. Epidemiology of Hepatitis B Among Paciﬁc People in Auckland: Outcome of a Hepatitis B Screening Programme. Auckland Regional Public Economic models may assist the assessment of ‘value for money’ Health Service, Auckland. of proposed intervention options. In New Zealand, the presence of Herman, A., Bullen, C., et al., 2006. Mobilising Paciﬁc people for health: Insights endemic CHB infection in distinct ethnic groups, along with the from a Hepatitis B Screening Programme in Auckland. Pac. Health Dialog 13 (2), 9–15. modelled forecasts of effect, supports the use of such disease- Lavanchy, D., 2004. Hepatitis B virus epidemiology, disease burden, treatment, and speciﬁc infection control programmes. current and emerging prevention and control measures. J. Viral Hepat. 11 (2), 97–107. Lok, A.S.F., McMahon, B.J., 2003. AASLD Practice Guidelines: Chronic Hepatitis B, References pp. 1–19. Maddrey, W., 2000. Hepatitis B: an important public health issue. J. Med. Virol. 61, 362–366. Barwick, H., 2000. Improving Access to Primary Care for Maori, and Paciﬁc Peoples Medley, G.F., Lindop, N.A., et al., 2001. Hepatitis-B virus endemicity: heterogeneity, Wellington. Health Funding Authority. catastrophic dynamics and control. Nat. Med. 7 (5), 619–624. Beutels, P., Edmunds, W.J., et al., 2002. Economic evaluation of vaccination Ministry of Health, 2001. An Integrated Approach to Infectious Disease: Priorities programmes: a consensus statement focusing on viral hepatitis. Pharmacoe- for Action 2002–2006. Wellington, Ministry of Health. conomics 20 (1), 1–7. Ministry of Health, 2002. Immunisation Handbook 2002. Wellington, Ministry of Blakely, T., Salmond, C., et al., 1998. Hepatitis B virus carrier prevalence in New Health. Zealand: population estimates using the 1987 police and customs personnel Ministry of Health, 2006. Hepatitis B. Immunisation Handbook 2006. Wellington, survey. N. Z. Med. J. 111 (1064), 142–144. Ministry of Health. Chen, H., Chang, M., et al., 1996. Seroepidemiology of hepatitis B virus infection in North Health, 1996. North Health’s 1996 Immunisation Coverage Survey. North children: ten years of mass vaccination in Taiwan. JAMA 276 (11), 906–908. Health, Auckland, pp. 1–6. Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical epidemiology of infectious Roberts, M.G., Tobias, M.I., 2000. Predicting and preventing measles epidemics diseases: model building, analysis and interpretation. Wiley, Chichester. in New Zealand: application of a mathematical model. Epidemiol. Infect. 124, Edmunds, W.J., Medley, G.F., et al., 1993. The inﬂuence of age on the development 279–287. of the hepatitis B carrier state. Proc. R. Soc. Lond.—Ser. B: Biol. Sci. 253 (1337), Robinson, T., Bullen, C., et al., 2005. The New Zealand Hepatitis B Screening 197–201. Programme: screening coverage and prevalence of chronic hepatitis B infection Edmunds, W.J., Medley, G.F., et al., 1996. Epidemiological patterns of hepatitis B [see comment]. N. Z. Med. J. 118 (1211), U1345. virus (HBV) in highly endemic areas. Epidemiol. Infect., 313–325. Statistics New Zealand, 2001. Tongan People in New Zealand. Statistics New Edmunds, W.J., Medley, G.F., et al., 1999. Evaluating the cost-effectiveness Zealand, Wellington. of vaccination programmes: a dynamic perspective. Stat. Med. 18 (23), Tobias, M., Christie, S., et al., 1997. Predicting the next measles epidemic. The New 3263–3282. Zealand Public Health Report 4, No. 1, pp. 1–3. Please cite this article as: Thornley, S., et al., Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control.... J. Theor. Biol. (2008), doi:10.1016/j.jtbi.2008.06.022

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