1.1 Research gap

Although many mathematical models have been developed to study hepatitis B transmission [18–25], existing models often lack the detailed compartmentalization required to capture the subtle interactions between various stages of infection, immunity, and reactivation, especially within the vertical and horizontal transmission pathways. This model integrates both vertical and horizontal transmission dynamics, including acute hepatitis B infection, chronic infection, chronic infection with complications, and HBV reactivation. A notable gap in existing models is the omission of HBV reactivation among recovered individuals. This model uniquely incorporates reactivation, representing a novel contribution to hepatitis B transmission modeling. Furthermore, the model addresses a critical aspect by considering the effects of vaccination not just in infants but also in adults, reflecting real-world vaccination coverage scenarios.

By providing a more detailed and realistic representation of hepatitis B transmission dynamics, this model aims to offer valuable insights for public health policymakers. It can help design more effective vaccination strategies and intervention programs, contributing to the elimination goal set by the World Health Organization. Therefore, this model serves as a crucial tool in combating hepatitis B, particularly in regions with high endemicity like Indonesia.

The remainder of this paper is structured as follows: Sect 2 describes the materials and methods used in this study. Sect 3 presents the results. Sect 4 provides a discussion of the findings, and Sect 5 concludes the study.

2.1 Model assumptions

To construct the model, it is necessary to state the assumptions used in this study as follows:

1. The population is considered well-mixed, meaning that individuals have an equal likelihood of encountering any other member of the population.
2. There are no sex-based differences in the risk of infection or disease progression.
3. There is no migration into or out of the population occurs during the study period.
4. Individuals who recover from hepatitis B infection can be either fully recovered or recovered with a chance of HBV reactivation, potentially leading to illness again.
5. Vaccination reduces susceptibility to infection but does not guarantee complete protection.
6. Newborns who test positive for HBV within 24 hours after birth are considered to be infected through vertical transmission, while those testing positive thereafter are classified as horizontally infected.
7. Both recovered and vaccinated individuals will not have permanent immunity but long-term immunity, which will decline over time.
8. Individuals who experience reactivation of HBV and become ill again will move directly into the chronic infection compartment.

The assumption that newborns testing positive for HBV within 24 hours of birth are considered to be infected through vertical transmission is based on both clinical evidence and practical modeling considerations of vertical transmission, occurring intrauterinely or perinatally [7]. However, horizontal transmission can also occur shortly after birth through contact with infected blood or body fluids. To minimize ambiguity between vertical and horizontal transmission routes, the model classifies infections detected within 24 hours as vertical transmissions. This approach accounts for the diagnostic time frame and ensures accurate representation of hepatitis B dynamics.

The SIVRM model incorporates for vaccination coverage, enabling analysis of its impact on disease transmission dynamics. A schematic diagram of the model is shown in Fig 1.

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Fig 1. Schematic diagram of the model in (1).

This transmission diagram illustrates the interactions among the compartments, including infection, reactivation, and transition processes.

https://doi.org/10.1371/journal.pone.0341120.g001

2.2 Model formulation

The transmission diagram in Fig 1 illustrates the interactions among the 14 compartments of the SIVRM model. Solid arrows represent the flow of individuals between compartments. susceptibility, infected, vaccinated, recovered, and mortality. It is a deterministic model formulated using a system of nonlinear ordinary differential equations (ODEs). In this research, we developed the modified SIVRM model considering three infection statuses and vertical-horizontal transmission. This model stratifies the entire population into 14 compartments based on key epidemiological characteristics: susceptible (S), acutely infected via a horizontal transmission (I_AH), chronically infected via a horizontal transmission (I_CH), chronically infected with complications via a horizontal transmission (I_CCH), recovered individuals from horizontal transmission infection with a chance of HBV reactivation (RR_H), fully recovered individuals from horizontal transmission (FR_H), mortality due to horizontal transmission infection (M_H), vaccinated (V), acutely infected via vertical transmission (), chronically infected via a vertical transmission (), chronically infected with complications via a vertical transmission (), recovered individuals from vertical transmission infection with a chance of HBV reactivation (), fully recovered individuals from vertical transmission (), and mortality due to vertical transmission infection ().

The model construction based on Fig 1 is described as follows. As previously mentioned, the disease can be transmitted through newborns. We use Λ to denote the constant recruitment rate of the population, while ρ represents the proportion of newborns born as infected individuals. Hence, represents newborns born with HBV, who enter the compartment . On the other hand, represents newborns born as susceptible individuals. The model also takes into account a vaccination strategy for newborns with a constant proportion . Therefore, represents susceptible newborns who have not yet been vaccinated, while represents vaccinated susceptible newborns. In addition to newborn vaccination, we also incorporate vaccination for non-newborn individuals at a constant rate . Hence, there is a transition from the susceptible compartment S to the vaccinated compartment V at rate . Since the HBV vaccine does not provide lifetime immunity, vaccinated individuals eventually lose vaccine efficacy after a period of .

In our model, β denotes the infection rate of HBV. Accordingly, the incidence of new infections among susceptible individuals is given by . Since the vaccine does not provide complete protection, vaccinated individuals may still become infected with HBV. Therefore, new infections among vaccinated individuals are modeled by , where represents the vaccine efficacy.

As previously mentioned, the infected compartments in our model are divided into two categories, namely infections due to vertical transmission (newborn infection) and infections due to horizontal transmission (contact between S or V and I_AH). Each infection category is further divided into three stages: acute infection, chronic infection, and chronic infection with complications. We use and to represent the transition rates from acute to chronic infection and from chronic infection to chronic infection with complications, respectively. Furthermore, we assume that individuals in each infected compartment may recover at a constant rate ω. Hence, transitions occur from and I_CCH to the fully recovered compartments FR_H or . On the other hand, recovery may also be incomplete, in which case individuals enter the partially recovered compartments, denoted by and RR_H, and may later experience reactivation to become chronically infected individuals. Finally, we assume that all compartments experience the same natural death rate, denoted by , while HBV-induced mortality occurs only in infected compartments at a constant rate .

Based on the above model construction, we formulate a mathematical model of HBV infection that incorporates newborn and adult vaccination, vertical and horizontal transmission, and three stages of infection, as follows.

(1)

The parameters used in model (1) and their descriptions are presented in Table 1. All parameters are assumed to be non-negative and will be estimated in the next section.

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Table 1. Description of the parameters in model (1).

https://doi.org/10.1371/journal.pone.0341120.t001

2.3 Ethical approval

This study is part of a comprehensive research project titled “Modeling Progress Toward Hepatitis B Elimination in Indonesia: A Comprehensive Assessment Study” which was approved by the Research and Community Engagement Ethical Committee - Faculty of Public Health - Universitas Indonesia, under approval number Ket- 559/UN2.F10.D11/PPM.00.02/2024, dated September 2, 2024. Ethical approval covers the entire research project, including the current study, which represents the second part of the comprehensive research.

2.4 Parameter estimation

The birth rate (Λ) and natural mortality rate () were assumed to be constant over time for the purpose of parameter estimation in MATLAB. Although real-world populations often exhibit temporal variations due to migration, changing fertility rates, or age-specific mortality patterns, the assumption of a stable population provides a reasonable approximation within the scope of this study. This approach is particularly suitable for deterministic models, as it aligns with the underlying mathematical modeling framework, ensures consistency in parameter estimation, and reduces the complexity of fitting the model to available data [26].

Vaccination coverage for adults () was assumed to be 0.00001, reflecting the currently very low adult vaccination uptake in Indonesia. In contrast, vaccination coverage for newborns () was assumed to be 0.80, based on the average proportion of newborns who receive all doses of the hepatitis B vaccine. These assumptions align with Indonesia’s current vaccination status based on data from the Indonesian Ministry of Health and were incorporated into the model to reflect realistic immunization trends. The estimated parameters include ρ, , , , , , α, β, , , , and δ.

Parameter estimation was conducted using data from the Social Security Administrator for Health (BPJS Kesehatan) registrants from January 2019 to December 2023. BPJS Kesehatan is Indonesia’s national health insurance agency, established under Law No. 24 of 2011, which administers social security. program, including the National Health Insurance (JKN) program [27]. The JKN program aims to provide universal health coverage by ensuring access to healthcare services for all Indonesian citizens, including hepatitis B prevention and treatment programs [28].

Cumulative infected cases, in this study were derived from the sum of infected compartments in the model, representing different stages of infection within the population. Specifically, the cumulative infected cases were calculated as the sum of the following compartments:

A constrained nonlinear least-squares fitting approach was applied to calibrate the model by minimizing the sum of squared differences between observed data and model output. This optimization was implemented in MATLAB using the fmincon function with the interior-point algorithm.

All parameters were estimated within lower and upper bounds based on literature values to ensure numerical stability and realistic outcomes. Initial conditions were assigned based on real data that reflected the observed population state at the start of the simulation. An initial guess for each parameter was randomly generated within its specified bounds using equation (2).

(2)

where lb and ub represent the lower and upper bounds, respectively.

The initial conditions for the model simulation are were as follows: S(0) = 2800000, I_AH(0) = 51639, I_CH(0) = 69870, I_CCH(0) = 580, RR_H(0) = 118656, FR_H(0) = 2797, M_H(0) = 1598, V(0) = 647774, I_AV(0) = 90, I_CV(0) = 101, I_CCV(0) = 0, RR_V(0) = 179, FR_V(0) = 10, and M_V = 1.

The estimation results are illustrated in Fig 2, which presents comparison between the observed data and the model’s output.

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Fig 2. Validation of model output against observed data (2019–2023).

This figure compares the model output with observed data collected between 2019 and 2023. The blue line represents the model output, while the black line represents the observed data, and the shaded area represents the 95% confidence interval.

https://doi.org/10.1371/journal.pone.0341120.g002

Fig 2 shows the number of infected individuals (). The black line represents the observed data on infected individuals from 2019 to 2023. The blue line indicates the estimated number of infected individuals (model output) over the same period. The parameters that are assumed and estimated are listed in Table 2.

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Table 2. Model parameters values.

https://doi.org/10.1371/journal.pone.0341120.t002

This table presents the estimated values of the parameters in the SIVRM model, derived from validating the model outputs against observed data.

The Root Mean Square Error (RMSE) of the model fitting was 7735.87, with a total of 122,280 infected individuals. This corresponds to a relative RMSE of approximately 6.33%. Visual inspection of the fitted curve also confirm that the model accurately captures the epidemic trends. These results validate the model’s reliability for subsequent analysis, scenario projections, and policy evaluations related to hepatitis B transmission dynamics.

2.5 Mathematical analysis

This section outlines the mathematical analysis of the SIVRM model, focusing on the derivation of the disease-free equilibrium (DFE) and the basic reproduction number .

2.6 Disease-free equilibrium (DFE)

The disease-free equilibrium (DFE) represents the state of the population in which there are no infected individuals. It is obtained by setting all infected compartments in the model to zero and solving the resulting system of equations [29]. The DFE for the model is given in equation (3) as:

(3)

where and .

3.1 Basic reproduction number ()

The basic reproduction number is determined as the spectral radius of the NGM. Since the NGM is a matrix, the basic reproduction number of the proposed model is given in equation (10):

(10)

Following the framework of van den Driessche and Watmough [31], the model satisfies all required conditions for the next-generation matrix approach. Hence, the disease-free equilibrium is locally asymptotically stable whenever and unstable when .

3.2 Interpretation

The basic reproduction number () represents the average number of secondary infections generated by a single infected individual in a fully susceptible population. Its value determines whether the infection can persist:

• if , the infection is expected to spread, as each infected individual transmits the disease to more than one new individual on average.
• If , the infection cannot sustain itself in the population and will eventually die out.

Based on equation (10), the basic reproduction number () obtained from the parameter estimation for this model is , indicating the average number of secondary infections generated by a single infected individual in a fully susceptible population. This value suggests that transmission of the hepatitis B virus remains persistent, highlighting the need for enhanced elimination strategies.

4.1 Model predictions for HBV transmission dynamics

In this section, the predicted dynamics of the hepatitis B transmission model are described. The model consists of 14 compartments, representing various population states, including and . The sum of infected compartments is also analyzed to provide a comprehensive overview of the disease progression and control measures.

Fig 3 shows the cumulative infected cases and cumulative numbers for all 14 compartments from 2019 to 2023, calibrated using observed data, along with predictions for these cumulative numbers from 2024 to 2030.

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Fig 3. Model predictions for hepatitis B cumulative cases (2024–2030):

The figure shows the predicted cumulative infected cases, followed by the cumulative numbers for all 14 compartments of the SIVRM model, from the period 2024 to 2030, based on model simulations calibrated using data from 2019 to 2023. Cumulative infected cases refer to the combined total of cases in the infected compartments (I_AH, I_CH, I_CCH, I_AV, I_CV, and I_CCV). The blue line represents the model simulation, and the red line represents the predicted trend.

https://doi.org/10.1371/journal.pone.0341120.g003

As shown in Fig 3, from 2024 to 2030, the model predicts a steady increase in cumulative infected cases under current assumptions and intervention strategies. The cumulative number of horizontally infected individuals in the compartments I_AH, I_CH, and I_CCH has steadily increased over time. Fig 3 also shows the predicted cumulative vertically infected cases (, , and ) based on model outputs and projections. The results indicate a continued rise in cases during the forecast period. Notably, there is no significant increase in the number of infected individuals in the compartment from 2024 to 2030, as the values remain relatively stable. The trend line appears nearly straight during this period, indicating a potential saturation point in the number of acutely infected individuals through vertical transmission.

The compartments of recovered individuals from horizontal (RR_H) and vertical () transmission with a chance of HBV reactivation demonstrate a steady increase over time.

Fig 3 illustrates the cumulative number of deaths due to horizontal (M_H) and vertical () infection as modelled and predicted under current intervention strategies and assumptions. For the M_H compartment, the cumulative deaths caused by horizontal infection demonstrate a steady increase over the forecast period. For the compartment, a similar increasing trend is observed, although at a smaller scale compared to horizontal transmission.

The prediction analysis based on the model demonstrates a potential increase in hepatitis B transmission rates in Indonesia under current intervention conditions. Model projections suggest that, without improvements in the prevention and treatment efforts, transmission is likely to increase over time. The numerical simulations show that when the estimated is above one (4.39), persistent infection levels are observed. This behaviour is consistent with the theoretical interpretation of as a threshold parameter, thereby supporting the validity of the numerical findings in this study [32].To address this issue, scenario analysis were conducted to determine the optimal vaccination coverage for both adults and newborns to reduce transmission and achieve the hepatitis B elimination goal.

4.2 Impact of different vaccination coverage scenarios

The success of hepatitis B elimination efforts depends on achieving sufficient vaccination coverage in both newborns and adults. This section examines various vaccination scenarios to determine the optimal coverage levels necessary to reduce the basic reproduction number () below 1, thereby ensuring disease elimination. The baseline scenario, representing the current hepatitis B situation in Indonesia, assumes newborn vaccination coverage of 0.80 and adult vaccination coverage of 0.00001, resulting in a baseline of 4.39. The impact of adjusting vaccination coverage rates is analyzed with consideration of economic feasibility and public health priorities.

Given Indonesia’s healthcare budget constraints, a reduction in newborn vaccination coverage from the current level of 0.80 to 0.70 is proposed as a cost-balancing measure. This adjustment represents a strategic approach to optimizing resource allocation within Indonesia’s healthcare system. A systematic review conducted by Liang et al. (2018) highlighted a modeling study from the Gambia, which emphasized that hepatitis B elimination is feasible even with infant vaccination coverage below 70% [33]. This finding supports the premise that maintaining a 70% coverage level could still contribute to hepatitis B control. Adult vaccination coverage is systematically increased by 200% in each scenario relative to its baseline value until falls below 1, thereby addressing potential immunity gaps. The optimal strategy is identified as the scenario in which elimination of the infection is achieved while maintaining a balance between epidemiological effectiveness and economic feasibility.

Since changes in newborn and adult vaccination coverage primarily affect the horizontal transmission, the vertical transmission compartments remain unchanged. This is because the vertical transmission equations do not include vaccination coverage for adults () or newborns (). Therefore, Fig 4 presents only the affected compartments, focusing on the effects of vaccination adjustments. It illustrates the effect of increasing adult vaccination coverage in reducing , thereby demonstrating the potential for hepatitis B elimination. Table 3 provides a summary of the corresponding values for each scenario, enabling a comparative analysis of different vaccination strategies and their effectiveness in reducing disease transmission.

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Fig 4. Impact of adjusting newborn and adult vaccination coverage on hepatitis B transmission dynamics.

This figure shows the effect of different vaccination coverage levels for adults and newborns on the basic reproductive number () using the SIVRM model. Multiple simulation scenarios were conducted to evaluate the impact of changes in vaccination coverage on hepatitis B transmission and the potential for elimination. A basic reproduction number () below 1 indicates that disease transmission may eventually cease, suggesting successful elimination.

https://doi.org/10.1371/journal.pone.0341120.g004

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Table 3. Basic reproduction number () under different vaccination scenarios.

https://doi.org/10.1371/journal.pone.0341120.t003

This table presents the basic reproduction number () values under different vaccination scenarios. Various levels of adult and newborn vaccination coverage were analyzed to assess their impact on the potential for hepatitis B elimination, aiming to identify coverage levels that reduce below 1.

As shown in Fig 4 and Table 3, the study examines how adjusting newborn and adult vaccination coverage affects hepatitis B transmission dynamics. Different scenarios were analyzed to determine the optimal combination of adult vaccination coverage () and newborn vaccination coverage () that could lead to disease elimination. The basic reproduction number () was used as a key indicator to evaluate the effectiveness of each scenario in reducing disease transmission. In all scenarios, the value of was kept constant at 0.70, except for the baseline scenario, which was set at 0.80.

In Scenarios 1 to 5, as adult vaccination coverage () increased from 0.000030 to 0.002430, the basic reproduction number () initially showed a slight decrease, falling from 4.77 in Scenario 1 to 4.40 in scenario 5 but remaining slightly above the baseline value of 4.39, indicating that the increase in adult vaccination coverage under these scenarios had a limited impact on reducing disease transmission.

In Scenario 6, where , the basic reproduction number () declined to 3.82, reflecting a more substantial reduction, though transmission persisted. Scenarios 7 and 8 introduced further increases in adult vaccination coverage. When was raised to 0.021870, the reproduction number dropped to 2.83, indicating a more pronounced decline. A further increase to resulted in = 1.81, suggesting that the virus’s transmission potential was significantly weakened. Nevertheless, complete elimination was not achieved at this stage. In Scenario 9, adult vaccination coverage was increased to 0.196830, resulting in a significant reduction in to 1.18.

Finally, in Scenario 10, further increases in adult vaccination coverage resulted in an even lower value of , with yielding . This is the first scenario in which falls below the epidemic threshold of 1, indicating that hepatitis B transmission would eventually be eliminated from the population. The theoretical threshold of the basic reproduction number indicates that when , the disease persists, whereas if , the disease will be eliminated. This theoretical rule is consistent with the numerical findings of this study; under the baseline condition (), persistent infection is observed, while in Scenario 10 (), elimination is achieved. Thus, the computational findings are directly supported by the rigorous threshold rule, in agreement with established epidemiological theory [32].

This scenario represents a special case, demonstrating that under the model assumptions, achieving at least 59% vaccination coverage among adults, combined with 70% coverage among newborns, is the optimal strategy for driving hepatitis B toward elimination.

5.1 Implication of the basic reproduction number on HBV control strategies

The basic reproduction number () is estimated to be 4.39, indicating a substantial potential for disease persistence if effective control measures are not implemented. The value in this study for Indonesia is higher than that reported for Ethiopia ( = 3.7) [43] and China ( = 2.41) [44]. Since , HBV remains endemic in the population, necessitating targeted interventions to curb its spread. The relatively high observed in this study emphasizes the urgent need to enhance prevention strategies, strengthen early detection programs, and improve access to treatment to effectively reduce transmission and advance toward hepatitis B elimination.

5.2 Insights and implications of model predictions on HBV transmission and control strategies

The predictions generated by the SIVRM model provide critical insights into the future trajectory of hepatitis B transmission under current intervention strategies. By incorporating the dynamics of horizontal and vertical transmission, recovery, and mortality, the model offers a robust framework for understanding the evolving epidemic. These findings reveal a concerning trend, as the cumulative number of hepatitis B cases is projected to rise steadily, with a significant increase in new infections in later years if current intervention strategies remain unchanged.

The model’s predictions emphasize the urgent need to strengthen preventive measures, expand screening programs, and improve access to treatment. These interventions are essential for altering the epidemic trajectory and achieving the World Health Organization’s (WHO) elimination goal. Without such efforts, the burden of disease is likely to escalate, posing significant public health challenges.

The model highlights the progression of hepatitis B infection acquired through vertical transmission. While acute infections from vertical transmission stabilize over time, suggesting the effectiveness of maternal screening and antiviral treatment during pregnancy, the number of chronically infected individuals, including those with complications, continues to rise. This stability indicates that current prevention measures have succeeded in reducing new acute infections, but additional efforts are needed to prevent chronic infection in newborns. As emphasized by Maria A. Corcorran et al. (2024), maternal screening and antiviral treatment play a crucial role in reducing perinatal transmission [45]. Nevertheless, the persistent increase in chronic HBV infections and related complications is largely due to the high risk of chronic progression in vertically infected newborns, whose immature immune systems struggle to clear the virus effectively. These findings suggest that while current preventive measures focus on reducing new infections, they do not adequately address those already chronically infected. Therefore, strengthening prevention strategies, expanding maternal screening programs, and implementing targeted treatment are essential to reducing the long-term burden of chronic infection.

In addition to transmission dynamics, the model underscores the increasing number of individuals recovering from hepatitis B, including those at risk of virus reactivation. This finding highlights the importance of long-term patient management strategies. As demonstrated by Yang et al. (2024), while antiviral treatment can effectively reduce HBV viremia, the risk of reactivation remains a persistent challenge [46]. To address this, treatment adherence programs, regular monitoring of at-risk individuals, and early intervention for reactivation cases are critical. These measures can help mitigate the risk of disease resurgence and improve long-term outcomes for recovered individuals.

The model’s projections also reveal a concerning rise in hepatitis B-related mortality, driven by both horizontal and vertical transmission pathways. Without significant intervention, the cumulative number of deaths is expected to increase substantially in the coming years. This escalating mortality burden underscores the necessity of timely diagnosis, comprehensive treatment access, and strengthened healthcare infrastructure.

5.3 Optimal vaccination coverage for hepatitis B elimination: Scenario testing approach

This study highlights the delicate balance between epidemiological effectiveness and economic feasibility in designing hepatitis B vaccination strategies for Indonesia. The findings demonstrate that while reducing newborn vaccination coverage from 80% to 70% may serve as a potential cost-balancing measure, achieving hepatitis B elimination under this scenario requires a substantial increase in adult vaccination coverage. The analysis indicates that increasing adult vaccination leads to a steady decline in the basic reproduction number (), reinforcing the critical role of adult vaccination in compensating for lower newborn coverage. While newborn vaccination remains a cornerstone of virus control, these findings suggest that prioritizing adult vaccination could serve as an effective complementary strategy within Indonesia’s public health framework.

These results align with previous studies emphasizing the importance of comprehensive vaccination strategies. For instance, a modeling study on hepatitis B transmission dynamics in China found that increasing vaccination rates across multiple age groups is crucial for effective disease control [47]. Similarly, research evaluating national immunization programs in Korea demonstrated significant reductions in hepatitis B incidence among adolescents following the expansion of vaccination efforts [48]. Moreover, a modeling study conducted in the Gambia indicated that hepatitis B elimination is feasible even with newborn vaccination coverage below 70% [33], supporting the rationale for Indonesia’s adjusted vaccination strategy. However, differences in epidemiological contexts, such as transmission patterns and healthcare infrastructure, must be carefully considered when interpreting these findings.

From an economic perspective, adjusting newborn vaccination coverage requires careful evaluation of its long-term impact. While reducing coverage from 80% to 70% may alleviate immediate financial constraints, this study suggests that achieving under this scenario necessitates a substantial increase in adult vaccination coverage. Previous studies have demonstrated the cost-effectiveness of universal adult vaccination, showing that it can substantially reduce hepatitis B incidence and associated healthcare costs [49]. Therefore, any reduction in newborn vaccination must be offset by a well-planned expansion of adult immunization to sustain progress toward elimination of the disease.

Based on the findings of this study using the SIVRM model, increasing adult vaccination coverage to 59% while reducing newborn vaccination coverage to 70% is suggested as a feasible strategy considering Indonesia’s economic constraints. This reflects the fact that while national vaccination programs have achieved high newborn coverage, a substantial portion of the adult population remains unprotected. Expanding adult vaccination through workplace clinics, community health centers (Puskesmas), and other healthcare services could help close this immunity gap. Although budget constraints may require reducing some resources for newborn coverage, this trade-off could be justified if catch-up vaccination of adults successfully reduces to below 1. Therefore, this integrated strategy offers a practical and context-sensitive pathway toward eliminating the hepatitis B virus in Indonesia by 2030, balancing epidemiological effectiveness with financial and operational feasibility.

6.1 Strengths of the model

This study presents a comprehensive mathematical model describing the transmission dynamics of hepatitis B virus infection. The key strengths of the model are as follows:

1. Incorporation of dual transmission pathways
   The model explicitly accounts for both vertical and horizontal transmission, capturing the complete scope of virus spread within the population.
2. Integration of vaccination strategies and loss of immunity
   The model incorporates vaccination coverage for both newborns and adults, accounting for waning vaccine-induced immunity over time and the loss of naturally acquired immunity after recovery. By including these factors, the model offers a more realistic representation of hepatitis B immunity dynamics and provides valuable insights into the long-term effectiveness of vaccination programs, their role in reducing hepatitis B transmission, and their potential contribution to elimination efforts.
3. Differentiation between acute and chronic infection
   The model distinguishes between acute and chronic HBV infections, which is critical for understanding disease progression, long-term complications, and treatment strategies. This distinction enhances the model’s ability to assess the burden of the disease in the population.
4. Incorporation of HBV reactivation
   Unlike many existing models, this model includes a compartment for HBV reactivation, where previously recovered individuals may transition back to an infectious state. This feature enhances the model’s ability to predict long-term disease persistence and potential recurrence, making it more applicable to real-world epidemiological conditions.

6.2 Limitations of the model

Despite its strengths, the model has several limitations that should be considered when interpreting its results:

1. Assumption of homogeneous mixing
   The model assumes that individuals mix homogeneously within each compartment, meaning that every susceptible individual has an equal probability of contact with infectious individuals. In reality, infection transmission is influenced by social behaviors, demographic factors, and healthcare access, which create heterogeneous contact patterns.
2. Exclusion of migration effects
   The model does not account for population mobility, including international migration, which can significantly influence virus transmission dynamics. Since migration can introduce new infections or alter transmission patterns, its exclusion may limit the model’s ability to fully capture disease spread in dynamic populations.
3. Limited consideration of demographic variability
   Although the model includes birth and mortality rates, it does not explicitly incorporate age-structured or sex-specific transmission dynamics. Since the risk of infection transmission varies by age and sex (e.g., higher rates of mother-to-child transmission or increased risk among at-risk adult populations), an age-stratified or sex-specific model could enhance accuracy and provide more detailed public health insights.