During the 20th century, deaths from a range of serious infectious diseases decreased dramatically due to the development of safe and effective vaccines. However, infant immunization coverage has increased only marginally since the 1960s, and many people remain susceptible to vaccine-preventable diseases. “Catch-up vaccination” for age groups beyond infancy can be an attractive and effective means of immunizing people who were missed earlier. However, as newborn vaccination rates increase, catch-up vaccination becomes less attractive: the number of susceptible people decreases, so the cost to find and vaccinate each unvaccinated person may increase; additionally, the number of infected individuals decreases, so each unvaccinated person faces a lower risk of infection. This paper presents a general framework for determining the optimal time to discontinue a catch-up vaccination program. We use a cost-effectiveness framework: we consider the cost per quality-adjusted life year gained of catch-up vaccination efforts, as a function of newborn immunization rates over time and consequent disease prevalence and incidence. We illustrate our results with the example of hepatitis B catch-up vaccination in China. We contrast results from a dynamic modeling approach with an approach that ignores the impact of vaccination on future disease incidence. The latter approach is likely to be simpler for decision makers to understand and implement because of lower data requirements.

During the 20th century, deaths from a range of serious infectious diseases such as smallpox, measles, polio, diphtheria, pneumococcal disease, and bacterial meningitis decreased dramatically due to the development of safe and effective vaccines against these pathogens. However, infant immunization coverage has increased only marginally since the 1960s, and many people remain susceptible to vaccine-preventable diseases.^{1} The World Health Organization (WHO) reports that almost 20% of children born in 2007 did not receive complete routine vaccination.^{2} Newborns and young children may not receive recommended vaccinations because they lack access to health care, face social barriers, or have parents who are unaware of or unmotivated about vaccination.^{3} Vaccine price may also be an impediment, especially in the developing world.^{4, 5}

Because infant immunization is incomplete, there may be a need for recurring annual campaigns to reach children missed in infancy. The WHO recommends expanding immunization to every eligible person, including those in age groups beyond infancy.^{1} Such “catch-up vaccination” can be an attractive and effective means of immunizing people who were missed earlier.^{6–8} Although newborn vaccination is the most cost-effective strategy for preventing disease,^{9, 10} catch-up vaccination for individuals missed by newborn vaccination can be highly cost-effective (e.g.,^{10}). Many successful catch-up vaccination programs have been implemented around the globe, including immunization programs for measles,^{11, 12} haemophilus influenza type B (which causes bacterial meningitis),^{13} and polio.^{14} An ongoing program in China aims to provide catch-up vaccination for hepatitis B virus to at least 500,000 children.^{8}

For many vaccine-preventable diseases, newborn vaccination rates are increasing.^{4, 15, 16} This affects the cost-effectiveness of catch-up vaccination: the number of susceptible individuals decreases, so the cost to find and vaccinate each unvaccinated person may increase; additionally, the number of infected individuals decreases, so each unvaccinated person faces a lower risk of infection. This paper presents a general framework for determining the optimal time to discontinue catch-up vaccination programs. We focus specifically on control of chronic diseases (as opposed to diseases such as influenza where patients either recover or die^{17}). We consider an ongoing catch-up vaccination program where individuals of any given age(s) who missed newborn vaccination are vaccinated each year. We use a cost-effectiveness framework: we consider the cost per quality-adjusted life year (QALY) gained^{18} of catch-up vaccination efforts, as a function of newborn immunization rates over time and consequent disease prevalence and incidence. Much previous theoretical research on controlling vaccine-preventable diseases has focused on disease eradication.^{19–24} However, disease eradication can be extremely costly and, for most vaccine-preventable diseases, is not a realistic goal for the foreseeable future. Other studies focus on levels of vaccination to achieve herd immunity.^{25–27} However, it may still be valuable from a public health perspective to vaccinate even if coverage levels to achieve herd immunity are not possible, and it may also be valuable to vaccinate ^{18} Other research has examined the long-term impact of infection reduction for chronic diseases, but has focused on newborn vaccination only and has not used a cost-effectiveness framework.^{28–30}

Many studies that evaluate the cost-effectiveness of vaccination use simple Markov cohort models that do not take into account the dynamics of the epidemic,^{10, 31–36} yet studies have shown that the impact of vaccination on herd immunity can be important, particularly when the vaccination program can affect a substantial fraction of the population.^{37–40} However, studies of the impact of herd immunity often focus on mass infant vaccination for diseases such as measles and varicella that have relatively short infectious periods.^{38, 39}

We solve the catch-up vaccination problem with an approach that captures the dynamics of infection transmission, and then contrast results from this approach with an approach that ignores the impact of vaccination on disease incidence. We allow for age dependency (as opposed to a homogeneous population^{17}) to account for the fact that younger children may receive more benefits from the intervention and/or harms from infection. Age is often an important inclusion criteria for catch-up vaccination guidelines.^{41} Finally, unlike previous research which assumes constant marginal costs of immunization,^{19–21, 23, 24, 42, 43} we allow for increasing marginal costs.

In Section 2 we formulate our model. In Section 3 we focus on a single age group and ask, “What should our vaccination coverage goal be this year for each age group?” This can help establish the upper age limit at which one would want to start catch-up vaccination. In Section 4 we look over time and ask, “When should we stop catch-up vaccination programs for different age groups?” We illustrate our results in Section 5 with the example of hepatitis B catch-up vaccination in China. Section 6 concludes with discussion.

We consider a population of individuals stratified by age and disease state. All notation is shown in

We consider the decision of whether to perform catch-up vaccination among eligible individuals in age group _{a}_{0}(

To model the dynamics of infection transmission, we use an age-structured deterministic SIR (Susceptible, Infected, Recovered) epidemic model with homogeneous mixing.^{22} This model is similar to others used to predict long-term prevalence of chronic infectious diseases.^{28} (An SICR model incorporating an additional Carrier state could also be used, but because the infectious state can be short, we elected to use an SIR model). In this model, the infection rate at time ^{22} In our age-structured model, the infection rate for individuals in age group

The dynamics of the model are as follows. We distinguish between newborns (

We assume that initial compartment sizes are known for non-newborns (_{a}_{a,}_{0}, _{a}_{a,}_{0}, _{a}_{a,}_{0}. _{a}^{44}

The model is illustrated schematically in _{1}(_{1}(_{1}(_{1}(_{1}(_{1}(

We solve the optimal vaccination problem by decomposing it. We first solve the problem for a single age group in a single time period (Section 3) and then consider the long-term problem of catch-up vaccination target coverage levels for different age groups over time and when to discontinue such efforts (Section 4).

We first focus on a single time period and single age group _{0}(^{45} and Gold^{18} provide details on the appropriateness of this as a value measure). We assign a monetary value ^{18, 45} There is not always agreement on what this value of ^{46, 47}) is one to three times a country’s per capita GDP.^{48, 49}

The problem of maximizing the net present monetary benefit of vaccinating a fraction _{α}

The first term in (_{α}

_{v}_{a}_{a}_{a}_{a}_{a}_{a}_{a}

Research on vaccine program size suggests that the average cost of vaccinating individuals begins to rise when vaccination coverage reaches high levels^{50} because the last individuals requiring vaccination may be very difficult to find and reach.^{51, 52} A systematic review found that average costs of vaccination programs operating from fixed facilities initially decline with the scale of vaccination programs because of high fixed costs, but costs may later increase when the programs expand outside of dense urban areas into more rural and remote regions.^{53} Another review found that the average cost per immunized child is often minimized at a coverage level of about 50–60% and then increases noticeably at about 80% population coverage.^{50}

P1 is a complex nonlinear problem (the objective function is governed by the nonlinear dynamics of the epidemic, (_{a}_{a}_{a}

To simplify P1, we assume that future incidence (the chance that an unvaccinated, susceptible person acquires the infection in the future) does not depend on _{α}

The only interactions between age cohorts in _{α}

Given the above assumption, and letting _{a,d}_{a,d}

The first term in the objective function is the net present monetary value of the health benefits minus changes in health care costs accruing from successfully vaccinated individuals. The second term is the vaccination cost. To solve P2, we must know _{α,R}_{α,S}_{α,R}_{α,S}_{α,R}_{α,R}^{10, 54} The quantities _{α,S}_{α,S}_{a}_{a}_{α,S}_{α,S}

It is straightforward to find the optimal level of vaccination

In the approximate single-period problem P2, the infection risk (incidence) is a determinant of the expected health effects and health care costs for susceptible individuals, _{α,S}_{α,S}

The risk of infection for a susceptible individual of age

In this section we discuss four ways of estimating (

The risk of infection (_{a}_{a}

One simple estimate of future incidence is to assume that it is the same as current incidence; that is, _{a}_{a}

Another estimate of future incidence can be obtained by assuming that the current age-infection distribution remains the same over time, except for assuming that individuals born at time 0 and afterward will not become infected (as though 100% of newborns are vaccinated with a 100% effective vaccine):

This estimate is similar to “cutting off” the tail of (or zeroing-out) the age-infection distribution corresponding to younger ages. This may be reasonable if the age-infection distribution is in a steady state to begin with and, except for vaccination of the young, it is expected that the steady-state infection distribution will continue.

Another estimate of future incidence can be obtained by assuming no new infections or resolved infections, and also assuming that individuals born at time 0 and afterward will not become infected (as though 100% are vaccinated with a 100% effective vaccine). This takes the current population of infected individuals with their current infection prevalence and “ages them out”. This implies, for example, that the infection prevalence for age ^{55} If we assume Type I survivorship^{22} (everyone lives exactly

We now examine the problem of when to stop catch-up vaccination in each age group _{a}

We assume that catch-up vaccination at the level determined at time 0 for each age group,

The problem can be written as:

This formulation allows us to consider catch-up vaccination in any subset of age groups; for age groups not considered for catch-up vaccination, we can set

We can solve P3 using numerical methods. To do so, we start at time _{a}_{a}_{a}

We illustrate our models with the example of hepatitis B catch-up vaccination for children and adolescents in China. Approximately one-third of the world’s 350 million cases of hepatitis B infection occur in China,^{56, 57} where the disease is a generalized epidemic and an estimated 7.4% of the population is chronically infected.^{56, 58} Approximately 1% of those under 5 are infected, 2.5% of those between 5 and 14 are infected, and 8.5% of those 15 and older are infected.^{58} The most common routes of transmission are neonatal infection and horizontal transmission during early childhood.^{56, 59}

The Chinese government recommended hepatitis B vaccination in 1992, and in 2002 the vaccine was made free for newborns. These policy changes helped newborn vaccination coverage rise from 70.7% in 1997 to 89.8% in 2003.^{59} However, even with recent dramatic increases in newborn vaccination rates, an estimated 150 million children in China are still unprotected from hepatitis B.^{4, 58} Infection in children is a particular problem because the younger the age at infection, the more likely it is that the infection will become chronic (lifelong).^{60} Left untreated, approximately one in four chronically infected individuals will die from liver disease related to hepatitis B.^{44, 61, 62} Because it is effective and cost-effective (and likely to be cost-saving^{10}), hepatitis B catch-up vaccination for school-age children was made free by the government in 2009.^{63} At the same time, extensive public health efforts have led to steady increases in newborn vaccination rates across the country.^{4} Given the increases in newborn vaccination rates, when is catch-up vaccination for different age groups no longer cost-effective?

Parameter values for the model (^{10} that study modeled the effects of chronic hepatitis B infection, but only considered the cost-effectiveness of current catch-up vaccination efforts (without considering when to stop catch-up vaccination), and did not consider costs of vaccination above baseline (thus, the optimal catch-up vaccination level for any age group was either 0% or 100%). To model the epidemic, we used a slightly more sophisticated model of disease than the model given by (^{10} In this model of hepatitis B, the basic reproductive number is calculated to be 1.1 which is similar to that found in other studies of hepatitis B.^{64, 65} We used this model of disease when solving both P1 and P2. We used a health system perspective and included all lifetime health care costs for individuals in the population. The current per capita GDP in China is about $4500,^{66–68} so an incremental cost-effectiveness ratio between $4,500 and $13,500would be considered cost-effective, and a ratio less than $4500 would be considered highly cost-effective according to WHO criteria.^{48, 49} In our base case analyses we assumed

We conducted sensitivity analysis on the recovery rate to see how these general conclusions might apply for diseases with shorter infectious periods. If the length of infection is much shorter (2–10 years) the results using a static model of incidence versus a dynamic model diverge (

We now examine the problem of how many years into the future to perform catch-up vaccination. We first consider programs that would vaccinate children at school entry, either at age 5 when entering kindergarten or at age 12 when entering middle school. We assumed 90% newborn vaccination coverage each year. We assumed different maximum feasible levels of catch-up vaccination, ranging from 50% to 100%. We solved P3 with

We next consider a program that would provide catch-up vaccination to children at both points of school entry, ages 5 and 12.

The model can also be used to determine the “best case” scenario for catch-up vaccination among children, which would occur if all eligible children were to receive 100% catch-up vaccination.

In China, remote rural regions have significantly lower birth-dose vaccination coverage than urban areas.^{4, 58}

We conducted sensitivity analysis on the rate of disease transmission, the awareness of serostatus, the discount rate, length of infection, and threshold cost-effectiveness ratio. If the rates of disease transmission are significantly lower, the catch-up vaccination program becomes less valuable and would not be continued as long, but the different infection risk estimates would still lead to similar conclusions (

Catch-up vaccination can be a cost-effective (or even cost-saving) health intervention, but catchup vaccination programs can become less cost-effective as newborn vaccination rates increase. Our models can be used to determine the optimal fraction of each age group to vaccinate, and when to stop such catch-up vaccination. Such information can help decision makers make the best use of limited health care resources now, and can assist with future public health planning.

We have shown that simple analyses, which ignore changes in future disease incidence caused by catch-up vaccination, can provide good solutions to the catch-up vaccination problem. This is particularly true for diseases such as hepatitis B; that is, incurable diseases with long infectious periods for which there is a stable infection reservoir in the population. Our simple model of when to stop catch-up vaccination (P3 with the approximate subproblem P2 and a simple estimate of constant future disease incidence) could be readily used by decision makers, as such a model requires significantly less data than a full dynamic model and can easily be implemented in a spreadsheet. The simple model requires estimates of vaccination cost as a function of how many people are vaccinated (_{a}_{a}_{a,R}_{a,S}_{a,R}_{a,S}_{a}_{a}

These results complement prior research examining vaccination programs using static and dynamic models of infection. Edmunds et al.^{38} and Brisson and Edmunds^{39} note that constant-force-of-infection models may be appropriate if mass immunization does not substantially alter herd immunity. We have found this to be the case for catch-up vaccination for a disease with a large, stable infection reservoir.

We illustrated our ideas using the example of hepatitis B catch-up vaccination in China. We showed that, even with 90% newborn vaccination coverage, it is still cost-effective to provide catch-up vaccination to preschool age children for decades into the future, particularly if the catch-up vaccination programs cannot reach all susceptible children. If only selected groups can be reached by catch-up vaccination, or if newborn vaccination coverage is lower than 90%, it is cost-effective to perform catchup vaccination even longer.

Our analysis has several limitations. Our dynamic model of infection (which we calibrated to observed hepatitis B incidence in the Chinese population for our example) assumes homogenous mixing. Since this is unlikely to be the case, the quantitative results should be interpreted with caution if used to inform decision making. For some diseases such as hepatitis B, there is little hard evidence on how population mixing affects disease spread other than in limited instances such as transmission from mother to child. However, it may be possible that a population exhibits preferential mixing patterns where certain age groups have higher infectious contact with certain other age groups: for example, young adults could preferentially transmit an infection through sexual contact with other young adults, or young children could preferentially transmit a bloodborne infection to other children through childhood cuts and scrapes. It is straightforward to modify our dynamic model given by (^{69, 70} waning immunity may be important to incorporate for other diseases. Waning immunity would lead to a longer continuation of catch-up vaccination efforts. Future analyses could incorporate waning immunity into the model.

Many countries also face a high burden of disease from hepatitis B. For example, Vietnam has chronic hepatitis B prevalence and death rates that are approximately twice as high as in China.^{62, 71} Our models could be used to determine cost-effective catch-up vaccination levels for these settings. Our methods may also be applicable to other long-term chronic infectious diseases. For example, human papillomavirus (HPV) is an incurable, vaccine-preventable infection that may have a long infectious period. Thus, as for hepatitis B, a catch-up vaccination program is unlikely to have a major impact on the number of people currently infected, which means that it is unlikely to have a major impact on the infection risk. However, because the major mode of transmission of HPV is sexual, a dynamic transmission model with preferential mixing patterns might be more accurate than the model given by (

Our sensitivity analyses suggest that static models may be acceptable for evaluating catch-up vaccination programs that are supplemental and unlikely to have a large impact on the course of the epidemic. For an epidemic that is not in a steady state or when the catch-up vaccination program is likely to have an appreciable impact on the overall course of the epidemic, then a dynamic model may be needed to provide more accurate results.

No vaccines currently exist for human immunodeficiency virus (HIV) and hepatitis C virus although a number of trials for potential vaccines against these diseases are currently underway.^{72, 73} If and when vaccines are developed for these diseases, the dissemination of these vaccines would likely follow a path similar to that of the hepatitis B vaccine: first to high-risk groups, then to young age groups and, finally, catch-up vaccination. If the disease spread is generalized (not concentrated in certain risk groups), then an analysis such as ours might be helpful once vaccine production levels are sufficient to allow for catch-up vaccination.

Newborn vaccination rates are insufficient to protect children from many diseases, making catchup vaccination an important and cost-effective health intervention. We have shown that simple models may be sufficient to give policymakers insight into the appropriate levels of catch-up vaccination and guidance as to how long catch-up vaccination should be continued. Such models can be especially helpful for diseases such as hepatitis B that are endemic and have a large infection reservoir.

David Hutton was supported by a Stanford Graduate Fellowship, Grant Number R18PS000830 from the US Centers for Disease Control and Prevention, and Grant Number R01-DA15612 from the National Institute on Drug Abuse. Margaret Brandeau was supported by Grant Number R01-DA15612 from the National Institute on Drug Abuse.

We characterize the optimal level of vaccination
_{α}_{α}_{α}_{α}_{α}

The optimal vaccination level may be zero,
_{α}_{α}

Given the above expression for the optimal vaccination level, it is straightforward to establish the following.

The optimal vaccination level obtained by solving P2 is:

nonincreasing in the newborn vaccination level _{0}(·) (i.e., the level of prior immunity);

nonincreasing in cohort age

nondecreasing in the disease incidence _{a}

nonincreasing in vaccination cost _{α}_{α}

As the newborn vaccination level _{0}((·) increases, the number of individuals who are immune but unaware of their immunity _{a}_{R}_{,}_{a}_{a,R}_{a,S}_{a,R}_{a,S}_{a,R}_{a,S}_{a,R}_{a,S}_{α}_{α}

The model of hepatitis B used in our example analysis includes additional health states for the “infected” health state, as shown in

To estimate incidence of acute hepatitis B infection in unprotected individuals, we evaluated how prevalence has evolved over time in the population of children in China. We first calculated what we would expect prevalence of chronic disease to be at birth given expected chronic disease prevalence of 5% at birth without vaccination^{85} and using observed birth-dose vaccination coverage and vaccine efficacy. We then used this along with information about the likelihood of acute infections becoming chronic to estimate the number of chronic infections and probability of immunity to hepatitis B in early childhood (ages 1–4 and 5–14). By varying the incidence of acute infection, we could see what incidence level would most closely match the observed prevalence of chronic infection and immunity. Knowing the incidence of acute infection and current prevalence in the entire population of China (7.4%) enabled us to calculate the rate of contact that is sufficient to transmit the infection,

Diagram of model of hepatitis B infection and progression*. This model is used in both P1 and P2. It is used to calculate the dynamic health effects and costs in P1 and used to calculate the long-term health effects and costs, _{a,d}_{a,d}

* ALT = alanine aminotransferase. Circles represent health states. Lines represent transitions between those states. Although not shown, individuals with decompensated cirrhosis can also have other complications such as variceal bleeding, ascites, or encephalopathy. Additionally, the model is age-structured, so each disease state is indexed by age

Projected incidence starting in a steady state epidemic (Appendix Figure 2a) and a declining epidemic (Appendix Figure 2b), under the status quo and in the presence of catch-up vaccination. The “northwest-southeast” diagonal lines represent the benefit projected using a static model. The “northeast-southwest” diagonal lines represent the benefit projected using the dynamic model.

Example of the form of the benefits, costs and net monetary benefit of vaccinating a fraction _{a}

Hepatitis B example: Sensitivity analysis of net monetary benefit for a single susceptible individual as a function of rate of recovery from chronic hepatitis B infection.

Hepatitis B example: Results for a lower transmission scenario (β at 1/10 of its initial value and the risk of mother-to-child transmission halved).

Hepatitis B example: Sensitivity to changes in awareness of hepatitis B serostatus.

Hepatitis B example: Sensitivity to changes in discount rate.

Hepatitis B example: Impact of rate of recovery from chronic hepatitis B infection on length of time to continue a catch-up vaccination program for all children ages 1–19.

Hepatitis B example: Impact of willingness-to-pay threshold on the length of time to continue a vaccination program for both 5-year-olds and 12-year-olds.

Hepatitis B model parameters.

Parameter | Value | Source |
---|---|---|

Compliance with vaccine intervention | 70% | Assumed |

Percent chronically infected who are aware of infection | 50% | ^{83}, ^{84} |

Percent aware who receive medical management | 50% | Assumed |

Chronic infections that have elevated ALT | 2.0% | Assumed |

Already immune (given no chronic infection) | 50% | Assumed |

Aware of immunity (previous vaccination) | 75% | Assumed |

Protected by three doses of HBV vaccine | 95% | ^{81, 82, 86} |

Annual voluntary vaccination | 0.5% | Assumed |

Annual acute HBV infection incidence | 1000/100,000 | ^{56, 87–91} |

Asymptomatic acute infections | 90% | ^{80, 92–95} |

Symptomatic acute infections that require hospitalization | 12% | ^{80, 92, 93} |

Hospitalized cases that are fulminant | 4% | ^{80, 92, 93} |

Fulminant cases that result in death | 70% | ^{80, 92, 93} |

Normal ALT to elevated ALT | 0.15% | ^{96} |

Normal ALT to HCC | 0.34% | ^{98} |

Durable virologic response while on treatment | 15% | ^{79, 99–102} |

Chronic HBV infection with elevated ALT to compensated cirrhosis | 3.8% | ^{76, 79} |

Chronic HBV infection with elevated ALT to HCC | 1.5% | ^{76, 79} |

Durable response relapse to elevated ALT | 7% | ^{79, 103, 104} |

Durable response to HCC | 0.34% | ^{98} |

Compensated cirrhosis to decompensated cirrhosis | 7% | ^{76, 79} |

Mortality from compensated cirrhosis | 4.8% | ^{76, 79} |

Mortality from decompensated cirrhosis | 17.3% | ^{76, 79} |

Cirrhosis to HCC | 3.3% | ^{76, 79, 105} |

Cirrhosis to cirrhosis with ascites | 68% | ^{79} |

Cirrhosis to cirrhosis with variceal bleeding | 14.6% | ^{79} |

Cirrhosis to cirrhosis with encephalopathy | 10% | ^{79} |

Receiving a liver transplant while in decompensated cirrhosis | 1.5 % | ^{76, 79, 106–108} |

Mortality from HCC | 40.0% | ^{76, 79, 109, 110} |

Mortality from HCC while on medical management (due to early detection) | 20% | ^{110} |

Receiving a liver transplant while in HCC | 0.1% | ^{76, 79, 108, 111–115} |

Mortality first year after liver transplantation | 15% | ^{76, 79} |

Mortality second and subsequent years after liver transplantation | 1.5% | ^{76, 79} |

Vaccine (per dose) | 0.34 | ^{15, 74, 1163} |

Vaccine administration (per dose) | 0.60 | ^{116} |

30,000 | ^{76, 77} | |

Fraction of patients on drug therapy while in durable response4 | 50% | Assumed |

Drugs | 2000 | ^{76, 78} |

Regular health monitoring | 250 | ^{75, 77} |

Cirrhosis | 2000 | ^{75–77} |

Ascites | 2500 | ^{75–77} |

Encephalopathy | 2500 | ^{75–77} |

Variceal hemorrhage | 2500 | ^{75–77} |

HCC | 5000 | ^{75–77} |

Transplantation followup | 3000 | ^{76} |

Annual normal health care costs | 118 | ^{117} |

Discount rate | 3% | ^{118} |

Acute HBV infection | 0.94 | ^{80} |

Chronic HBV infection, normal ALT | 1.00 | ^{80, 99} |

Chronic HBV infection, elevated ALT | 0.99 | ^{76, 80, 99} |

Durable response | 1.00 | ^{79} |

Compensated cirrhosis | 0.80 | ^{76, 79, 99} |

Decompensated cirrhosis | 0.60 | ^{76, 79, 99} |

HCC | 0.73 | ^{76, 79, 99} |

Liver transplant | 0.86 | ^{76, 79, 99} |

HBV – hepatitis B virus; ALT – alanine aminotransferase

Percentage of all chronically infected individuals who are aware of their infection

This value was estimated from ^{96}, and then calibrated to yield approximately 25% mortality from untreated liver disease ^{86, 97}.

Personal communication with physicians in China

Some therapies are discontinued if the therapy suppresses the virus. This parameter is the fraction of patients who continue on drug therapy after the therapy has suppressed the virus into a “durable response.”

Schematic of age-structured SIR model, showing transitions that occur from time period

Hepatitis B example: Effect of different incidence estimates on the net monetary benefit of vaccinating a single susceptible individual, the first two terms in the objective function of P2 (which excludes secondary infections). The top sets of lines are for willingness-to-pay values of $13500 per QALY, the second set of lines are for willingness-to-pay values of $4500 per QALY, and the bottom set of lines are for willingness-to-pay values of $0 per QALY.

Hepatitis B example: Optimal fraction of each age group to vaccinate, obtained by solving P1 (which includes secondary infections) and P2 (which excludes secondary infections) using the assumption of constant incidence and assuming

Hepatitis B example: Policy space of cost-effectiveness for various ages and levels of prior vaccination coverage for different incidence estimates, obtained by solving P2 (which excludes secondary infections) with

Hepatitis B example: Number of years for which it is cost-effective to perform catch-up vaccination for hepatitis B in children ages 5 and 12, obtained by solving P3, and assuming

Hepatitis B example: Number of years for which it is cost-effective to perform catch-up vaccination for hepatitis B in children age 19 and younger, obtained by solving P3, and assuming

Model notation.

Variable | Description |
---|---|

Time index; | |

Index for age groups; | |

_{a} | Fraction of eligible individuals of age _{0}( |

_{a} | Number of susceptible individuals in age group _{a}_{a,}_{0}, |

_{a} | Number of infected individuals in age group _{a}_{a,}_{0}, |

_{a} | Number of immune individuals in age group _{a}_{a,}_{0}, |

_{a} | Total number of individuals in age group _{a}_{a}_{a}_{a} |

Total number of individuals in the population in time period | |

_{a} | Disease sufficient contact rate for individuals of age |

_{a} | Risk of infection to a susceptible individual of age _{a} |

_{a} | Fertility rate for individuals in age group |

_{0} | Chance that an unvaccinated child is infected perinatally and develops chronic infection |

_{a} | Chance that an acute infection in a person of age |

Rate of recovery from the chronic infection, | |

_{a}_{,}_{d} | Fraction of individuals of age |

Effectiveness of the vaccine at inducing immunity in a susceptible individual | |

_{I}_{,}_{a} | Fraction of infected individuals in age group |

_{R}_{,}_{a} | Fraction of immune individuals in age group |

_{ad} | Health care cost per unit time for a person of age |

_{ad} | Quality-of-life multiplier for a person of age |

_{a,d} | Expected net present health care costs for a person of age |

_{a,d} | Expected net present health effects measured in quality-adjusted life years (QALYs) for a person of age |

Fixed cost of the catch-up vaccination program; | |

_{v} | Baseline marginal cost to vaccinate one person |

_{a}_{a} | Cost above baseline per person vaccinated, as a function of the fraction _{a}_{a} |

_{a}_{a} | Total cost of vaccinating a fraction _{a} |

Monetary value of health effects | |

Discount rate for costs and health benefits |

Key parameter values for numerical example of hepatitis B in China.

Parameter | Base Value | Source |
---|---|---|

Newborn vaccination coverage, _{0} ( | 90% | ^{4, 58} |

Maximum allowable catch-up vaccination level,
| 1 | Assumed |

Fixed cost of vaccination program, | $0 | Assumed |

Baseline marginal cost of vaccination, _{v} | $2.82 | ^{10, 74} |

Cost of vaccination above baseline, _{a}_{a} | 0 for _{a}_{v}_{a}_{a} | ^{50–53} |

Monetary value of a year of full health, | $4500/QALY | Based on per capita GDP^{66–68} |

Population size, | 1.3 billion | ^{68} |

Health care cost per year, _{a}_{,}_{d} | $118–$5,000 | ^{10, 75–78} |

Quality multiplier for health states, _{a}_{,}_{d} | 0.6–1.0 | ^{10, 33, 76, 79, 80} |

Vaccine effectiveness, | 0.95 | ^{10, 44, 81, 82} |

Fraction of infected who are unaware of their infection, _{I}_{,}
_{a} | 50% | ^{10, 83, 84} |

Fraction of immune who are unaware of their immunity, _{R}_{,}_{a} | 25% | ^{10} |

Sufficient contact rate, _{a} | 0.135 | Based on calculations from ^{10, 58} |

Discount rate, | 3% | ^{18} |

Costs vary by disease state (see