In the context of cervical cancer prevention, several models have been developed, each associated with strengths and limitations, and each equipped with features suited to address different policy questions. In general, the choice of model type is a function of several considerations including: 1) the specific question(s) being addressed; 2) the features of natural history and disease that are important to capture for the specific problem being analyzed; 3) the data available to parameterize, calibrate and validate the model; 4) the familiarity of the analyst with different modeling techniques; 5) the time requirements for model development; and 6) the ease and speed of running the model and conducting the analysis [14
There are modeling challenges that are of particular relevance to HPV-16/18 vaccination [11
]. First, as with many vaccines, there are potentially complex epidemiological consequences at the population level due to herd immunity effects. Second, there is a long lag time between the intervention and the ultimate health benefits, as an HPV-16/18 vaccine delivered in adolescence is intended to prevent cancers in adulthood several decades later. Third, current vaccines prevent only two high-risk HPV types, introducing challenges related to the modeling of potential type interactions. Fourth, cervical cancer prevention strategies that are currently available in most developed countries have different mechanisms of action, considerably complicating the amount of detail required in a model that includes them all. For example, cytology is aimed at detecting abnormal cervical cells resulting from HPV infection with both low- and high-risk types; current HPV DNA tests used for screening detect the presence of any of 13 high-risk types; the bivalent vaccine targets high-risk types HPV-16 and 18; and the quadrivalent vaccine also targets two low-risk types, HPV-6 and 11. Because it is very difficult to capture all of these complexities to the fullest degree possible, as a general rule, all models involve tradeoffs.
depicts a general model of cervical carcinogenesis. The natural history of disease in an individual woman is represented as a sequence of transitions between mutually-exclusive health states. Health states in the model, descriptive of the underlying true health, are generally defined to include HPV infection status, grade of cervical intraepithelial neoplasia (CIN) and stage of cancer. Health states are stratified to reflect different levels of detail, depending on the nature of the question, type of model and data availability. For example, health states reflecting HPV infection may be stratified by specific individual HPV types (e.g., type -16, type -18), or by categories of HPV types (e.g., high-risk types, low-risk types), while those reflecting CIN may be stratified by specific grade (e.g., CIN 1, CIN 2,3). Health states may also be stratified according to dimensions important to the analysis (e.g., detected and undetected cancer) or according to classification systems for which there are data (e.g., stage 1-4 cancer, or local, regional and distant cancer).
Model schematic of cervical cancer natural history
As models evolve in complexity, the requirement for parameter values quickly multiplies and input values are rarely available for all parameters. Calibration techniques aimed at fitting uncertain model parameters to observed, epidemiologic data (e.g., age-related prevalence of type-specific HPV and CIN, age-related incidence of invasive cancer, and the distribution of HPV types within CIN and cancer) are increasingly being used [15
]. Models utilize a wide range of approaches to estimate uncertain or unknown parameters and evaluate uncertainty and variability [23
Models can be generally classified along several dimensions depending on whether they possess generic attributes, such as 1) whether populations within the model can interact or not (dynamic versus static models); 2) whether populations are allowed to enter the model or not (open versus closed models); 3) whether transition rates are fixed (deterministic) or subject to chance (stochastic); and 4) whether the population's behavior in a model is simulated using values reflecting population averages (aggregate) or at a micro level where the behaviors of individuals in the population are tracked (individual-based) (). Models used to assess type-specific HPV vaccination can be further characterized by whether they reflect one or more HPV types, include other HPV-related non-cervical outcomes, and according to the degree to which they can represent screening, diagnostic and treatment protocols.
Attributes of mathematical models used to evaluate HPV vaccinationa
2.1 Static versus dynamic
In a static model of cervical cancer, the force of HPV infection may change as a function of age or other individual-based factors, but is constant over time, and explicit interactions between individuals are not modeled; for example, HPV incidence may be parameterized as an age-specific probability of infection that is calibrated to observed epidemiologic patterns. In contrast, in a dynamic model, the probability of an individual acquiring an HPV infection is parameterized as a function of three inputs: 1) the sexual contact patterns of that individual with others, 2) the transmissibility (i.e., infectiousness) of the HPV type; and 3) the type-specific prevalence of infection within the population. While static models may be acceptable to comparatively assess screening strategies, vaccination in such models protects a proportion of the population from infection, but those who are not vaccinated do not receive any protection or benefit. In dynamic transmission models, because of the explicit interactions between individuals in a population, vaccination is likely to change the risk of infection for unvaccinated individuals in the community; therefore, dynamic models are required to address questions which involve widescale vaccination of girls and boys.
To date, all dynamic models of HPV infection have involved only heterosexual transmission. Most have stratified the population into different levels of sexual activity by age, with corresponding numbers of sexual partnerships, and then modeled transmission as a probability per partnership. Sexual mixing can occur between people in different age groups, as well as in different sexual activity levels. Model inputs directly inform how sexual partnerships form between females and males, during which HPV can be transmitted between partners. Barnabas RV et al.
], Kim JJ et al.
] and Choi YH et al.
] used behavioral survey data to inform initial model inputs (e.g., stratification of the population
into low, medium, or high sexual activity levels and corresponding number of new partners per year, by age) and then calibrated the transmission probabilities per partnership to match population-based estimates of HPV prevalence.
2.2 Open versus closed
An open model allows individuals to enter and exit the model over time, while a closed model does not allow for new entrances over time. The most common closed model used for cervical cancer prevention is a single birth cohort simulation using a Markov model. The most common open models are dynamic transmission models that allow for the entry of “susceptible” (or uninfected) individuals into the model (e.g., births over time) replenishing the susceptible compartment. A microsimulation model that does not allow for interactions between individuals (i.e., is not dynamic) can also be open by simulating multiple birth cohorts over time. Open models are most appropriate for addressing questions that involve modeling temporal trends, and for interventions targeted to different age groups within a population and that may vary by calendar year (e.g., comparative ages to target catch-up vaccination).
2.3 Aggregate (or population-average) versus individual-based
In an aggregate model, individuals are assigned to health states, and movement between them depends on health status or other relevant variables. Individuals in each health state move according to parameter values at the aggregate level (i.e., averages of the individuals belonging to a compartment or the population as a whole), and the model records the number of individuals in each health state over time. Many characteristics could be modeled in aggregate but increasing their number makes population average models more complex and unwieldy. Most dynamic transmission models that have been published for HPV-related cervical cancer are of this type.
In contrast, an individual-based model (or microsimulation model) keeps track (memory) of each individual's behavior, allowing for heterogeneity in behavior to be adequately explored. The ability to keep track of the individual life experiences of women (and indeed men), which may impact future risk of infection or future screening schedule, is a great advantage when behaviors are very diverse (as is the case for sexual mixing) or when the intervention is tailored to individuals (as is the case for screening). For example, if a woman has had treatment for CIN, she may be followed more frequently than the average woman in her birth cohort. An example related to vaccination is in modeling an individual woman's likelihood of screening patterns given her particular vaccination status.
2.4 Deterministic versus stochastic
While in a deterministic model, all events occur according to fixed parameter values, a stochastic model allows for events to occur by chance (randomly). Most dynamic models of HPV transmission have been deterministic; since microsimulation randomly samples individuals with their own sets of assigned attributes, microsimulation models are naturally stochastic. Even with fixed parameter values in a microsimulation model, the individual realization of each transition may differ from person to person due to chance. It is important to note that the variance associated with individual sampling (first-order uncertainty) in microsimulation is different from the uncertainty related to the parameter values (second-order uncertainty). In contrast, variability refers to the often “known” heterogeneity across subgroups or in a population (e.g., age or sex).
Different types of models can be coupled to leverage different model attributes for a given analysis. For example, dynamic models of HPV (which represent transmission and herd immunity benefits of vaccination of vaccine-targeted types) have been coupled with static models (which incorporate other HPV types and detailed screening strategies) [25
]. One advantage of using multiple models is that projected results can be compared using independently structured models, which can greatly enhance evaluating the impact of model structure on cost-effectiveness results. However, this approach usually involves making a number of approximations and requires the influence of one model's results on the other to be handled carefully.
2.5. Natural history data required to develop a mathematical model of HPV and cervical carcinogenesis
Because of the epidemiological variation in age-related HPV, cervical cancer incidence and proportion of cancer attributable to HPV-16 and -18, primary prevention strategies may have differential impact in different settings. These differences will be most pronounced in countries without effective screening [28
]. Therefore, irrespective of the type of model chosen, it is important to tailor the models to each particular setting using country-specific data, where possible. To simulate a particular country's burden of cervical cancer and prevention policies, a substantial amount of data is required to inform parameter inputs, either directly or, in the absence of empirical data, through model fitting or calibration. For example, estimates of age- and type-specific HPV incidence from longitudinal studies can be used directly as input into static models, whereas in a dynamic model, direct inputs for HPV incidence would require data on sexual behavior, such as number of new sexual partners, and transmission probability of HPV infection.
HPV prevalence, on the other hand, is a function of a number of transitions occurring simultaneously in the model, such as HPV incidence, clearance and progression, and therefore, cannot be input directly. In this situation, empirical data on HPV prevalence in the population can be used as a calibration target, which is used to infer other uncertain parameters in the model such that good model fit to the empirical data can be achieved. A wide array of model calibration techniques have been employed [15
] and range from fitting a limited number of model parameters and comparing model results to data visually or by using more statistically rigorous techniques, to fitting multiple model parameters to data and retaining those combinations of input parameter values whose results are within certain target ranges for key outputs [16
]. Calibration target data in these analyses have included HPV-16 seroprevalence, age-specific prevalence of HPV and CIN, incidence of cervical cancer (by type and overall), and HPV type-distribution among CIN and cervical cancer cases. The process of calibration enables the analyst to identify multiple good-fitting parameter values, or sets of parameter values; when the analyses are repeated using multiple good-fitting values, they can serve as uncertainty analyses over the natural history parameters.
Another important piece of information in evaluating the long-term prospects of the HPV vaccine in a particular setting is the burden of the vaccine-targeted HPV types in cervical lesions and invasive cancer (including adenocarcinomas), and in the case of the quadrivalent vaccine, in genital warts. Estimates of type-specific HPV prevalence in different countries show regional variation and have been reported and discussed extensively in previous publications [4
Although vaccine efficacy against vulvar and vaginal lesions have been reported in clinical trial data [30
] the impact of vaccines on other HPV-16/18 related diseases and conditions (e.g., penile cancer, anal cancers, oral cancers) has not yet been observed and is less certain. While HPV-16/18 are estimated to contribute to roughly 32% vulvar/vaginal cancer, 25% penile cancer, 83% anal cancers, and 3-11% oral cancers [31
], there are also less natural history data on these diseases than in the case of cervical cancer. Similarly, while infections with HPV-6/11 are associated with recurrent respiratory papillomatosis, the ultimate impact of the quadrivalent vaccine on this condition is not yet known. Although the burden of these conditions varies by setting, to the extent that the vaccines will avert additional morbidity and mortality due to these diseases (and their associated costs), the cost-effectiveness of vaccination strategies could improve. Better data on the natural history and burden of these other HPV-related conditions and interventions (and ideally, direct reporting of these outcomes in clinical trials) will be useful for revising current models to incorporate the potential benefits of the vaccine on non-cervical cancer outcomes.
While model parameterization and calibration are necessary steps required for model development, good modeling practice also requires that models undergo validation exercises to assess that model predictions are consistent with observed data that were not used to inform input parameters [33
]. Most models report face validity against recent estimates of cancer incidence and mortality in the presence of screening, such as from the Surveillance Epidemiology and End Results- (SEER) cancer registry [16
2.6 Model outcomes
2.6.1 Health benefits
The most common outcomes generated by models to express the health benefits associated with vaccination are life expectancy, quality-adjusted life expectancy, reductions in lifetime risk of cervical cancer and reductions in prevalence of HPV infection and CIN over time. In developing countries often disability-adjusted life years (DALYs) averted are reported instead of quality-adjusted life years (QALYs) [28
]. Analyses evaluating the quadrivalent vaccine also reported estimates of reductions in genital warts [25
], and one analysis also reported potential vaccine benefits related to other HPV-16/18 associated cancers and HPV-6/11 associated juvenile onset recurrent respiratory papillomatosis (JORRP) [25
]. Two analyses reported the number needed to vaccinate to avert one cervical cancer case [35
For outcomes expressed in QALYs, utility weights are assigned to different health states to reflect decrements in quality of life associated with residing in a particular state. In analyses that reported QALYs, utility weights associated with different stages of cancer (applied multiplicatively to baseline, age-specific utility weights) were most common. Some analyses also included temporary disutility associated with a positive screening test result, diagnosis of or treatment for precancerous lesions [26
], and development of genital warts [25
2.6.2 Costs and cost-effectiveness
All analyses that reported costs included direct medical costs of the interventions. Direct non-medical costs, such as cost of transportation, and patient time costs were included only in Goldie SJ et al.
], Kim JJ et al.
], and Goldhaber-Fiebert JD et al.
]. Vaccine costs (assuming three doses) ranged from $200 to $500, which included different cost components in the different analyses; all else equal, analyses assuming lower vaccination costs would make vaccination strategies more attractive. Roughly half of the studies explicitly included costs associated with administration of the vaccine program [25
]. In principle, analyses should include the same component costs for both vaccination and screening interventions.
Using the models, total lifetime costs and health benefits are estimated for each strategy. Comparisons across strategies are expressed using the incremental cost-effectiveness ratio (ICER), calculated as the incremental costs divided by the incremental benefits of a strategy compared to the next less costly strategy. Strategies that are more costly and less effective than another strategy are considered “strongly dominated”, and strategies that are less cost-effective (i.e., have higher cost-effectiveness ratios) than a more costly alternative are considered “weakly dominated”; both types of dominated strategies are excluded from the final calculations of the cost-effectiveness analysis. Consistent with published guidelines on cost-effectiveness analysis in health and medicine [46
], life expectancy (unadjusted and adjusted for quality) and lifetime costs were discounted at rates ranging from 1.5% to 5% per year; Bergeron C et al.
] discounted costs and benefits differentially at 3.5% and 1.5% per year, respectively. At higher discounting rates, cervical cancer screening strategies tend to be favored compared to vaccination since vaccination costs accrue immediately, yet benefits accrue over a longer time horizon.
2.6.3 Interpretation of results
The cost-effectiveness of a strategy measures its value for money, or how much benefit a strategy provides for each additional dollar, compared to other strategies. There is no consensus on the appropriate threshold for cost-effectiveness, and indeed, different countries have adopted different thresholds. Some countries use specific thresholds as guides for resource allocation decisions; for example, the UK cites a threshold of (x020A4)30,000 per QALY gained [50
]. Other countries, such as the USA rely on an implicit threshold, such as $50,000 or $100,000 per QALY gained [51
], using it as a benchmark to denote efficiency, but rarely as sole justification for decision-making around health coverage. The Commission on Macroeconomics in Health and Medicine has suggested that interventions with cost-effectiveness ratios less than the country's per-capita Gross Domestic Product (GDP) should be considered very cost-effective
It is important to note that an analysis of cost-effectiveness is different from an analysis evaluating affordability. While a cost-effectiveness analysis measures efficient use of resources (value for money), an assessment of affordability seeks to make the impact on a specific budget transparent and to estimate the real-time financial costs to implement and sustain a program. While information on affordability, in conjunction with cost-effectiveness, is often requested by decision makers in developing countries, where budget limitations are often profound, this information is increasingly being requested by a range of health care payors in higher-income countries.