WJE’s partner works for GSK. This does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.
Conceived and designed the experiments: HCJ WJE. Performed the experiments: HCJ. Analyzed the data: HCJ KME. Wrote the paper: HCJ.
Quantifying rates governing the clearance of Human Papillomavirus (HPV) and its progression to clinical disease, together with viral transmissibility and the duration of naturallyacquired immunity, is essential in estimating the impact of vaccination programmes and screening or testing regimes. However, the complex natural history of HPV makes this difficult. We infer the viral transmissibility, rate of waning natural immunity and rates of progression and clearance of infection of 13 highrisk and 2 nononcogenic HPV types, making use of a number of rich datasets from Sweden. Estimates of viral transmissibility, clearance of initial infection and waning immunity were derived in a Bayesian framework by fitting a susceptibleinfectiousrecoveredsusceptible (SIRS) transmission model to age and typespecific HPV prevalence data from both a crosssectional study and a randomised controlled trial (RCT) of primary HPV screening. The models fitted well, but overestimated the prevalence of four highrisk types with respect to the data. Three of these types (HPV33, 35 and 58) are among the most closely related phylogenetically to the most prevalent HPV16. The fourth (HPV45) is the most closely related to HPV18; the second most prevalent type. We suggest that this may be an indicator of crossimmunity. Rates of progression and clearance of clinical lesions were additionally estimated from longitudinal data gathered as part of the same RCT. Our estimates of progression and clearance rates are consistent with the findings of survival analysis studies and we extend the literature by estimating progression and clearance rates for non16 and non18 highrisk types. We anticipate that such typespecific estimates will be useful in the parameterisation of further models and in developing our understanding of HPV natural history.
Persistent infection with highrisk human papillomavirus (HPV) has been shown to be the necessary precursor of progression to cervical cancer
We have aimed to infer the viral transmissibility, rate of waning natural immunity and rates of progression and clearance of infection of 13 highrisk HPV types and for HPV6 and 11, making use of two rich epidemiological datasets and a comprehensive survey of sexual activity from Sweden. Estimates of viral transmissibility, clearance of initial infection and waning immunity were derived in a Bayesian framework by fitting a susceptibleinfectiousrecoveredsusceptible (SIRS) transmission model to age and typespecific HPV prevalence data from both a crosssectional study and a randomised controlled trial (RCT) of primary HPV screening. Rates of progression and clearance of clinical lesions were estimated from longitudinal data gathered as part of the same RCT. Where possible we compare our estimates of these parameters with those derived from similar studies.
We developed a deterministic model of heterosexual partnership formation, monotype HPV transmission and the potential progression to cervical cancer for each of 13 highrisk HPV types (16, 18, 31, 33, 35, 39, 45, 51, 52, 56, 58, 59 and 66) and for HPV6 and 11. We model an age and sexual activity structured population with two sexual activity groups for each of three age brackets (14–29; 30–60 and 60+) where the size of the high sexual activity group is a fixed proportion (15%) of the overall population, regardless of age. Contact rates for each of these six age/sexual activity groups were derived from the Swedish sexual activity data gathered as part of the
We assumed an SIRS structure for HPV infection in both men and women, allowing for an SIR structure in the limit where the rate of waning natural immunity, estimated within this analysis, tended to zero. The infected (I) compartment for females was divided into five subcompartments, corresponding to the progression of HPV infection to disease: Initial infection, CIN1, CIN2, CIN3 and Cancer (where CIN stands for
Susceptible individuals become infected at a rate proportional to the force of infection λ. Following infection, they may progress in a stepwise fashion to neoplastic lesions of increasing severity (CIN1, CIN2, CIN3) and then to cancer (with rates ). Alternatively, they may spontaneously clear the infection from any precancerous stage (with rates ). Highgrade clinical lesions (CIN2 and CIN3) may be identified by cytological screening and successfully treated (at a rate π). Following viral clearance or successful treatment of lesions, women retain an immunity to reinfection with the same HPV type. This immunity wanes at rate κ, precipitating a return to the Susceptible compartment.
It was assumed that only the initial infection with HPV and lowergrade cervical disease (CIN1 and CIN2type lesions) are infectious. Also, the force of infection was calculated assuming that viral transmissibility
The state variables used in the model are summarised in the Supplementary Material.
and
The force of infection acting on males depends on the number of females in each of the three infectious
At demographic equilibrium,
Males  Females  
Age  Low sexual activity  High sexual activity  Low sexual activity  High sexual activity 
14–29  0.54  4.11  0.25  2.89 
30–59  0.05  0.64  0.07  1.47 
60+  0.01  0.21  0.02  0.11 
Mixing in the model is determined on a proportionate random basis i.e. without accounting for either assortativity on the basis of either age or sexual activity group. In order to balance the number of partnerships between males and females we assume that females adjust their behaviour completely to the number of partnerships sought by males. This leads to the simplistic conclusion that the overall rate of partnerships formed between females of age
Cervical screening was incorporated into the model according to the Swedish screening protocol. In Sweden, women are screened at 3yearly intervals between the ages of 23 and 50 and at 5yearly intervals between 51 and 60. Uptake is 79% (J. Dillner, personal communication) and we made the limiting assumption that this rate is independent of sexual behaviour. Treatment is provided for screened women with CIN2+ lesions and cancer, resulting in the clearance of infection; the combined rate for screening and subsequent successful treatment is π.
We assumed that the rates of waning immunity and clearance of initial infection are independent of age, gender, sexual activity and personal history of HPV infection.
Sexual behaviour in the model (e.g. partnership formation rates) was parameterised using data from a comprehensive Swedish sexual activity survey (n = 2810). Complete details of the survey have been published in ‘


Within normal limits  Within normal limits 
Koilocytosis  
CIN1  ASCUS/LSIL 
CIN2/3  HSIL 
As described above, the rate of partner acquisition,
Given a certain annual rate of partner acquisition
We consider males and females, and each age group, separately. Defining
Since we have data on the number of new partners per year, stratified by age
In the parameterisation of this model, we considered two sexual activity groups, the high sexual activity group,
The proposal densities were standard Normal distributions with standard deviation dependent on the sexual activity group:
Since the higher sexual activity group has a smaller population, it is more difficult to fit the likelihood than for the larger low sexual activity group. For this reason we specified a prior function of exponential form:
The consequence of this was to penalise higher contact rates, essentially attempting to describe the data with rates as low as possible. The resultant estimated annual rates of new partner acquisition are shown in
The median predicted prevalence for each type is represented by the solid curve and the turquoise bands represent the 50%, 70%, 90% and 95% posterior intervals.
In order to estimate the perpartnership viral transmissibility
Swedescreen is a prospective randomised controlled trial of 12,527 women, aged 32–38, attending populationbased invitational cervical screening in Sweden. Women were allocated in a 1∶1 ratio to either receive an HPV test as well as a cytology test (intervention) or a cytology test alone (control). HPV positive women were invited for a second HPV test at least twelve months later and women with persistent infections were invited to colposcopy. Women were then followed with comprehensive registrybased followup. Further details of the study have been published elsewhere
ANNUAL RATE  16  18  31  33  OHR 
Progression of HPV to CIN1  0.026 (0.007, 0.045)  0.058 (0.000, 0.179)  0.039 (0.008, 0.073)  0.046 (0.000, 0.147)  0.020 (0.004, 0.040) 
Progression of CIN1 to CIN2  0.042 (0.010, 0.087)  0.111 (0.000, 0.255)  0.069 (0.000, 0.182)  0.182 (0.000, 0.406)  0.111 (0.048, 0.192) 
Progression of CIN2 to CIN3  0.124 (0.024, 0.235)  –  0.136 (0.024, 0.240)  0.105 (0.000, 0.357)  0.169 (0.072, 0.276) 
Progression of CIN3 to Cancer  0.026 (0.000, 0.080)  –  –  –  – 
Clearance of CIN1  1.468 (0.893, 2.043)  1.257 (0.286, 2.197)  1.136 (0.531, 1.672)  1.386 (0.365, 2.197)  1.212 (0.766, 1.645) 
Clearance of CIN2+  1.082 (0.710, 1.438)  0.884 (0.148, 1.695)  0.496 (0.191, 0.833)  0.788 (0.201, 1.299)  0.606 (0.393, 0.822) 
The data from the Swedish voluntary Chlamydia screening programme provided information on the prevalence of each of the 13 highrisk HPV types, by age in women aged between 14 and 50 (N = 32,693). Although attendance rates for Chlamydia screening in Sweden are high (J. Dillner, personal communication), we anticipate that the very young (<17) and older women (>35) who attend elective screening are more likely to be at higher risk. We accommodated this in the model by specifying that a higher proportion of the high than the low sexual activity groups attended Chlamydia screening.
Six typespecific HPV progression and clearance rates of CIN1+ lesions were estimated from longitudinal data gathered as part of a large prospective randomised controlled trial of HPV testing and screening in Sweden (Swedescreen, see
(A) The estimated rates of progression of infection with HPV16, HPV18 and other highrisk (OHR) HPV types to CIN1type lesions. We compare the rates estimated from the longitudinal Swedescreen data with the modelbased estimates of Jit et al.
These parameters were informed by a corresponding vector
where
The vertical line signifies the median of the median rates for all fifteen types; horizontal bars represent the 95% posterior intervals.
As outlined in Bogaards et al.
There is an art to deciding after what time lag
One limitation of the Swedescreen data was that each woman received only one HPV test as part of the Swedescreen trial – at baseline. As a consequence of this, it was necessary to assume that her history of infection and disease progression was attributable to the HPV type(s) that were identified at her baseline HPV test. Also, since it was not therefore possible to demonstrate clearance of infection, we assumed that a reversion of abnormal to normal cytology was an indicator of clearance of HPV.
Within the Swedescreen study, histological tests were only conducted if the results of a cytological test were positive. To paint a fuller picture of disease progression, we decided to include both cytological and histological data in our estimation of rates from longitudinal data. To do so, we considered equivalence between the grades shown in
(A) Comparison of the estimated annual rates of clearance of initial HPV16 infection in this and six further studies. The vertical line signifies the median of the median values of all seven studies. (B) Comparison of the estimated annual rates of clearance of initial HPV18 infection in this and five further studies. The vertical line signifies the median of the median values of all six studies. (C) Comparison of the estimated annual rates of clearance of initial HPV31 infection in four further studies. The vertical line signifies the median of the median values of all five studies.
In cases of a cancer diagnosis, we always relied on the definition from the histological test.
We assumed that development of disease was attributable to a hierarchy of HPV types: if a woman was infected with HPV16 this was considered the cause of any cytological abnormalities; if she was infected with HPV18 but not HPV16, HPV18 was considered the cause. To estimate rates for the other highrisk types (OHR) independently, we considered all women who tested positive for the type in question and negative for 16 and 18. This meant that we included in the estimation of OHR type progression and clearance, women who were coinfected with a number of OHR types. This was deemed necessary since many of the rarer types were only found in cases of coinfection with several OHR types.
HPV type  Median  95% posterior interval  
Lower  Upper  
6  0.755  0.295  1.000 
11  0.741  0.256  1.000 
16  0.718  0.286  1.000 
18  0.736  0.292  1.000 
31  0.743  0.292  1.000 
33  0.752  0.268  1.000 
35  0.746  0.264  1.000 
39  0.755  0.233  1.000 
45  0.739  0.279  1.000 
51  0.755  0.233  1.000 
52  0.745  0.317  1.000 
56  0.743  0.253  1.000 
58  0.758  0.311  1.000 
59  0.745  0.252  1.000 
66  0.757  0.275  1.000 
In order to increase the effective sample size of women progressing from one state to another, we considered the first occurrence of a certain histological state for each woman to be her ‘baseline’ test at that level. For example, to estimate the rate of progression from CIN2 to CIN3 (
Typespecific estimates of the perpartnership transmissibility
The likelihood function for the Chlamydia screening programme HPV prevalence data,
The function
where
Thus we are able to jointly consider the ‘success’ probability of being in the susceptible state, the initial infection state, the CIN1 state and so forth.
Each of the 13 highrisk HPV types was modelled separately. Estimates of the three natural history parameters were derived from posterior distributions of the Markov Chain Monte Carlo (MCMC) chains (chain length = 20,000) using the MetropolisHastings algorithm
In each iteration of the algorithm, a proposal for
For each type the algorithm was run for a chain of length n = 20,000. Visual inspection confirmed that the MetropolisHastings algorithm converged swiftly to the posterior distribution and parameter estimates were made rejecting a ‘burnin’ period of 100 iterations. Examples of the MCMC traces are shown in Figures S1 and S2 for HPV16 and HPV45 respectively. We report 95% posterior intervals for the posterior distributions of all MCMCestimated parameters. These are obtained from the 2.5^{th} and 97.5^{th} percentiles of all iterates after the burnin period.
The observed agedependent prevalence of highrisk HPV infection was accurately replicated by the appropriate typespecific model for each type (see
The median estimated rates of progression and clearance of CIN1type lesions in four highrisk HPV types, each with a 95% adjusted bootstrap confidence interval, are given in
The progression and clearance rates did not differ significantly between HPV types. However, HPV16 appeared to progress from initial infection to CIN1 type lesions at an annual rate of 0.026 (0.007–0.045), somewhat more slowly than HPV18 which had an annual rate of 0.058 (0–0.179) but at a similar rate to other highrisk types (OHR) which had an annual rate of 0.02 (0.004–0.04).
The rate of clearance of typespecific initial HPV infection was well characterised by the posterior distribution of the MCMC chain, with a median posterior/prior ratio of 0.320 across all HPV types. The median values for the rate of clearance of initial infection did not differ significantly between types (see
Although few estimates have been made of the annual clearance rates of some of the rarer highrisk HPV types, more cohort studies have concentrated on estimates for HPV16, 18 and the grouped other highrisk (OHR) types. In
The median estimated rates of transmissibility in the thirteen highrisk HPV types, each with a 95% adjusted bootstrap confidence interval, are given in
The estimated rates of waning natural immunity (or resistance to subsequent infection) are shown in
We have made estimates for the transmissibility, clearance and progression rates and rates of waning natural immunity for 13 highrisk types of HPV by using Swedish behavioural and epidemiological data to fit an ensemble of deterministic models of HPV transmission and progression. The datasets used included the ‘
It was observed that the models overestimated the observed prevalence of four of the highrisk types. We suggest that this is an artefact of modelling the HPV types independently. Three of these types (33, 35 and 58) are, in fact, three of the most closely phylogeneticallyrelated to HPV16; the most prevalent type. HPV45 is the most closely related to the second most prevalent type, HPV18. It is feasible that this similarity leads to crossimmunity between the types, effectively reducing the number of young women susceptible to infection with types 33, 35, 45 and 58.
We found variation in the rates of clearance of initial infection to be a very informative parameter in the fitting of typespecific models of HPV prevalence to data, indicating that this may be an important determinant in the differing prevalences between types. The lowest rates of clearance of initial infection were observed in HPV16, 18 and 31, the most prevalent of the highrisk HPV types. This would again support the theory that higher prevalence is associated with persistence of infection, though we would agree with Bogaards et al.
Our estimates of transmissibility did not differ significantly between types although there remained a large degree of uncertainty. Our inferred values were in line with those estimated in previous studies: the median perpartnership transmissibility of HPV16 was estimated to be 0.718 with a 95% posterior interval of 0.286–0.999) compared with 0.8 estimated by Hughes et al.
The rate of waning immunity was estimated to be very low; the 5year cumulative proportions of clearance varied between 6%–12% across the 13 types. This would imply that few women become susceptible to reinfection with any given HPV type and that the observed decrease in the force of infection in the early twenties is more likely to be due to immunity than to decreased sexual activity. Indeed, our estimates of the rate of waning immunity would imply that the protection conferred by infection with HPV is effectively close to lifelong.
We have also further contributed to the literature estimating the rates of progression and clearance of clinical lesions due to HPV, extending the field by deriving typespecific estimates of progression rates up to CIN3grade lesions for type 31 and 33 and clearance thereof. Our results implied that the rate of progression of other highrisk types from CIN1 to higher grade lesions was significantly lower than for HPV16 and HPV18, indicating that their less prominent role in the cause of cervical cancer may be due not solely to lower prevalence but also to an inherently lower oncogenicity. Our estimates are similar to others in the literature although they are subject to a large degree of uncertainty, highlighting the benefit that would be derived from pooling data on disease progression from
Although this analysis provides some useful new insight into HPVrelated parameters, it has some limitations. Firstly, the nature of deterministic compartmental models means that it is not possible to consider diversity in individuallevel behaviour. For example, we estimate here the perpartnership transmissibility of HPV infection. In order to derive from this a peract transmissibility requires us to assume an ‘average’ partnership between a susceptible and an infected partner. Also, we assume that all women have an equivalent approach to screening and subsequent treatment. Such an assumption belies the true underlying heterogeneity; in fact, it is those women who do not attend screening who are more likely to go on to develop highgrade clinical lesions and cervical cancer. In considering the more subtle questions policymakers face in terms of vaccination and screening programmes it is common for complex individualbased models to be used. However, the fitting of these models is similarly complex due to the long run times and inherent stochasticity of these models. The typespecific estimates we have made here for HPV transmissibility and rates of progression and clearance will facilitate the parameterisation of such individualbased models by, for example, providing prior ranges for natural history parameters. These models will then, in turn, allow a more thorough understanding of the complex natural history of HPV infection and its implications for the efficacy of vaccine and screening programmes in this new era of HPV prevention.
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We would like to thank Joakim Dillner, Pontus Naucler and Anna Söderlund Strand for providing Swedish epidemiological data and for their advice, Simopekka Vänskä for his advice on estimating contact rates and Mark Jit for helpful discussion.