The Kirby Institute, University of New South Wales, Sydney, NSW 2052, Australia

Regional World Health Organization Human Papillomavirus Laboratory Network, Department of Microbiology and Infectious Diseases, The Royal Women’s Hospital, 3052, Melbourne, VIC, Australia

Department of Obstetrics and Gynaecology, University of Melbourne, 3052, Melbourne, VIC, Australia

Murdoch Childrens Research Institute, 3052, Melbourne, VIC, Australia

Abstract

Background

Seroreactivity, processes of seroconversion and seroreversion, in the context of HPV infection has been investigated in numerous studies. However, the data resulting from these studies are usually not accounted for in mathematical transmission models of various HPV types due to gaps in our understanding of the nature of seroreactivity and its implications for HPV natural history.

Methods

In this study we selected a number of simple but plausible compartmental transmission models of HPV-16, differing in assumptions regarding the relation between seropositivity and immunity, and attempted to calibrate them to Australian HPV seroprevalence data for females and males, as well as DNA prevalence data for females, using a Bayesian model comparison procedure. We ranked the models according to both their simplicity and ability to be fitted to the data.

Results

Our results demonstrate that models with seroreversion where seropositivity indicates only a partial or very short-term full protection against re-infection generate age-specific HPV DNA prevalence most consistent with the observed data when compared with other models.

Conclusions

Models supporting the notion that seropositive individuals are fully immune to reinfection demonstrated consistently inferior fits to the data than other models making no such assumption.

Background

Genital human papillomaviruses (HPV) are viral sexually transmitted infections (STIs) with around 40 types having tropism for the anogenital region. High-risk (oncogenic) HPV types 16 and 18 are more virulent than others and associated with about 70-76% of cervical cancers

Because cancer generally develops long after initial infection with HPV, the actual impact of vaccination programs for cancer prevention will not be known for decades after these programs have commenced. Mathematical models have therefore been commonly employed to predict the potential population-level impact of vaccination under different vaccination scenarios and assumptions regarding vaccine properties.

Mathematical transmission models can be constructed in a number of ways but deterministic compartmental models are commonly used due to their relative simplicity and tractability

A necessary element of every modelling study is model calibration. Calibration is performed by adjustment of parameter values to ensure that the model predictions, which are intrinsically uncertain, are consistent with available real-life data. The accuracy of the calibration process can be iteratively improved as more data become available. Often HPV models are calibrated to HPV incidence or prevalence data collected in a particular country or jurisdiction. In view of increasing availability of data related to seroreactivity (production of antibodies in response to infection, known as seroconversion, and their decay, or seroreversion), it is timely to investigate the present possibilities to use them for model calibration. A number of studies (for example,

In this study, we develop eight compartmental models based on types SIS (Susceptible-Infected-Susceptible), SIR (Susceptible-Infected-Recovered) and SIRS (Susceptible-Infected-Recovered-Susceptible)

Methods

Modelled population

Since we intended to calibrate our models to Australian data, it was important to ensure that the population we modelled was a reasonably accurate representation of the sexually active heterosexual Australian population. We defined the modelled population as a set of non-overlapping groups of individuals stratified by gender, age, sexual activity and infection state. In compartmental models (sometimes referred to as population-based models, in contrast with individual-based models), each of these groups (“compartments”) is assumed to be large enough to behave independently of individual stochastic effects. Throughout this paper, when we refer to an “individual” from a particular compartment, we actually mean a descriptor representing the whole population in that compartment, whose attributes are averaged attributes of that population. The age structure of the population was represented by 48 one-year age groups in the range 12 to 59 years of age. This was motivated by the following factors: 1) to model HPV we need to model only the sexually active Australian population, which excludes those younger than a certain age: in our models individuals start sexual activity at 15, but we also included 12–14 year olds, to allow for possible extension of the model should sexual behaviour data for this age group become available; 2) the sexual behaviour data we used

Sexual activity was described by four groups defined by the annual number of new sexual partners. These groups are numbered 1 to 4 in order of increasing activity and contain 60%, 27%, 11% and 2% of the modelled population, respectively

Sexual mixing

Sexual behaviour in the Australian population is described in our models by means of a mixing matrix which quantifies the rate of new partner acquisition by males and females based on their age and level of sexual activity. Our implementation of the mixing matrix is as previously employed and described in

Natural history of HPV-16

Differences between the models we evaluate here are in terms of what is assumed in regard to naturally acquired immunity, i.e. immunity acquired as a result of exposure to infection. In this study we evaluate two general scenarios: either individuals cannot become reinfected while being seropositive, or they can. The implementations of seroreactivity, by model, are briefly summarized in Table

**Model**

**Effect of seropositivity on transmission**

**Seroconversion before clearance**

**Seroreversion**

**DIC score**

**Rank**

**No reinfection while seropositive**

**Reduced risk of re-infection while seropositive**

**Risk of re-infection while seropositive is unchanged**

SIS_{1}

–

✓

–

–

–

−101.3

3

SIS_{2}

–

✓

–

–

✓

−108.8

1

SIR_{1}

✓

–

–

–

–

−28.6

8

SIR_{2}

✓

–

–

✓

–

−31.2

7

SIRS_{1}

✓

–

–

–

✓

−40.4

6

SIRS_{2}

✓

–

–

✓

✓

−45.7

5

SIRS_{3}

–

–

✓

–

–

−93.4

4

SIRS_{4}

–

–

✓

–

✓

−105.6

2

The models we refer to as SIS (see Figure _{1} does not incorporate seroreversion (the decay or loss of antibodies detectable by current assays in an individual): seropositive individuals remain seropositive for life. On the other hand, in SIS_{2} they are assumed to be losing antibodies at a constant rate while in the state S+, so there is a steady migration of individuals from S+ to S- as they lose whatever degree of immunity they had.

SIS models: SIS_{1 }(left) and SIS_{2 }(right); “+” and “-” denote seropositivity and seronegativity, respectively

**SIS models: SIS**
_{
1
}
**(left) and SIS**
_{
2
}
**(right); “+” and “-” denote seropositivity and seronegativity, respectively.**

In our SIS models females who cleared infection are ensured a degree of natural immunity (see prior distributions in Table

**Parameter description**

**Symbol**

**Prior**

**Source**

All durations are in years, and all rates are per capita annual rates.

Per-partnership probability of transmission from female to male used to calculate the force of infection

_{
m
}

U(0.10-1.00)

Per-partnership probability of transmission from male to female used to calculate the force of infection

_{
f
}

U(0.10-1.00)

Average duration of infection for males

T_{in,m}

U(0.60,1.70)

Average duration of infection for females

T_{in,f}

U(0.75,1.50)

Average rate of loss of immunity for males; defined as 1/T_{im,m} i.e. the inverse of the average duration of natural immunity for males

r_{li,m}

U(0.01,0.33) (SIRS_{1}, SIRS_{2}); U(0.01,1.0) (SIRS_{3}, SIRS_{4});

_{1}, SIRS_{2}); Not available (SIRS_{3}, SIRS_{4});

Average rate of loss of immunity for females; defined as 1/Tim,f i.e. the inverse of the of natural immunity for females

r_{li,f}

U(0.01,0.33) (SIRS_{1}, SIRS_{2}); U(0.01,1.0) (SIRS_{3}, SIRS_{4});

_{1}, SIRS_{2}); Not available (SIRS_{3}, SIRS_{4});

Probability of seroconversion for males

p_{m}

U(0.01,0.30)

Probability of seroconversion for females

p_{f}

U(0.40,0.70)

Average rate of seroreversion for males

r_{sr,m}

U(0.01,0.10)

Average rate of seroreversion for females

r_{sr,f}

U(0.10,1.00)

Average degree of immunity for seropositive males

s_{m}

U(0.00,1.00)

Not available

Average degree of immunity for seropositive females

s_{f}

U(0.10,1.00)

Average time to seroconversion for males (a proportion of T_{in,m})

T_{sc,m}

U(0.50,0.95)

Average time to conversion for females (a proportion of T_{in,f})

T_{sc,f}

U(0.50,0.95)

Degree of assortativity by age group

_{
a
}

U(0.10,0.90)

Not available

Degree of assortativity by sexual activity group

_{
r
}

U(0.10,0.90)

Not available

The key feature of SIR models (Figure _{1}). In SIR_{2}, individuals can test seropositive and still remain infected for some time.

SIR models: **SIR**_{1 }(left) and **SIR**_{2 }(right); “+” and “-” denote seropositivity and seronegativity, respectively

**SIR models: SIR**
_{
1
}
**(left) and SIR**
_{
2
}
**(right); “+” and “-” denote seropositivity and seronegativity, respectively.**

In SIRS models (Figure

SIRS models: SIRS_{1 }(top left), SIRS_{2 }(top right), SIRS_{3 }(bottom left) and SIRS_{4 }(bottom right); “+” and “-” denote seropositivity and seronegativity, respectively

**SIRS models: SIRS**
_{
1
}
**(top left), SIRS**
_{
2
}
**(top right), SIRS**
_{
3
}
**(bottom left) and SIRS**
_{
4
}
**(bottom right); “+” and “-” denote seropositivity and seronegativity, respectively.**

Model SIRS_{1} is essentially SIR_{1} with waning immunity. Note that we assume no difference in the rates of loss of immunity between seropositive (R+) and seronegative (R-) immune individuals, and losing immunity is equivalent to losing seropositive status. Similarly, SIRS_{2} is an extension of SIR_{2}. Model SIRS_{3}, just like SIR_{1} or SIRS_{1}, assumes that clearance and seroconversion are synchronous, but seropositivity is not an indication of immunity. Consequently, seropositive individuals lose immunity at the same rate as seronegative ones and then can become infected while testing seropositive. Seropositive status is life-long.

Finally, SIRS_{4} is SIRS_{3} with seroreversion. Both susceptible and immune individuals who seroconverted due to previous infection are losing antibodies at a constant rate which is different for males and females.

Note that in order to limit complexity, we chose not to model the scenario whereby an individual can serorevert while infected, since in the infected state the level of antibodies can be assumed to be high - there is, however, no evidence to convincingly support this hypothesis.

Model parameters are gender specific, which allows for possible differences in HPV-16 natural history between females and males. The ordinary differential equations describing the models included in this comparison are provided in the Additional file

**Technical Appendix.**

Click here for file

Model comparison and calibration

According to the Bayesian approach we adopted, it is necessary to formulate our beliefs about each of the model parameters, before any data produced by the models have been observed, as probability distributions. These distributions are known as prior distributions or “priors”. The priors we used in this study are given in Table _{
a
} and _{
r
} describing assortativity by age and sexual activity group (see the Additional file

Results

The DIC values, calculated for each tested model, are presented in Table _{1}, SIRS_{1}, SIR_{2}, SIRS_{2}) are clearly inferior to the other models. Allowing seroconversion prior to clearance of infection in SIR_{1}, and SIRS_{1} (which turn them into SIR_{2} and SIRS_{2}, respectively) somewhat improves their scores, but these are still not competitive. Calibration plots for all models can be found in the Additional file

Here we would like to briefly comment on some of the inferred parameter values for the two “best” models SIS_{2} and SIRS_{4}. Firstly, we observe that in SIS_{2} the per-partnership transmission probability from male to female (βf, posterior median 0.806 and the 95% Highest Posterior Density (HPD) interval, i.e. the shortest interval in parameter space which contains 95% of the distribution, (0.514-0.999)) is higher than that from female to male (βm , posterior median 0.59, 95% HPD interval 0.248-0.961)). This is also the case for SIRS_{4}, where the posterior mean for βf is 0.885 against 0.695 for βm. These values are consistent with the values predicted in other modelling studies: for example, β (assumed to be the same for female to male and male to female) was estimated at 0.8 (median) with the 95% posterior interval (0.6, 0.99) in _{2} and 1.48 (males) and 1.367 (females) in SIRS_{4}. In SIS_{2} the probability of seroconversion for males pm is low (median at 0.135), and for females (pf) it is not higher than the values reported in literature. In particular, its posterior median is at 0.494 while the 95% HPD interval is (0.4-0.654), which is in agreement with 0.5-0.6 suggested in _{4}. The inferred values for the degree of immunity for males do not let us make any meaningful conclusions regarding whether or not males are protected, because sm appears to have little influence on the model performance, which is evident from its nearly flat posterior and 95% HPD interval (0.001-0.903). In contrast, the degree of protection for females, sf, has a non-flat posterior, and its 95% HPD interval (0.100-0.810) suggests that we can at least be reasonably confident that it is certainly not complete and does not exceed 0.81, which is an important implication. Another modelling study _{4}) were high for both males and females, 95% HPD interval for males is (0.365-0.999) and for females (0.403-1.0). These indicate very short average durations of natural immunity, namely, 1–2.74 years for males and 1–2.48 years for females. Finally, the rates of seroreversion under SIS_{2} are low but higher for females than for males (median 0.08 against 0.03). Under SIRS_{4} these are very similar (median 0.079 for females and 0.03 for males).

Discussion

The results we obtained show that models assuming that seropositive individuals are fully and permanently protected from reinfection with HPV-16 are clearly inferior to the other models making no such strong assumptions. This conclusion is based on DIC scores. It is important to realise that DIC does not detect a ‘correct’ model in terms of HPV-16 transmission mechanism. Instead, it provides a quantitative model ranking which discourages complexity and is based on the ability of models under consideration (among which the ‘correct’ model may not even be present) to be fitted to the data. Hence, if a simpler model can be calibrated to the data at least as well as a more complex model, it will get a better DIC score. To receive a better DIC ranking, a more complex model would have to justify its complexity by producing a notably better fit than its simpler competitors. To further clarify the context in which our results should be viewed, we mention that our results can be meaningfully interpreted only if we completely rely on the available data – should these be extended or replaced, our results would inevitably change too. Another important aspect is that the DIC ranking factors in how well the models can be fitted to all data at once, for both males and females. If we, for instance, restricted ourselves to only calibrating the models to HPV seroprevalence, the resulting model ranking would likely be different.

As is evident from Table _{2}, closely followed by SIRS_{4}. The difference in DIC scores between the two models is not substantial and hence does not imply that SIS_{2} is clearly preferable. We should note that the reason why SIS_{2} outscored SIS_{1} is inclusion of seroreversion. Indeed, it is the only difference between the models. Seroreversion in SIS_{2} is implemented with the help of two additional parameters (rsr,m and rsr,f), as compared with SIS_{1}, and nonetheless, it improved the fit substantially enough to overcome penalisation for extra parameters and get ahead of SIS_{1} by 8.5 points. The benefits of seroreversion in SIS_{1} are predictable since without it, SIS_{1} can not capture declining seroprevalence in older females. For the same reason, SIRS_{4} provided a significant improvement over SIRS_{3}. We see that seroreversion in SIS and SIRS models is crucial in terms of improving the fit to data, even though the rate of seroreversion is low.

Although the highest ranking models SIS_{2} and SIRS_{4} have different structures, as we mentioned in Results, the fitted durations of full natural immunity in SIRS_{4} are very short. Hence, this model is approaching a limit case when it almost becomes SIS_{2} (see Figure

It is our view, given what is currently known about immunity (in particular, the reported association between seropositivity and reductions in the number of incident infections in seropositive individuals _{2} may be a more realistic representation of naturally acquired protective immunity than a short but full immunity as in SIRS_{4}.

It is important to note that nearly all information available regarding the possible association of seropositivity with protective immunity has come from studies of females. The only study of males in this context that we are aware of _{2} was not sensitive to variations in the degree of immunity in males. To increase sensitivity, the amount of data used for model specification and calibration and/or their accuracy should be increased, which we expect to happen in future, when, for example, HPV DNA seroprevalence data for males become available.

Our models have a number of limitations. In particular, we assumed the duration of immunity to be the same for all ages, which is unlikely to be true in reality, and the probability of seroconversion to be independent of an individual’s age though there is some evidence to the contrary

Conclusions

In conclusion, the models which provided the optimal combination of parsimony and goodness of fit to the currently available Australian data are these where seropositivity indicates only a partial (or very short full) immunity against re-infection and seroreversion is assumed to be taking place. Future studies will no doubt provide greater insight into the nature of acquired immunity and its association with seropositivity, enabling us to build more accurate models.

Competing interests

The authors of this manuscript do not perceive any direct conflict of interest in relation to the research described. However, in the interest of full disclosure, we declare the following:

The research described in this manuscript was funded by an Australian Research Council Linkage Project (LP0883831). CSL Limited, the distributor of Gardasil® in Australia and New Zealand, is a Partner Organization on this project.

Igor Korostil’s salary is funded by the above mentioned grant (LP0883831).

Professor Suzanne Garland has received advisory board fees and grant support from CSL Ltd and GlaxoSmithKline (GSK), and lecture fees from Merck, GSK and Sanofi Pasteur; in addition, she has received funding through her institution to conduct HPV vaccine studies for Merck and GSK. She is a member of the Merck Global Advisory Board as well as the Merck Scientific Advisory Committee for HPV.

Dr David Regan has received honoraria from CSL Ltd for advisory board participation and for presenting his work at sponsored symposia. He has also received grant support from CSL Ltd for other HPV-related research projects.

Authors' contribution

MGL and SMG participated in study design, and manuscript preparation. DGR coordinated the study and participated in study design, model parameterisation and manuscript preparation. IAK conceived of the study, developed and implemented the models, analysed results and drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

We thank Julia Brotherton for her valuable help with the HPV-16 prevalence data extraction from the WHINURS study database.

This work was supported by an Australian Research Council Linkage Project [LP0883831] and a National Health and Medical Research Council Program Grant [568971]. The Kirby Institute receives funding from the Australian Government Department of Health and Ageing. The views expressed in this publication do not necessarily represent the position of the Australian Government. The Kirby Institute is affiliated with the Faculty of Medicine, University of New South Wales.

Pre-publication history

The pre-publication history for this paper can be accessed here: