We developed a dynamical model for HPV transmission in a heterosexual population to assess the impact of vaccination against HPV infections on the incidence of cervical cancer in France. We considered current vaccination policies and compared several vaccination scenarios. We confirmed the effectiveness of vaccination against HPV to prevent infections and cervical cancers using deterministic modeling 
. However, the impact of vaccination differed according to the various scenarios considered in our paper. Vaccination coverage among young females under 14 and higher vaccination coverage in currently targeted females (in France) improved the prevention of infections and cervical cancers due to HPV 16/18 significantly.
The use of a deterministic model in our study allowed us to take into account herd immunity, which corresponds to a decrease of HPV 16/18 infections and cervical cancers in non-vaccinated subpopulations of females due to vaccination coverage of other individuals.
In the fitting procedure, probabilities of HPV 16/18 transmission for women and men were estimated to be of 0.25 and 0.20 respectively. The values that were yielded from our model are close to published estimates from fitted deterministic models for HPV 
: in the susceptible-infected-susceptible (SIS) deterministic model developed by Taira et al 
, age-specific probabilities ranged from 0.15 to 0.35.
Screening of precancerous lesions of the cervix and cervical cancer using cervical smear tests is recommended in France. Screening coverage has been estimated to be 58.7% in women of 25 to 65 years old in 2005; however, significant discrepancies were observed between age-groups 
. Screening was not directly integrated in our model as we did not stratify on screening status. Some authors distinguished between regularly screened women and never-screened women 
. Nonetheless, epidemiological data regarding screening of cervical cancer and cervix precancerous lesions are limited in France. Therefore, our model was fitted on incidence of cervical cancer in France and considered regression of cervix precancerous lesions due to spontaneous regression or treatment after screening. Screening represents a complementary tool to prevent cervical cancers, especially in case of precancerous lesions due to other high-risk HPV types than 16/18 whereas current available vaccines protect against HPV16/18, which are responsible for 70% of cervical cancers.
The study could not take into consideration future changes in screening programs, using new screening policies or new screening technologies on the market. In France, in 2010 Public Health Agency (HAS) proposed to organize cervical cancer screening to improve screening coverage. Moreover, use of new screening tools may also modify current epidemiological data on HPV infections and related cervical cancers. In the USA, the ACOG guidelines (2009) and the American Cancer Society (2003) recommended HPV-plus-pap testing in women aged 30 and older 
. However, in France, Public Health policies explicitly recommend the use of HPV testing only in cases of abnormal cytology (www.has-sante.fr
Efficacy of vaccination was assumed to last lifelong. Consequently, we did not include the need for vaccine booster shots in our model. The protective effect of the vaccines is known to last at least several years 
and the need for boosters is currently unknown.
The sophistication of deterministic models by adding strata corresponding to uncertain data may lead to computational difficulties and increase the uncertainty of outputs and parameters. Thus, screening and use of booster shots were not included in our model.
In sensitivity analyses that investigated the effect of an increase of the sexual mixing parameter (ε
), we observed a decrease of HPV 16/18 prevalence after using a high value of ε
. This result is consistent with that reported in Garnett and Anderson paper 
. We observed that in the presence of frequent completely random sexual mixing (when the value of ε
is far from the assortative sexual mixing corresponding to ε
0), the infection is less likely to persist endemically 
. An explanatory mechanism regarding this finding suggests that individuals in the higher sexually active group transmit the infection to individuals in other sexual behavior groups which are less sexually active and less likely to transmit infection.
This analysis shows that the value of ε used in our base case modeling is consistent with endemic prevalence of HPV infections. Sexual mixing seems to be preferentially close to assortative case. The modeling of sexual mixing that we developed in this paper could be used in other models corresponding to the French sexual behavior.
Efficacy of both vaccines against HPV has been evaluated in several randomized controlled trials 
. While vaccines are highly effective in women who have never been infected with HPV 16/18, efficacy of vaccines could drop in women who have been infected prior to vaccination. We considered in the base case scenario a 90% efficacy of the vaccine against HPV 16/18 infection and 60% in sensitivity analysis. Sensitivity analysis showed that if vaccine efficacy is poor, impact of vaccination on the reduction of diseases linked to HPV 16/18 will be attenuated.
In previous papers assessing the impact of HPV vaccination, assumed vaccine coverage had very high reaching values - between 80% and 100%. Current vaccine coverage in France has been estimated to be low, under 30%, and diminishing over time 
. Consequently, the impact of vaccination has to be assessed using real life coverage data. In the first scenario, we considered the female vaccine coverage (30%) that is currently reached in France. In the other scenarios of female vaccination (scenarios 3 and 5) we considered realistic alternatives. Finally, we studied scenarios in which males were also vaccinated (scenario 2 and 4).
In the first scenario, we estimated the impact of vaccination on HPV prevalence in male and female, on cervical cancer and on specific mortality due to cervical cancer in the horizon of 10, 20, 50 and 100 years. We considered the vaccine coverage that was observed in France at the beginning of the vaccination campaign (in 2007 and 2008). This scenario predicts a decrease in cervical cancer incidence due to HPV 16/18 by one third 20 years after introduction of vaccine and a halving of deaths by cervical cancer. In this scenario, we considered vaccine coverage to be constant in time. However, female vaccine coverage in France is decreasing. While 33.3% of girls aged 14 in 2007 were vaccinated with 3 doses of the vaccine, only 23.7% and 5.4% of girls aged 14 were vaccinated respectively in 2008 and 2009 
. Therefore, the expected effect of the first scenario is probably over-estimated in this paper.
In Scenario 3, we considered a high vaccine coverage for women and young girls (80%), which yielded a better impact of vaccination than scenario 1. In countries where school-based vaccination programs have been implemented (The United Kingdom and Australia), these vaccine coverage rates have been reached 
. It may be difficult to implement such a program in France, as a previous school-based program of vaccination against hepatitis B failed due to controversies 
In scenario 5, young girls could be vaccinated before 14. In Europe, many countries recommend vaccination for females aged 12 
. Vaccination of young girls before 14 may improve vaccine coverage and impact of vaccination in France. A better compliance with complete vaccination (using 3 shots) is observed in young females 
who are little likely to be infected with HPV prior to vaccination. However, the need of booster shots is still unknown 
. Nevertheless, if the long term efficacy of the vaccine is to be confirmed, vaccination coverage among younger girls will certainly improve the impact of vaccination. The French committee of vaccination recommended the vaccine for girls aged 14 in 2006 
. We also investigated the possible vaccination of boys and young men 
. In scenarios 2 and 4, we assessed the impact of an additional male vaccination on the reduction of HPV 16/18 prevalence (in males and females) and of cervical cancer incidence in females. Performing vaccination in men improves moderately the effect on HPV 16/18 prevalence and cervical cancer incidence compared to vaccination strategies in females only.
In this paper, we did not assess the impact of vaccination on other cancers or diseases due to HPV 16/18 infections (Recurrent respiratory papillomatosis, cancers of the anus, penis, vagina, vulva, and head and neck). Vaccination against HPV 16/18 infection is likely to reduce incidence of other cancers in the anogenital area in males and female. Reduction of HPV-16/18 prevalence in men due to female vaccination could reduce the incidence of some male cancers. These additional potential benefits of HPV 16/18 vaccination have to be investigated by developing specific dynamic models that consider the natural history of the diseases.
Finally, some recent studies have shown the bivalent vaccine's protective effect against high-risk HPV types, which are not targeted by it (HPV types 31 and 45) 
. In this case, the model may underestimate the benefit of vaccination on the reduction of cervical cancer incidence as we did not take into account cross-protection in our model.
In conclusion, we developed a deterministic model of HPV heterosexual transmission and progression to cervical cancer to assess the epidemiologic impact of HPV vaccination in France.
While current vaccination policies may decrease cervical cancer incidence in France significantly, other complementary strategies could be employed for females to improve vaccination efficacy.