While cervical cancer is the fourth most frequent female cancer globally (there were 661,021 new cervical cancer cases and 348,189 fatalities globally in the year 2022 alone (Bray et al. 2024)), it is the third most frequent in the 15-44-year female age group in South Korea (there were 3,218 new cervical cancer cases and 1,018 fatalities in South Korea in the year 2020 (Bruni et al. 2023)). Cervical cancer is one of the top three cancers in females under the age of 45 globally, and the average ages for cervical cancer diagnosis and death are 53 years and 59 years, respectively (Arbyn et al. 2020). The burden of the cervical cancer disproportionately affects resource-limited countries that have lack of availability and access to screening, vaccination, and treatment measures (Bray et al. 2024).

Virtually all cervical cancer cases are caused by HPV(Human Papillomavirus) (Walboomers et al. 1999), which is transmitted primarily through skin-to-skin or mucosa-to-mucosa sexual contacts (Schiffman et al. 2007). Among the more than 200 serotypes of HPV, the most prevalent serotypes are HPV-16 and HPV-18, which are high-risk oncogenic types that are found in 70% of cervical cancer cases (Smith et al. 2007). The oncogenic types HPV-31/33/35/45/52/58 are the next most common serotypes that account for additional 20% of cervical cancers globally (Clifford et al. 2006). These high-risk oncogenic types cause the most pre-cancerous lesions and cancers in humans. On the other hand, HPV-6 and HPV-11 are low-risk non-oncogenic types that are responsible for 90% of genital warts (Walboomers et al. 1999). Although 70-90% of sexually active adults contract HPV at least once during their lifetime (Bosch and de Sanjose 2003; Chesson et al. 2014), 90% of HPV infections clear naturally within 2 years (Schiffman et al. 2007). HPV persists in the remaining 10%, increasing the risk of developing pre-cancerous lesions and cancer in these individuals (Bulkmans et al. 2007; Schiffman et al. 2007). Empirical studies in the US and Netherlands show that males recover faster from HPV infection than females (Giuliano et al. 2008a; Moscicki et al. 2004, 1998), and persistent infection is more common in females than in males (Van Doornum et al. 1994). This explains, in part, the higher HPV prevalence and number of HPV-related cancer cases in females than in males in South Korea (National Cancer Center Korea Central Cancer Registry 2022; Shin et al. 2004). The pre-cancer stage of the natural history of HPV is referred to as intraepithelial neoplasia. The cervical intraepithelial neoplasia (CIN) stage is divided into three grades, namely CIN1, CIN2, and CIN3, depending on the extent of the abnormalities (IARC 1995), and the risk of developing invasive cervical cancer is higher for higher grades of CIN (IARC 1995; Schiffman et al. 2007). Similarly, intraepithelial neoplasia in males (INM) is divided into three stages (INM1, INM2, and INM3), and leads to various cancers in males (such as anal and penile cancers) (IARC 1995).

Although the risk posed by the cervical cancer has been the major driving force of the studies on the HPV dynamics, HPV is also responsible for 88% of anal cancer, 50% of penile cancer, and 30.8% of oropharyngeal cancer cases globally, exerting burden on both females and males (de Martel et al. 2017). In South Korea, HPV-related male cancer cases tripled in the past 20 years, reaching 573 new cases in 2020 (National Cancer Center Korea Central Cancer Registry 2022). An empirical study in the US, Mexico, and Brazil reported that 65.2% of asymptomatic males in the age group of 18-70 have HPV DNA in the genital area (Giuliano et al. 2008b). Major risk factors for HPV infection include anal intercourse and contraction of other sexually-transmitted infections (STIs) (Chin-Hong Peter et al. 2004; Hernandez et al. 2008; Kyo et al. 1994; Lee et al. 2016; Marra et al. 2018), hence men who have sex with men (MSM) is one of the main risk groups of HPV infection among sexually active individuals (Chin-Hong Peter et al. 2004; Lee et al. 2016; Marra et al. 2018).

Three HPV vaccines have been approved for use in females of age 9-45 and males of age 9-26 in South Korea; the bivalent (Cervarix^®, preventing HPV 16/18), quadrivalent (Gardasil^®, preventing HPV 16/18/6/11), and 9-valent (Gardasil9^®, preventing HPV 16/18/6/11/31/33/45/52/58) vaccines, which were approved in 2008, 2007, and 2016, respectively. The 9-valent vaccine (Gardasil9^®) provides the broadest coverage of HPV types, covering 9 HPV types, and costs about $287.54 per dose in the United States in 2024 (CDC 2024) (it costs about $150 per dose in South Korea (Choi et al. 2022)). All three vaccines protect against HPV infection with over 90% efficacy if used prior to HPV exposure, and offer protection against precancerous lesions (Cheng et al. 2020). People of age 9-14 are recommended to get 2 doses, while those of age 15-26 are recommended to get 3 doses of the vaccine (World Health Organization 2022). In South Korea, the HPV National Immunization Program (NIP) began in June 2016, offering 2 doses of bivalent(Cervarix^®) or quadrivalent(Gardasil^®) vaccines to girls of age 12. The routine HPV immunization program in Korea was expanded to cover girls of ages 12-17 in 2022. In 2016, before the onset of the NIP, 23.1% of Korean females aged 9-59 years received the first dose of HPV vaccine (Choi et al. 2022). In 2018 (two years after the implementation of the NIP), however, 82.3% of 12-year-old girls received the first dose of HPV vaccine in Korea (Kim et al. 2018). This is still relatively low compared to the 91-98% coverage of infant vaccination series against childhood diseases covered by the NIP in South Korea (KCDC Center for Infectious Disease Control 2023).

Papanicolaou test (Pap test) is widely used as the primary method of screening for cervical cancer in countries with organized screening programs, allowing for early detection and treatment of pre-cancerous lesions, thereby reducing the incidence and mortality of cervical cancer (Devesa et al. 1989; Lim and Yoo 2019; Min et al. 2015). Some empirical studies suggest the utility of using HPV DNA test (such as, Hybrid Capture 2 and PCR-based methods) as an alternative or adjunctive screening method to the Pap test (Reid et al. 1991; Denny and Wright 2005). However, South Korea does not recommend it as a standalone test, due to its low specificity, high cost, and high variability of its performance (Lim and Yoo 2019). In contrast, Pap test (conventional Pap smear or liquid-based cytology) is implemented in South Korea, with the Korean Society for Cytopathology entrusted with managing the quality control of its nationwide implementation (Lim and Yoo 2019; Lee et al. 2008). South Korea started offering free Pap screening, as a part of the National Cancer Screening Program (NCSP), to females over age 30 every 2 years in 2002, and then extended to females over age 20 since 2016 (Shin et al. 2022). The national cervical cancer screening guideline in South Korea (2015) suggests that females over age 20 receive Pap test every 3 years, with the option of combining Pap test with HPV test in consideration of clinical decision for individual risk and preference (Min et al. 2015). The guideline also states that the screening can terminate at the age of 74 if more than 3 consecutive cytology results are negative within 10 years (Min et al. 2015). The Korean National Cancer Screening Survey reported that the Pap screening coverage in Korea during the period 2005-2020 was 61% on average (Shin et al. 2022).

Numerous mathematical models, typically in the form of a deterministic system of nonlinear differential equations, have been developed and used to study the natural history of HPV (Elbasha and Galvani 2005; Gao et al. 2021) and its progression to associated cancers (Alsaleh and Gumel 2014; Choi and Shim 2024; Elbasha et al. 2007; Goldie et al. 2007, 2008; Javame and Gumel 2016; Malik et al. 2013; Milwid et al. 2018). For instance, Elbasha and Galvani (Elbasha and Galvani 2005) examined the impact of vaccination on the dynamics of two types of HPV in a community. (Gao et al. 2021) developed and analyzed a model for HPV dynamics that includes vaccination in both the heterosexual and the MSM populations. (Alsaleh and Gumel 2014) developed and analyzed a 29-dimensional risk-structured model for HPV dynamics, computing optimal vaccination coverage level. Some mathematical models were also developed and used to investigate the cost-effectiveness of different HPV vaccination strategies (Choi and Shim 2024; Cody et al. 2021; Elbasha et al. 2007; Goldie et al. 2007, 2008). Furthermore, while (Javame and Gumel 2016) investigated the effects of Pap screening on the prevalence of HPV and related cancers, other studies (such as those in (Malik et al. 2013; Milwid et al. 2018)) examined the effects of both the vaccination and Pap screening programs against HPV and related cancers in a community.

The purpose of the current study is to extend some of the previous modeling studies to quantify the combined population-level impacts of vaccination and Pap screening against HPV and HPV-related cancers in South Korea. Specifically, the main objective of this study is to assess whether or not the existing control resources (i.e., vaccination and Pap screening, under current levels of coverage and efficacy) will be sufficient to significantly curtail the incidence and burden of HPV and HPV-related cancers in the main high-risk groups in South Korea. The effectiveness of various control strategies will also be theoretically assessed. The paper is organized as follows. The vaccination and Pap screening model for the transmission of HPV is formulated in Section 2. Detailed qualitative analysis (with respect to the existence and asymptotic stability of its disease-free and endemic equilibria) of the model are carried out in this section. In order to calibrate the model and estimate its unknown parameters, a simplified version of the model without Pap screening and with the three CIN and INM stages lumped into one compartment, is considered in Section 3. The calibration and qualitative analysis of this simplified version of the model are also described and presented in this section. Detailed global sensitivity analysis for the full model with vaccination and Pap screening (based on the values of the unknown parameters estimated from the fitting of the reduced model) is carried out in Section 4 to determine the most influential parameters of the full vaccination and Pap screening model. Numerical simulations of this model are carried out in Section 5. Concluding remarks and discussion are presented in Section 6.

In this section, a mechanistic model for the heterosexual transmission of HPV will be developed and used to assess the combined population-level impact of routine HPV vaccination and Pap cytology screening on the incidence and burden of HPV and HPV-related cancers in South Korea. In addition to incorporating the life history features of HPV and associated dysplasia, the model to be developed stratifies the total sexually active population (which is assumed to be of age 17 or older (Nakazawa et al. 2023; Shin et al. 2004)) based on gender (male or female). Specifically, the total sexually active population of South Korea at time t, denoted by N(t), is split into the total number of females at time t (denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_f(t)$$\end{document}Nf(t)), and the total number of males at time t (denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_m(t)$$\end{document}Nm(t)), so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(t)=N_f(t)+N_m(t)$$\end{document}N(t)=Nf(t)+Nm(t). Furthermore, the population of sexually active females is stratified into the 15 mutually exclusive compartments of unvaccinated susceptible (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_f(t)$$\end{document}Sf(t)), vaccinated susceptible (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_f(t)$$\end{document}Vf(t)), exposed/pre-symptomatic infectious (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_f(t)$$\end{document}Ef(t)), symptomatic infectious (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_f(t)$$\end{document}If(t)), persistently infected (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_f(t)$$\end{document}Pf(t)) females, females having undetected CIN (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{iu}$$\end{document}Qiu, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2,3$$\end{document}i=1,2,3 accounting for the three CIN stages or grades), detected CIN (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{id}, i=1,2,3$$\end{document}Qid,i=1,2,3), undetected cervical cancer (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_u(t)$$\end{document}Cu(t)), detected cervical cancer (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_d(t)$$\end{document}Cd(t)), females who recovered from cervical cancer (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_f^C(t)$$\end{document}RfC(t)), and females who recovered from HPV infection before developing cervical cancer (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_f(t)$$\end{document}Rf(t)). Hence,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_f(t)= & S_f(t)+V_f(t)+E_f(t)+I_f(t)+P_f(t)\\ & +{\sum _{i=1}^3\left( Q_{iu}(t)+Q_{id}(t)\right) }+C_u(t)+C_d(t)+R_f^C(t)+R_f(t). \end{aligned}$$\end{document}Nf(t)=Sf(t)+Vf(t)+Ef(t)+If(t)+Pf(t)+∑i=13Qiu(t)+Qid(t)+Cu(t)+Cd(t)+RfC(t)+Rf(t).Similarly, the total population of sexually active males is split into the 11 mutually exclusive compartments of unvaccinated susceptible (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_m(t)$$\end{document}Sm(t)), vaccinated susceptible (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_m(t)$$\end{document}Vm(t)), exposed/pre-symptomatic infectious (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_m(t)$$\end{document}Em(t)), symptomatic infectious (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_m(t)$$\end{document}Im(t)), persistently infected (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_m(t)$$\end{document}Pm(t)) males, males having INM (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{im},$$\end{document}Jim, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ i=1,2,3$$\end{document}i=1,2,3 accounting for the three INM stages), males having HPV-related cancers (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_m$$\end{document}Cm), males who recovered from HPV-related cancers (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_m^C$$\end{document}RmC), and males who recovered from HPV infection before developing HPV-related cancers (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_m$$\end{document}Rm), so that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}N_m(t)=S_m(t) + V_m(t) + E_m(t) + I_m(t) + P_m(t) + \sum _{i=1}^3 J_{im}(t) + C_m(t) + R_m^C(t) + R_m(t).\end{aligned}$$\end{document}Nm(t)=Sm(t)+Vm(t)+Em(t)+Im(t)+Pm(t)+∑i=13Jim(t)+Cm(t)+RmC(t)+Rm(t).The population of sexually active unvaccinated susceptible females (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_f$$\end{document}Sf) and males (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_m$$\end{document}Sm) acquire HPV infection at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{mf}$$\end{document}λmf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{fm}$$\end{document}λfm, respectively, given by:2.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{mf}&= \left( \beta _{mf} c_{mf}\right) \left[ \frac{I_m + \eta _m E_m + \theta _m [P_m + J_{1m} + \kappa _m(J_{2m} + J_{3m})]}{N_m}\right] , \\ \lambda _{fm}&= \left( \beta _{fm} c_{fm}\right) \left[ \frac{I_f + \eta _{f}E_f + \theta _{f} [P_f + Q_{1u} + \kappa _{f}(Q_{2u} + Q_{3u}) + \nu \{Q_{1d} + \kappa _{fd}(Q_{2d} + Q_{3d})\}]}{N_f}\right] . \nonumber \end{aligned}$$\end{document}λmf=βmfcmfIm+ηmEm+θm[Pm+J1m+κm(J2m+J3m)]Nm,λfm=βfmcfmIf+ηfEf+θf[Pf+Q1u+κf(Q2u+Q3u)+ν{Q1d+κfd(Q2d+Q3d)}]Nf.In (2.1), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{mf}$$\end{document}βmf is the probability of HPV transmission from symptomatic infectious males to unvaccinated susceptible females per partnership, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{mf}$$\end{document}cmf is the average number of male sexual partners a sexually active female has per unit time (for this study, time is measured in years). The modification parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \eta _m < 1$$\end{document}0≤ηm<1 accounts for the assumed reduced transmission probability of HPV by exposed/pre-symptomatic infectious males, in comparison to the transmission probability of symptomatic infectious males. Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \theta _m < 1$$\end{document}0≤θm<1 measures the assumed reduction of infectiousness of males who are at, or past, the persistent HPV infection stage, in comparison to the infectiousness of symptomatic infectious males, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _m \ge 1$$\end{document}κm≥1 is the modification parameter accounting for the assumed increased probability of HPV transmission from males with INM2 and INM3, in comparison to males with INM1 and persistent HPV infection. Similarly, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{fm}$$\end{document}βfm is the transmission probability from symptomatic infectious females to unvaccinated susceptible males per partnership, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{fm}$$\end{document}cfm represents the average number of female sexual partners a male has per unit time. The modification parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \eta _{f} < 1$$\end{document}0≤ηf<1 accounts for the assumed reduced probability of HPV transmission by exposed/pre-symptomatic infectious females, in comparison to the HPV transmission by symptomatic infectious females. Furthermore, the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \theta _{f} < 1$$\end{document}0≤θf<1 represents the assumed reduction of the infectiousness of females that are at, or past, the persistent HPV infection stage, in comparison to the infectiousness of symptomatic infectious females, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{f} \ge 1$$\end{document}κf≥1 accounts for the assumed increased probability of HPV transmission by females with undetected CIN2 and CIN3, in comparison to females with undetected CIN1 and persistent HPV infection. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \nu < 1$$\end{document}0≤ν<1 accounts for the assumed reduction in the number of sexual contacts by females detected with pre-cancerous lesions, in comparison to females with undetected pre-cancerous lesions and persistent infection. Finally, the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{fd} \ge 1$$\end{document}κfd≥1 represents the assumed increase in the probability of HPV transmission by females with detected CIN2 and CIN3, in comparison to females with detected CIN1.

It should be stated that, since the model to be developed in this section only considers heterosexual transmission of HPV (i.e., transmission only occurs via sexual contacts between males and females), the following conservation law of total number of sexual partnerships per unit time must hold (i.e., the total number of sexual partnerships sexually active females have (with males) in the community must balance those by sexually active males (with females)):2.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_{mf}N_f(t) = c_{fm}(t)N_m(t). \end{aligned}$$\end{document}cmfNf(t)=cfm(t)Nm(t).It is assumed that females can always find enough number of male partners in the community to have sexual contacts with, so \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{mf}$$\end{document}cmf is set to be a constant (this assumption is justified by the fact that males generally have a higher desired number of female sexual partners than females (Fenigstein and Preston 2007), and tend to overreport their lifelong number of female sexual partners (Brewer et al. 2000), suggesting a higher demand among males for multiple female sexual partners. In addition, an empirical study using the famous online dating app, Tinder, shows that women attain a larger number of men matches in a short time (Tyson et al. 2016), which supports the claim that it is easier for women to find sexual partners than men). Hence, based on all this, the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{fm}$$\end{document}cfm (for the average number of female sexual partners a male has) can be computed from (2.2) as (i.e., the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{fm}$$\end{document}cfm depends on the ratio of the total sub-populations of males and females at each time)2.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_{fm}(t) = \frac{c_{mf}N_f(t)}{N_m(t)}. \end{aligned}$$\end{document}cfm(t)=cmfNf(t)Nm(t).Using (2.2) (with (2.3)), it follows that the expressions for the forces of infection, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{mf}$$\end{document}λmf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{fm}$$\end{document}λfm, given in (2.1), can now be re-written, respectively, as:2.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{mf}&= \left( \beta _{mf} c_{mf}\right) \left[ \frac{I_m + \eta _m E_m + \theta _m [P_m + J_{1m} + \kappa _m(J_{2m} + J_{3m})]}{N_m}\right] , ~\textrm{and}\\ \lambda _{fm}&= \left( \beta _{fm} c_{mf}\right) \left[ \frac{I_f + \eta _{f}E_f + \theta _{f} [P_f + Q_{1u} + \kappa _{f}(Q_{2u} + Q_{3u}) + \nu \{Q_{1d} + \kappa _{fd}(Q_{2d} + Q_{3d})\}]}{N_m}\right] . \nonumber \end{aligned}$$\end{document}λmf=βmfcmfIm+ηmEm+θm[Pm+J1m+κm(J2m+J3m)]Nm,andλfm=βfmcmfIf+ηfEf+θf[Pf+Q1u+κf(Q2u+Q3u)+ν{Q1d+κfd(Q2d+Q3d)}]Nm.Based on the above formulation, derivations (for the conservation law (2.3) and forces of infection (2.4)), and assumptions, the sex-structured model for the heterosexual transmission of HPV and related cancers, in the presence of HPV vaccination and Pap screening, in South Korea is given by the following deterministic system of nonlinear differential equations:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{Females} \left\{ \begin{array}{lcl} \dot{S}_f & =& (1-q_f)\pi _f - (\lambda _{mf} + \xi _f + \mu ) S_f, \vspace{2mm} \\ \dot{V}_f & =& q_f \pi _f + \xi _f S_f- (1-\varepsilon _v) \lambda _{mf} V_f - \mu V_f, \vspace{2mm} \\ \dot{E}_f & =& \lambda _{mf} S_f + (1-\varepsilon _v) \lambda _{mf} V_f - (\sigma _f + \mu ) E_f, \vspace{2mm} \\ \dot{I}_f & =& \sigma _f E_f - (\psi _f + \mu ) I_f, \vspace{2mm}\\ \dot{P}_f & =& (1-b_f)\psi _f I_f - (\alpha _f + \mu )P_f, \vspace{2mm}\\ \dot{Q}_{1u} & =& (1-k_f) \alpha _f P_f + h_{u2}y_f Q_{2u} -(g_f +\mu ) Q_{1u}, \vspace{2mm}\\ \dot{Q}_{2u} & =& (1-d_{u1}-d_{u3}) g_f Q_{1u} +q_{u2}z_fQ_{3u} -(y_f +\mu ) Q_{2u}, \vspace{2mm}\\ \dot{Q}_{3u} & =& (1-h_{u1}-h_{u2}-h_{u3})y_f Q_{2u}-(z_f+\mu ) Q_{3u}, \vspace{2mm}\\ \dot{Q}_{1d} & =& d_{u3}g_f Q_{1u} + h_{d2} \zeta _2 Q_{2d} -(\zeta _1 +\mu ) Q_{1d}, \vspace{2mm}\\ \dot{Q}_{2d} & =& h_{u3}y_f Q_{2u} +(1-d_{d1})\zeta _1 Q_{1d} + q_{d2}\zeta _3 Q_{3d} -(\zeta _2+\mu ) Q_{2d}, \vspace{2mm}\\ \dot{Q}_{3d} & =& q_{u3} z_f Q_{3u} + (1-h_{d1}-h_{d2})\zeta _2 Q_{2d} -(\zeta _3+\mu ) Q_{3d}, \vspace{2mm}\\ \dot{C}_u & =& (1-q_{u1}-q_{u2}-q_{u3})z_f Q_{3u} -(w_f + r_u + \delta + \mu ) C_u, \vspace{2mm} \\ \dot{C}_d & =& (1-q_{d1}-q_{d2})\zeta _3 Q_{3d} + w_f C_u -(r_d + \delta _d + \mu ) C_d, \vspace{2mm} \\ \dot{R}_f^C & =& r_u C_u + r_d C_d -\mu R_f^C, \vspace{2mm}\\ \dot{R}_f & =& b_f \psi _f I_f + k_f \alpha _f P_f + d_{u1}g_f Q_{1u} + h_{u1}y_f Q_{2u} + q_{u1}z_f Q_{3u} + d_{d1} \zeta _1 Q_{1d} \\ & & + h_{d1}\zeta _2 Q_{2d} + q_{d1} \zeta _3 Q_{3d} -\mu R_f, \end{array} \right. \\\\ \end{aligned}$$\end{document}FemalesS˙f=(1-qf)πf-(λmf+ξf+μ)Sf,V˙f=qfπf+ξfSf-(1-εv)λmfVf-μVf,E˙f=λmfSf+(1-εv)λmfVf-(σf+μ)Ef,I˙f=σfEf-(ψf+μ)If,P˙f=(1-bf)ψfIf-(αf+μ)Pf,Q˙1u=(1-kf)αfPf+hu2yfQ2u-(gf+μ)Q1u,Q˙2u=(1-du1-du3)gfQ1u+qu2zfQ3u-(yf+μ)Q2u,Q˙3u=(1-hu1-hu2-hu3)yfQ2u-(zf+μ)Q3u,Q˙1d=du3gfQ1u+hd2ζ2Q2d-(ζ1+μ)Q1d,Q˙2d=hu3yfQ2u+(1-dd1)ζ1Q1d+qd2ζ3Q3d-(ζ2+μ)Q2d,Q˙3d=qu3zfQ3u+(1-hd1-hd2)ζ2Q2d-(ζ3+μ)Q3d,C˙u=(1-qu1-qu2-qu3)zfQ3u-(wf+ru+δ+μ)Cu,C˙d=(1-qd1-qd2)ζ3Q3d+wfCu-(rd+δd+μ)Cd,R˙fC=ruCu+rdCd-μRfC,R˙f=bfψfIf+kfαfPf+du1gfQ1u+hu1yfQ2u+qu1zfQ3u+dd1ζ1Q1d+hd1ζ2Q2d+qd1ζ3Q3d-μRf,2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{Males} \left\{ \begin{array}{lcl} \dot{S}_m & =& (1-q_m)\pi _{m} - (\lambda _{fm} +\xi _m +\mu ) S_m, \vspace{2mm}\\ \dot{V}_m & =& q_m\pi _m +\xi _mS_m - (1-\varepsilon _v) \lambda _{fm} V_m - \mu V_m, \vspace{2mm}\\ \dot{E}_m & =& \lambda _{fm} S_m+ (1-\varepsilon _v) \lambda _{fm} V_m - (\sigma _m +\mu )E_m, \vspace{2mm} \\ \dot{I}_m & =& \sigma _m E_m - (\psi _m + \mu ) I_m, \vspace{2mm}\\ \dot{P}_m & =& (1-b_m)\psi _m I_m - (\alpha _m + \mu )P_m, \vspace{2mm}\\ \dot{J}_{1m} & =& (1-k_m)\alpha _m P_m + h_{m2}y_m J_{2m}- (g_m+\mu )J_{1m}, \vspace{2mm} \\ \dot{J}_{2m} & =& (1-d_m)g_m J_{1m} + q_{m2}z_m J_{3m}- (y_m +\mu )J_{2m}, \vspace{2mm} \\ \dot{J}_{3m} & =& (1-h_{m1}-h_{m2})y_m J_{2m}-(z_m+\mu ) J_{3m}, \vspace{2mm} \\ \dot{C}_m & =& (1-q_{m1}-q_{m2})z_m J_{3m} -(r_m +\mu ) C_m, \vspace{2mm} \\ \dot{R}_m^C & =& r_m C_m -\mu R_m^C, \vspace{2mm}\\ \dot{R}_m & =& b_m \psi _m I_m + k_m \alpha _m P_m + d_mg_m J_{1m} + h_{m1}y_m J_{2m} + q_{m1}z_m J_{3m}- \mu R_m. \end{array} \right. \end{aligned} \end{aligned}$$\end{document}MalesS˙m=(1-qm)πm-(λfm+ξm+μ)Sm,V˙m=qmπm+ξmSm-(1-εv)λfmVm-μVm,E˙m=λfmSm+(1-εv)λfmVm-(σm+μ)Em,I˙m=σmEm-(ψm+μ)Im,P˙m=(1-bm)ψmIm-(αm+μ)Pm,J˙1m=(1-km)αmPm+hm2ymJ2m-(gm+μ)J1m,J˙2m=(1-dm)gmJ1m+qm2zmJ3m-(ym+μ)J2m,J˙3m=(1-hm1-hm2)ymJ2m-(zm+μ)J3m,C˙m=(1-qm1-qm2)zmJ3m-(rm+μ)Cm,R˙mC=rmCm-μRmC,R˙m=bmψmIm+kmαmPm+dmgmJ1m+hm1ymJ2m+qm1zmJ3m-μRm.In the model (2.5), the population of sexually active unvaccinated susceptible females (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_f$$\end{document}Sf) is generated by the recruitment of new sexually active individuals into the community at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-q_f)\pi _f$$\end{document}(1-qf)πf (where the proportion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_f$$\end{document}qf is vaccinated against HPV), and is reduced by the acquisition of HPV infection at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{mf}$$\end{document}λmf, administration of the catch-up HPV vaccination (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _f$$\end{document}ξf), and natural death (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}μ; individuals in all epidemiological class are assumed to die naturally at this rate). The population of sexually active vaccinated susceptible females (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_f$$\end{document}Vf) is increased by the cohort (at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_f\pi _f$$\end{document}qfπf) and catch-up (at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _f$$\end{document}ξf) vaccination. Breakthrough infection occurs at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\varepsilon _v)\lambda _{mf}$$\end{document}(1-εv)λmf, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \varepsilon _v \le 1$$\end{document}0≤εv≤1 is the efficacy of the vaccine to protect against the acquisition of HPV infection. Exposed/pre-symptomatic infectious females (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_f$$\end{document}Ef class) develop clinical symptoms of HPV at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _f$$\end{document}σf. Symptomatic infectious females (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_f$$\end{document}If class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _f$$\end{document}ψf, a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document}bf, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f\psi _f$$\end{document}bfψf), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-b_f$$\end{document}1-bf, progress to persistent HPV infection (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-b_f)\psi _f$$\end{document}(1-bf)ψf). Females with persistent HPV infection (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_f$$\end{document}Pf class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _f$$\end{document}αf, a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_f$$\end{document}kf, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_f\alpha _f$$\end{document}kfαf), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-k_f$$\end{document}1-kf, move to the undetected CIN1 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-k_f)\alpha _f$$\end{document}(1-kf)αf).

Females with undetected CIN1 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{1u}$$\end{document}Q1u class) transition out of this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_f$$\end{document}gf, and a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{u1}$$\end{document}du1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{u1}g_f$$\end{document}du1gf), another proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{u3}$$\end{document}du3, are detected via Pap screening and move to the detected CIN1 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{u3}g_f$$\end{document}du3gf), and, finally, the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(d_{u1}+d_{u3})$$\end{document}1-(du1+du3), progress to the undetected CIN2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-d_{u1}-d_{u3})g_f$$\end{document}(1-du1-du3)gf). Females with undetected CIN2 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{2u}$$\end{document}Q2u class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_f$$\end{document}yf. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{u1}$$\end{document}hu1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{u1}y_f$$\end{document}hu1yf), another proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{u2}$$\end{document}hu2, regress to the undetected CIN1 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{u2}y_f$$\end{document}hu2yf), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{u3}$$\end{document}hu3, are detected via Pap screening and move to the detected CIN2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{u3}y_f$$\end{document}hu3yf), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(h_{u1}+h_{u2}+h_{u3})$$\end{document}1-(hu1+hu2+hu3), progress to the undetected CIN3 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-h_{u1}-h_{u2}-h_{u3})y_f$$\end{document}(1-hu1-hu2-hu3)yf). Females with undetected CIN3 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{3u}$$\end{document}Q3u class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_f$$\end{document}zf. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{u1}$$\end{document}qu1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{u1}z_f$$\end{document}qu1zf), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{u2}$$\end{document}qu2, regress to the undetected CIN2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{u2}z_f$$\end{document}qu2zf), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{u3}$$\end{document}qu3, are detected via Pap screening and move to the detected CIN3 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{u3}z_f$$\end{document}qu3zf), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(q_{u1}+q_{u2}+q_{u3})$$\end{document}1-(qu1+qu2+qu3), progress to the undetected cervical cancer class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-q_{u1}-q_{u2}-q_{u3})z_f$$\end{document}(1-qu1-qu2-qu3)zf). Females with detected CIN1 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{1d}$$\end{document}Q1d class) transition out of this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _1$$\end{document}η1. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{d1}$$\end{document}dd1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{d1}\eta _1$$\end{document}dd1η1), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-d_{d1}$$\end{document}1-dd1, progress to the detected CIN2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-d_{d1})\eta _1$$\end{document}(1-dd1)η1). Females with detected CIN2 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{2d}$$\end{document}Q2d class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _2$$\end{document}η2. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{d1}$$\end{document}hd1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{d1}\eta _2$$\end{document}hd1η2), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{d2}$$\end{document}hd2, regress to the detected CIN1 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{d2}\eta _2$$\end{document}hd2η2), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(h_{d1}+h_{d2})$$\end{document}1-(hd1+hd2), progress to the detected CIN3 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-h_{d1}-h_{d2})\eta _2$$\end{document}(1-hd1-hd2)η2). Females with detected CIN3 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{3d}$$\end{document}Q3d class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _3$$\end{document}η3. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{d1}$$\end{document}qd1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{d1}\eta _3$$\end{document}qd1η3), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{d2}$$\end{document}qd2, regress to the detected CIN2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{d2}\eta _3$$\end{document}qd2η3), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(q_{d1}+q_{d2})$$\end{document}1-(qd1+qd2), progress to the detected cervical cancer class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-q_{d1}-q_{d2})\eta _3$$\end{document}(1-qd1-qd2)η3). Females with undetected cervical cancer (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_u$$\end{document}Cu class) recover from cancer at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_u$$\end{document}ru, are detected via Pap screening and move to the detected cervical cancer class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_f$$\end{document}wf, or die of cancer at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}δ. Females with detected cervical cancer (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_d$$\end{document}Cd class) recover from cervical cancer at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_d$$\end{document}rd, or die of cancer at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _d$$\end{document}δd.

The population of sexually active unvaccinated susceptible males (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_m$$\end{document}Sm) is increased by recruitment at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-q_m)\pi _m$$\end{document}(1-qm)πm (where the proportion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_m$$\end{document}qm is vaccinated against HPV), and is decreased by the acquisition of HPV infection (at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{fm}$$\end{document}λfm), catch-up vaccination (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _m$$\end{document}ξm), and natural death (at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}μ). The population of vaccinated susceptible males (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_m$$\end{document}Vm) is increased by the cohort (at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_m\pi _m$$\end{document}qmπm) and catch-up (at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _m$$\end{document}ξm) vaccination. Breakthrough infection occurs at the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\epsilon _v)\lambda _{mf}$$\end{document}(1-ϵv)λmf. Exposed/pre-symptomatic infectious males (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_m$$\end{document}Em class) develop clinical symptoms of HPV at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _m$$\end{document}σm. Symptomatic infectious males (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_m$$\end{document}Im class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _m$$\end{document}ψm. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_m$$\end{document}bm, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_m\psi _m$$\end{document}bmψm), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-b_m$$\end{document}1-bm, progress to persistent infection (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-b_m)\psi _m$$\end{document}(1-bm)ψm). Males with persistent HPV infection (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_m$$\end{document}Pm class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _m$$\end{document}αm. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_m$$\end{document}km, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_m\alpha _m$$\end{document}kmαm), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-k_m$$\end{document}1-km, move to the undetected CIN1 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-k_m)\alpha _m$$\end{document}(1-km)αm). Males with INM1 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{1m}$$\end{document}J1m class) transition out of this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_m$$\end{document}gm. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{a}$$\end{document}da, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_mg_m$$\end{document}dmgm), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-d_m$$\end{document}1-dm, progress to the INM2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-d_m)g_m$$\end{document}(1-dm)gm). Males with INM2 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{2m}$$\end{document}J2m class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_m$$\end{document}ym. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m1}$$\end{document}hm1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m1}y_m$$\end{document}hm1ym), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m2}$$\end{document}hm2, regress to the INM1 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{m2}y_m$$\end{document}hm2ym), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(h_{m1}+h_{m2})$$\end{document}1-(hm1+hm2), progress to the INM3 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-h_{m1}-h_{m2})y_m$$\end{document}(1-hm1-hm2)ym). Males with INM3 (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{3m}$$\end{document}J3m class) leave this class at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_m$$\end{document}zm. A proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{m1}$$\end{document}qm1, of these individuals recover (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{m1}z_m$$\end{document}qm1zm), a proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{m2}$$\end{document}qm2, regress to the INM2 class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{m2}z_m$$\end{document}qm2zm), and the remaining proportion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-(q_{m1}+q_{m2})$$\end{document}1-(qm1+qm2), progress to the HPV-related cancer class (at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-q_{m1}-q_{m2})z_m$$\end{document}(1-qm1-qm2)zm). Males with HPV-related cancer (i.e., those in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_m$$\end{document}Cm class) recover from cancer at a rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_m$$\end{document}rm.

The main assumptions made in the formulation of the model (2.5) include (inter alia): (i)

Homogeneous mixing (i.e., every sexually active heterosexual individual is equally likely to mix with, and have sexual contact, with every other sexually active heterosexual individual)