Analysis and simulation for a two-sex transmission model of HPV infection in Xinjiang of China

Abstract

This study develops a two-sex deterministic compartmental model to investigate the transmission dynamics of Human Papillomavirus (HPV) and the subsequent progression to cervical cancer within a heterosexual population. The model stratifies the population by gender, tracking females through susceptible, HPV-infected, precancerous lesion, and cervical cancer states, while males are categorized as susceptible or infected. The mathematical analysis provides the expression of the basic reproduction number R₀, and obtain that when R₀ ≤ 1, the disease is eventually extinct, while when R₀ > 1, the disease is continuously prevalent. Calibrated with epidemiological data from Xinjiang, China, over the period 2009 to 2019, the model estimates a basic reproduction number of R₀ = 1.5023, confirming the endemic nature of HPV. Sensitivity analysis identifies the male-to-female transmission rate (β_mf) and the female HPV clearance rate (γ_1f) as the most influential parameters. Intervention simulations reveal that while single-parameter strategies are insufficient, a combined approach-reducing male-to-female transmission by 50%, enhancing male clearance by 30%, and modulating precancerous progression parameters can lead to the elimination of cervical cancer by 2074. Our results demonstrate that sustainable control requires a dual-gender strategy, integrating behavioral prevention to reduce transmission with clinical screening and treatment to block cancer development.

1. Introduction

Cervical cancer remains a significant global public health challenge, particularly in low- and middle-income countries (Bruni et al., 2018; Ferlay et al., 2020). Substantial epidemiological evidence has established that persistent infection with high-risk human papilloma virus (HPV) types serves as the necessary cause of cervical cancer, accounting for over 99% of cases worldwide (WHO. HPV, 2021). Although existing HPV transmission models and prevention strategies predominantly focus on female populations, male HPV infection warrants equal attention. Men act not only as HPV carriers but also play a crucial role in viral transmission through sexual contact. High-risk HPV types (e.g., HPV-16 and HPV-18) demonstrate substantial prevalence among males, potentially serving as persistent infection sources for females. Furthermore, HPV is associated with various cancers beyond cervical cancer, including anal, penile, and oropharyngeal cancers, highlighting the importance of understanding HPV transmission dynamics across both genders (WHOa; Green, 2018).

Cervical cancer ranks as the fourth most common cancer among women globally. According to the latest 2022 global statistics, approximately 604,000 new cases and 342,000 deaths were recorded annually, with nearly 90% of these occurring in developing regions where access to prevention and treatment services remains limited.Data from the National Cancer Center report an incidence rate of 11.4 per 100,000 and a mortality rate of 3.4 per 100,000, both showing an increasing trend (Arie; WHO, 2020; Sung et al., 2021). In China, cervical cancer has emerged as a growing health concern with distinctive epidemiological patterns.In 2020, the number of new cervical cancer cases in China is 109,741 and the number of deaths is 59,060. Combination of a substantial population and relatively high rates of incidence and death results in China contributing 18.2% to the total global burden of cervical cancer (Bray et al., 2018; Guo et al., 2021; Duan et al., 2022; WHOb). As shown in Fig. 1, the global burden distribution highlights significant regional disparities. Fig. 2 demonstrates the temporal trends in incidence and mortality rates in China from 1990 to 2021. The Xinjiang of China deserves particular attention due to its multi-ethnic composition and unique geographical setting, demonstrating higher incidence rates compared to national averages and significant disparities among different ethnic groups and between urban and rural areas (Ailawaer et al., 2024; Rifhat et al., 2023). Current HPV prevention strategies include vaccination, surgical excision of lesions, sexual abstinence, and condom use. Vaccination has demonstrated significant efficacy in preventing infections with high-risk HPV strains. Consequently, the WHO recommends vaccination for girls aged 9 to 14 (WHO. HPV, 2021). By October 2019, more than 100 nations had integrated HPV vaccines into their national immunization programs (Rifhat et al., 2023). Moreover, cervical cancer screening methods (including Pap smears and HPV testing) remain crucial for early detection and timely intervention. Despite these available preventive measures, the global burden of HPV-related cancers persists at alarmingly high levels, particularly in regions with limited healthcare infrastructure and resource constraints (Elbasha & Dasbach, 2010; Lei et al., 2020; WHOc).

Fig. 1.

Age-standardized incidence rate of cervical cancer in countries worldwide, 2022 (Source: https://www.int/zh/news-room/fact-sheets/detail/cervical-cancerWHO).

Fig. 2.

Temporal trends in cervical cancer incidence and mortality rates in China (1990–2021) (Data source: https://vizhub.healthdata.org/gbd-results/).

However, in Xinjiang of China, influenced by traditional concepts, backward culture, multi-ethnicity and other factors, although the Chinese government has encouraged young women to get the HPV vaccine since 2019, the effect has not been satisfactory so far. Many people do not understand and reject it, resulting in a very low coverage rate. Therefore, in Xinjiang, the main ways to prevent and control the spread of HPV infection and secondary cervical cancer still are as follows: First, strengthen various sexuality protective measures and health education and publicity, such as promoting the use of condoms, drug intervention, reducing sexual partners and unsafe sexual behaviors, etc., with the aim of reducing the disease transmission rate between men and women; Second, enhance medical screening and intensify treatment efforts to reduce the number of female and male infected with HPV, women precancerous lesions, and then secondary cervical cancer patients.

Mathematical modeling has become an indispensable tool for understanding HPV transmission dynamics and evaluating intervention strategies. Numerous modeling approaches have been developed to address various aspects of HPV transmission and control. Drolet et al. (Drolet et al., 2021) constructed a transmission-dynamic model (HPV-ADVISE LMIC) to evaluate HPV vaccination strategies in LMICs, and found that two-dose vaccination of girls aged 9-14 years is the most efficient and cost-effective approach to prevent cervical cancer, with potential for elimination in some countries. Peng et al. (Peng et al., 2018) developed an age-structured ordinary differential equation model calibrated to Texas and US data, and predicted that cervical cancer incidence would decline more rapidly in Texas than nationally, with greater sensitivity to changes in vaccination coverage and sexual behavior. Brisson et al. (Brisson et al., 2020) developed a multi-model comparative framework (HPV-ADVISE, Harvard, Policy1-Cervix) to assess HPV vaccination and screening strategies in 78 LMICs, and found that girls-only vaccination with 90% coverage could eliminate cervical cancer in most countries by 2100, while adding twice-lifetime screening accelerated elimination by 11-31 years and averted an additional 12.1 million cases. Zhong et al. (Zhong et al., 2021) constructed an age-structured ordinary differential equation (ODE) model to simulate HPV transmission and oropharyngeal cancer (OPC) progression in Texas, and found that a 50% increase in HPV vaccination uptake could prevent 4403 cumulative OPC cases over 100 years with an ICER of $94,518 per QALY gained. Ren et al. (Ren et al., 2025) constructed a two-sex, age-structured dynamic model with optimal control to analyze HPV transmission and cervical cancer progression in China, and found that prioritizing vaccination for 14-year-old girls and boys, followed by catch-up vaccination for women aged 15 to 44, minimizes the basic reproduction number to 0.244 and is the most cost-effective strategy under limited vaccine supply. Chauhan et al. (Chauhan et al., 2020) built a decision-analytic Markov model to compare the cost-effectiveness of different cervical cancer screening intervals and HPV vaccination. The results indicated that screening frequency should be tailored to HPV vaccination coverage levels, with higher coverage allowing for less frequent screening (e.g., every 10 years). Xia et al. (Xia et al., 2021) built an age-structured transmission-economic model to compare multiple intervention scenarios for cervical cancer elimination. They concluded that synchronizing vaccine switching (bivalent to nonavalent) with tailored screening intervals maximizes health benefits and leads to net cost savings after 2036. Gao et al. (Gao et al., 2021) constructed a dynamic SIS-type compartmental model that incorporates heterosexual populations and men who have sex with men (MSM) to evaluate HPV vaccination strategies. Their study found that prioritizing vaccination in MSM is the most effective strategy to reduce overall HPV prevalence, especially in populations with high female vaccination coverage.

Inspired by these foundational models, this study aims to develop a novel dynamic multi-compartmental model that integrates both biological and behavioral factors influencing HPV transmission. The model will specifically examine parameter variations representing potential vaccination, sexuality protective measures, screening, treatments, and behavioral interventions on HPV transmission dynamics, with particular focus on cross-gender transmission pathways. By investigating inter-gender transmission model, this research seeks to provide a more comprehensive understanding of HPV spread, ultimately contributing to more effective prevention strategy formulation. The research objectives include gaining deeper insights into HPV dynamics (from infection to precancerous lesions and cancer progression) and evaluating the potential impact of different intervention parameter configurations on preventing HPV-related cancers.

This paper is structured as follows: In Section 2, we propose a two-sex HPV transmission dynamical model, providing the basic properties of the model and the biological description of the model parameters. Section 3 presents the mathematical analysis, where the basic reproduction number is calculated and the threshold criteria on the main dynamical behavior of the model are established. In Section 4, the data fitting and and parameters estimation are presented. Section 5 is the sensitivity analysis on the basic reproduction number and model parameters. In Section 6, some control strategies involving model parameters are discussed. Lastly, In Section 7, a conclusion is given.

2. Model formulation

In this study, the modeling of the male population is restricted to the risk of HPV acquisition and omits carcinogenic transformation. This decision is based on the fact that, although males are susceptible to HPV-associated pathologies (e.g., genital warts, anal and throat cancers), the burden is markedly lower than that of cervical cancer, which remains the primary focus of global public health interventions. Furthermore, considering the reality that the HPV vaccination rate in Xinjiang region is very low, in our modeling, we will not specifically set up a compartment for vaccine recipients. Instead, the preventive and control effect produced by vaccination will be regarded as a reduction in the disease transmission rate and decreasing the susceptible population involved in the spread of diseases.

In this section, we will formulate a two-sex deterministic model to study the transmission of HPV infection in a heterosexually active population. The population is divided into two groups, namely, heterosexual females and heterosexual males (for simplicity, we will use females and males below), and subscripts f and m are used to denote them. In female group, the population is divided into 4 classes: susceptible (S_f), HPV infected I_HPVf, premalignant cervical lesion I_CIN and cervical cancer patient I_CC. In male group, the population is divided into 2 classes: susceptible (S_m), HPV infected I_HPVm. The interaction mechanism between them is represented in Fig. 3. The total female and male groups have the natural mortality rates μ_f and μ_m, respectively. Additionally, we denote by σ_f the mortality rate of cervical cancer for female, and by σ_m the HPV-infection-related mortality for male.

Fig. 3.

The dynamical diagram of HPV infection and secondary cervical cancer.

Based on the current actual situation where young people in China have reached sexual maturity, we assume that the young men and women who reach the age of 16 each year enter the susceptible compartments S_f and S_m with the recruitment rates Δ_f and Δ_m.

Susceptible female and male individuals are infected with HPV at infection rates ξ_f and ξ_m, and enter the HPV-infected compartments I_f and I_m. Although most primary HPV infections can clear HPV from their bodies and return to susceptible individuals through their own immunity, here we record this conversion rate for men and women as γ_m and γ_1f, respectively. However, there are still some women infected with HPV who develop severe HPV infection after persistent infection, that is, they will enter the precancerous lesion compartment (see [47,48]), here we record the conversion rate of entering precancerous lesions as γ_2f. For female patients with precancerous lesions, after a series of effective treatments, they can transform into mild HPV infection or even recover and become susceptible. Here, we denote these two conversion rates as η_1f and η_2f. However, if precancerous lesions delay treatment or the treatment is ineffective, their condition will further deteriorate and eventually turn into cervical cancer patients. Here, we denote this conversion rate as η_3f.

Therefore, in this paper based on the above compartmental diagram we can directly establish the following model of ordinary differential equation on the dynamics of HPV infection and secondary cervical cancer:

$$\left\{\begin{array}{ll}& S_f^{\prime }={\mathrm{\Delta }}_f-{\xi }_f{S}_f+{\gamma }_{1f}{I}_{\mathit{HPV\; f}}+{\eta }_{1f}{I}_{\mathit{CIN}}-{\mu }_f{S}_f,\\ & I_{\mathit{HPV\; f}}^{\prime }={\xi }_f{S}_f+{\eta }_{2f}{I}_{\mathit{CIN}}-({\mu }_f+{\gamma }_f)I_{\mathit{HPV\; f}},\\ & I_{\mathit{CIN}}^{\prime }={\gamma }_{2f}{I}_{\mathit{HPV\; f}}-({\mu }_f+{\eta }_f)I_{\mathit{CIN}},\\ & I_{CC}^{\prime }={\eta }_{3f}{I}_{\mathit{CIN}}-({\mu }_f+{\sigma }_f)I_{CC},\\ & S_m^{\prime }={\mathrm{\Delta }}_m-{\xi }_m{S}_m+{\gamma }_m{I}_m-{\mu }_m{S}_m,\\ & I_{\mathit{HPV\; m}}^{\prime }={\xi }_m{S}_m-({\mu }_m+{\gamma }_m+{\sigma }_m)I_{\mathit{HPV\; m}}.\end{array}\right)$$ (1)

The forces of infection ξ_f and ξ_m are given by

$${\xi }_f=\frac{{\beta }_{mf}{I}_{\mathit{HPV\; m}}}{N_f},{\xi }_m=\frac{{\beta }_{fm1}{I}_{\mathit{HPV\; f}}+{\beta }_{fm2}{I}_{\mathit{CIN}}}{N_m},$$

where N_f = S_f+I_HPVf+I_CIN+I_CC and N_m = S_m+I_HPVm.

For model (1), we denote by N = N_f+N_m the total population of female and male, and assume that the parameters Δ_f, Δ_m, μ_f, μ_m, β_mf, β_fm1,β_fm2, γ_2f, γ_m, α_1f and η_3f are positive constants, and γ_1f, η_1f and η_2f are nonnegative constants. All parameters in model (1) are listed in Table 1.

Table 1.

Description of parameters in model (1).

Symbol	Description
Δf	Recruits into sexually active female
Δm	Recruits into sexually active male
μf (μm)	natural mortality rate for female (male)
βmf	Transmission rate from males to females
βfm1	Transmission rate from HPV infected female to male
βfm2	Transmission rate from precancerous lesions female to male
γ1f	Conversion rate from HPV infected female to susceptible female
γ2f	Conversion rate from HPV infected female to precancerous lesions female
γm	Conversion rate from HPV infected male to susceptible male
η1f	Conversion rate from precancerous lesions female to susceptible female
η2f	Conversion rate from precancerous lesions female to HPV infected female
η3f	Conversion rate from precancerous lesions female to cervical cancer female
σf	Mortality rate due to cervical cancer for female
σm	Mortality rate due to HPV for male

From the biological considerations, the initial conditions for any solution of model (1) are given as

$$\begin{array}{ll}& S_k(0)=S_{k0}>0,k=f,m,I_{\mathit{HPV\; f}}(0)=I_{\mathit{1f0}}\ge 0,\\ & I_{\mathit{CIN}}(0)=I_{\mathit{CIN0}}\ge 0,I_{CC}(0)=I_{\mathit{CC0}}\ge 0,I_{\mathit{HPV\; m}}(0)=I_{\mathit{HPV\; m0}}\ge 0,\end{array}$$ (2)

Based on the fundamental theory of ordinary differential equations, it can be easily proved that the solution u(t, u₀) = (u_f(t, u₀), u_m(t, u₀)) of model (1) satisfying the initial conditions (2) uniquely exists and is nonnegative for all t ≥ 0, where u_f(t, u₀) = (S_f(t, u₀), I_HPVf(t, u₀), I_CIN(t, u₀), I_CC(t, u₀)), u_m(t, u₀) = (S_m(t, u₀), I_HPVm(t, u₀)), u₀ = (S_f0, I_HPVf0, I_CIN0, I_CC0, S_m0, I_HPVm0).

3. Mathematical analysis of model

In model (1) we see that the variable I_CC does not appear in the first three equations. Therefore, in the following discussion we can consider the sub-model of model (1):

$$\left\{\begin{array}{ll}& S_f^{\prime }={\mathrm{\Delta }}_f-{\xi }_f{S}_f+{\gamma }_{1f}{I}_{\mathit{HPV\; f}}+{\eta }_{1f}{I}_{\mathit{CIN}}-{\mu }_f{S}_f,\\ & I_{\mathit{HPV\; f}}^{\prime }={\xi }_f{S}_f+{\eta }_{2f}{I}_{\mathit{CIN}}-({\mu }_f+{\gamma }_f)I_{\mathit{HPV\; f}},\\ & I_{\mathit{CIN}}^{\prime }={\gamma }_{2f}{I}_{\mathit{HPV\; f}}-({\mu }_f+{\eta }_f)I_{\mathit{CIN}},\\ & S_m^{\prime }={\mathrm{\Delta }}_m-{\xi }_m{S}_m+{\gamma }_m{I}_{\mathit{HPV\; m}}-{\mu }_m{S}_m,\\ & I_{\mathit{HPV\; m}}^{\prime }={\xi }_m{S}_m-({\mu }_m+{\gamma }_m+{\sigma }_m)I_{\mathit{HPV\; m}}.\end{array}\right)$$ (3)

The initial conditions for any solution of model (3) are given as

$$\begin{array}{ll}& S_k(0)=S_{k0}>0(k=f,m),I_{\mathit{HPV\; f}}(0)=I_{\mathit{HPV\; f0}}\ge 0,\\ & I_{\mathit{CIN}}(0)=I_{\mathit{CIN0}}\ge 0,I_{\mathit{HPV\; m}}(0)=I_{\mathit{HPV\; m0}}\ge 0.\end{array}$$ (4)

We easily prove that model (3) has a unique nonnegative solution u(t, u₀) = (u_f(t, u₀), u_m(t, u₀)) satisfying the initial conditions (4) and defined for all t ≥ 0, where u_f(t, u₀) = (S_f(t, u₀), I_HPVf(t, u₀), I_CIN(t, u₀)), u_m(t, u₀) = (S_m(t, u₀), I_HPVm(t, u₀)) and u₀ = (S_f0, I_HPVf0, I_CIN0, S_m0, I_HPVm0). And then, we can easily prove that the region

$$\begin{array}{ll}H=& \{(S_f,I_{\mathit{HPV\; f}},I_{\mathit{CIN}},S_m,I_{\mathit{HPV\; m}})\in {\mathcal{R}}_{+}^5:\\ & S_f+I_{\mathit{HPV\; f}}+I_{\mathit{CIN}}\le \frac{{\mathrm{\Delta }}_f}{{\mu }_f}≔N_f^{\ast },S_m+I_{\mathit{HPV\; m}}\le \frac{{\mathrm{\Delta }}_m}{{\mu }_m}≔N_m^{\ast }\}.\end{array}$$

is globally attractive and positively invariant for model (3). Thus, in this section our discussions can be stated for model (3) only in the region H.

We define

$$R_{0,fm}=\frac{{\beta }_{fm1}({\mu }_f+{\eta }_f)+{\gamma }_{2f}{\beta }_{fm2}}{({\mu }_f+{\gamma }_f)({\mu }_f+{\eta }_f)-{\gamma }_{2f}{\eta }_{2f}},R_{0,mf}=\frac{{\beta }_{mf}}{{\mu }_m+{\gamma }_m+{\sigma }_m}.$$

R_0,mf represents the average number of susceptible women infected by a single infected male over the course of his entire infectious period in the early stage of the disease. By contrast, R_0,fm represents the average number of susceptible men infected by an infected woman (with HPV infection and precancerous lesions) over her entire infectious period. Using the next generation approach (Rifhat et al., 2025), we derive that the basic reproduction number for model (3) is

$$R_0=\sqrt{R_{0,mf}{R}_{0,fm}}.$$

R₀ represents the number of susceptible individuals who are infected by one infected person throughout the entire infection period in the early stage of the disease epidemic.

The model (3) always has a unique disease-free equilibrium (DFE) $E^0=(S_f^0,0,0,S_m^0,0)$ with $S_f^0=N_f^{\ast }$ and $S_m^0=N_m^{\ast }$. On the stability of DFE, we have the following results.

Theorem 3.1

When R₀ < 1, the DFE E₀ is locally asymptotically stable; when R₀ > 1, the DFE E₀ is unstable.

Theorem 3.2

When R₀ ≤ 1, then DFE E₀ is globally asymptotically stable.

The detailed proofs of Theorem 3.1, Theorem 3.2 are given in Appendix A.

On the existence of endemic equilibrium, by the detailed calculation we can easily obtain the following conclusion.

Theorem 3.3

When R₀ > 1, model (3) has a unique endemic equilibrium (EE) $E^{\ast }=(S_f^{\ast },I_{\mathit{HPV\; f}}^{\ast },I_{\mathit{CIN}}^{\ast },S_m^{\ast },I_{\mathit{HPV\; m}}^{\ast })$, where

$$\begin{array}{ll}{S}_f^{\ast }=& \frac{\tilde{D}[{\gamma }_m{\mathrm{\Delta }}_f({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})+N_m({\mu }_m+{\gamma }_m){\mu }_m\tilde{C}]}{({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})\tilde{B}},\\ I_{\mathit{HPV\; f}}^{\ast }=& \frac{({\mu }_f+{\gamma }_f)({\mu }_f+{\eta }_f){\mu }_f{\mu }_m\tilde{D}(R_0^2-1)N_m}{({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})\tilde{B}},I_{\mathit{CIN}}^{\ast }=\frac{{\gamma }_{2f}}{{\mu }_f+{\eta }_f}{I}_{\mathit{HPV\; f}}^{\ast },\\ S_m^{\ast }=& \frac{N_f{N}_m({\mu }_m+{\gamma }_m){\mu }_f{\mu }_m+{\beta }_{mf}{N}_m({\mu }_m+{\gamma }_m){\mathrm{\Delta }}_f\tilde{C}}{{\beta }_{mf}{N}_m({\mu }_m+{\gamma }_m){\mu }_m\tilde{C}+{\mathrm{\Delta }}_f{\mu }_m{N}_f({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})},\\ I_{\mathit{HPV\; m}}^{\ast }=& \frac{({\mu }_m+{\gamma }_m){\mu }_f{\mu }_m\tilde{D}(R_0^2-1)N_f{N}_m}{{\beta }_{mf}{N}_m({\mu }_m+{\gamma }_m){\mu }_m\tilde{C}+{\mathrm{\Delta }}_f{\mu }_m({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})N_m}.\end{array}$$

where $\tilde{D}=({\mu }_f+{\gamma }_f)({\mu }_f+{\eta }_f)-{\gamma }_{2f}{\eta }_{2f}$, $\tilde{C}=({\mu }_f+{\gamma }_f)({\mu }_f+{\eta }_f)-{\eta }_{2f}{\gamma }_{2f}-{\eta }_{1f}{\gamma }_{2f}-({\mu }_f+{\eta }_f){\gamma }_{1f}$ and $\tilde{B}=\frac{{\beta }_{mf}}{N_f}{\mathrm{\Delta }}_m\tilde{C}+\tilde{D}{\mu }_f{\mu }_m$.

Remark 3.4

Notably, demonstrating the local asymptotic stability of the endemic equilibrium E∗ through linearization techniques, characteristic equation analysis, and the Routh-Hurwitz criterion proves to be highly challenging. Likewise, attempts to directly establish the global asymptotic stability of E∗ by employing Lyapunov functions in conjunction with LaSalle’s invariance principle have thus far been unsuccessful. Consequently, these questions remain unresolved at present and will not be explored further in this study.

Now, we continue to discuss the uniform persistence of the disease for model (3) by using the persistence theory of dynamical systems.

Theorem 3.5

Assume R₀ > 1. Then model (3) is uniformly persistent. That is, there exists a constant ɛ > 0 such that for any solution (S_f, I_HPVf, I_CIN, S_m, I_HPVm) of model (3) with initial values S_f(0) > 0, I_HPVf(0) ≥ 0, I_CIN(0) ≥ 0, S_m(0) > 0, I_HPVm(0) ≥ 0, and I_HPVf(0) +I_CIN(0) +I_HPVm(0) > 0, one has

$${\mathrm{lim\; inf}}_{t\to \infty }(S_f(t),I_{\mathit{HPV\; f}}(t),I_{\mathit{CINf}}(t),S_m(t),I_{\mathit{HPV\; m}}(t))\ge (ϵ,ϵ,ϵ,ϵ,ϵ).$$

The detailed proof of Theorem 3.5 is given in Appendix A.

Remark 3.6

From the conclusions established in Theorems 1-4 we easily see that for the transmission of HPV infection described by using model (3), the basic reproduction number is a sharp threshold value to control the spread of HPV infection, and also a core indicator for evaluating HPV’s ability to sustain transmission in a population. Specifically, when the basic reproduction number R₀ < 1 then the HPV infection will be controlled, and eventually to die out, otherwise, when R₀ > 1 , then the spread of HPV infection can not be efficaciously controlled, and even will break out.

4. Data and parameter fitting

This section presents the parameter estimation for model (1) using epidemiological data on cervical cancer and HPV infection cases recorded from January 2009 to December 2019 in Xinjiang of China. The data is obtained from major regional medical institutions in Xinjiang, including the Affiliated Hospitals of Xinjiang Medical University and other tertiary hospitals.

Model parameters and initial conditions are estimated through an iterative fitting process using Markov Chain Monte Carlo (MCMC) methodology. We implemented an adaptive MCMC approach, specifically the Delayed Rejection Adaptive Metropolis (DRAM) algorithm in MATLAB R2023b, conducting 50,000 iterations with a burn-in period of 45,001 cycles. Convergence is assessed using the Geweke diagnostic, with values approaching 1 indicating successful convergence. The fitting results, presented in Fig. 4, demonstrate close agreement between model simulations and empirical data.

Fig. 4.

(A) The results for fitting new annual cases of HPV infected from 2009 to 2019; (B) The results for fitting the new annual cases of diagnosed CC patients from 2009 to 2019; The solid line and dot represent the simulated curve and the actual data, respectively. The gray area signifies the ranges of 95% CI.

The parameter values in model (1) are derived from demographic and epidemiological sources:

• (i)Initial population sizes are back-calculated from 1994 census data, yielding N_f(0) = 5,778,500 and N_m(0) = 6,113,900 after accounting for 15 years of exponential decay at the estimated mortality rate (Rifhat et al., 2025);
• (ii)Initial infectious states were set to I_HPVf(0) = 16,785 (HPV infections)and I_CC(0) = 237 (cervical cancer cases) based on 2009 hospital surveillance data.

The parameter estimation provides mean values, standard deviations, and 95% credible intervals for all system parameters and the basic reproduction number R₀, as detailed in Table 2.

Table 2.

Estimated values of parameters in model (1).

Parameters	Mean value	Std	95% CI	Reference
Δf	157245.91	-	-	Rifhat et al. (2025)
Δm	164287.27	-	-	Rifhat et al. (2025)
μf (μm)	1/74.42	-	-	Rifhat et al. (2025)
βmf	0.4512	0.114	[0.4480,0.4544]	MCMC
βfm1	0.75579	0.17947	[0.7508,0.7608]	MCMC
βfm2	0.61528	0.15576	[0.6110,0.6196]	MCMC
γ1f	0.48597	0.096447	[0.4833,0.4886]	MCMC
γ2f	0.28476	0.014857	[0.2833,0.2852]	MCMC
γm	0.32959	0.018221	[0.3291,0.3301]	MCMC
η1f	0.16563	0.045308	[0.1644,0.1669]	MCMC
η2f	0.032596	0.016932	[0.0321,0.0331]	MCMC
η3f	0.058668	0.0040189	[0.0586,0.0588]	MCMC
σf	0.3333	0.19007	[0.3280,0.3386]	MCMC
σm	0.027088	0.014536	[0.0267,0.0275]	MCMC
Sf(0)	7.222 × 106	3.3573 × 105	[7.2136 × 106, 7.2323 × 106]	MCMC
IHPVf(0)	6785	-	-	(ii)
ICIN(0)	10826	776.92	[1.0805 × 104, 1.0848 × 104]	MCMC
ICC(0)	237	-	-	(ii)
Sm(0)	6.9992 × 106	4.391 × 105	[6.9870 × 106, 7.0114 × 106]	MCMC
IHPVm(0)	8164	613	[7.9003 × 103, 9.009 × 103]	MCMC

The fitting results of the newly reported cases are shown in Fig. 4. As shown in the figure, the model fits the data well. Fig.(A) indicates that the incidence of HPV infection in Xinjiang shows a stable trend, and Fig.(B) shows that the incidence of cervical cancer in Xinjiang is increasing year by year. Based on the parameter estimation given in Table 2, the basic reproduction number R₀ of the model (1) is calculated as

$$R_0=1.5023(95\%CI:1.3583\sim 1.6464),$$

as illustrated in Fig. 5(A). According to Theorem 3.5, this implies that the disease is uniformly persistent, underscoring that cervical cancer remains a significant public health concern in Xinjiang. The distribution of R₀, shown in Fig. 5(B), approximately follows a normal distribution with a slight positive skew. This near-normality allows straightforward estimation of the confidence interval and mean value of R₀.

Fig. 5.

(A) The Markov chain for the last 5000 samples of ${\mathcal{R}}_0$, with ${\mathcal{R}}_0$ values indicated with blue dots; (B) The posterior frequency distribution (histogram) of ${\mathcal{R}}_0$

5. Sensitivity analysis

Fig. 6 presents the sensitivity of the basic reproduction number $({\mathcal{R}}_0)$ to various model parameters. From this figure, two key observations emerge clearly: ${\mathcal{R}}_0$ shows greater sensitivity to parameters η_1f, γ_1f, γ_m, σ_m, β_mf, β_fm1, and β_fm2, meaning small fluctuations in these parameters can significantly shift ${\mathcal{R}}_0$ and thus alter HPV transmission risk; in contrast, it exhibits minor sensitivity to η_2f, η_3f, γ_2f, and σ_f, indicating adjustments to these parameters have little impact on ${\mathcal{R}}_0$ and overall transmission potential. Additionally, Fig. 6 clarifies the direction of correlation between parameters and ${\mathcal{R}}_0$: transmission related parameters β_mf, β_fm1, β_fm2 and γ_2f are positively correlated with ${\mathcal{R}}_0$ higher values of these parameters directly increase ${\mathcal{R}}_0$ by boosting transmission or weakening convertion; all other parameters (η_1f,η_2f, η_3f γ_1f, γ_m, σ_m) are negatively correlated with ${\mathcal{R}}_0$, such that increases in these parameters (e.g., stronger immune clearance via γ_1f or γ_m) reduce ${\mathcal{R}}_0$ and inhibit HPV spread.

Fig. 6.

The PRCC results for dependence of${\mathcal{R}}_0$on different parameters, with the light-gray area signifying statistically insignificant PRCC values, while the dark-gray areas represent those with moderate relationships. The asterisk (∗) indicates a significant impact at the 0.05 significance level.

To systematically assess the sensitivity of parameters in Model (1), we adopted a combined approach of Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) (Rifhat et al., 2025) a rigorous framework that balances comprehensive parameter variability capture and accurate correlation quantification. For all input parameters (η_1f, η_2f, η_3f, γ_1f, γ_2f, γ_m, β_mf, β_fm1, β_fm2, σ_f, σ_m), we assumed a normal distribution (with mean and standard deviation values detailed in Table 2), a standard practice in epidemiological modeling that reflects realistic parameter uncertainty. Using LHS, we generated 2000 stratified samples of these parameters stratification ensures even coverage of each parameter's range, minimizing sampling bias and enhancing the reliability of sensitivity results compared to simple random sampling. We focused on two clinically meaningful output variables: new HPV infections (I_HPVf) and new cervical cancer cases (I_CC), as these directly quantify the model's ability to predict disease burden. PRCC values were interpreted to reflect correlation direction: a positive PRCC indicates that an increase in the input parameter corresponds to an increase in the output variable (e.g., higher β_mf leading to more I_HPVf), while a negative PRCC denotes an inverse relationship (e.g., higher γ_1f resulting in fewer I_HPVf). The full sensitivity results for all parameters are visualized in Fig. 7, facilitating intuitive comparison of parameter impacts on I_HPVf and I_CC.

Fig. 7.

PRCC sensitivity analysis of model parameters. (A) Sensitivity of parameters to HPV infection prevalence over time; (B) Sensitivity of parameters to cervical cancer incidence over time; The color scale represents the magnitude and direction of PRCC values: red color indicates positive correlation, blue color indicates negative correlation. |PRCC| > 0.2 indicates statistical significance (p < 0.05).

The sensitivity analysis of 11 key parameters shows their temporal evolution from January 2009 to December 2019 directly affects new HPV infections (I_HPVf) and new cervical cancer cases (I_CC). LHS results identify critical influences: β_mf(male to female transmission rate) demonstrates the most pronounced impact on I_HPVf, serving as the dominant driver of HPV transmission, though its effect on I_CC is comparatively modest. β_fm1 and β_fm2 both correlate positively with new diagnoses, with β_fm2 exhibiting notably stronger and rapidly increasing influence. η_1f exhibits a strong and progressively increasing negative effect on both I_HPVf and I_CC, indicating its growing importance in reducing disease burden over time. η_2f shows divergent temporal patterns: while maintaining a significant negative effect on I_CC, its positive effect on I_HPVf shows a marked declining trend throughout the study period. η_3f maintains a significant positive influence on I_CC, though this effect demonstrates a gradual attenuation over time. γ_1f consistently shows significant and increasingly strong negative effects on both I_HPVf and I_HPVf, confirming the crucial role of enhanced viral clearance through immune responses or interventions. But γ_2f displays negligible effects on both outcomes, suggesting limited influence on disease dynamics. Similarly, γ_m shows minimal impact on model outputs. Transmission rates (β_mf, β_fm1, β_fm2) correlate positively with new diagnoses; reducing them (e.g., via condom use) curbs HPV spread and CC incidence. HPV-convertion rates (γ_f, γ_m) strongly impact BRN and disease prevalence, validating early effective treatment and screening important role in boosting clearance to lower infection and CC risk. The mortality rate σ demonstrates minimal contribution to disease dynamics across all time points.

The high sensitivity of model outputs to β_mf, γ_1f, and η_f points to two essential control mechanisms: boosting host clearance and curbing transmission. This supports public health strategies that combine barrier protection to reduce transmission with interventions that enhance natural immunity to accelerate clearance, thereby directly targeting the core transmission and progression pathways identified in the model.

The preceding sensitivity analysis systematically identifies the key parameters governing HPV transmission dynamics, with β_mf, γ_1f, and η_f emerging as the most influential drivers of disease spread. This directly translates to actionable public health strategies: controlling β_mf (male to female transmission) necessitates interventions targeting males, such as promoting condom use and reducing partner concurrency (FitzGerald et al., 2018; Marshall et al., 2022; Steiner et al., 2021); enhancing γ_1f (HPV clearance) involves boosting population immunity through vaccination and health education (Laganà et al., 2023; Marshall et al., 2022; McBride & Singh, 2018; Thanasas et al., 2020); and modulating η_f (CIN-convertion) relies on scaling up cervical screening and treatment of precancerous lesions (Wentzensen et al., 2021).

Building on these findings, Fig. 8 reveals a concerning trajectory: under current intervention levels, both HPV infections and cervical cancer incidence are projected to maintain an upward trend through 2084. This persistent growth underscores the insufficiency of existing control measures and highlights the urgent need for enhanced strategies. Crucially, our sensitivity results provide a strategic road map by targeting the high impact parameters identified,we can design more precise and effective interventions. The subsequent analysis will leverage these insights to simulate optimized control scenarios, aiming to identify the most implementable and sustainable approaches for reversing the current trajectory and achieving long-term disease reduction.

Fig. 8.

(A) Fitted and projected trends of HPV infection cases from 2009 to 2084; (B) Fitted and projected trends of cervical cancer cases from 2009 to 2084; The solid line represents the simulated curve, dots indicate observed data, the gray area denotes the 95% credible interval.

6. Control strategies analysis

Based on the sensitivity analysis, we systematically evaluate intervention strategies by modulating key parameters in the model, simulating their impact from 2024 to 2084. Through single and combined parameter analyses, we identify optimal intervention strategies by examining transmission rates, convertion parameters, and their synergistic effects on disease outcomes.

6.1. Single-parameter control strategies

6.1.1. Effect of transmission parameters (β_fm1, β_fm2, β_mf)

Single-parameter analysis reveals that transmission parameters significantly influence disease dynamics, with the male-to-female transmission rate (β_mf) exhibiting the most pronounced effect.

As shown in Fig. 11, reducing β_mf by 50% (to 0.5β_mf) is projected to reduce new cervical cancer cases by 99.97% compared to the baseline by the time of elimination in 2071, preventing approximately 3319 cervical cancer cases. This underscores that interventions targeting male-related transmission vectors-such as promoting consistent condom use and reducing sexual partner concurrency among men-constitute the single most potent lever for epidemic control.

Fig. 11.

The effect of the transmission parameters β_mf on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the transmission parameter β_mf. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

Further reduction of β_mf to 0.2β_mf accelerates the elimination timeline to 2047 and achieves a 99.96% reduction in new cases compared to the baseline, preventing 1978 cases due to earlier interruption of disease transmission. In contrast, the 0.8β_mf reduction strategy achieves incomplete control. Despite reducing new cases by 97.27% in 2084, it does not lead to disease elimination within the studied period, suggesting that more aggressive intervention is required for complete eradication.

In contrast, Fig. 9, Fig. 10 demonstrate that reductions in female-to-male transmission rates (β_fm1 and β_fm2) have more subdued individual effects. Even substantial reductions (e.g., 80%) in these parameters are insufficient to achieve disease elimination alone. This asymmetric impact highlights the critical importance of prioritizing interventions that target the male reservoir of infection within comprehensive HPV control strategies.

Fig. 9.

The effect of the transmission parameters β_fm1 on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the transmission parameter β_fm1. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

Fig. 10.

The effect of the transmission parameters β_fm2 on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the transmission parameter β_fm2. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

6.1.2. Effect of HPV conversion parameters (γ_1f, γ_2f, γ_m)

Enhancing HPV clearance rates represents a crucial immunological intervention strategy achievable through both biomedical and behavioral approaches.

As shown in Fig. 12, increasing the female clearance rate γ_1f produces progressive reductions in disease burden. enhancing the female clearance rate γ_1f shows progressive efficacy. We see that increasing γ_1f by 20% (to 1.2γ_1f) reduces cervical cancer by 62.29%, preventing 9886 cancer cases. At 50% enhancement, reductions improve to 91.64%, averting 14544 cases. The maximum enhancement 80% will achieve 97.83% reductions.

Fig. 12.

The effect of the γ_1f on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the γ_1f. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

From Fig. 13, we can see that although reducing the conversion rate γ_2f can lower the number of precancerous lesions over a relatively long period of time, and then further reduce the number of cervical cancer patients. However, due to the accumulation of more HPV-infected individuals, the disease will rebound in the long term and will return to its original scale after 2075.

Fig. 13.

The effect of the γ_2f on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the γ_2f. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

Fig. 14 shows a more obvious effect on increasing the clearance rate γ_m of men. We see that, increasing γ_m by 20%, the number of cervical cancer cases could be reduced by 70.02% by 2084, and approximately 11,158 cancer cases could be avoided. It was further increased to 50%, with a reduction of 95.36% respectively, preventing 15,195 cancer cases. The densest scene (80% enhancement) will result in a 99.08% reduction.

Fig. 14.

The effect of the γ_m on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the γ_m. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

These immunological enhancements correspond to practical interventions including HPV vaccination, smoking cessation, nutritional optimization, and lifestyle modifications that bolster innate immune function. The consistently superior performance of male-targeted interventions underscores the importance of comprehensive male engagement in HPV prevention, which provides dual protection through direct immunity and transmission disruption. While both strategies demonstrate significant efficacy, neither single-parameter intervention achieves complete disease elimination within the simulation timeframe.

6.1.3. Effect of CIN conversion parameters (η_1f, η_2f, η_3f)

The analysis of metastasis parameters of precancerous lesions revealed different intervention approaches and different effects. In particular, improving screening and treatment to increase the regression rate of precancerous lesions η_1f or reduce the conversion rate of cancer η_3f will demonstrate some short-term benefits.

It can be clearly seen from Fig. 15 that increasing η_1f by 80%, although the cases of HPV infection and cervical cancer decreased in the early stage, especially the decline in cervical cancer cases was significant, the cases of HPV infection and cervical cancer will rebound after 2044, and will exceed the original level after 2060. Therefore, increasing η_1f merely delays the time of the disease outbreak.

Fig. 15.

The effect of the η_1f on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the η_1f. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

In Fig. 17, the reduction of progression to invasive cancer η_3f indicates a clear trade-off between cancer prevention and the burden of HPV infection. While a reduction of 80% in η_3f can lower the incidence of cervical cancer by 61.09% in the short term. However, as the individuals with precancerous lesions treated enter the HPV-infected or susceptible individuals, the number of HPV-infected individuals is further increased instead. Therefore, as time goes by, the disease will rebound, and will return to its previous scale after 2060. This indicates that reducing η_3f can only have a short-term effect.

Fig. 17.

The effect of the η_3f on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the η_3f. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

Fig. 16 shows that the parameter η_2f has a very small impact on both HPV infection dynamics and cervical cancer incidence, and is insufficient to produce meaningful reductions in either HPV transmission and cervical cancer cases.

Fig. 16.

The effect of the η_2f on the disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females as influenced by the variations in the η_2f. (B) The number of new cervical cancer (CC) cases predicted under the same parameter variations.

Therefore, the observed limitations of individual parameter controls-whether incomplete efficacy or unintended consequences underscore the necessity for integrated approaches that simultaneously address multiple intervention points within the disease progression pathway.

6.2. Combined parameter analysis

6.2.1. Combined transmission control

Targeted control of female-to-male transmission parameters (β_fm1, β_fm2) alone shows measurable but limited effectiveness. Simulation results indicate that even with a substantial reduction of both parameters by 80%, disease elimination is only achieved by 2080. While this strategy averts 11333 cervical cancer cases-an 99.99% reduction compared to the baseline-the relatively late elimination year underscores the insufficiency of unidirectional transmission control in rapidly interrupting viral circulation within the population (Fig. 18).

Fig. 18.

The combined effect of controlling the transmission parameters β_fm1, β_fm2 on disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females under joint control of β_fm1 and β_fm2. (B) The number of new cervical cancer (CC) cases in females under joint control of β_fm1, β_fm2.

When the strategy is expanded to include the reduction of male-to-female transmission (β_mf), a marked synergistic effect is observed. Under the combined reduction of all three transmission parameters (β_fm1, β_fm2, β_mf) by 50%, disease elimination is accelerated to 2085, with 16192 cancer cases averted-corresponding to a 99.99% reduction relative to the baseline. This demonstrates that concurrently controlling transmission from both sexes, particularly by reducing the role of males as a key transmission reservoir, significantly enhances overall intervention efficacy (Fig. 19).

Fig. 19.

The combined effect of controlling the transmission parameters β_fm1, β_fm2, and β_mf on disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females under joint control of β_fm1, β_fm2, and β_mf. (B) The number of new cervical cancer (CC) cases in females under joint control of β_fm1, β_fm2, and β_mf.

Although high-intensity, multi-directional transmission control can theoretically achieve disease elimination, its practical implementation remains highly challenging. A sustainable elimination strategy should therefore integrate moderate transmission control with biomedical interventions like vaccination and screening.

Response surface analysis (Fig. 20, Fig. 21) reveals the relative importance of male to female and female to male transmission rates in determining the disease reproduction number R₀. While both parameters of female transmission rates exert considerable influence as shown in Fig. 20, the contour plot in Fig. 21(B) demonstrates a more pronounced effect of β_mf: R₀ falls below 1 when β_mf is reduced to approximately 0.4, regardless of the value of β_fm1.

Fig. 20.

Response surface analysis of β_fm1 and β_fm2 effects on disease transmission.

Fig. 21.

Response surface analysis of β_fm1 and β_mf effects on disease transmission.

6.2.2. Combined HPV conversion control

The simulation results reveal distinct outcomes between female-only and gender-inclusive immunological strategies, as shown in Fig. 22, Fig. 23. Strategies that combine enhanced female clearance γ_1f with reduced disease progression γ_2f, even at high intensity levels, consistently fail to achieve disease extinction before 2084. Specifically, scenarios with γ_1f increased by 20%, 50%, and 80% coupled with corresponding reductions in γ_2f all maintain endemic transmission throughout the simulation period (see Fig. 24).

Fig. 22.

The combined effect of controlling the female HPV transition parameters γ_1f, γ_2f on disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females under joint control of γ_1f and γ_2f. (B) The number of new cervical cancer (CC) cases in females under joint control of γ_1f and γ_2f.

Fig. 23.

The combined effect of controlling the female HPV transition parameters γ_1f, γ_2f, and γ_m on disease progression predicted over the period 2024-2084. (A) The number of new HPV infections in females under joint control of γ_1f, γ_2f, and γ_m. (B) The number of new cervical cancer (CC) cases in females under joint control of γ_1f, γ_2f, and γ_m.

Fig. 24.

Interaction effects of γ_1f and γ_2f parameters on disease clearance.

The epidemiological trajectory changes fundamentally when male clearance enhancement γ_m is incorporated into the intervention strategy. A tripartite approach increasing γ_1f by 80%, decreasing γ_2f by 50%, and increasing γ_m by 80% successfully achieves disease extinction by 2074. This coordinated strategy prevents 8229 cervical cancer cases, representing a 99.99% reduction compared to baseline projections. These findings demonstrate that female-focused immunological interventions alone cannot achieve disease elimination. Effective control requires gender-synchronized strategies that enhance clearance mechanisms in both sexes while simultaneously slowing disease progression in women.

Response surface analysis (Fig. 25) elucidates the relative impact of clearance parameters on the disease reproduction number R₀ and reveals nonlinear interactions among them. The visualization identifies a pivotal finding: R₀ drops below 1 when the male clearance rate γ_m exceeds approximately 0.85, a threshold that remains robust across variations in female clearance rates. This quantitatively confirms γ_m as a primary leverage point for intervention.

Fig. 25.

Interaction effects of γ_2f and γ_m parameters on disease clearance.

6.2.3. Combined CIN conversion control

The simulation results presented in Fig. 26 demonstrate that coordinated control of precancerous transition parameters has inherent limitations. Strategies combining enhanced regression rates η_1f, modified inter-stage conversion η_2f, and reduced progression to cancer η_3f consistently fail to achieve disease extinction before 2084, regardless of intervention intensity. Even under maximum parameter modulation increasing η_1f and η_2f by 80% while decreasing η_3f by 80%, the maximum achievable reduction in cervical cancer incidence reaches only 92.11% by 2084.

Fig. 26.

The combined effect of controlling the precancerous transition parameters η_1f, η_2f and η_3f on disease progression predicted over the period 2024-2084.

Response surface analysis (Fig. 27) quantitatively confirms the dominant role of the progression rate from precancer to cancer (η_3f) in determining overall cancer burden. This underscores that containing cervical cancer is most effectively achieved by interrupting the final step of oncogenic progression through robust screening and treatment of precancerous lesions.

Fig. 27.

Effects of η parameters on disease clearance.

The analysis of all combined parameter strategies demonstrate that no single category of intervention can independently eliminate cervical cancer.

6.3. Hybrid control strategy

Our comprehensive simulation analysis systematically evaluates multiple integrated intervention strategies to identify the most effective and feasible combination for cervical cancer elimination. We examine four distinct parameter combinations that synergize transmission control, immune enhancement, and clinical management:

Case 1

In this paper, β_fm is used as a combined control variable to collectively denote β_fm1 and β_fm2 for brevity. 60% reduction in β_fm, 50% increase in γ_m, and 20% increase in η_1f, η_2f.

As shown in Fig. 28, this combination fails to achieve extinction with moderate parameter adjustments but succeeds under intensive intervention, reaching extinction by 2077 with 99.99% reduction (9351 cervical cancer cases averted) and preventing 167,172 HPV infections. The required intensity reflects the inherent limitations of targeting female-to-male transmission as the primary control mechanism, though it yields substantial reductions in HPV infections.

Case 2

70% reduction in β_fm, 50% increase in γ_m, and 20% reduction in η_3f.

As shown in Fig. 29, this combination demands even higher transmission control than Case 1, achieving extinction by 2078 with 99.99% reduction (9969 cervical cancer cases averted) and preventing 166,372 HPV infections. The substantial β_fm reduction requirement (70%) makes this approach particularly challenging to implement in practice, though it produces significant dual benefits for both cervical cancer and HPV prevention.

Case 3

50% reduction in β_mf, 50% increase in γ_m, and 30% increase in η_1f and η_2f.

As shown in Fig. 30, this combination demonstrates superior efficiency, achieving extinction by 2082 with 99.99% reduction (13055 cervical cancer cases averted) and preventing 214,213 HPV infections. Compared to other strategies requiring 60-70% reductions in β, this approach achieves disease elimination with only a 50% reduction. Such moderate parameter adjustments enhance its feasibility for implementation in resource-limited settings while delivering the highest HPV prevention benefit among all cases.

Case 4

60% reduction in β_mf, 60% increase in γ_m, and 30% reduction in η_3f.

As shown in Fig. 31, this configuration achieves the highest impact with extinction by 2085 and 99.99% reduction (15313 cervical cancer cases averted), and prevents 247,097 HPV infections-the highest HPV prevention among all strategies. However, it requires the most intensive interventions across all parameters, particularly the 60% increase in male vaccination coverage.

The comparative analysis reveals distinct optimization strategies under different resource scenarios. When resources are sufficient, Case 1 represents the optimal choice for rapid elimination, achieving extinction by 2077 with 9351 cervical cancer cases averted and 167,172 HPV infections prevented. The prioritization of rapid elimination justifies the intensive interventions required, particularly the 60% reduction in β_fm. Conversely, under resource-constrained conditions, Case 3 emerges as the superior strategy, balancing moderate intervention requirements with substantial public health impact. With only a 50% reduction in β_mf-compared to the 60-70% reductions needed in other cases-this approach achieves elimination by 2082 while averting 13,055 cervical cancer cases and preventing 214,213 HPV infections, offering the best HPV prevention per unit of intervention intensity. The 30% increase in γ_m and improvement in screening parameters represent feasible enhancements to existing healthcare infrastructure, making this combination particularly suitable for implementation in regions like Xinjiang where resource allocation must balance competing health priorities.

Fig. 32 illustrates the number of avoided HPV infections and cervical cancer cases from 2025 to 2085 under the Case 3 intervention strategy, which reduces male-to-female transmission by 50%, increases male clearance by 50%, and improves screening and treatment efficiency by 30%. Over the entire projection period (2024-2086), this strategy achieves a 99.09% reduction in HPV infections and a 90.37% reduction in cervical cancer cases compared to the baseline, with approximately 1,013,161 HPV infections and 93,752 cervical cancer cases avoided. These results demonstrate that the integrated strategy not only provides immediate annual benefits but also sustains a near-elimination level of disease burden in the long term, supporting its potential as a feasible and sustainable pathway for cervical cancer elimination in resource-limited settings.

Fig. 33 shows the additional cervical cancer cases that could be prevented by launching the Case 3 intervention strategy one year earlier (2023 vs. 2024). By advancing implementation, an estimated 964 additional cervical cancer cases could be avoided by 2085. This demonstrates that earlier intervention significantly reduces cumulative disease burden, reinforcing the public health principle that earlier action yields greater benefit. Prompt implementation of integrated control strategies is therefore essential to maximize health gains and accelerate progress toward cervical cancer elimination.

Fig. 28.

The combined effect of controlling the parameters β_fm, η_1f,η_2f and γ_m on disease progression predicted over the period 2024-2084.

Fig. 29.

The combined effect of controlling the parameters β_fm, η_3f and γ_m on disease progression predicted over the period 2024-2084.

Fig. 30.

The combined effect of controlling the parameters β_mf, η_1f, η_2f and γ_m on disease progression predicted over the period 2024-2084.

Fig. 31.

The combined effect of controlling the parameters β_mf, η_3f and γ_m on disease progression predicted over the period 2024-2084.

Fig. 32.

Reduction in HPV infections and cervical cancer cases under Case 3 intervention strategy over the period 2025-2085.

Fig. 33.

Additional cervical cancer cases avoided by advancing Case 3 intervention from 2024 to 2023.

6.4. Conclusions

This study establishes a gender-stratified deterministic dynamical model to investigate HPV transmission dynamics and cervical cancer development within heterosexual populations. The model incorporates distinct clinical pathways for both genders, with females categorized into susceptible, HPV-infected, precancerous lesions, and cervical cancer states, while males are classified as susceptible or infected compartments.

The mathematical analysis provides the expression of the basic reproduction number R₀ of the model, and it is found that when R₀ ≤ 1, the disease is eventually extinct, while when R₀ > 1, the disease is continuously prevalent. In this way, the mathematical criteria for determining whether the disease is prevalent or not have been fully established.

The fitting results based on the actual medical data of HPV infection and cervical cancer cases in Xinjiang of China show that the basic reproduction number R₀ = 1.5023(95%CI : 1.3583 ∼ 1.6464), which confirms the endemic persistence of HPV infection and cervical cancer in Xinjiang region.

Sensitivity analysis identifies transmission rates (β_mf, β_fm1 and β_fm2) and disease conversion parameters (γ_1f, γ_2f and γ_m) as the most influential factors governing epidemiological dynamics, highlighting the necessity of integrated interventions targeting both transmission reduction and conversion delay. In conclusion, this study underscores that the effective containment of HPV and cervical cancer hinges on integrated strategies that simultaneously impede viral transmission through robust vaccination and education, enhance natural clearance via immune optimization and smoking cessation, and halt oncogenic progression via scalable screening and precancer treatment.

Numerical simulation analyses of various control strategies show that coordinated intervention strategies yield superior disease control outcomes compared to single-parameter interventions. Specifically, a combined strategy reducing the male-to-female transmission rate (β_mf) by 50%, enhancing the male vaccination coverage (γ_m) by 50%, and increasing the screening efficacy (η_1f and η_2f) by 30% represents the most feasible path to cervical cancer elimination under resource constraints, achieving disease extinction by 2082 with over 99.99% reduction in new cases (12,784 cervical cancer cases and 214,213 HPV infections averted).

However, to significantly reduce the number of cervical cancer patients in the short term, it is an effective strategy to increase the conversion rate (η_1f) from precancerous lesions to susceptible populations and decrease the conversion rate (η_3f) from precancerous lesions to cervical cancer when funds are limited.

Looking forward, subsequent research will explore the epidemiological impact and cost-effectiveness of various vaccination strategies within diverse sexual networks, including MSM populations, to refine intervention portfolios and guide the design of efficient, sustainable cervical cancer elimination programs.

CRediT authorship contribution statement

He Zhao: Software, Data curation, Conceptualization. Shayidan Abuduwaili: Software, Formal analysis, Data curation. Zhidong Teng: Writing – review & editing, Writing – original draft, Conceptualization. Lei Wang: Writing – review & editing. Ramziya Rifhat: Writing – review & editing, Software, Methodology, Conceptualization. Zihan Zhang: Data curation.

Availability of data and materials

Data sharing is not applicable because the current study fails to generate or analyze datasets.

Funding

This research is supported by the project of Top-notch Talents of Technological Youth of Xinjiang (Grant No. 2024TSYCCX0081),the National Natural Science Foundation of China(Grant Nos. 12101529, 12371504), Xinjiang Tianshan Innovation Team Project (Grant No. 2025D14017),Xinjiang Medical University College Students’ Innovation and Entrepreneurship Training Program Project (Grant No. X202510760463).

Conflict of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Handling Editor: Dr Daihai He

Appendix A.

Proof of Theorem 3.1

Proof. It is easy to calculate that the characteristic equation of linearized system for model (3) at equilibrium $E^0=(S_f^0,0,0,S_m^0,0)$ is

$$(\lambda +{\mu }_f)(\lambda +{\mu }_m)[{\lambda }^3+d_1{\lambda }^2+d_2\lambda +d_3]=0,$$

where d₁ = a₁₁+a₂₂+a₃₃, d₂ = a₁₁a₂₂ − a₁₂a₂₁+a₁₁a₃₃ − a₁₃a₃₁+a₂₂a₃₃ and d₃ = a₁₁a₂₂a₃₃ − a₁₂a₂₁a₃₃ − a₁₃a₂₁a₃₂ − a₁₃a₂₂a₃₁ with a₁₁ = μ_f+γ_f, a₁₂ = η_2f, a₁₃ = β_mf, a₂₁ = γ_2f, a₂₂ = μ_f+η_f, a₃₁ = β_fm1, a₃₂ = β_fm2 and a₃₃ = μ_m+γ_m. By calculating, we easily imply that d₁ > 0, d₂ > 0 and

$$\begin{array}{l}{d}_3=(a_{11}{a}_{22}-a_{12}{a}_{21})a_{33}\left[1-\frac{a_{13}(a_{21}{a}_{32}+a_{22}{a}_{31})}{(a_{11}{a}_{22}-a_{12}{a}_{21})a_{33}}\right]\\ =(a_{11}{a}_{22}-a_{12}{a}_{21})a_{33}[1-R_0^2]>0.\end{array}$$ (5)

Obviously, we have a₁₁a₂₂ − a₁₂a₂₁ > 0, and

$$\sqrt{\frac{a_{13}{a}_{31}}{a_{11}{a}_{33}}}<R_0=\sqrt{\frac{a_{13}(a_{22}{a}_{31}+a_{21}{a}_{32})}{a_{33}(a_{11}{a}_{22}-a_{12}{a}_{21})}}<1\iff a_{13}{a}_{31}<a_{11}{a}_{33},$$ (6)

From (5) and (6), by the detailed calculation we get

$$\begin{array}{ll}{d}_1{d}_2-d_3=& (a_{11}+a_{22})(a_{11}{a}_{22}-a_{12}{a}_{21})+(a_{11}+a_{22}+a_{33})a_{11}{a}_{33}\\ & -(a_{11}+a_{33})a_{13}{a}_{31}+(a_{11}+a_{22}+a_{33})a_{22}{a}_{33}+a_{13}{a}_{21}{a}_{32}\\ & >(a_{11}+a_{22})(a_{11}{a}_{22}-a_{12}{a}_{21})+a_{22}{a}_{11}{a}_{33}\\ & +(a_{11}+a_{22}+a_{33})a_{22}{a}_{33}+a_{13}{a}_{21}{a}_{32}>0.\end{array}$$

Therefore, by the Routh-Hurwitz criterion [60], the DFE E₀ is locally asymptotically stable. If R₀ > 1, then d₄ < 0. This implies that E₀ is unstable.

Proof of Theorem 3.2

Proof. It is clear that we only need to prove the conclusion in region H. We define the following Lyapunov function

$$L=S_f-S_f^0-S_f^0\ln \frac{S_f}{S_f^0}+I_{\mathit{HPV\; f}}+I_{\mathit{CIN}}+C{R}_{0,mf}\left(S_m-S_m^0-S_m^0\ln \frac{S_m}{S_m^0}+I_{\mathit{HPV\; m}}\right),$$

where

$$C=\max \left\{1,\frac{({\mu }_f+{\eta }_f)({\gamma }_{1f}+{\mu }_f)}{Q},\frac{{\gamma }_{2f}({\eta }_{1f}+{\mu }_f)}{Q}\right\},$$

Q = (μ_f+γ_f)(μ_f+η_f) − γ_2fη_2f and $S_k^0=N_k^{\ast }$ for k = f, m. It is clear that L is radially unbounded and positive definite in ${\mathcal{R}}_{+}^5$. The derivative of L along any solution of model (3) yields

$$\begin{array}{ll}\dot{L}=& \left(1-\frac{S_f^0}{S_f}\right)({\mathrm{\Delta }}_f-{\xi }_f{S}_f+{\gamma }_{1f}{I}_{\mathit{HPV\; f}}+{\eta }_{1f}{I}_{\mathit{CIN}}-{\mu }_f{S}_f)+\left[{\xi }_f{S}_f+{\eta }_{2f}{I}_{\mathit{CIN}}\right)\\ & \left(-({\mu }_f+{\gamma }_f)I_{\mathit{HPV\; f}}\right]+[{\gamma }_{2f}{I}_{\mathit{HPV\; f}}-({\mu }_f+{\eta }_f)I_{\mathit{CIN}}]+C{R}_{0,mf}\left\{\left(1-\frac{S_m^0}{S_m}\right)\right)\\ & \left(\times ({\mathrm{\Delta }}_m-{\xi }_m{S}_m+{\gamma }_m{I}_{\mathit{HPV\; m}}-{\mu }_m{S}_m)+[{\xi }_m{S}_m-({\mu }_m+{\gamma }_m)I_{\mathit{HPV\; m}}]\right\}.\end{array}$$

Using the equilibrium conditions ${\mathrm{\Delta }}_k={\mu }_k{S}_k^0$ and $\frac{S_k^0}{S_k}\ge 1$ for k = f, m, we obtain

$$\begin{array}{ll}\dot{L}=& \left(1-\frac{S_f^0}{S_f}\right){\mu }_f(S_f^0-S_f)+{\xi }_f{S}_f^0-\left(\frac{S_f^0}{S_f}{\gamma }_{1f}+{\mu }_f\right)I_{\mathit{HPV\; f}}-\left(\frac{S_f^0}{S_f}{\eta }_{1f}+{\mu }_f+{\eta }_{1f}\right)\\ & \left(+{\eta }_{3f}\right)I_{\mathit{CIN}}+C{R}_{0,mf}\left\{\left(1-\frac{S_m^0}{S_m}\right){\mu }_m(S_m^0-S_m)+{\xi }_m{S}_m^0-\left(\frac{S_m^0}{S_m}{\gamma }_m+{\mu }_m\right)I_{\mathit{HPV\; m}}\right\}\\ & \le -\frac{{\mu }_f}{S_f}{(S_f^0-S_f)}^2+{\beta }_{mf}{I}_{\mathit{HPV\; m}}-({\gamma }_{1f}+{\mu }_f)I_{\mathit{HPV\; f}}-({\eta }_{1f}+{\mu }_f+{\eta }_{1f}+{\eta }_{3f})I_{\mathit{CIN}}\\ & +C{R}_{0,mf}\left\{-\frac{{\mu }_m}{S_m}{(S_m^0-S_m)}^2+{\beta }_{fm1}{I}_{\mathit{HPV\; f}}+{\beta }_{\mathit{fm2}}{I}_{\mathit{CIN}}-({\gamma }_m+{\mu }_m)I_{\mathit{HPV\; m}}\right\}\\ & \le -\frac{{\mu }_f}{S_f}{(S_f^0-S_f)}^2-C{R}_{0,mf}\frac{{\mu }_m}{S_m}{(S_m^0-S_m)}^2-({\eta }_{1f}+{\eta }_{3f})I_{\mathit{CIN}}\\ & +({\gamma }_{1f}+{\mu }_f)\left[\frac{({\mu }_f+{\eta }_f)}{H}{\beta }_{fm1}{R}_{0,mf}-1\right]I_{\mathit{HPV\; f}}+({\eta }_{1f}+{\mu }_f)\left[\frac{{\gamma }_{2f}}{H}{\beta }_{fm2}{R}_{0,mf}-1\right]I_{\mathit{HPV\; f}}\\ & \le -\frac{{\mu }_f}{S_f}{(S_f^0-S_f)}^2-C{R}_{0,mf}\frac{{\mu }_m}{S_m}{(S_m^0-S_m)}^2-({\eta }_{1f}+{\eta }_{3f})I_{\mathit{CIN}}\\ & +({\gamma }_{1f}+{\mu }_f)[R_0^2-1]I_{\mathit{HPV\; f}}+({\eta }_{1f}+{\mu }_f)[R_0^2-1]I_{\mathit{CIN}}.\end{array}$$

Obviously, when R₀ ≤ 1, we have that $\dot{L}\le 0$ and the set $\mathrm{\Omega }=\{(S_f,I_{\mathit{HPV\; f}},I_{\mathit{CIN}},S_m,I_{\mathit{HPV\; m}})\in H:\dot{L}=0\}=\{S_f=S_f^0,I_{\mathit{CIN}}=0,S_m=S_m^0\}$. We easily prove that the maximal invariant set in set Ω is DFE E₀. Therefore, by LaSalle's invariance principle [61] and Theorem 3.1, the DFE E₀ is globally asymptotically stable.

Proof of Theorem 3.5

Proof. Define the set X = {(S_f, I_HPVf, I_CIN, S_m, I_HPVm) ∈ H: S_k > 0, k ∈ {f, m}}, and X₀ = {(S_f, I_HPVf, I_CIN, S_m, I_HPVm) ∈ X: I_HPVfI_CINI_HPVm > 0}. We have

$$\partial X_0=X\setminus X_0=\{(S_f,I_{\mathit{HPV\; f}},I_{\mathit{CIN}},S_m,I_{\mathit{HPV\; m}})\in X:I_{\mathit{HPV\; f}}{I}_{\mathit{CIN}}{I}_{\mathit{HPV\; m}}=0\}.$$

Firstly, from model (3) we can get that

$$N_f^{\prime }(t)\ge {\mathrm{\Delta }}_f-({\mu }_f+{\sigma }_f)N_f(t),N_m^{\prime }(t)\ge {\mathrm{\Delta }}_m-({\mu }_m+{\sigma }_m)N_m(t).$$

By the comparison principle, it follows that

$${\mathrm{lim\; inf}}_{t\to \infty }{N}_f(t)\ge \frac{{\mathrm{\Delta }}_f}{{\mu }_f+{\sigma }_f}≔N_f^{\ast },{\mathrm{lim\; inf}}_{t\to \infty }{N}_m(t)\ge \frac{{\mathrm{\Delta }}_m}{{\mu }_m+{\sigma }_m}≔N_m^{\ast }.$$

Without loss of generality, we can assume that $N_f(t)\ge N_f^{\ast }$ and $N_m(t)\ge N_m^{\ast }$ for all t ≥ 0. Then, from the equations of S_f and S_m in model (3) we further derive

$$S_f^{\prime }(t)\ge {\mathrm{\Delta }}_f-\left({\mu }_f+\frac{{\beta }_{mf}{N}_m^{\ast }}{N_f^{\ast }}\right)S_f(t)$$

and

$$S_m^{\prime }(t)\ge {\mathrm{\Delta }}_m-\left({\mu }_m+\frac{N_f^{\ast }}{N_m^{\ast }}({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})\right)S_m(t),$$

which implies that

$${\mathrm{lim\; inf}}_{t\to \infty }{S}_f(t)\ge \frac{{\mathrm{\Delta }}_f}{{\mu }_f+\frac{{\beta }_{mf}{N}_m^{\ast }}{N_f^{\ast }}},{\mathrm{lim\; inf}}_{t\to \infty }{S}_m(t)\ge \frac{{\mathrm{\Delta }}_m}{{\mu }_m+\frac{N_f^{\ast }}{N_m^{\ast }}(c_{1,fm}{\beta }_{fm}+c_{2,fm}{\beta }_{fm})}.$$

Without loss of generality, we can assume S_f(t) ≥ M_s and S_m(t) ≥ M_s for all t ≥ 0, where $M_s=\min \{\frac{{\mathrm{\Delta }}_f}{{\mu }_m+\frac{N_m^{\ast }}{N_f^{\ast }}{\beta }_{mf}},\frac{{\mathrm{\Delta }}_m}{{\mu }_m+\frac{N_f^{\ast }}{N_m^{\ast }}({\beta }_{\mathit{fm1}}+{\beta }_{\mathit{fm2}})}\}$. This shows that S_f and S_m are uniformly persistent in model (3). We define the set

$$\begin{array}{ll}{M}_{\partial }=& \left\{(S_f(0),I_{\mathit{HPV\; f}}(0),I_{\mathit{CIN}}(0),S_m(0),I_{\mathit{HPV\; m}}(0))\in X:\right)\\ & \left((S_f(t),I_{\mathit{HPV\; f}}(t),I_{\mathit{CIN}}(t),S_m(t),I_{\mathit{HPV\; m}}(t))\in \partial X_0,\forall t\ge 0\right\}.\end{array}$$ (7)

We first claim that

$$M_{\partial }=\{(S_f,0,0,S_m,0):S_k>0,k=f,m\}.$$ (8)

In fact, for any initial point (S_f0, 0, 0, S_m0, 0) with S_f0 > 0 and S_m0 > 0. Let S_f(t) and S_m(t) be the solutions of the following equations

$$S_f^{\prime }(t)={\mathrm{\Delta }}_f-{\mu }_f{S}_f(t),S_m^{\prime }(t)={\mathrm{\Delta }}_m-{\mu }_m{S}_m(t)$$

satisfying the initial value S_f(0) = S_f0 and S_m(0) = S_m0, respectively. Obviously, we have S_f(t) > 0 and S_m(t) > 0 for all t ≥ 0, and (S_f(t), 0, 0, S_m(t), 0) also is the solution of model (3) with the initial value (S_f0, 0, 0, S_m0, 0). Since (S_f(t), 0, 0, S_m(t), 0) ∈ ∂X₀ for all t ≥ 0, we hence have (S_f0, 0, 0, S_m0, 0) ∈ M_∂. This shows that {(S_f, 0, 0, S_m, 0) : S_k > 0, k = f, m}⊆ M_∂.

Next, we prove that M_∂ ⊆{(S_f, 0, 0, S_m, 0) : S_k > 0, k = f, m}. Let the initial point (S_f(0), I_HPVf(0), I_CIN(0), S_m(0), I_HPVm(0)) ∈ M_∂. We have (S_f(t), I_HPVf(t), I_CIN(t), S_m(t), I_HPVm(t)) ∈ ∂X₀ for all t ≥ 0. Hence, I_HPVf(t)I_CIN(t)I_HPVm(t) ≡ 0 for all t ≥ 0. Now, we need to prove that

$$I_{\mathit{HPV\; f}}(t)=I_{\mathit{CIN}}(t)=I_{\mathit{HPV\; m}}(t)\equiv 0,\forall t\ge 0.$$ (9)

Suppose that there is a t₀ ≥ 0 such that I_HPVf(t₀) +I_CIN(t₀) +I_HPVm(t₀) > 0. If I_HPVf(t₀) > 0, then from the second equation of model (3) we can obtain that $I_{\mathit{HPV\; f}}(t)\ge I_{\mathit{HPV\; f}}(t_0)e^{-({\mu }_f+{\gamma }_f)(t-t_0)}>0$ for all t ≥ t₀. Then, from third equation of model (3) we further can obtain that $I_{\mathit{CIN}}(t)>I_{\mathit{CIN}}(t_0)e^{-({\mu }_f+{\eta }_f)(t-t_0)}\ge 0$ for all t ≥ t₀. Further, from the last equation of model (3) and the uniform persistence of S_m we can obtain that $I_{\mathit{HPV\; m}}(t)>I_{\mathit{HPV\; m}}(t_0)e^{-({\mu }_m+{\gamma }_m)(t-t_0)}\ge 0$ for all t ≥ t₀. Thus, we finally imply that I_HPVf(t)I_CIN(t)I_HPVm(t) > 0 for all t > t₀ which leads to a contradiction. Similarly, if I_CIN(t₀) > 0 or I_HPVm(t₀) > 0, we also can imply that I_HPVf(t)I_CIN(t)I_HPVm(t) > 0 for all t > t₀ which leads to the contradiction.

From (9), we obtain I_HPVf(0) = I_CIN(0) = I_HPVm(0) = 0, and hence (S_f(0), I_HPVf(0), I_CIN(0), S_m(0), I_HPVm(0)) = (S_f(0), 0, 0, S_m(0), 0). This shows that M_∂ ⊆{(S_f, 0, 0, S_m, 0) : S_k > 0, k = f, m}.

From (8), restricted to set M_∂, model (3) degrades into

$$S_f^{\prime }(t)={\mathrm{\Delta }}_f-{\mu }_f{S}_f(t),S_m^{\prime }(t)={\mathrm{\Delta }}_m-{\mu }_m{S}_m(t).$$

Obviously, we have ${\lim }_{t\to \infty }{S}_f(t)=S_f^0$ and ${\lim }_{t\to \infty }{S}_m(t)=S_m^0$. This shows that the set

$$\begin{array}{ll}\mathrm{\Omega }=& \bigcup \left\{\omega (S_f(0),I_{\mathit{HPV\; f}}(0),I_{\mathit{CIN}}(0),S_m(0),I_{\mathit{HPV\; m}}(0)):\right)\\ & \left((S_f(0),I_{\mathit{HPV\; f}}(0),I_{\mathit{CIN}}(0),S_m(0),I_{\mathit{HPV\; m}}(0))\in M_{\partial }\right\}=\{E_0\},\end{array}$$ (10)

where ω(S_f(0), I_HPVf(0), I_CIN(0), S_m(0), I_HPVm(0)) is the omega limit set of solution (S_f(t), I_HPVf(t), I_CIN(t), S_m(t), I_HPVm(t)) of model (3). It is easily seen that Ω = {E₀} is isolated and acyclic.

Finally, the proof will be done if E₀ is a weak repeller for X₀, i.e., W^S(E₀)⋂X₀ = ∅, where W^S(E₀) is the stable manifold of E₀. Suppose it is not true, then there exists a solution (S_f(t), I_HPVf(t), I_CIN(t), S_m(t), I_HPVm(t)) in X₀, such that as t → ∞,

$$S_f(t)\to S_f^0,I_{\mathit{HPV\; f}}(t)\to 0,I_{\mathit{CIN}}(t)\to 0,S_m(t)\to S_m^0,I_{\mathit{HPV\; m}}(t)\to 0.$$ (11)

From R₀ > 1, we can choose an enough small ϵ > 0 such that

$$\begin{array}{ll}{R}_0(ϵ)=& \frac{{\beta }_{mf}\frac{{\mu }_f}{{\mathrm{\Delta }}_f}\left(\frac{{\mathrm{\Delta }}_f}{{\mu }_f}-ϵ\right)}{({\mu }_m+{\gamma }_m)}\\ & \times \frac{({\mu }_f+{\eta }_f){\beta }_{fm1}+{\gamma }_{2f}{\beta }_{fm2}}{({\mu }_f+{\gamma }_f)({\mu }_f+{\eta }_f-(1-{\alpha }_{1f})(1-{\rho }_{1f})(1-{\rho }_{2f}){\gamma }_f{\eta }_f)}\frac{{\mu }_m}{{\mathrm{\Delta }}_m}\left(\frac{{\mathrm{\Delta }}_m}{{\mu }_m}-ϵ\right)>1.\end{array}$$

For this ϵ > 0, by (11), there exists a T > 0 such that $S_f^0-ϵ<S_f(t)<S_f^0+ϵ$ and $S_m^0-ϵ<S_m(t)<S_m^0+ϵ$ for all t ≥ T. Define a Lyapunov function V(t) = I_HPVf(t) +c₁I_CIN(t) +c₂I_HPVm(t), where

$$c_1=-\frac{\frac{{\beta }_{mf}{\mu }_f}{{\mathrm{\Delta }}_f({\mu }_m+{\gamma }_m)}\left(\frac{{\mathrm{\Delta }}_f}{{\mu }_f}-ϵ\right)\frac{{\beta }_{fm1}{\mu }_m}{{\mathrm{\Delta }}_m}\left(\frac{{\mathrm{\Delta }}_m}{{\mu }_m}-ϵ\right)-({\mu }_f+{\gamma }_f)}{{\gamma }_{2f}},c_2=\frac{{\beta }_{mf}{\mu }_f}{{\mathrm{\Delta }}_f({\mu }_m+{\gamma }_m)}\left(\frac{{\mathrm{\Delta }}_f}{{\mu }_f}-ϵ\right).$$

The derivative of V(t) along the solution (S_f(t), I_HPVf(t), I_CIN(t), S_m(t), I_HPVm(t)) is given by

$$\begin{array}{ll}\frac{dV(t)}{dt}=& \left[\frac{{\beta }_{mf}{\mu }_f}{{\mathrm{\Delta }}_f}{S}_f-c_2({\mu }_m+{\gamma }_m)\right]I_{\mathit{HPV\; m}}+\left[c_2\frac{{\beta }_{fm1}{\mu }_m}{{\mathrm{\Delta }}_m}{S}_m+c_1{\gamma }_{2f}-({\mu }_f+{\gamma }_f)\right]I_{\mathit{HPV\; f}}\\ & +\left[c_2\frac{{\beta }_{fm2}{\mu }_m}{{\mathrm{\Delta }}_m}{S}_m+{\eta }_{2f}-c_1({\mu }_f+{\eta }_f)\right]I_{\mathit{CIN}}\\ & \ge \left[\frac{{\beta }_{mf}{\mu }_f}{{\mathrm{\Delta }}_f}\left(\frac{{\mathrm{\Delta }}_f}{{\mu }_f}-ϵ\right)-c_2({\mu }_m+{\gamma }_m)\right]I_{\mathit{HPV\; m}}+\left[c_2\frac{{\beta }_{fm1}{\mu }_m}{{\mathrm{\Delta }}_m}\left(\frac{{\mathrm{\Delta }}_m}{{\mu }_m}-ϵ\right)+c_1{\gamma }_{2f}\right)\\ & \left(-({\mu }_f+{\gamma }_f)\right]I_{\mathit{HPV\; f}}+\left[c_2\frac{{\beta }_{fm2}{\mu }_m}{{\mathrm{\Delta }}_m}\left(\frac{{\mathrm{\Delta }}_m}{{\mu }_m}-ϵ\right)+{\eta }_{2f}-c_1({\mu }_f+{\eta }_f)\right]I_{\mathit{CIN}}\\ & \ge \frac{({\mu }_f+{\gamma }_f)({\mu }_f+{\eta }_f)-(1-{\alpha }_{1f})(1-{\rho }_{1f})(1-{\rho }_{2f}){\eta }_f{\gamma }_f}{{\gamma }_{2f}}(R_0(ϵ)-1)I_{\mathit{CIN}}>0,\end{array}$$

for all t ≥ T. On that account, V(t) is increasing for t ≥ T. Therefore, V(t) does not tend to zero as t → ∞, which leads to a contradiction.

Finally, based on the persistence theory of dynamical systems, we can obtain that model (3) is uniformly permanent. This finishes the proof.