\(N(t)\) represents the total population. It is further divided into five classes such as vaccinated \(V(t)\), susceptible \(S\left( t \right)\), infected \(I\left( t \right)\), cervical patient\(C(t)\), and recovered \(R(t)\). The variables and parameters of the system are all non-negative.Modelling the awareness campaignIn this model, the awareness campaign is represented by the parameter \(\sigma\) (0 ≤ \(\sigma\) ≤ 1), which captures the proportion of the susceptible population that modifies their behavior in response to public health messaging. Specifically, the campaign affects the S to I transmission pathway: aware individuals reduce high-risk sexual contact and are more likely to seek vaccination, thereby lowering their effective probability of acquiring HPV. This is incorporated into the model through the modified transmission term \({\beta _1}\left( {1 - \sigma } \right)\). When \(\sigma =0\), no awareness exists and transmission proceeds at the baseline rate \({\beta _1}\). As \(\sigma \to 1\), the campaign approaches maximum effectiveness and transmission is suppressed. This formulation follows established approaches in behavioral epidemiology, where awareness or education interventions are modeled as a multiplicative reduction in the contact/transmission rate.Table 1 Parameters Description and its values of the HPV model.Full size tableBased on the description above, an HPV disease model can be constructed as follows:$$\left. \begin{gathered} \frac{{dV}}{{dt}}=\phi S+{\nu _1}R - \left( {{\kappa _1}+{\omega _1}} \right)V, \hfill \\ \frac{{dS}}{{dt}}=\pi +{\omega _1}V+{\psi _1}R - \left( {\phi +{\kappa _1}} \right)S - {\beta _1}\left( {1 - \sigma } \right)SI, \hfill \\ \frac{{dI}}{{dt}}={\beta _1}\left( {1 - \sigma } \right)SI - \left( {\tau +{\rho _1}+{\kappa _1}} \right)I, \hfill \\ \frac{{dC}}{{dt}}={\rho _1}I - \left( {{\kappa _1}+\xi } \right)C, \hfill \\ \frac{{dR}}{{dt}}=\tau I - \left( {{\kappa _1}+{\nu _1}+{\psi _1}} \right)R. \hfill \\ \end{gathered} \right\}.$$(1)Initial conditions for model (1):$$V\left( 0 \right) \ge 0,{\mkern 1mu} {\mkern 1mu} S\left( 0 \right) \ge 0,{\mkern 1mu} {\mkern 1mu} I\left( 0 \right) \ge 0,{\mkern 1mu} {\mkern 1mu} C\left( 0 \right) \ge 0,R\left( 0 \right) \ge 0.$$(2)Positive invariant setThe solution of the HPV model is uniformly bounded in a proper subset \(\Pi \in \Im _{5}^{+}\)At any given time t total population is given as follow:$$N(t)=V(t)+S(t)+I(t)+C\left( t \right)+R(t)$$(3)$$\frac{{dN\left( t \right)}}{{dt}}=\frac{{dV\left( t \right)}}{{dt}}+\frac{{dS\left( t \right)}}{{dt}}+\frac{{dI\left( t \right)}}{{dt}}+\frac{{dC\left( t \right)}}{{dt}}+\frac{{dR\left( t \right)}}{{dt}}$$(4)$$\frac{{dN\left( t \right)}}{{dt}}=\pi - {\kappa _1}N - \xi C$$(5)The feasible solution set in the model (1) will be:$$\Pi _{1} = \left\{ {(V,{\mkern 1mu} S,{\mkern 1mu} I,{\mkern 1mu} C,R) \in \Im ^{5} ,{\mkern 1mu} 0 \le N \le \frac{\pi }{{\kappa _{1} }}} \right\}.$$(6)Disease-free equilibrium pointsDisease-free point of the model (1), when the HPV infection does not exist in the population, is determined by putting \(I=C=0\)42,43.$$DFE=\left\{ {\frac{{\pi \phi }}{{{\kappa _1}\left( {\phi +{\kappa _1}+{\omega _1}} \right)}},\frac{{\pi \left( {{\kappa _1}+{\omega _1}} \right)}}{{{\kappa _1}\left( {\phi +{\kappa _1}+{\omega _1}} \right)}},0,0,0} \right\}.$$(7)Endemic equilibrium pointsTo determine the endemic equilibrium points of model (1), we set the right-hand side of the system in (1) equal to zero. After simplifying the resulting algebraic equations, we obtain:$$\left. \begin{gathered} V=\frac{{ - \phi a_{2}^{2}{a_3}+\tau {a_2}{a_4}{\nu _1} - \pi \tau {\beta _1}{\nu _1}+\pi \sigma \tau {\beta _1}{\nu _1}+\tau \phi {a_2}{\psi _1}}}{{\left( {\sigma - 1} \right){\beta _1}\left( {{a_1}{a_2}{a_3} - \tau {a_1}{\psi _1} - \tau {\nu _1}{\omega _1}} \right)}}, \hfill \\ S=\frac{{{a_2}}}{{\left( {1 - \sigma } \right){\beta _1}}}, \hfill \\ I=\frac{{{a_3}\left( {{a_1}{a_2}{a_4} - \pi {a_1}{\beta _1}+\pi \sigma {a_1}{\beta _1} - \phi {a_2}{\omega _1}} \right)}}{{\left( {\sigma - 1} \right){\beta _1}\left( {{a_1}{a_2}{a_3} - \tau {a_1}{\psi _1} - \tau {\nu _1}{\omega _1}} \right)}}, \hfill \\ C=\frac{{{a_3}{\rho _1}\left( {{a_1}{a_2}{a_4} - \pi {a_1}{\beta _1}+\pi \sigma {a_1}{\beta _1} - \phi {a_2}{\omega _1}} \right)}}{{\left( {\sigma - 1} \right){\beta _1}\left( {\xi +{\kappa _1}} \right)\left( {{a_1}{a_2}{a_3} - \tau {a_1}{\psi _1} - \tau {\nu _1}{\omega _1}} \right)}}, \hfill \\ R \to \frac{{\tau \left( {{a_1}{a_2}{a_4} - \pi {a_1}{\beta _1}+\pi \sigma {a_1}{\beta _1} - \phi {a_2}{\omega _1}} \right)}}{{\left( {\sigma - 1} \right){\beta _1}\left( {{a_1}{a_2}{a_3} - \tau {a_1}{\psi _1} - \tau {\nu _1}{\omega _1}} \right)}}. \hfill \\ \end{gathered} \right\}$$(8)Where$${a_1}={\kappa _1}+{\omega _1},\,\,\,\,\,{a_2}=\left( {\tau +{\rho _1}+{\kappa _1}} \right),\,\,\,\,\,\,{a_3}=\left( {{\kappa _1}+{\nu _1}+{\psi _1}} \right),\,\,\,\,\,{a_4}=\left( {\phi +{\kappa _1}} \right)$$Reproduction numberThe basic reproduction number is a crucial quantity in epidemic modelling that determines whether a disease will spread or eliminated from the population. If \({R_0}1\) then disease become persist in the population. The next-generation matrix method is employed to derive the reproduction number for the HPV model (1)44,45. Consider the infected compartments given below to calculate the reproduction number.$$\left. \begin{gathered} \frac{{dI}}{{dt}}={\beta _1}\left( {1 - \sigma } \right)SI\left( {\tau +{\rho _1}+{\kappa _1}} \right)I, \hfill \\ \frac{{dC}}{{dt}}={\rho _1}i - \left( {{\kappa _1}+\xi } \right)C, \hfill \\ \end{gathered} \right\}.$$(9)Let$$F=\left( {\begin{array}{*{20}{c}} {{\beta _1}\left( {1 - \sigma } \right)SI} \\ 0 \end{array}} \right),$$$$V=\left( {\begin{array}{*{20}{c}} {\left( {\tau +{\rho _1}+{\kappa _1}} \right)I} \\ { - {\rho _1}I+\left( {{\kappa _1}+\xi } \right)C} \end{array}} \right)$$The Jacobian matrices of F and V by at \({E_0}\) then:$$F=\left( {\begin{array}{*{20}{c}} {\frac{{\partial {F_1}}}{{\partial I}}}&{\frac{{\partial {F_1}}}{{\partial C}}} \\ {\frac{{\partial {F_2}}}{{\partial I}}}&{\frac{{\partial {F_2}}}{{\partial C}}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} {S\left( {1 - \sigma } \right){\beta _1}}&0 \\ 0&0 \end{array}} \right)$$(10)$$V=\left( {\begin{array}{*{20}{c}} {\frac{{\partial {V_1}}}{{\partial I}}}&{\frac{{\partial {V_1}}}{{\partial C}}} \\ {\frac{{\partial {V_2}}}{{\partial I}}}&{\frac{{\partial {V_2}}}{{\partial C}}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} {\tau +{\kappa _1}+{\rho _1}}&0 \\ {{\rho _1}}&{\xi +{\kappa _1}} \end{array}} \right)$$(11)Where$${V^{ - 1}}=\left( {\begin{array}{*{20}{c}} {\frac{1}{{\tau +{\kappa _1}+{\rho _1}}}}&0 \\ {\frac{{{\rho _1}}}{{\left( {\xi +{\kappa _1}} \right)\left( {\tau +{\kappa _1}+{\rho _1}} \right)}}}&{\frac{1}{{\xi +{\kappa _1}}}} \end{array}} \right)$$(12)Then$$FV^{{ - 1}} = \left( {\begin{array}{*{20}c} {\frac{{\pi \left( {\kappa _{1} + \omega _{1} } \right)}}{{\kappa _{1} \left( {\phi + \kappa _{1} + \omega _{1} } \right)}}\left( {1 - \sigma } \right)\beta _{1} } & 0 \\ 0 & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\frac{1}{{\tau + \kappa _{1} + \rho _{1} }}} & 0 \\ {\frac{{\rho _{1} }}{{\left( {\xi + \kappa _{1} } \right)\left( {\tau + \kappa _{1} + \rho _{1} } \right)}}} & {\frac{1}{{\xi + \kappa _{1} }}} \\ \end{array} } \right)$$(13)$$F{V^{ - 1}}=\left( {\begin{array}{*{20}{c}} {\frac{{S\left( {1 - \sigma } \right){\beta _1}}}{{\tau +{\kappa _1}+{\rho _1}}}}&0 \\ 0&0 \end{array}} \right)$$(14)Therefore, the basic reproduction number\({R_0}\) of the HPV model (1) is determined as the spectral radius of the next-generation matrix \(F{V^{ - 1}}\) given as follows:$${R_0}=\frac{{\pi \left( {1 - \sigma } \right){\beta _1}\left( {{\kappa _1}+{\omega _1}} \right)}}{{{\kappa _1}\left( {\tau +{\kappa _1}+{\rho _1}} \right)\left( {\phi +{\kappa _1}+{\omega _1}} \right)}}.$$(15)Local stability analysis of disease-free equilibriaTheoremWhen\({R_0}