HPV Screening and Vaccination Strategies in an Unscreened Population: A Mathematical Modeling Study

Abstract

Human papillomavirus (HPV), a sexually transmitted infection, is the necessary cause of cervical cancer, the third most common cancer affecting women worldwide. Prevention and control strategies include vaccination, screening, and treatment. While HPV prevention and control efforts are important worldwide, they are especially important in low-income areas with a high infection rate or high rate of cervical cancer. This study uses mathematical modeling to explore various vaccination and treatment strategies to control for HPV and cervical cancer while using Nepal as a case study. Two sets of deterministic models were created with the goal of understanding the impact of various prevention and control strategies. The first set of models examines the relative importance of screening and vaccination in an unscreened population, while the second set examines various screening scenarios. Partial rank correlation coefficients confirm the importance of screening and treatment in the reduction of HPV infections and cancer cases even when vaccination uptake is high. Results also indicate that less expensive screening technologies can achieve the same overall goal as more expensive screening technologies.

Appendix

Equations representing the models found in Fig. 2:

The equations for the no-intervention model are:

$$\begin{aligned} \begin{array}{lcl} S' &{} = &{} \lambda + \gamma _{A0} A_0+ (\gamma _{I0}+\psi \gamma _{I0}) I_0 -{\frac{F_{\beta } S }{N}} - dS \\ A_0' &{} = &{} {\frac{F_{\beta } S}{N}} - (\gamma _{A0} + \alpha _{A0} +d) A_0 + \gamma _{A1} A_1 \\ A_1' &{} = &{} \alpha _{A0} A_0 - (\gamma _{A1} + \alpha _{A1} + d) A_1 + \gamma _{A2} A_2 \\ A_2' &{} = &{} \alpha _{A1} A_1 - (\gamma _{A2} + \alpha _{A2} + d) A_2 \\ I_0' &{} = &{} - (\gamma _{I0} +\psi \gamma _{I0}+ \alpha _{I0} + d) I_0 + (\gamma _{I1}+\psi \gamma _{I1}) I_1 \\ I_1' &{} = &{} - (\gamma _{I1} +\psi \gamma _{I1}+ \alpha _{I1} + d) I_1 + (\gamma _{I2}+\psi \gamma _{I2} )I_2 +\alpha _{I0} I_0 \\ I_2' &{} = &{} - (\gamma _{I2} +\psi \gamma _{I2}+ \alpha _{I2} + d) I_2 + (\gamma _{C1}+\psi \gamma _{C1}) C_1 +\alpha _{I1} I_1 \\ C_1' &{} = &{} \alpha _{A2} A_2 + \alpha _{I2} I_2 - (\gamma _{C1}+\psi \gamma _{C1} + \alpha _{C1} + d ) C_1 \\ C_2' &{} = &{} \alpha _{C1} C_1 - ( d + \delta ) C_2 \\ F_{\beta } &{} = &{} \begin{array}{l}\beta ((A_{0} + A_{1} + A_{2}) + q_{I}(I_{0} + I_{1} +I_{2}))\\ \end{array} \end{array} \end{aligned}$$

(2)

The equations that represent the screening only case are:

$$\begin{aligned} \begin{array}{ccl} S' &{} = &{} \lambda + \gamma _{A0} A_0+ (\gamma _{I0}+\psi \gamma _{I0}) I_0 -{F_{\beta } S \over N} - dS \\ A_0' &{} = &{}{\frac{ F_{\beta } S}{N}} - (\gamma _{A0} + \alpha _{A0} + \zeta \phi +d) A_0 + \gamma _{A1} A_1 \\ A_1' &{} = &{} \alpha _{A0} A_0 - (\gamma _{A1} + \alpha _{A1} + \zeta \phi + d) A_1 + \gamma _{A2} A_2 \\ A_2' &{} = &{} \alpha _{A1} A_1 - (\gamma _{A2} + \alpha _{A2} + \zeta \phi + d) A_2 \\ I_0' &{} = &{} \zeta \phi (A_0) -(\gamma _{I0} +\psi \gamma _{I0}+ \alpha _{I0} + d) I_0 + (\gamma _{I1}+\psi \gamma _{I1}) I_1 \\ I_1' &{} = &{} \zeta \phi (A_1) -(\gamma _{I1}+\psi \gamma _{I1} + \alpha _{I1} + d) I_1 + (\gamma _{I2}+\psi \gamma _{I2}) I_2 +\alpha _{I0} I_0 \\ I_2' &{} = &{} \zeta \phi (A_2) - (\gamma _{I2} + \psi \gamma _{I2}+\alpha _{I2} + d) I_2 + (\gamma _{C1}+\psi \gamma _{C1}) C_1 +\alpha _{I1} I_1 \\ C_1' &{} = &{} \alpha _{A2} A_2 + \alpha _{I2} I_2 - (\gamma _{C1}+\psi \gamma _{C1} + \alpha _{C1} + d ) C_1 \\ C_2' &{} = &{} \alpha _{C1} C_1 - ( d + \delta _{C_2}) C_2 \\ F_{\beta } &{}=&{} \begin{array}{l}\beta (q_{A}(A_{0} + A_{1} + A_{2}) + q_{I}(I_{0} + I_{1} +I_{2}))\\ \end{array} \end{array} \end{aligned}$$

(3)

The equations that represent the vaccination only case are:

$$\begin{aligned} \begin{array}{ccl} S' &{} = &{} \lambda + \gamma _{A0} A_0+ (\gamma _{I0}+\psi \gamma _{I0}) I_0 -{F_{\beta } S \over N} -\nu \rho \epsilon S - dS \\ A_0' &{} = &{} {\frac{F_{\beta } S}{N}} - (\gamma _{A0} + \alpha _{A0}+d) A_0 + \gamma _{A1} A_1 \\ A_1' &{} = &{} \alpha _{A0} A_0 - (\gamma _{A1} + \alpha _{A1} + d) A_1 + \gamma _{A2} A_2 \\ A_2' &{} = &{} \alpha _{A1} A_1 - (\gamma _{A2} + \alpha _{A2} + d) A_2 \\ I_0' &{} = &{} - (\gamma _{I0} +\psi \gamma _{I0}+ \alpha _{I0} + d) I_0 + (\gamma _{I1}+\psi \gamma _{I1})I_1 \\ I_1' &{} = &{} - (\gamma _{I1} +\psi \gamma _{I1}+ \alpha _{I1} + d) I_1 + (\gamma _{I2}+\psi \gamma _{I2}) I_2 +\alpha _{I0} I_0 \\ I_2' &{} = &{} - (\gamma _{I2} +\psi \gamma _{I2}+ \alpha _{I2} + d) I_2 +(\gamma _{C1}+\psi \gamma _{C1}) C_1 +\alpha _{I1} I_1 \\ C_1' &{} = &{} \alpha _{A2} A_2 + \alpha _{I2} I_2 + \alpha _{V2} V_2 - (\gamma _{C1} + \psi \gamma _{C1}+\alpha _{C1} + d ) C_1 \\ C_2' &{} = &{} \alpha _{C1} C_1 - ( d + \delta ) C_2 \\ S_V' &{} = &{} \nu \rho \epsilon S +\gamma _{V0} V_0 - {\frac{q F_{\beta } S_V}{N}} - d S_V \\ V_0' &{} = &{} {\frac{q F_{\beta } S_V}{N}} - (\gamma _{V0} + \alpha _{V0} + d) V_0 + \gamma _{V1} V_1 \\ V_1' &{} = &{} \alpha _{V0} V_0 - (\gamma _{V1} +\alpha _{V1} + d) V_1 + \gamma _{V2} V_2 \\ V_2' &{} = &{} \alpha _{V1} V_1 - (\gamma _{V2} +\alpha _{V2} + d) V_2 \\ F_{\beta } &{}=&{} \begin{array}{l}\beta ((A_{0} + A_{1} + A_{2}) + q_{I}(I_{0} + I_{1} +I_{2})+q_{V}(V_{0} + V_{1} +V_{2}))\\ \end{array} \end{array} \end{aligned}$$

(4)

The equations that represent the screening and vaccination case are:

$$\begin{aligned} \begin{array}{ccl} S' &{} = &{} \lambda + \gamma _{A0} A_0+ (\gamma _{I0}+\psi \gamma _{I0} )I_0 -{F_{\beta } S \over N} -\nu \rho \epsilon S - dS \\ A_0' &{} = &{} {\frac{F_{\beta } S}{N}} - (\gamma _{A0} + \alpha _{A0} + \zeta \phi +d) A_0 + \gamma _{A1} A_1 \\ A_1' &{} = &{} \alpha _{A0} A_0 - (\gamma _{A1} + \alpha _{A1} + \zeta \phi + d) A_1 + \gamma _{A2} A_2 \\ A_2' &{} = &{} \alpha _{A1} A_1 - (\gamma _{A2} + \alpha _{A2} + \zeta \phi + d) A_2 \\ I_0' &{} = &{} \zeta \phi A_0+\zeta _2\phi V_0 - (\gamma _{I0} +\psi \gamma _{I0}+ \alpha _{I0} + d) I_0 + (\gamma _{I1}+\psi \gamma _{I1}) I_1 \\ I_1' &{} = &{} \zeta \phi A_1 +\zeta _2\phi V_1 - (\gamma _{I1} +\psi \gamma _{I1} + \alpha _{I1} + d) I_1 + (\gamma _{I2}+\psi \gamma _{I2}) I_2 +\alpha _{I0} I_0 \\ I_2' &{} = &{} \zeta \phi A_2 +\zeta _2\phi V_2 - (\gamma _{I2} + \psi \gamma _{I2}+\alpha _{I2} + d) I_2 + (\gamma _{C1}+\psi \gamma _{C1}) C_1 +\alpha _{I1} I_1 \\ C_1' &{} = &{} \alpha _{A2} A_2 + \alpha _{I2} I_2 + \alpha _{V2} V_2 - (\gamma _{C1} +\psi \gamma _{C1}+\alpha _{C1} + d ) C_1 \\ C_2' &{} = &{} \alpha _{C1} C_1 - ( d + \delta ) C_2 \\ S_V' &{} = &{} \nu \rho \epsilon S + \gamma _{V0}V_0 - {\frac{q F_{\beta } S_V}{N}} - d S_V \\ V_0' &{} = &{} {\frac{ q F_{\beta } S_V}{N}} - (\gamma _{V0} + \alpha _{V0} + \zeta _2\phi + d) V_0 + \gamma _{V1} V_1 \\ V_1' &{} = &{} \alpha _{V0} V_0 - (\gamma _{V1} +\alpha _{V1} + \zeta _2 \phi + d) V_1 + \gamma _{V2}V_2 \\ V_2' &{} = &{} \alpha _{V1} V_1 - (\gamma _{V2} +\alpha _{V2} +\zeta _2 \phi + d) V_2 \\ F_{\beta } &{}=&{} \begin{array}{l}\beta ((A_{0} + A_{1} + A_{2}) + q_{I}(I_{0} + I_{1} +I_{2})+q_{V}(V_{0} + V_{1} +V_{2})\\ \end{array} \end{array} \end{aligned}$$

(5)

Equations representing the models found in Fig. 3:

The equations that represent the screen and treat model are:

$$\begin{aligned} \begin{array}{ccl} S' &{} = &{} \lambda +\gamma _{A0}A_0-{\frac{F_\beta S}{N}} -dS\\ A_0' &{} =&{} {\frac{F_\beta S}{N}}+\gamma _{A1}A_1-(\gamma _{A0}+\alpha _{A0}+\phi \zeta +d)A_0\\ A_1' &{} =&{}\alpha _{A0}A_0+\gamma _{A2}A_2-(\gamma _{A1}+\alpha _{A1}+\phi \zeta +d) A_1\\ A_2' &{} = &{}\alpha _{A1}A_1-(\gamma _{A2}+\alpha _{A2}+\phi \zeta +d) A_2\\ C_1' &{} = &{} \alpha _{A2}A_2-(\alpha _{C1}+\gamma _{C1}+d)C_1\\ C_2' &{} = &{}\alpha _{C1}C_1-(d+\delta )C_2\\ I_{0p}' &{} = &{}\phi \zeta A_0-(\kappa +d)I_{0p}\\ I_{1p}' &{} = &{}\phi \zeta A_1 -(\kappa +d)I_{1p}\\ I_{2p}' &{} = &{}\phi \zeta A_2- (\kappa +d)I_{2p}+\gamma _{C1}C_1\\ R' &{} = &{}(I_{0p}+I_{1p}+I_{2p})\kappa -dR\\ F_\beta &{}=&{} \begin{array}{l}\beta ((A_0 +A_1 +A_2)+q_{Ip}(I_{0p}+I_{1p}+I_{2p}))\\ \end{array} \end{array} \end{aligned}$$

(6)

The equations that represent the model in which only those individuals who have a high risk type of HPV are treated are:

$$\begin{aligned} \begin{array}{ccl} S' &{} = &{}\lambda +\gamma _{A0}A_0+\gamma _{I0p}I_{0p}-{\frac{F_\beta S}{N}}-dS\\ A_0' &{} =&{}{\frac{F_\beta S}{N}} +\gamma _{A1}A_1-(\gamma _{A0}+\alpha _{A0}+\zeta \phi +d)A_0\\ A_1' &{} =&{}\alpha _{A0}A_0+\gamma _{A2}A_2-(\gamma _{A1}+\alpha _{A1}+\phi \zeta +d) A_1\\ A_2' &{} = &{}\alpha _{A1}A_1-(\gamma _{A2}+\alpha _{A2}+\phi \zeta +d) A_2\\ C_1' &{} = &{} \alpha _{A2}A_2+\alpha _{I2p}I_{2p}-(\alpha _{C1}+\gamma _{C1}+d)C_1\\ C_2' &{} = &{}\alpha _{C1}C_1-(d+\delta )C_2\\ I_{0p}' &{} = &{}\phi \zeta A_0+\gamma _{I1p}I_{1p}-(\gamma _{I0p}+\alpha _{I0p}+\eta +d)I_{0p}\\ I_{1p}' &{} = &{}\phi \zeta A_1+\gamma _{I2p}I_{2p}+\alpha _{I0p}I_{0p}-(\gamma _{I1p}+\alpha _{I1p}+\eta +d)I_{1p}\\ I_{2p}' &{} = &{}\phi \zeta A_2 +\alpha _{I1p}I_{1p}-(\gamma _{I2p}+\alpha _{I2p}+\eta +d)I_{2p}\\ I_{0pT}' &{} = &{}\eta I_{0p}-(\kappa +d)I_{0pT}\\ I_{1pT}' &{} = &{}\eta I_{1p}-(\kappa +d)I_{1pT}\\ I_{2pT}' &{} = &{}\eta I_{2p}+\gamma _{C1}C_1-(\kappa +d)I_{2pT}\\ R' &{} = &{}(I_{0pT}+I_{1pT}+I_{2pT})\kappa -dR\\ F_\beta &{} = &{} \begin{array}{l}\beta ((A_0 +A_1 +A_2)+q_Ip(I_{0p}+I_{1p}+I_{2p})\\ +q_{IpT}(I_{0pT}+I_{1pT}+I_{2pT}))\\ \end{array} \end{array} \end{aligned}$$

(7)

The equations that represent the model in which there is co-testing with a cytology test and cobas4800 are:

$$\begin{aligned} S'= & {} \lambda +\gamma _{A0}A_0+\gamma _{I0p}I_{0p}+\gamma _{I0pT}I_{0pT}-{\frac{F_\beta S}{N}}-dS\nonumber \\ A_0'= & {} {\frac{F_\beta S}{N}} +\gamma _{A1}A_1-(\gamma _{A0}+\alpha _{A0}+\zeta \phi +d)A_0\nonumber \\ A_1'= & {} \alpha _{A0}A_0+\gamma _{A2}A_2-(\gamma _{A1}+\alpha _{A1}+\phi \zeta +d) A_1\nonumber \\ A_2'= & {} \alpha _{A1}A_1-(\gamma _{A2}+\alpha _{A2}+\phi \zeta +d) A_2\nonumber \\ C_1'= & {} \alpha _{A2}A_2+\alpha _{I2p}I_{2p}+\alpha _{I2pT}I_{2pT}-(\alpha _{C1}+\gamma _{C1}+d)C_1\nonumber \\ C_2'= & {} \alpha _{C1}C_1-(d+\delta )C_2\nonumber \\ I_{0p}'= & {} \phi \zeta A_0+\gamma _{I1p}I_{1p}-(\gamma _{I0p}+\alpha _{I0p}+\eta +d)I_{0p}\nonumber \\ I_{1p}'= & {} \phi \zeta A_1+\gamma _{I2p}I_{2p}+\alpha _{I0p}I_{0p}-(\gamma _{I1p}+\alpha _{I1p}+\eta +d)I_{1p}\nonumber \\ I_{2p}'= & {} \phi \zeta A_2 +\alpha _{I1p}I_{1p}-(\gamma _{I2p}+\alpha _{I2p}+\eta +d)I_{2p}\nonumber \\ I_{0pT}'= & {} \eta I_{0p}+\gamma _{I1pT}I_{1pT}-(\gamma _{I0pT}+\alpha _{I0pT}+\eta _2\phi _2+d)I_{0pT}\nonumber \\ I_{1pT}'= & {} \eta I_{1p}+\gamma _{I2pT}I_{2pT}+\alpha _{I0pT}I_{0pT}-(\gamma _{I1pT}+\alpha _{I1pT}+\eta _2 \phi _2+d)I_{1pT}\nonumber \\ I_{2pT}'= & {} \eta I_{2p}+\alpha _{I1pT}I_{1pT}+\gamma _{C1}C_1-(\gamma _{I2pT}+\alpha _{I2pT}+\eta _2 \phi _2+d)I_{2pT}\nonumber \\ I_{0pTy}'= & {} \eta _2\phi _2I_{0pT}-(\kappa +d)I_{0pTy}\nonumber \\ I_{1pTy}'= & {} \eta _2\phi _2I_{1pT}-(\kappa +d)I_{1pTy}\nonumber \\ I_{2pTy}'= & {} \eta _2\phi _2I_{2pT}-(\kappa +d)I_{2pTy}\nonumber \\ R'= & {} (I_{0pTy}+I_{1pTy}+I_{2pTy})\kappa -dR\nonumber \\ F_\beta= & {} \begin{array}{l}\beta ((A_0 +A_1 +A_2)+q_Ip(I_{0p}+I_{1p}+I_{2p})+q_{IpT}(I_{0pT}+I_{1pT}+I_{2pT})\\ +q_{IpTy}(I_{0pTy}+I_{1pTy}+I_{2pTy}))\\ \end{array}\nonumber \\ \end{aligned}$$

(8)

The equations that represent co-testing with a pap smear and cobas4800 are:

$$\begin{aligned} \begin{array}{ccl} S' &{} = &{}\lambda +\gamma _{A0}A_0+\gamma _{I0pap}I_{0pap}-{\frac{F_\beta S}{N}}-dS\\ A_0' &{} =&{}{\frac{F_\beta S}{N}} +\gamma _{A1}A_1-(\gamma _{A0}+\alpha _{A0}+\zeta \phi _{1a}+d)A_0\\ A_1' &{} =&{}\alpha _{A0}A_0+\gamma _{A2}A_2-(\gamma _{A1}+\alpha _{A1}+\phi _{1a} \zeta +d) A_1\\ A_2' &{} = &{}\alpha _{A1}A_1-(\gamma _{A2}+\alpha _{A2}+\phi _{1b} \zeta +d) A_2\\ C_1' &{} = &{} \alpha _{A2}A_2+\alpha _{I2pap}I_{2pap}-(\alpha _{C1}+\gamma _{C1}+d)C_1\\ C_2' &{} = &{}\alpha _{C1}C_1-(d+\delta )C_2\\ I_{0pap}' &{} = &{}\phi _{1a} \zeta A_0+\gamma _{I1pap}I_{1pap}-(\gamma _{I0pap}+\alpha _{I0pap}+\phi _2\eta +d)I_{0pap}\\ I_{1pap}' &{} = &{}\phi _{1a} \zeta A_1+\gamma _{I2pap}I_{2pap}+\alpha _{I0pap}I_{0pap}\\ &{}&{}-(\gamma _{I1pap}+\alpha _{I1pap}+\phi _2\eta +d)I_{1pap}\\ I_{2pap}' &{} = &{}\phi _{1b} \zeta A_2 +\alpha _{I1pap}I_{1pap}-(\gamma _{I2pap}+\alpha _{I2pap}+\phi _2\eta +d)I_{2pap}\\ I_{0pTy}' &{} = &{}\phi _2\eta I_{0pap}-(\kappa +d)I_{0pTy}\\ I_{1pTy}' &{} = &{}\phi _2\eta I_{1pap}-(\kappa +d)I_{1pTy}\\ I_{2pTy}' &{} = &{}\phi _2\eta I_{2pap}+\gamma _{C1}C_1-(\kappa +d)I_{2pTy}\\ R' &{} = &{}(I_{0pTy}+I_{1pTy}+I_{2pTy})\kappa -dR\\ F_\beta &{}=&{} \beta ((A_0 +A_1 +A_2)+q_Ip(I_{0pap}+I_{1pap}+I_{2pap})\\ &{}&{}+q_{IpT}(I_{0pTy}+I_{1pTy}+I_{2pTy}))\\ \end{array} \end{aligned}$$

(9)

The next generation method was used to calculate the $R_0$ for the models in Fig. 2. $R_0$ can be calculated as the spectral radius of $FV^{-1}$. We derive the $R_0$ for the no-intervention model only as the derivation generalizes to the larger screening and vaccination models. $R_0$ can be calculated from the following F and V matrices:

$$\begin{aligned} F= & {} \left( \begin{array}{cccccccc} \beta &{} \beta &{} \beta &{} \beta q_II&{} \beta q_II&{} \beta q_II&{}0&{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ \end{array} \right) \\ V_{(i,i)}= & {} \left( \begin{array}{c} {\gamma }_{{{ A0}}}+\alpha _{{{ A0}}}+d\\ {\gamma }_{{{ A1}}}+\alpha _{{{ A1}}}+d\\ {\gamma }_{{{ A2}}}+\alpha _{{{ A2}}}+d\\ {\gamma }_{{{ I0}}}+{\gamma }_{{{ I0}}}\psi +\alpha _{{{ I0}}} +d\\ {\gamma }_{{{ I1}}}+\psi \,{\gamma }_{{{ I1}}}+\alpha _{{{ I1}}}+d\\ {\gamma }_{{{ I2}}}+\psi \,{\gamma }_{{{ I2}}}+\alpha _{{{ I2}}}+d\\ {\gamma }_{{{ C1}}}+\psi \,{\gamma }_{{{ C1}}}+\alpha _{{{ C1}}}+d\\ d+\delta \end{array} \right) \\ V_{(i,i-1)}= & {} \left( \begin{array}{cc} -\alpha _{{{ A0}}}\\ -\alpha _{{{ A1}}}\\ 0\\ -\alpha _{{{ I0}}}\\ -\alpha _{{{ I1}}}\\ -\alpha _{{{ I2}}}\\ -\alpha _{{{ C1}}}\\ \end{array} \right) \\ V_{(i,i+1)}= & {} \left( \begin{array}{cc} -{\gamma }_{{{ A1}}}\\ -{\gamma }_{{{ A2}}}\\ 0\\ -{\gamma }_{{{ I1}}}-\psi \,{\gamma }_{{{ I1}}}\\ -{\gamma }_{{{ I2}}}-\psi \,{\gamma }_{{{ I2}}}\\ -{\gamma }_{{{ C1}}}-\psi \,{\gamma }_{{{ C1}}}\\ 0\\ \end{array} \right) \end{aligned}$$

and $V(i,j)=0$ otherwise.