Computational modeling of human papillomavirus with impulsive vaccination

Abstract

In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.

Keywords: Global attractivity; Human papillomavirus; Impulsive vaccination; Uniform persistence.

Fig. 1

This figure shows the time series of the susceptible unvaccinated females and the phase portrait of $I_{\mathrm{f}}$ versus $S_{\mathrm{uf}}$ with parameter values $T=5$, ${\mathrm{\Pi }}_{\mathrm{f}}=5$, ${\mathrm{\Pi }}_{\mathrm{m}}=5$, ${\beta }_{\mathrm{f}}=0.004$, ${\beta }_{\mathrm{m}}=0.0004$, ${\theta }_1={\theta }_2=0.5$ and other parameters are as in Table 1

Fig. 2

This figure shows the time series of the susceptible unvaccinated males and the phase portrait of $I_{\mathrm{m}}$ versus $S_{\mathrm{um}}$ with parameter values $T=5$, ${\mathrm{\Pi }}_{\mathrm{f}}=5$, ${\mathrm{\Pi }}_{\mathrm{m}}=5$, ${\beta }_{\mathrm{f}}=0.004$, ${\beta }_{\mathrm{m}}=0.0004$, ${\theta }_1={\theta }_2=0.5$ and other parameters are as in Table 1

Fig. 3

Time evolution of infected individuals before and after initiation of periodic vaccination with the parameters $T=5$, ${\mathrm{\Pi }}_{\mathrm{f}}=5$, ${\mathrm{\Pi }}_{\mathrm{m}}=5$, ${\beta }_{\mathrm{f}}=0.004$, ${\beta }_{\mathrm{m}}=0.0004$, ${\theta }_1={\theta }_2=0.5$ and other parameters are as in Table 1

Fig. 4

This figure shows the time series of the susceptible unvaccinated females and the phase portrait of $I_{\mathrm{f}}$ versus $S_{\mathrm{uf}}$ with parameter values $T=5$, ${\mathrm{\Pi }}_{\mathrm{f}}=50$, ${\mathrm{\Pi }}_{\mathrm{m}}=50$, ${\beta }_{\mathrm{m}}=0.001$, ${\beta }_{\mathrm{f}}=0.01$, ${\theta }_1={\theta }_2=0.5$ and other parameters are as in Table 1

Fig. 5

This figure shows the time series of the susceptible unvaccinated males and the phase portrait of $I_{\mathrm{m}}$ versus $S_{\mathrm{um}}$ with parameter values $T=5$, ${\mathrm{\Pi }}_{\mathrm{f}}=50$, ${\mathrm{\Pi }}_{\mathrm{m}}=50$, ${\beta }_{\mathrm{m}}=0.001$, ${\beta }_{\mathrm{f}}=0.01$, ${\theta }_1={\theta }_2=0.5$ and other parameters are as in Table 1

Fig. 6

This figure shows the time series of the vaccinated females and males with parameter values $T=5$, ${\mathrm{\Pi }}_{\mathrm{f}}=50$, ${\mathrm{\Pi }}_{\mathrm{m}}=50$, ${\beta }_{\mathrm{m}}=0.001$, ${\beta }_{\mathrm{f}}=0.01$, ${\theta }_1={\theta }_2=0.5$ and other parameters are as in Table 1