Article Contents
Article Contents

# A male-female mathematical model of human papillomavirus (HPV) in African American population

• * Corresponding author: Najat Ziyadi
• We introduce mathematical human papillomavirus (HPV) epidemic models (with and without vaccination) for African American females (AAF) and African American males (AAM) with "fitted" logistic demographics and use these models to study the HPV disease dynamics. The US Census Bureau data of AAF and AAM of 16 years and older from 2000 to 2014 is used to "fit" the logistic demographic models. We compute the basic reproduction number, $\mathcal{R}_0$ , and use it to show that $\mathcal{R}_0$ is less than 1 in the African American (AA) population with or without implementation of HPV vaccination program. Furthermore, we obtain that adopting a HPV vaccination policy in the AAF and AAM populations lower $\mathcal{R}_0$ and the number of HPV infections. Sensitivity analysis is used to illustrate the impact of each model parameter on the basic reproduction number.

Mathematics Subject Classification: 92B05.

 Citation:

• Figure 1.  The logistic demographic model "fits" $2000 - 2014$ US Census Bureau data of AAF population of 16 years and older, while constant recruitment Model (1) over estimates the AAF population

Figure 2.  The logistic demographic model "fits" $2000 - 2014$ US Census Bureau data of AAM population of 16 years and older, while constant recruitment Model (1) over estimates the AAM population

Figure 3.  Normalized sensitivity indices of $\mathcal{R}_{0}$ are evaluated at values of the parameters of Table (3). The most sensitive parameters for $\mathcal{R}_{0}$ are the clearance rate, $\delta$, the infection rate of the AAF population, $\sigma_f$, and the infection rate of the AAM population, $\sigma_m$. While the least sensitive parameters are the intrinsic growth rate for AAF population, $r_f$, the intrinsic growth rate for AAM population, $r_m$, the death rate of AAF population, $\mu_f$, and the death rate for AAM population, $\mu_m$

Figure 4.  African American male and female HPV model simulations

Figure 5.  Normalized sensitivity indices of $\mathcal{R}_{0}^v$ are evaluated at values of model parameters. The most sensitive parameters for $\mathcal{R}_{0}^v$ are the clearance rate, $\delta$, the infection rate of the AAF population, $\sigma_f$, and the infection rate of the AAM population, $\sigma_m$, the success rate of HPV vaccine, $\tau$, and the proportion of HPV vaccinated females, $p_f$. While the least sensitive parameters are the intrinsic growth rate for AAF population, $r_f$, the intrinsic growth rate for AAM population, $r_m$, the death rate of AAF population, $\mu_f$, and the death rate for AAM population, $\mu_m$

Figure 6.  African American male and female HPV model with vaccination simulations

Figure 7.  Susceptible AAF population

Figure 8.  Susceptible AAM population

Figure 9.  HPV Infected AAF population

Figure 10.  HPV Infected AAM population

Table 1.  2000 to 2014 US Census Bureau AAF population data

 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 AAF population 16 years and older [12,16] 13,825,055 14,041,520 14,259,413 14,473,927 14,707,490 14,952,963 15,224,330 15,486,244 15,743,096 15,992,822 16,176,048 16,471,449 16,696,303 16,918,225 17,139,986 AAF total population [12,16] 18,787,192 19,013,351 19,229,855 19,434,349 19,653,829 19,882,081 20,123,789 20,374,894 20,626,043 20,868,282 21,045,595 21,320,013 21,543,051 21,767,521 21,988,307

Table 2.  2000 to 2014 US Census Bureau AAM population data

 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 AAM population 16 years and older [12,16] 11,909,507 12,124,810 12,332,791 12,518,252 12,756,370 12,996,123 13,266,163 13,517,841 13,765,707 14,006,594 14,181,655 14,490,027 14,724,637 14,950,933 15,176,189 AAM total population [12,16] 17,027,514 17,249,678 17,454,795 17,631,747 17,856,753 18,079,607 18,319,259 18,560,639 18,803,371 19,033,988 19,260,298 19,487,042 19,719,238 19,945,997 20,169,931

Table 3.  Table of model parameters

 Parameter (per day) Description Reference $\mu_f=0.007266$ Death rate for AAF population [6] $\mu_m=0.008227$ Death rate for AAM population [6] $\delta= 0.9$ Clearance rate [11] $r_f=0.028564978$ Intrinsic growth rate for AAF population Estimated $r_m=0.028104926$ Intrinsic growth rate for AAM population Estimated $K_f=71,036,484$ Carrying capacity for AAF population Estimated $K_m=140,659,009$ Carrying capacity for AAF population Estimated $\sigma_f=0.5$ Infection rate for AAF population [1] $\sigma_m=0.4$ Infection rate for AAM population [1]

Table 4.  Initial conditions for HPV Model (10)

 $S_f(0)$ = 8,618,960 $S_m(0)$ = 7,119,370 $I_f(0)$ = 5,422,560 $I_m(0)$ = 5,005,440

Table 5.  Normalized sensitivity indices and order of importance of $\mathcal{R}_{0}$ to the nine parameters in Table (3)

 Parameter Sensitivity index of $\mathcal{R}_{0}$ Order of Importance $\delta$ -0.9915 1 $\sigma_f$ 0.5000 2 $\sigma_m$ 0.5000 3 $K_f$ 0.1526 4 $K_m$ -0.1526 5 $r_m$ -0.0631 6 $\mu_m$ 0.0586 7 $\mu_f$ -0.0561 8 $r_f$ 0.0520 9

Table 6.  Initial conditions for HPV Model (12)

 $S_f(0)$ = 5,257,566 $S_f^v(0)$ = 3,361,394 $S_m(0)$ = 5,667,019 $S_m^v(0)$ = 1,452,351 $I_f(0)$ = 5,086,421 $I_f^v(0)$ = 336,139 $I_m(0)$ = 4,860,205 $I_m^v(0)$ = 145,235

Table 7.  Normalized sensitivity indices and order of importance of $\mathcal{R}_{0}^v$ to model parameters

 Parameter Sensitivity index of $\mathcal{R}_{0}^v$ Order of Importance $\delta$ -0.9915 1 $\sigma_f$ 0.5000 2 $\sigma_m$ 0.5000 3 $\tau$ -0.3829 4 $p_f$ -0.2704 5 $K_f$ 0.1526 6 $K_m$ -0.1526 7 $p_m$ -0.1124 8 $r_m$ -0.0631 9 $\mu_m$ 0.0586 10 $\mu_f$ -0.0561 11 $r_f$ 0.0520 12

Table 8.  Initial conditions for HPV model

 $p_f=39\%$$p_m=20.4\% p_f=50\%$$p_m=50\%$ $p_f=70\%$$p_m=70\%$ $S_f(0)$ 5,257,566 4,309,480 2,585,688 $S_f^v(0)$ 3,361,394 4,309,480 6,033,272 $S_m(0)$ 5,667,019 3,559,685 2,135,811 $S_m^v(0)$ 1,452,351 3,559,685 4,983,559 $I_f(0)$ 5,086,421 4,991,612 4,819,233 $I_f^v(0)$ 336,139 430,948 603,327 $I_m(0)$ 4,860,205 4,649,472 4,507,084 $I_m^v(0)$ 145,235 355,969 498,356
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