# American Institute of Mathematical Sciences

January  2017, 14(1): 339-358. doi: 10.3934/mbe.2017022

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

 Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

Received  November 25, 2015 Accepted  April 22, 2016 Published  October 2016

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.

Citation: Najat Ziyadi. A male-female mathematical model of human papillomavirus (HPV) in African American population. Mathematical Biosciences & Engineering, 2017, 14 (1) : 339-358. doi: 10.3934/mbe.2017022
##### References:
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##### References:
 [1] A. Alsaleh and A. B. Gumel, Analysis of risk-structured vaccination model for the dynamics of oncogenic and warts-causing HPV types, Bulletin of Mathematical Biology, 76 (2014), 1670-1726.  doi: 10.1007/s11538-014-9972-4.  Google Scholar [2] Black male statistics, Available from: http://blackdemographics.com/black-male-statistics/. Accessed 4/12/2016. Google Scholar [3] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Texts in Applied Mathematics, Springer, New York, NY, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar [4] J. Cariboni, D. Gatelli, R. Liska and A. Saltelli, The role of sensitivity analysis in ecological modelling, Ecological modelling, 203 (2007), 167-182.  doi: 10.1016/j.ecolmodel.2005.10.045.  Google Scholar [5] Centers for disease control and prevention, Genital HPV infection: CDC fact sheet Available from: http://www.cdc.gov/std/HPV/STDFact-HPV.htm. Accessed 4/12/2016. Google Scholar [6] Centers for disease control and prevention, National Vital Statistics Reports, Volume 64, Number 2. Google Scholar [7] Centers for Disease Control and Prevention, Human Papillomavirus (HPV), What is HPV Available from: http://www.cdc.gov/hpv/whatishpv.html. Accessed 4/12/2016. Google Scholar [8] Centers for Disease Control and Prevention, Morbidity and mortality weekly report, CDC grand rounds: Reducing the burden of HPV-associated cancer and disease, MMWR, Weekly, 63 (2014), 69-72. Google Scholar [9] Center for Disease Control and Prevention, Morbidity and Mortality Weekly Report, Weekly / Vol. 64 / No. 29. http://www.cdc.gov/mmwr/pdf/wk/mm6429.pdf. Accessed on 4/8/2016. Google Scholar [10] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameter in the spread of malaria through the sensitivity analysis of mathematical model, Bulletin of Mathematical Biology, 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar [11] S. Hariri, E. Dunne, M. Saraiya, E. Unger and L. Markowitz, Chapter 5: Human papillomavirus, VPD Surveillance Manual, 5th Edition, 2011. Available from: http://www.cdc.gov/vaccines/pubs/surv-manual/chpt05-hpv.pdf. Accessed 4/12/2016. Google Scholar [12] [13] S. Lee and A. Tameru, A mathematical model of human papillomavirus (HPV) in the United States and its impact on cervical cancer, Journal of Cancer, 3 (2012), 262-268.  doi: 10.7150/jca.4161.  Google Scholar [14] L. Ribassin-Majed, R. Lounes and S. Clemencon, Deterministic modelling for transmission of human papillomavirus 6/11: Impact of vaccination, Math Med Biol, 31 (2014), 125-149.  doi: 10.1093/imammb/dqt001.  Google Scholar [15] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Amer. Math. Soc., Rhode Island, 1995.  Google Scholar [16] US Census Bureau, Available from: http://factfinder.census.gov/faces/tableservices/jsf/pages/productview.xhtml?pid=ACS_11_1YR_B01001B&prodType=table. Accessed 8/28/2016. Google Scholar [17] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [18] N. Ziyadi, Local and global sensitivity analysis of $\mathcal{R}_0$ in a male and female human papillomavirus (HPV) epidemic model of Moroccans, Journal of Evolution Equations 9 (2016), Accepted. Google Scholar [19] N. Ziyadi and A.-A. Yakubu, Local and global sensitivity analysis in a discrete-time seis epidemic model, Advances in Dynamical Systems and Applications, 11 (2016), 15-33.   Google Scholar
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
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
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, $\mu_m$
African American male and female HPV model simulations
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, $\mu_m$
African American male and female HPV model with vaccination simulations
Susceptible AAF population
Susceptible AAM population
HPV Infected AAF population
HPV Infected AAM population
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
 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
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
 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 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]
 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]
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
 $S_f(0)$ = 8,618,960 $S_m(0)$ = 7,119,370 $I_f(0)$ = 5,422,560 $I_m(0)$ = 5,005,440
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
 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
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
 $S_f(0)$ = 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
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
 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
Initial conditions for HPV model
 $p_f=39\%$$p_m=20.4\% p_f=50\%$$p_m=50\%$ $p_f=70\%$$p_m=70\% S_f(0) 5,257,566 4,309,480 2,585,688 S_f^v(0) 3,361,394 4,309,480 6,033,272 S_m(0) 5,667,019 3,559,685 2,135,811 S_m^v(0) 1,452,351 3,559,685 4,983,559 I_f(0) 5,086,421 4,991,612 4,819,233 I_f^v(0) 336,139 430,948 603,327 I_m(0) 4,860,205 4,649,472 4,507,084 I_m^v(0) 145,235 355,969 498,356  p_f=39\%$$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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