# American Institute of Mathematical Sciences

October  2013, 18(8): 2151-2174. doi: 10.3934/dcdsb.2013.18.2151

## Qualitative analysis of an age- and sex-structured vaccination model for human papillomavirus

 1 Department of Applied Mathematics and Sciences, Khalifa University of Science, Technology and Research, PO Box 127788, Abu Dhabi 2 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2 3 Merck Research Laboratories, UG1C-60, PO Box 1000, North Wales, PA 19454-1099

Received  December 2011 Revised  February 2013 Published  July 2013

A new model for the transmission dynamics of human pappilomavirus (HPV) is designed and analysed. The model, which stratifies the total population in terms of age and gender, incorporates an imperfect anti-HPV vaccine with some therapeutic benefits. Rigorous qualitative analysis of the resulting age-structured model, which takes the form of a deterministic system of non-linear partial differential equations with separable transmission coefficients, shows that the disease-free equilibrium of the model is locally-asymptotically stable whenever the effective reproduction number (denoted by $\mathcal{R}_v$) is less than unity. It is shown to be globally-asymptotically stable if certain additional conditions hold. Furthermore, it is shown that the model has at least one endemic equilibrium when $\mathcal{R}_v$ exceeds unity. Hence, the effective control of HPV spread in a community, using a vaccine, is governed by the threshold quantity $\mathcal{R}_v$ (the use of the vaccine will lead to effective disease control or elimination only if it reduces the threshold quantity to a value less than unity; and the use of such vaccine will not lead to effective disease control if it fails to make the threshold quantity to be less than unity).
Citation: Tufail Malik, Abba Gumel, Elamin H. Elbasha. Qualitative analysis of an age- and sex-structured vaccination model for human papillomavirus. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2151-2174. doi: 10.3934/dcdsb.2013.18.2151
##### References:
 [1] M. Al-arydah and R. J. Smith, An age-structured model of human papillomavirus vaccination, Mathematics and Computers in Simulation, 82 (2011), 629-642. doi: 10.1016/j.matcom.2011.10.006. [2] R. Barnabas and G. Garnett, The potential public health impact of vaccines against human papillomavirus, in "The Clinical Handbook of Human Papillomavirus" (eds. Prendiville and Davies), Taylor and Francis, (2004), 61-79. [3] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci., 151 (1998), 135-154. doi: 10.1016/S0025-5564(98)10016-0. [4] "Genital HPV Infection - CDC Fact Sheet,", Centers for Disease Control and Prevention (CDC)., , (). [5] H. W. Chesson, D. U. Ekwueme, M. Saraiya and L. E. Markowitz, Cost-effectiveness of human papillomavirus vaccination in the United States, Emerg. Infect. Dis., 14 (2008), 244-251. doi: 10.3201/eid1402.070499. [6] M. E. Cruickshank, L. Sharp, G. Chambers, L. Smart and G. Murray, Persistent infection with human papillomavirus following the successful treatment of high grade cervical intraepithelial neoplasia, BJOG, 109 (2002), 579-581. [7] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," John Wiley & Sons Ltd., England, 2000. [8] E. H. Elbasha, E. J. Dasback and R. P. Insinga, Model for assessing human papillomavirus vaccination strategies, Emerg. Infect. Dis., 13 (2007), 28-41. doi: 10.3201/eid1301.060438. [9] E. H. Elbasha, Global stability of equilibria in a two-sex HPV vaccination model, Bull. Math. Biol., 70 (2008), 894-909. doi: 10.1007/s11538-007-9283-0. [10] E. H. Elbasha, E. J. Dasbach and R. P. Insinga, A multi-type HPV transmission model, Bull. Math. Biol., 70 (2008), 2126-2176. doi: 10.1007/s11538-008-9338-x. [11] E. H. Elbasha and E. J. Dasbach, Impact of vaccinating boys and men against HPV in the United States, Vaccine, 28 (2010), 6858-6867. doi: 10.1016/j.vaccine.2010.08.030. [12] A. Ferenczy, Persistent human papillomavirus infection and cervical neoplasia, The Lancet Oncology, 3 (2002), 11-16. doi: 10.1016/S1470-2045(01)00617-9. [13] D. Greenhalgh, Threshold and stability results for an epidemic model with an age-structured meeting rate, IMA J. Math. Appl. Med. Biol., 5 (1988), 81-100. doi: 10.1093/imammb/5.2.81. [14] H. W. Hethcote, Age-structured epdiemiology models and expressions for $R_0$, in "Mathematical Understanding of Infectious Disease Dynamics" (eds. S. Ma and Y. Xia), World Sci. Publ., Hackensack, NJ, (2009), 91-128. doi: 10.1142/9789812834836_0003. [15] J. Hughes, G. Garnett and L. Koutsky, The theoretical population level impact of a prophylactic human papillomavirus vaccine, Epidemiology, 13 (2002), 631-639. [16] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. [17] W. Kaplan, "Advanced Calculus," Addison-Wesley Mathematics Series, 1984. [18] J. J. Kim and S. J. Goldie, Cost effectiveness analysis of including boys in a human papillomavirus vaccination programme in the United States, BMJ, 339 (2009), 1-10. doi: 10.1136/bmj.b3884. [19] C. J. N. Lacey, C. M. Lowndes and K. V. Shah, Burden and management of non-cancerous HPV-related conditions: HPV-6/11 disease, Vaccine, 24 (2006), 35-41. doi: 10.1016/j.vaccine.2006.06.015. [20] X. Li, G. Gupur and G. Zhu, Threshold and stability results for an age-structured SEIR epidemic model, Comput. Math. Appl., 42 (2001), 883-907. doi: 10.1016/S0898-1221(01)00206-1. [21] X. Li and J. Liu, Stability of an age-structured epidemiological model for hepatitis C, J. Appl. Math. Comput., 27 (2008), 159-173. doi: 10.1007/s12190-008-0060-5. [22] X. Li, J. Liu and M. Martcheva, An age-structured two-strain epidemic model with super-infection, Math. Biosci. Engg., 7 (2009), 123-147. doi: 10.3934/mbe.2010.7.123. [23] A. T. Newall et al., Cost-effectiveness analyses of human papillomavirus vaccination, The Lancet Infectious Diseases, 7 (2007), 289-296. [24] D. M. Parkin and F. Bray, The burden of HPV-related cancers, Vaccine, 23 (2006), 11-25. doi: 10.1016/j.vaccine.2006.05.111. [25] D. M. Parkin, The global health burden of infection-associated cancers in the year 2002, Int. J. Cancer, 118 (2006), 3030-3044. doi: 10.1002/ijc.21731. [26] M. A. Safi et al., Qualitative analysis of an age-structred SEIR epidemic model with treatment, Appl. Math. Comput., 219 (2013), 10627-10642. doi: 10.1016/j.amc.2013.03.126. [27] N. F. Schlecht et al., Persistent human papillomavirus infection as a predictor of cervical intraepithelial neoplasia, JAMA, 286 (2001), 3106-3114. [28] L. L. Villa, R. L. R. Costa, C. A. Petta et al., Prophylactic quadrivalent human papillomavirus (types 6, 11, 16, and 18) L1 virus-like particle vaccine in young women: A randomised double-blind placebo-controlled multicentre phase II efficacy trial, Lancet Oncol., 6 (2005), 271-278. doi: 10.1016/S1470-2045(05)70101-7. [29] L. Zou, S. Ruan and W. Zhang, An age-structured model for the transmission dynamics of Hepatitis B, SIAM J. Appl. Math., 70 (2010), 3121-3139. doi: 10.1137/090777645.

show all references

##### References:
 [1] M. Al-arydah and R. J. Smith, An age-structured model of human papillomavirus vaccination, Mathematics and Computers in Simulation, 82 (2011), 629-642. doi: 10.1016/j.matcom.2011.10.006. [2] R. Barnabas and G. Garnett, The potential public health impact of vaccines against human papillomavirus, in "The Clinical Handbook of Human Papillomavirus" (eds. Prendiville and Davies), Taylor and Francis, (2004), 61-79. [3] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci., 151 (1998), 135-154. doi: 10.1016/S0025-5564(98)10016-0. [4] "Genital HPV Infection - CDC Fact Sheet,", Centers for Disease Control and Prevention (CDC)., , (). [5] H. W. Chesson, D. U. Ekwueme, M. Saraiya and L. E. Markowitz, Cost-effectiveness of human papillomavirus vaccination in the United States, Emerg. Infect. Dis., 14 (2008), 244-251. doi: 10.3201/eid1402.070499. [6] M. E. Cruickshank, L. Sharp, G. Chambers, L. Smart and G. Murray, Persistent infection with human papillomavirus following the successful treatment of high grade cervical intraepithelial neoplasia, BJOG, 109 (2002), 579-581. [7] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," John Wiley & Sons Ltd., England, 2000. [8] E. H. Elbasha, E. J. Dasback and R. P. Insinga, Model for assessing human papillomavirus vaccination strategies, Emerg. Infect. Dis., 13 (2007), 28-41. doi: 10.3201/eid1301.060438. [9] E. H. Elbasha, Global stability of equilibria in a two-sex HPV vaccination model, Bull. Math. Biol., 70 (2008), 894-909. doi: 10.1007/s11538-007-9283-0. [10] E. H. Elbasha, E. J. Dasbach and R. P. Insinga, A multi-type HPV transmission model, Bull. Math. Biol., 70 (2008), 2126-2176. doi: 10.1007/s11538-008-9338-x. [11] E. H. Elbasha and E. J. Dasbach, Impact of vaccinating boys and men against HPV in the United States, Vaccine, 28 (2010), 6858-6867. doi: 10.1016/j.vaccine.2010.08.030. [12] A. Ferenczy, Persistent human papillomavirus infection and cervical neoplasia, The Lancet Oncology, 3 (2002), 11-16. doi: 10.1016/S1470-2045(01)00617-9. [13] D. Greenhalgh, Threshold and stability results for an epidemic model with an age-structured meeting rate, IMA J. Math. Appl. Med. Biol., 5 (1988), 81-100. doi: 10.1093/imammb/5.2.81. [14] H. W. Hethcote, Age-structured epdiemiology models and expressions for $R_0$, in "Mathematical Understanding of Infectious Disease Dynamics" (eds. S. Ma and Y. Xia), World Sci. Publ., Hackensack, NJ, (2009), 91-128. doi: 10.1142/9789812834836_0003. [15] J. Hughes, G. Garnett and L. Koutsky, The theoretical population level impact of a prophylactic human papillomavirus vaccine, Epidemiology, 13 (2002), 631-639. [16] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. [17] W. Kaplan, "Advanced Calculus," Addison-Wesley Mathematics Series, 1984. [18] J. J. Kim and S. J. Goldie, Cost effectiveness analysis of including boys in a human papillomavirus vaccination programme in the United States, BMJ, 339 (2009), 1-10. doi: 10.1136/bmj.b3884. [19] C. J. N. Lacey, C. M. Lowndes and K. V. Shah, Burden and management of non-cancerous HPV-related conditions: HPV-6/11 disease, Vaccine, 24 (2006), 35-41. doi: 10.1016/j.vaccine.2006.06.015. [20] X. Li, G. Gupur and G. Zhu, Threshold and stability results for an age-structured SEIR epidemic model, Comput. Math. Appl., 42 (2001), 883-907. doi: 10.1016/S0898-1221(01)00206-1. [21] X. Li and J. Liu, Stability of an age-structured epidemiological model for hepatitis C, J. Appl. Math. Comput., 27 (2008), 159-173. doi: 10.1007/s12190-008-0060-5. [22] X. Li, J. Liu and M. Martcheva, An age-structured two-strain epidemic model with super-infection, Math. Biosci. Engg., 7 (2009), 123-147. doi: 10.3934/mbe.2010.7.123. [23] A. T. Newall et al., Cost-effectiveness analyses of human papillomavirus vaccination, The Lancet Infectious Diseases, 7 (2007), 289-296. [24] D. M. Parkin and F. Bray, The burden of HPV-related cancers, Vaccine, 23 (2006), 11-25. doi: 10.1016/j.vaccine.2006.05.111. [25] D. M. Parkin, The global health burden of infection-associated cancers in the year 2002, Int. J. Cancer, 118 (2006), 3030-3044. doi: 10.1002/ijc.21731. [26] M. A. Safi et al., Qualitative analysis of an age-structred SEIR epidemic model with treatment, Appl. Math. Comput., 219 (2013), 10627-10642. doi: 10.1016/j.amc.2013.03.126. [27] N. F. Schlecht et al., Persistent human papillomavirus infection as a predictor of cervical intraepithelial neoplasia, JAMA, 286 (2001), 3106-3114. [28] L. L. Villa, R. L. R. Costa, C. A. Petta et al., Prophylactic quadrivalent human papillomavirus (types 6, 11, 16, and 18) L1 virus-like particle vaccine in young women: A randomised double-blind placebo-controlled multicentre phase II efficacy trial, Lancet Oncol., 6 (2005), 271-278. doi: 10.1016/S1470-2045(05)70101-7. [29] L. Zou, S. Ruan and W. Zhang, An age-structured model for the transmission dynamics of Hepatitis B, SIAM J. Appl. Math., 70 (2010), 3121-3139. doi: 10.1137/090777645.
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