2013, 10(4): 1173-1205. doi: 10.3934/mbe.2013.10.1173

The impact of an imperfect vaccine and pap cytology screening on the transmission of human papillomavirus and occurrence of associated cervical dysplasia and cancer

1. 

Department of Applied Mathematics and Sciences, Khalifa University of Science, Technology and Research, PO Box 127788, Abu Dhabi, United Arab Emirates

2. 

Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, United Kingdom

3. 

Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

4. 

Merck Research Laboratories, UG1C-60, PO Box 1000, North Wales, PA 19454-1099

5. 

Community Health Sciences, University of Manitoba, Winnipeg, Manitoba, Canada

Received  April 2012 Revised  February 2013 Published  June 2013

A mathematical model for the natural history of human papillomavirus (HPV) is designed and used to assess the impact of a hypothetical anti-HPV vaccine and Pap cytology screening on the transmission dynamics of HPV in a population. Rigorous qualitative analysis of the model reveals that it undergoes the phenomenon of backward bifurcation. It is shown that the backward bifurcation is caused by the imperfect nature of the HPV vaccine or the HPV-induced and cancer-induced mortality in females. For the case when the disease-induced and cancer-induced mortality is negligible, it is shown that the disease-free equilibrium (i.e., equilibrium in the absence of HPV and associated dysplasia) is globally-asymptotically stable if the associated reproduction number is less than unity. The model has a unique endemic equilibrium when the reproduction threshold exceeds unity. The unique endemic equilibrium is globally-asymptotically stable for a special case, where the associated HPV-induced and cancer-induced mortality is negligible. Numerical simulations of the model, using a reasonable set of parameter values, support the recent recommendations by some medical agencies and organizations in the USA to offer Pap screening on a 3-year basis (rather than annually).
Citation: Tufail Malik, Jody Reimer, Abba Gumel, Elamin H. Elbasha, Salaheddin Mahmud. The impact of an imperfect vaccine and pap cytology screening on the transmission of human papillomavirus and occurrence of associated cervical dysplasia and cancer. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1173-1205. doi: 10.3934/mbe.2013.10.1173
References:
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M. Brisson, N. Van de Velde and M. C. Boily, Different population-level vaccination effectiveness for HPV types 16, 18, 6 and 11,, Sex. Transm. Infect., 87 (2011), 41.  doi: 10.1136/sti.2010.044412.  Google Scholar

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O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment,, Math. Biosci. Engrg., 5 (2008), 145.  doi: 10.3934/mbe.2008.5.145.  Google Scholar

[50]

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P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

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L. L. Villa et al., High sustained efficacy of a prophylactic quadrivalent human Papillomavirus types 6/11/16/18 virus-like particle vaccine through 5 years of follow-up,, British Journal of Cancer, 95 (2006), 1459.   Google Scholar

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show all references

References:
[1]

M. Llamazares and R. J. Smith, Evaluating human papillomavirus vaccination programs in Canada: Should provincial healthcare pay for voluntary adult vaccination?,, BMC Public Health, 8 (2008), 1.  doi: 10.1186/1471-2458-8-114.  Google Scholar

[2]

M. Al-arydah and R. J. Smith, An age-structured model of human papillomavirus vaccination,, Mathematics and Computers in Simulation, 82 (2011), 629.  doi: 10.1016/j.matcom.2011.10.006.  Google Scholar

[3]

F. Brauer, Backward bifurcations in simple vaccination models,, J. Math. Anal. Appl., 298 (2004), 418.  doi: 10.1016/j.jmaa.2004.05.045.  Google Scholar

[4]

M. Brisson, N. Van de Velde and M. C. Boily, Different population-level vaccination effectiveness for HPV types 16, 18, 6 and 11,, Sex. Transm. Infect., 87 (2011), 41.  doi: 10.1136/sti.2010.044412.  Google Scholar

[5]

V. Brown and K. A. J. White, The HPV vaccination strategy: Could male vaccination have a significant impact?,, Comput. Math. Methods. Med., 11 (2010), 223.  doi: 10.1080/17486700903486613.  Google Scholar

[6]

Canadian Cancer Society's Steering Committee, "Cancer Statistics 2009,", (2009). Available from: , (2009).   Google Scholar

[7]

J. Carr, "Applications of Centre Manifold Theory,", Springer-Verlag, (1981).   Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Engrg., 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[9]

B. Crawford and C. M. Kribs Zaleta, The impact of vaccination and coinfection on HPV and cervical cancer,, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 279.  doi: 10.3934/dcdsb.2009.12.279.  Google Scholar

[10]

L. A. Denny and J. T. C. Wright, Human papillomavirus testing and screening,, Best Practice & Research Clinical Obstetrics and Gynaecology, 19 (2005), 501.  doi: 10.1016/j.bpobgyn.2005.02.004.  Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[12]

J. Dushoff, H. Wenzhang and C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases,, J. Math. Biol., 36 (1998), 227.  doi: 10.1007/s002850050099.  Google Scholar

[13]

E. H. Elbasha and A. P. Galvani, Vaccination against multiple HPV types,, Math. Biosci., 197 (2005), 88.  doi: 10.1016/j.mbs.2005.05.004.  Google Scholar

[14]

E. H. Elbasha and A. B. Gumel, Theoretical assesment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits,, Bull. Math. Biol., 68 (2006), 577.   Google Scholar

[15]

E. H. Elbasha, E. J.Dasbach and R. P. Insinga, Model for assessing human papillomavirus vaccination strategies,, Emerg. Infect. Dis., 13 (2007), 28.  doi: 10.3201/eid1301.060438.  Google Scholar

[16]

E. H. Elbasha, Global stability of equilibria in a two-sex HPV vaccination model,, Bull. Math. Biol., 70 (2008), 894.  doi: 10.1007/s11538-007-9283-0.  Google Scholar

[17]

E. H. Elbasha, E. J. Dasbach and R. P. Insinga, A multi-type HPV transmission model,, Bull. Math. Biol., 70 (2008), 2126.  doi: 10.1007/s11538-008-9338-x.  Google Scholar

[18]

J. Ferlay, H. R. Shin, F. Bray, D. Forman, C. Mathers and D. M. Parkin, Estimates of worldwide burden of cancer in 2008: GLOBOCAN 2008,, Int. J. Cancer., 127 (2010), 2893.  doi: 10.1002/ijc.25516.  Google Scholar

[19]

E. L. Franco, S. M. Mahmud, J. Tota, A. Ferenczy and F. Coutle, The expected impact of HPV vaccination on the accuracy of cervical cancer screening: The need for a paradigm change,, Arch. Med. Res., 40 (2009), 478.  doi: 10.1016/j.arcmed.2009.06.003.  Google Scholar

[20]

E. L. Franco, E. Duarte-Franco and A. Ferenczy, Cervical cancer: Epidemiology, prevention and the role of human papillomavirus infection,, CMAJ., 164 (2001), 1017.   Google Scholar

[21]

S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics,, Math. Biosci., 215 (2008), 11.  doi: 10.1016/j.mbs.2008.05.002.  Google Scholar

[22]

, The GlaxoSmithKline Vaccine HPV-007 Study Group,, Sustained efficacy and immunogenicity of the human papillomavirus (HPV)-16/17 ASO4-adjuvanted vaccine: Analysis of a randomised placebo-controlled trial up to 6.4 years, ().   Google Scholar

[23]

S. J. Goldie et al., Projected clinical benefits and cost-effectiveness of a human papillomavirus 16/18 vaccine,, Journal of the National Cancer Institute, 96 (2004), 604.   Google Scholar

[24]

O. P. Günther et al., Protecting the next generation: What is the role of the duration of Human Papillomavirus vaccine-related immunity?,, J. Infect. Dis., 197 (2008), 1653.   Google Scholar

[25]

B. T. Hansen et al., Factors associated with non-attendance, opportunistic attendance and reminded attendance to cervical screening in an organized screening program: A cross-sectional study of 12,058 Norwegian women,, BMC Public Health, 11 (2011), 1.   Google Scholar

[26]

Health Canada, "Cervical Cancer Screening Activities in Canada,", (2012). Available from: , (2012).   Google Scholar

[27]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[28]

J. Hughes, G. Garnett and L. Koutsky, The theoretical population level impact of a prophylactic human papillomavirus vaccine,, Epidemiology, 13 (2002), 631.   Google Scholar

[29]

"International Agency for Research on Cancer" Working Group., Human papillomaviruses,, IARC Monographs on the Evaluation of the Carcinogenic Risks to Humans, (1995).   Google Scholar

[30]

O. Klungsøyr, M. Nygård, G. Skare, T. Eriksen and J. F Nygård, Validity of self-reported Pap smear history in Norwegian women,, J. Med. Screen, 16 (2009), 91.  doi: 10.1258/jms.2009.008087.  Google Scholar

[31]

L. Koutsky, Epidemiology of genital human papillomavirus infection,, American Journal of Medicine, 102 (1997), 3.   Google Scholar

[32]

C. Kribs-Zaleta and J. Velasco-Hernandez, A simple vaccination model with multiple endemic states,, Math Biosci., 164 (2000), 183.  doi: 10.1016/S0025-5564(00)00003-1.  Google Scholar

[33]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems,", Marcel Dekker, (1989).   Google Scholar

[34]

J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[35]

D. C. McCrory and D. B. Matchar, Evaluation of Cervical Cytology. Evidence Report/Technology Assessment No. 5;,, AHCPR Pub; No 99-E 010 1999., (1999).   Google Scholar

[36]

Z. Mukandavire, A. B. Gumel, W. Garira and J. M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection,, Math. Biosci. Engrg., 6 (2009), 333.  doi: 10.3934/mbe.2009.6.333.  Google Scholar

[37]

N. Munoz, F. X. Bosch, S. de Sanjose et al., Epidemiologic classification of human papillomavirus types associated with cervical cancer,, N. Engl. J. Med., 348 (2003), 518.   Google Scholar

[38]

N. Munoz et al., Against which human papillomavirus types shall we vaccinate and screen? The international perspective,, International Journal of Cancer, 111 (2004), 278.   Google Scholar

[39]

E. R. Myers et al., Mathematical model for the natural history of human papillomavirus infection and cervical carcinogenesis,, Am. J. Epidemiol., 151 (2000), 1158.   Google Scholar

[40]

S. L. Kulasingam and E. R. Myers, Potential health and economic impact of adding a human papillomavirus vaccine to screening programs,, JAMA., 290 (2003), 781.  doi: 10.1001/jama.290.6.781.  Google Scholar

[41]

J. M. Palefsky, Human papillomavirus-related disease in men: Not just a women's issue,, Journal of Adolescent Health, 46 (2010).  doi: 10.1016/j.jadohealth.2010.01.010.  Google Scholar

[42]

C. N. Podder and A. B. Gumel, Transmission dynamics of a two-sex model for Herpes Simplex Virus type 2,, Can. Appl. Math. Quarterly, 17 (2009), 339.   Google Scholar

[43]

Public Health Agency of Canada, "Cervical Cancer Screening in Canada,", (2013). Available from: , (2013).   Google Scholar

[44]

Public Health Agency of Canada, "Human Papillomavirus. HPV Purple Paper (bds),", (2010). Available from: , (2010).   Google Scholar

[45]

P. Sasieni and A. Castanon, Call and recall cervical screening programme: Screening interval and age limits,, Curr. Diagn. Pathol., 12 (2006), 114.   Google Scholar

[46]

D. Saslow et al., American cancer society, American society for colposcopy and cervical pathology, and American society for clinical pathology screening guidelines for the prevention and early detection of cervical cancer,, Am. J. Clin. Pathol., 137 (2012), 516.   Google Scholar

[47]

M. Schiffman and P. E. Castle, When to test women for human papillomavirus,, BMJ., 332 (2006), 61.  doi: 10.1136/bmj.332.7533.61.  Google Scholar

[48]

O. Sharomi, C. N. Podder, A. B. Gumel, E. H. Elbasha and J. Watmough, Role of incidence function in vaccine-induced backward bifurcation in some HIV models,, Math. Biosci., 210 (2007), 436.  doi: 10.1016/j.mbs.2007.05.012.  Google Scholar

[49]

O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment,, Math. Biosci. Engrg., 5 (2008), 145.  doi: 10.3934/mbe.2008.5.145.  Google Scholar

[50]

H. L. Smith and P. Waltman, "The Theory of the Chemostat,", Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[51]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[52]

N. Van de Velde, M. Brisson and M. C. Boily, Modeling human papillomavirus vaccine effectiveness: Quantifying the impact of parameter uncertainty,, Am. J. Epidemiol., 165 (2007), 762.   Google Scholar

[53]

L. L. Villa et al., High sustained efficacy of a prophylactic quadrivalent human Papillomavirus types 6/11/16/18 virus-like particle vaccine through 5 years of follow-up,, British Journal of Cancer, 95 (2006), 1459.   Google Scholar

[54]

J. M. M. Walboomers, Human papillomavirus is a necessary cause of invasive cervical cancer worldwide,, J. Pathol., 189 (1999), 12.  doi: 10.1002/(SICI)1096-9896(199909)189:1<12::AID-PATH431>3.0.CO;2-F.  Google Scholar

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