# American Institute of Mathematical Sciences

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:
 [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), 1-12. doi: 10.1186/1471-2458-8-114. [2] M. Al-arydah and R. J. Smith, An age-structured model of human papillomavirus vaccination, Mathematics and Computers in Simulation, 82 (2011), 629-652. doi: 10.1016/j.matcom.2011.10.006. [3] F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004), 418-431. doi: 10.1016/j.jmaa.2004.05.045. [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), 41-43. doi: 10.1136/sti.2010.044412. [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), 223-237. doi: 10.1080/17486700903486613. [6] Canadian Cancer Society's Steering Committee, "Cancer Statistics 2009," (2009). Available from: http://www.cancer.ca. [7] J. Carr, "Applications of Centre Manifold Theory," Springer-Verlag, New York, 1981. [8] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Engrg., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [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), 279-304. doi: 10.3934/dcdsb.2009.12.279. [10] L. A. Denny and J. T. C. Wright, Human papillomavirus testing and screening, Best Practice & Research Clinical Obstetrics and Gynaecology, 19 (2005), 501-515. doi: 10.1016/j.bpobgyn.2005.02.004. [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), 365-382. doi: 10.1007/BF00178324. [12] J. Dushoff, H. Wenzhang and C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227-248. doi: 10.1007/s002850050099. [13] E. H. Elbasha and A. P. Galvani, Vaccination against multiple HPV types, Math. Biosci., 197 (2005), 88-117. doi: 10.1016/j.mbs.2005.05.004. [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), 577-614. [15] E. H. Elbasha, E. J.Dasbach and R. P. Insinga, Model for assessing human papillomavirus vaccination strategies, Emerg. Infect. Dis., 13 (2007), 28-41. doi: 10.3201/eid1301.060438. [16] 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. [17] 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. [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), 2893-2917. doi: 10.1002/ijc.25516. [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), 478-485. doi: 10.1016/j.arcmed.2009.06.003. [20] E. L. Franco, E. Duarte-Franco and A. Ferenczy, Cervical cancer: Epidemiology, prevention and the role of human papillomavirus infection, CMAJ., 164 (2001), 1017-1025. [21] S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002. [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, (). [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), 604-615. [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), 1653-1661. [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), 1-13. [26] Health Canada, "Cervical Cancer Screening Activities in Canada," (2012). Available from: http://www.phac-aspc.gc.ca. [27] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [28] J. Hughes, G. Garnett and L. Koutsky, The theoretical population level impact of a prophylactic human papillomavirus vaccine, Epidemiology, 13 (2002), 631-639. [29] "International Agency for Research on Cancer" Working Group., Human papillomaviruses, IARC Monographs on the Evaluation of the Carcinogenic Risks to Humans, (1995). Lyon, France. [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), 91-97. doi: 10.1258/jms.2009.008087. [31] L. Koutsky, Epidemiology of genital human papillomavirus infection, American Journal of Medicine, 102 (1997), 3-8. [32] C. Kribs-Zaleta and J. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math Biosci., 164 (2000), 183-201. doi: 10.1016/S0025-5564(00)00003-1. [33] V. Lakshmikantham, S. Leela and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems," Marcel Dekker, Inc., New York and Basel, 1989. [34] J. P. LaSalle, "The Stability of Dynamical Systems," Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [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. [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), 333-362. doi: 10.3934/mbe.2009.6.333. [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), 518-527. [38] N. Munoz et al., Against which human papillomavirus types shall we vaccinate and screen? The international perspective, International Journal of Cancer, 111 (2004), 278-285. [39] E. R. Myers et al., Mathematical model for the natural history of human papillomavirus infection and cervical carcinogenesis, Am. J. Epidemiol., 151 (2000), 1158-1171. [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), 781-789. doi: 10.1001/jama.290.6.781. [41] J. M. Palefsky, Human papillomavirus-related disease in men: Not just a women's issue, Journal of Adolescent Health, 46 (2010), S12-S19. doi: 10.1016/j.jadohealth.2010.01.010. [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), 339-386. [43] Public Health Agency of Canada, "Cervical Cancer Screening in Canada," (2013). Available from: http://www.phac-aspc.gc.ca/publicat/ccsic-dccuac/pdf/chap_3_e.pdf. [44] Public Health Agency of Canada, "Human Papillomavirus. HPV Purple Paper (bds)," (2010). Available from: http://www.phac-aspc.gc.ca. [45] P. Sasieni and A. Castanon, Call and recall cervical screening programme: Screening interval and age limits, Curr. Diagn. Pathol., 12 (2006), 114-126. [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), 516-542. [47] M. Schiffman and P. E. Castle, When to test women for human papillomavirus, BMJ., 332 (2006), 61-62. doi: 10.1136/bmj.332.7533.61. [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), 436-463. doi: 10.1016/j.mbs.2007.05.012. [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), 145-174. doi: 10.3934/mbe.2008.5.145. [50] H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [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), 762-775. [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), 1459-1466. [54] J. M. M. Walboomers, Human papillomavirus is a necessary cause of invasive cervical cancer worldwide, J. Pathol., 189 (1999), 12-19. doi: 10.1002/(SICI)1096-9896(199909)189:1<12::AID-PATH431>3.0.CO;2-F.

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), 1-12. doi: 10.1186/1471-2458-8-114. [2] M. Al-arydah and R. J. Smith, An age-structured model of human papillomavirus vaccination, Mathematics and Computers in Simulation, 82 (2011), 629-652. doi: 10.1016/j.matcom.2011.10.006. [3] F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004), 418-431. doi: 10.1016/j.jmaa.2004.05.045. [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), 41-43. doi: 10.1136/sti.2010.044412. [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), 223-237. doi: 10.1080/17486700903486613. [6] Canadian Cancer Society's Steering Committee, "Cancer Statistics 2009," (2009). Available from: http://www.cancer.ca. [7] J. Carr, "Applications of Centre Manifold Theory," Springer-Verlag, New York, 1981. [8] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Engrg., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [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), 279-304. doi: 10.3934/dcdsb.2009.12.279. [10] L. A. Denny and J. T. C. Wright, Human papillomavirus testing and screening, Best Practice & Research Clinical Obstetrics and Gynaecology, 19 (2005), 501-515. doi: 10.1016/j.bpobgyn.2005.02.004. [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), 365-382. doi: 10.1007/BF00178324. [12] J. Dushoff, H. Wenzhang and C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227-248. doi: 10.1007/s002850050099. [13] E. H. Elbasha and A. P. Galvani, Vaccination against multiple HPV types, Math. Biosci., 197 (2005), 88-117. doi: 10.1016/j.mbs.2005.05.004. [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), 577-614. [15] E. H. Elbasha, E. J.Dasbach and R. P. Insinga, Model for assessing human papillomavirus vaccination strategies, Emerg. Infect. Dis., 13 (2007), 28-41. doi: 10.3201/eid1301.060438. [16] 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. [17] 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. [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), 2893-2917. doi: 10.1002/ijc.25516. [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), 478-485. doi: 10.1016/j.arcmed.2009.06.003. [20] E. L. Franco, E. Duarte-Franco and A. Ferenczy, Cervical cancer: Epidemiology, prevention and the role of human papillomavirus infection, CMAJ., 164 (2001), 1017-1025. [21] S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002. [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, (). [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), 604-615. [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), 1653-1661. [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), 1-13. [26] Health Canada, "Cervical Cancer Screening Activities in Canada," (2012). Available from: http://www.phac-aspc.gc.ca. [27] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [28] J. Hughes, G. Garnett and L. Koutsky, The theoretical population level impact of a prophylactic human papillomavirus vaccine, Epidemiology, 13 (2002), 631-639. [29] "International Agency for Research on Cancer" Working Group., Human papillomaviruses, IARC Monographs on the Evaluation of the Carcinogenic Risks to Humans, (1995). Lyon, France. [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), 91-97. doi: 10.1258/jms.2009.008087. [31] L. Koutsky, Epidemiology of genital human papillomavirus infection, American Journal of Medicine, 102 (1997), 3-8. [32] C. Kribs-Zaleta and J. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math Biosci., 164 (2000), 183-201. doi: 10.1016/S0025-5564(00)00003-1. [33] V. Lakshmikantham, S. Leela and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems," Marcel Dekker, Inc., New York and Basel, 1989. [34] J. P. LaSalle, "The Stability of Dynamical Systems," Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [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. [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), 333-362. doi: 10.3934/mbe.2009.6.333. [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), 518-527. [38] N. Munoz et al., Against which human papillomavirus types shall we vaccinate and screen? The international perspective, International Journal of Cancer, 111 (2004), 278-285. [39] E. R. Myers et al., Mathematical model for the natural history of human papillomavirus infection and cervical carcinogenesis, Am. J. Epidemiol., 151 (2000), 1158-1171. [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), 781-789. doi: 10.1001/jama.290.6.781. [41] J. M. Palefsky, Human papillomavirus-related disease in men: Not just a women's issue, Journal of Adolescent Health, 46 (2010), S12-S19. doi: 10.1016/j.jadohealth.2010.01.010. [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), 339-386. [43] Public Health Agency of Canada, "Cervical Cancer Screening in Canada," (2013). Available from: http://www.phac-aspc.gc.ca/publicat/ccsic-dccuac/pdf/chap_3_e.pdf. [44] Public Health Agency of Canada, "Human Papillomavirus. HPV Purple Paper (bds)," (2010). Available from: http://www.phac-aspc.gc.ca. [45] P. Sasieni and A. Castanon, Call and recall cervical screening programme: Screening interval and age limits, Curr. Diagn. Pathol., 12 (2006), 114-126. [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), 516-542. [47] M. Schiffman and P. E. Castle, When to test women for human papillomavirus, BMJ., 332 (2006), 61-62. doi: 10.1136/bmj.332.7533.61. [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), 436-463. doi: 10.1016/j.mbs.2007.05.012. [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), 145-174. doi: 10.3934/mbe.2008.5.145. [50] H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [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), 762-775. [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), 1459-1466. [54] J. M. M. Walboomers, Human papillomavirus is a necessary cause of invasive cervical cancer worldwide, J. Pathol., 189 (1999), 12-19. doi: 10.1002/(SICI)1096-9896(199909)189:1<12::AID-PATH431>3.0.CO;2-F.
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