Bulletin of Mathematical Biology

, Volume 78, Issue 1, pp 4–20 | Cite as

The Dynamics of HPV Infection and Cervical Cancer Cells

  • Tri Sri Noor Asih
  • Suzanne Lenhart
  • Steven Wise
  • Lina Aryati
  • F. Adi-Kusumo
  • Mardiah S. Hardianti
  • Jonathan FordeEmail author
Original Article


The development of cervical cells from normal cells infected by human papillomavirus into invasive cancer cells can be modeled using population dynamics of the cells and free virus. The cell populations are separated into four compartments: susceptible cells, infected cells, precancerous cells and cancer cells. The model system of differential equations also has a free virus compartment in the system, which infect normal cells. We analyze the local stability of the equilibrium points of the model and investigate the parameters, which play an important role in the progression toward invasive cancer. By simulation, we investigate the boundary between initial conditions of solutions, which tend to stable equilibrium point, representing controlled infection, and those which tend to unbounded growth of the cancer cell population. Parameters affected by drug treatment are varied, and their effect on the risk of cancer progression is explored.


HPV Cervical cancer Mathematical modeling 



Thanks to Tim Sparer, Louis J. Gross, Vitaly Ganusov, Jiang Jiang and Kelsey Bratton for useful discussion and assistance.


  1. Abdulkarim B, Sabri S, Deutsch E, Chagraoui H, Maggiorella L, Thierry J, Eschwege F, Vainchenker W, Chouaïb S, Bourhis J (2002) Antiviral agent cidofovir restores p53 function and enhances the radiosensitivity in HPV-associated cancers. Oncogene 21:2334–2346. doi: 10.1038/sj.onc.1206402 CrossRefGoogle Scholar
  2. Ault KA (2006) Clinical study, epidemiology and natural history of human papillomavirus infections in the female genital tract. Infect Dis Obstet Gynecol 40470. doi: 10.1155/IDOG/2006/40470
  3. Barnabas RV, Laukkanen P, Koskela P, Kontula O, Lehtinen M, Garnett GP (2006) Epidemiology of HPV 16 and cervical cancer in Finland and the potential impact of vaccination: mathematical modelling analyses. PLOS Med 3:0624–0632. doi: 10.1371/journal.pmed.0030138 CrossRefGoogle Scholar
  4. Bonhoeffer S, May RM, Shaw GM, Nowak MA (1997) Virus dynamics and drug therapy. Proc Natl Acad Sci 94:6971–6976CrossRefGoogle Scholar
  5. Bosch FX, Manos MM, Munoz N, Sherman M, Jansen AM, Peto J, Schiffman MH, Moreno V, Kurman R, Shah KV (1995) Prevalence of human papillomavirus in cervical cancer: a worldwide perspective. J Natl Cancer Inst 87:796–802. doi: 10.1093/jnci/87.11.796 CrossRefGoogle Scholar
  6. Bosch FX, Lorincz A, Munoz N, Meijer CJLM, Shah KV (2002) The causal relation between human papillomavirus and cervical cancer. J Clin Pathol 55:244–265CrossRefGoogle Scholar
  7. Brown V, White KA (2010) The HPV vaccination strategy: could male vaccination have an impact? Comput Math Methods Med 11:223–237. doi: 10.1080/17486700903486613 CrossRefMathSciNetzbMATHGoogle Scholar
  8. Brown VI, White KAJ (2011) The role of optimal control in assessing the most cost-effective implementation of a vaccination programme: HPV as a case study. Math Biosci 231:126–134. doi: 10.1016/j.mbs.2011.02.009 CrossRefMathSciNetzbMATHGoogle Scholar
  9. Clifford GM, Smith JS, Aguado T, Franceschi S (2003) Comparison of HPV type distribution in high-grade cervical lesions and cervical cancer: a meta-analysis. Br J Cancer 89:101–105. doi: 10.1038/sj.bjc.6601024 CrossRefGoogle Scholar
  10. Elbasha EH, Dasbach EJ, Insinga RP (2007) Model for assessing human papillomavirus vaccination strategies. Emerg Infect Dis 13:28–41. doi: 10.3201/eid1301.060438 CrossRefGoogle Scholar
  11. Elbasha EH (2008) Global stability of equilibria in a two-sex HPV vaccination model. Bull Math Biol 70:894–909. doi: 10.1007/s11538-007-9283-0 CrossRefMathSciNetzbMATHGoogle Scholar
  12. Forde J, Nelson P (2004) Application of Sturm sequence to bifurcation analysis of delay differential equation models. J Math Anal Appl 300:273–284. doi: 10.1016/j.jmaa.2004.02.063 CrossRefMathSciNetzbMATHGoogle Scholar
  13. Globocan 2008 (2010), Downloaded March 2012
  14. Goldhaber-Fiebert JD, Stout NK, Salomon JA, Kuntz KM, Goldie SJ (2008) Cost-effectiveness of cervical cancer screening with human papillomavirus DNA testing and HPV-16,18 vaccination. J Natl Cancer Inst 100:308–320. doi: 10.1093/jnci/djn019 CrossRefGoogle Scholar
  15. Goldie SJ, Kohli M, Grima D, Weinstein MC, Wright TC, Bosch FX, Franco E (2004) Projected clinical benefits and cost-effectiveness of a human papillomavirus 16/18 vaccine. J Natl Cancer Inst 96:604–615. doi: 10.1093/jnci/djh104 CrossRefGoogle Scholar
  16. Hostetler KY, Rought S, Aldern KA, Trahan J, Beadle JR, Corbeil J (2006) Enhanced antiproliferative effects of alkoxyalkyl esters of cidofovir in human cervical cancer cells in vitro. Mol Cancer Ther 5:156–159. doi: 10.1158/1535-7163.MCT-05-0200 CrossRefGoogle Scholar
  17. IARC (2005) IARC handbooks of cancer prevention, cervix cancer screening. IARC Press, LyonGoogle Scholar
  18. IARC (2007) IARC monographs on the evaluation of carcinogenic risk to humans, volume 90, human papillomavirus. IARC Press, LyonGoogle Scholar
  19. Kim JJ, Brisson M, Edmunds WJ, Goldie SJ (2008) Modeling cervical cancer prevention in developed countries. Vaccine 26:K76–K86. doi: 10.1016/j.vaccine.2008.06.009 CrossRefGoogle Scholar
  20. Kohli M, Ferko N, Martin A, Franco EL, Jenkins D, Gallivan S, Sherlaw-Johnson C, Drummond M (2007) Estimating the long-term impact of a prophylactic human papillomavirus 16/18 vaccine on the burden of cervical cancer in the UK. Br J Cancer 96:143–150. doi: 10.1038/sj.bjc.6603501 CrossRefGoogle Scholar
  21. Kulasingam SL, Myers ER (2003) Potential health and economic impact of adding a human papillomavirus vaccine to screening programs. JAMA 290:781–789. doi: 10.1001/jama.290.6.781 CrossRefGoogle Scholar
  22. Lee C, Laimins LA (2007) The differentiation-dependent life cycle of human papillomaviruses in keratinocytes. In: Garcea R, Di Maio D (eds) The papillomaviruses. Springer, New York, pp 45–67CrossRefGoogle Scholar
  23. Lee SL, Tameru AM (2012) A mathematical model of human papillomavirus (HPV) in the United States and its impact on cervical cancer. J Cancer 3:262–268. doi: 10.7150/jca.4161 CrossRefGoogle Scholar
  24. Lowy DR, Kirnbauer R, Schiller JT (1994) Genital human papillomavirus infection. Proc Natl Acad Sci USA 91:2436–2440CrossRefGoogle Scholar
  25. Motoyama S, Ladines-Llave CA, Villanueva SL, Maruo T (2004) The role of human papilloma virus in the molecular biology of cervical carcinoma. Kobe J Med Sci 50:9–19Google Scholar
  26. Mougin C, Dalstein V, Pretet JL, Gay C, Schall JP, Riethmuller D (2001) Epidemiology of cervical papillomavirus infections, recent knowledge. Presse Med 30:1017–1023Google Scholar
  27. Nowak M, Bangham CRM (1996) Population dynamics of immune responses to persistent viruses. Science 272:74–79CrossRefGoogle Scholar
  28. Nowak M, May R (2000) Virus dynamics, mathematical principles of immunology and virology. Oxford University Press, New YorkzbMATHGoogle Scholar
  29. Sanders GD, Taira AV (2003) Cost effectiveness of potential vaccine for human papillomavirus. Emerg Infect Dis 9:37–48. doi: 10.3201/eid0901.020168 CrossRefGoogle Scholar
  30. Schorge JO, Schaffer JI, Halvorson LM, Hoffman BL, Bradshaw KD, Chunningham FG (2008) Cervical cancer. In: Williams gynecology. McGraw-Hill, NYGoogle Scholar
  31. Taira AV, Neukersmans CP, Sanders GD (2004) Evaluating human papillomavirus vaccination programs. Emerg Infect Dis 10:915–1923. doi: 10.3201/eid1011.040222 CrossRefGoogle Scholar
  32. Trottier H, Franco EL (2006) The epidemiology of genital human papillomavirus infection. Vaccine 24:S1/4–S1/15. doi: 10.1016/j.vaccine.2005.09.054 CrossRefGoogle Scholar
  33. Woodman CBJ, Collins SI, Young LS (2007) The natural history of cervical HPV infection: unresolved issues. Nat Rev Cancer 7:11–22. doi: 10.1038/nrc2050 CrossRefGoogle Scholar
  34. Wright TC, Ferenczy A (2002) Anatomy and histology of the cervix. Blaustein’s pathology of the female genital tract, 5th edn. Springer, New York, pp 207–224Google Scholar

Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • Tri Sri Noor Asih
    • 1
    • 2
    • 3
  • Suzanne Lenhart
    • 4
  • Steven Wise
    • 4
  • Lina Aryati
    • 1
  • F. Adi-Kusumo
    • 1
  • Mardiah S. Hardianti
    • 1
  • Jonathan Forde
    • 3
    • 5
    Email author
  1. 1.Department of MathematicsGadjah Mada UniversityYogyakartaIndonesia
  2. 2.Semarang State UniversitySemarangIndonesia
  3. 3.National Institute for Mathematical and Biological Synthesis (NIMBioS)KnoxvilleUSA
  4. 4.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  5. 5.Department of Mathematics and Computer ScienceHobart and William Smith CollegesGenevaUSA

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