An age-structured model of the human papillomavirus dynamics and optimal vaccine control

Mo'tassem Al-Arydah, Tufail Malik

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7 Scopus citations


We formulate an age-structured model based on a system of nonlinear partial differential equations to assist the early and catch up female vaccination programs for human papillomavirus (HPV) types 6 and 11. Since these HPV types do not induce permanent immunity, the model, which stratifies the population based on age and gender, has a susceptible-infectious-susceptible (SIS) structure. We calculate the effective reproduction number Rv for the model and describe the local-asymptotic stability of the disease-free equilibrium using Rv. We prove the existence of an endemic equilibrium for Rv > 1 for the no vaccine case. However, analysis of the model for the vaccine case reveals that it undergoes the phenomenon of backward bifurcation. To support our theoretical results, we estimate the age and time solution with the given data for Toronto population, when an early and catch up female vaccine program is adopted, and when there is no vaccine. We show that early and catch up female vaccine program eliminates the infection in both male and female populations over a period of 30 years. Finally, we introduce the optimal control to an age-dependent model based on ordinary differential equations and solve it numerically to obtain the most cost-effective method for introducing the catch up vaccine into the population.

Original languageBritish English
Article number1750083
JournalInternational Journal of Biomathematics
Issue number6
StatePublished - 1 Aug 2017


  • age dependence
  • cervical cancer
  • Human papillomavirus
  • mathematical model
  • vaccine


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