Backward and Hopf bifurcations in a human papillomavirus–cancer model with imperfect vaccination and waning immunity

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Introduction Human papillomavirus (HPV) is the most common sexually transmitted infection and the cause of virtually all cervical cancer, yet elimination is complicated by two features of real vaccination programmes: available vaccines are imperfect, leaving vaccinated individuals partially susceptible, and naturally acquired immunity wanes. We ask whether reducing the basic reproduction number below unity is sufficient for elimination when both mechanisms are present. Methods We formulate and analyse a six-compartment ( SEIRVC ) deterministic model combining an imperfect vaccine and waning immunity with explicit progression to an HPV-related cancer class. We establish positivity, boundedness, and an invariant region; derive R 0 by the next-generation matrix method; characterize the equilibria and their stability using Routh–Hurwitz criteria, Lyapunov functions, and the geometric approach of Li and Muldowney; and analyse the bifurcation structure, supported by sensitivity indices and simulation. Results The reproduction number decomposes additively into a susceptible and a vaccine-breakthrough route, and the disease-free equilibrium is locally asymptotically stable precisely when R 0 < 1, and globally so under an explicit condition. Reducing the endemic equilibria to the roots of a quadratic shows that a leaky vaccine together with waning immunity produces a backward bifurcation, so that R 0 < 1 is necessary but not sufficient for elimination and control must clear a turning point R 0 c < 1 . Above threshold the unique endemic equilibrium can lose stability through a Hopf bifurcation, giving sustained oscillations whose amplitude is magnified in the cancer class, which varies by a factor of order 10 2 within a cycle under constant parameters. Both phenomena require both mechanisms and vanish when the vaccine becomes perfectly protective or immunity permanent, the limit in which global stability of the endemic equilibrium is proved; two-parameter continuation shows the two regimes to be disjoint alternatives. Discussion Sensitivity analysis gives sgnΥ β = sgn(θ−λ H ) so scaling up an insufficiently protective vaccine raises R 0 , while the waning rate has zero elasticity yet determines both non-classical regimes. Threshold quantities alone are therefore an unreliable guide to HPV control, and surveillance must track prevalence over time.

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Related topics: Mathematical and Theoretical Epidemiology and Ecology Models · Cervical Cancer and HPV Research · Mathematical Biology Tumor Growth