Imperfect Vaccines and Waning Immunity Can Undermine HPV Elimination, Model Warns

Mathematicians modeling human papillomavirus transmission have identified two distinct ways in which real-world vaccination programs may fail to eliminate the virus even when conventional thresholds are met. In a study published in Frontiers in Applied Mathematics and Statistics, researchers constructed a six-compartment model that tracks susceptible, exposed, infectious, recovered, vaccinated, and cancer-affected individuals, incorporating both an imperfect vaccine that leaves recipients partially susceptible and naturally acquired immunity that wanes over time. Their analysis reveals that the basic reproduction number R0, long treated as the decisive quantity in infectious disease control, is an unreliable guide when these two mechanisms operate together.

The model distinguishes itself from earlier HPV frameworks by combining three features in a single tractable system: a leaky vaccine whose recipients retain residual risk of infection, waning natural immunity that returns recovered individuals to the susceptible pool, and an explicit compartment for progression to HPV-related cancer. Using the next-generation matrix method, the team derived R0 in closed form and showed that it decomposes additively into two transmission routes—one through susceptible individuals and one through vaccine breakthrough infections. This decomposition explains why scaling up vaccination can paradoxically increase transmission when the vaccine is insufficiently protective.

The most consequential finding concerns what happens below the classical threshold. By reducing the endemic equilibria to the roots of a single quadratic equation, the researchers proved that the combination of a leaky vaccine and waning immunity produces a backward bifurcation at R0 = 1. In this regime, a stable endemic state coexists with the stable disease-free equilibrium over a range of reproduction numbers below unity, meaning that R0 < 1 is necessary but not sufficient for elimination. Control programs must instead push R0 below a lower turning point, or use transient intensification of screening and treatment to move prevalence across a separatrix that divides two basins of attraction.

Above the threshold, the model exhibits a second departure from classical behavior. The unique endemic equilibrium can lose stability through a Hopf bifurcation when the rate of immunity loss is sufficiently rapid, giving rise to sustained oscillations in prevalence. In numerical simulations with a representative parameter set, the infectious class oscillated between approximately 22 and 3,466 individuals around an equilibrium level of 736, with a period of roughly 194 years under constant parameters. Because the cancer class integrates the infectious class with a slow removal rate, the oscillation is amplified downstream: the modeled cancer burden varied by a factor of approximately 123 within a single cycle.

Both phenomena require the joint presence of the two mechanisms. When the vaccine is perfectly protective or immunity is permanent, the backward bifurcation and the Hopf bifurcation both vanish, and the researchers proved global stability of the endemic equilibrium using the geometric approach of Li and Muldowney. Two-parameter continuation in the vaccine leakiness and waning rate showed that the two regimes occupy disjoint regions of parameter space: a given vaccine and immunity profile may produce sub-threshold bistability, super-threshold oscillations, or neither, but never both simultaneously.

The sensitivity analysis added a methodological caution. The normalized elasticity of R0 with respect to the vaccination rate satisfies sgn Υβ = sgn(θ − λH), so increasing vaccination lowers R0 only when the vaccine is genuinely protective. When the breakthrough transmission coefficient exceeds that for susceptibles, scaling up vaccination raises R0. More strikingly, the waning rate has zero elasticity with respect to R0—because recovered individuals lie downstream of transmission and feed no new infections at the disease-free state—yet it determines both non-classical regimes. A parameter invisible to the reproduction number can therefore control whether elimination is achievable below threshold and whether the endemic state is a steady level or a large-amplitude cycle.

The researchers verified the robustness of the backward bifurcation through a Latin-hypercube sample of 200,000 parameter sets drawn over wide biologically plausible ranges, finding the phenomenon in approximately 41 percent of admissible sets. For the oscillatory regime, they confirmed that the Hopf threshold was reproduced independently in two implementations using different stiff solvers, agreeing to six significant figures in the critical waning rate, the spectrum at threshold, and the amplitude and period of the resulting cycle.

The public health implications are significant. In the backward regime, elimination is history-dependent: a population already in the endemic basin is not rescued by bringing R0 below unity, and control must either push R0 below the turning point or move prevalence across the separatrix. In the oscillatory regime, a transient intervention only displaces the trajectory along the cycle, and sustained pressure—through a more protective vaccine or a booster policy reducing the waning rate—is required. Because neither regime can be diagnosed from an estimate of R0 alone, the authors argue that surveillance must track prevalence over time rather than relying on a single threshold quantity.

The study was conducted by researchers at Khon Kaen University and Mahasarakham University in Thailand and was financially supported by Mahasarakham University. The authors note that their conclusions are drawn within a mean-field description assuming homogeneous mixing, constant parameters, and exponentially distributed sojourn times, and that relaxing the exponential-sojourn assumption would be particularly relevant to the Hopf threshold, which is governed by the delay in the reinfection loop. Future work includes computing the criticality of the Hopf bifurcation, formulating an optimal-control problem that targets the turning point rather than R0 = 1, and calibrating the model to age-stratified HPV and cervical-cancer data.

Subject of Research: Mathematical modeling of HPV transmission with imperfect vaccination and waning immunity

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

Article References: Prathumwan, D., Sunthonphong, K., Phakmee, S., Chaiya, I., & Trachoo, K. (2026). Backward and Hopf bifurcations in a human papillomavirus–cancer model with imperfect vaccination and waning immunity. Frontiers in Applied Mathematics and Statistics, 12, Article 1943761. https://doi.org/10.3389/fams.2026.1943761

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1943761

Keywords: HPV, backward bifurcation, Hopf bifurcation, imperfect vaccination, waning immunity, basic reproduction number, cervical cancer, mathematical epidemiology, bistability, sustained oscillations, sensitivity analysis, elimination threshold