Global Stability of Equilibria in a Two-Sex HPV Vaccination Model

Abstract

Human papillomavirus (HPV) is the primary cause of cervical carcinoma and its precursor lesions, and is associated with a variety of other cancers and diseases. A prophylactic quadrivalent vaccine against oncogenic HPV 16/18 and warts-causing genital HPV 6/11 types is currently available in several countries. Licensure of a bivalent vaccine against oncogenic HPV 16/18 is expected in the near future. This paper presents a two-sex, deterministic model for assessing the potential impact of a prophylactic HPV vaccine with several properties. The model is based on the susceptible-infective-removed (SIR) compartmental structure. Important epidemiological thresholds such as the basic and effective reproduction numbers and a measure of vaccine impact are derived. We find that if the effective reproduction number is greater than unity, there is a locally unstable infection-free equilibrium and a unique, globally asymptotically stable endemic equilibrium. If the effective reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable, and HPV will be eliminated.

The HPV Vaccination Strategy: Could Male Vaccination Have a Significant Impact?

“…This complements previous work ( [13,14,18], for example) and by using a simple structure, highlights key parameter groupings that are critical in understanding how HPV could be contained in a population, by a vaccination strategy aimed either only at females prior to their sexual debut or at both males and females.…”

Sex-Specific Immunization for Sexually Transmitted Infections Such as Human Papillomavirus: Insights from Mathematical Models

“…The projected reproduction number in a partly vaccinated population, R v , is related to the basic reproduction number R 0 without vaccination as follows [35]:Here, v f denotes the immunization coverage among females, and v m denotes the immunization coverage among males. This equation implies that it makes no difference whether the fraction of susceptible males or females is diminished in order to reduce the basic reproduction number.…”

“…Also note that low or waning vaccine efficacy may cause even complete coverage of a single sex to be insufficient for elimination. We refer to others for an analysis of conditions in which vaccination of both sexes may be needed to achieve R v <1 [35],[36].…”

Impact of coverage-dependent marginal costs on optimal HPV vaccination strategies

“…In the case where our model is symmetric (parameters for males and females are identical), our definition of ℛ * is the square of the traditional reproduction number ℛ 0 . According to the above definition, $${ℛ}_{*}=\frac{{\beta }_{fm}{\beta }_{mf}false(1-{\upsilon }_{\mathrm{m}}false)false(1-{\upsilon }_{\mathrm{f}}false)}{false(1+{\sigma }_{m}false)false(1+{\sigma }_{f}false)}.$$ If ℛ * ≤ 1, then the disease-free equilibrium is globally asymptotically stable [55], a situation known as herd immunity . On the other hand, if ℛ * > 1, then the disease-free equilibrium is unstable, and there is exactly one endemic equilibrium point, given by (recall that all rates were rescaled by γ) $$I_{\mathrm{m}}false({\upsilon }_{\mathrm{m}},{\upsilon }_{\mathrm{f}}false)=\frac{\overline{\beta }false(1-{\upsilon }_{\mathrm{f}}false)false(1-{\upsilon }_{\mathrm{m}}false)-\overline{s}}{{\beta }_{mf}\overline{s}+\overline{\beta }false(1+{\sigma }_{m}false)false(1-{\upsilon }_{\mathrm{f}}false)},I_{\mathrm{f}}false({\upsilon }_{\mathrm{m}},{\upsilon }_{\mathrm{f}}false)=\frac{\overline{\beta }false(1-{\upsilon }_{\mathrm{f}}false)false(1-{\upsilon }_{\mathrm{m}}false)-\overline{s}}{{\beta }_{fm}\overline{s}+\overline{\beta }false(1+{\sigma }_{f}false)false(1-{\upsilon }_{\mathrm{m}}false)},$$ where β̅ = β mf β fm , s̄ ≡ (1+σ f )(1+σ m ).…”

“…On the other hand, if ℛ * > 1, then the disease-free equilibrium is unstable, and there is exactly one endemic equilibrium point, given by (recall that all rates were rescaled by γ) $$I_{\mathrm{m}}false({\upsilon }_{\mathrm{m}},{\upsilon }_{\mathrm{f}}false)=\frac{\overline{\beta }false(1-{\upsilon }_{\mathrm{f}}false)false(1-{\upsilon }_{\mathrm{m}}false)-\overline{s}}{{\beta }_{mf}\overline{s}+\overline{\beta }false(1+{\sigma }_{m}false)false(1-{\upsilon }_{\mathrm{f}}false)},I_{\mathrm{f}}false({\upsilon }_{\mathrm{m}},{\upsilon }_{\mathrm{f}}false)=\frac{\overline{\beta }false(1-{\upsilon }_{\mathrm{f}}false)false(1-{\upsilon }_{\mathrm{m}}false)-\overline{s}}{{\beta }_{fm}\overline{s}+\overline{\beta }false(1+{\sigma }_{f}false)false(1-{\upsilon }_{\mathrm{m}}false)},$$ where β̅ = β mf β fm , s̄ ≡ (1+σ f )(1+σ m ). Furthermore, if ℛ * > 1, then this endemic equilibrium is globally asymptotically stable, which means that the endemic equilibrium exists and is globally asymptotically stable if and only if ℛ * > 1 [55]. …”