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Human papillomavirus (HPV) is often transmitted through sexual partnerships. However, many previous HPV transmission models ignore the existence of partnerships by implicitly assuming that each new sexual contact is made with a different person. Here, we develop a simplified pair model—based on the example of HPV—that explicitly includes sexual partnership formation and dissolution. We show that not including partnerships can potentially result in biased projections of HPV prevalence. However, if transmission rates are calibrated to match empirical pre-vaccine HPV prevalence, the projected prevalence under a vaccination program does not vary significantly, regardless of whether partnerships are included.

Human Papillomavirus (HPV) infections are responsible for several gynecologic diseases, including abnormal cervical cytology, cervical dysplasia, cervical cancer and genital warts [

About 75% of adults will have at least one type of HPV in their lifetime [

There are approximately 130 types of HPV, of which 30–40 are transmitted via sexual contact [

Carcinogenic HPV types are highly prevalent in Ontario women, infecting about 1 in 4 women aged 20–24 years [

HPV is unlike most other sexually transmitted infections (STIs), in that infection is not highly concentrated in small groups of highly sexually active people, referred to as “core groups” [

A multivalent vaccine that protects against infection by types 6, 11, 16 and 18 is now available [

Many mathematical models have examined the transmission of HPV and impact of various possible vaccination programs. They have generally found that a vaccination strategy of pre-adolescent females is both highly effective and highly cost-effective, in terms of reducing HPV-associated health burdens such as cervical cancer incidence. These have included compartmental models, deterministic models and stochastic models [

In an attempt to address this deficiency, a number of methods to incorporate sexual partnerships have been developed for modelling sexually transmitted infections, such as “pair models” and “pair approximations” [_{0} > 1, (2) the growth rate is lower and (3) the endemic equilibrium is higher, than in a model without partnership duration. It is also shown for models with nonzero partnership length that a single value of R_{0} can imply more than one possible epidemic growth rate and endemic equilibrium. (R_{0} is the basic reproductive ratio, _{0} cannot be estimated from empirical data on prevalence of a sexually transmitted infection without additional information on partnerships. Kretzschmar and Dietz also found that the transmission dynamics and R_{0} are affected by the assumed partnership dynamics for HIV, in particular.

Similarly, Dietz and Hadeler developed a simple SIS (Susceptible-Infected-Susceptible) model with pair dynamics [_{0} to determine the minimum intervention efforts required to eradicate an infection.

These previous models show that common insights based on classical epidemiological theory using homogeneous (non-pair) mixing models may no longer be valid once pair dynamics are included [

In this paper, we develop and analyze a pair model for HPV transmission and vaccination, assuming a Susceptible-Infectious-Recovered-Susceptible (SIRS) natural history. We analyze special cases of the model, considering the limit as the duration of partnerships goes to zero, in order to assess the impact of not including partnerships on model predictions. While the model is simplified in many respects, a simplified model is well suited to our objective of illustrating the impact of including

The HPV pair model describes single males and females forming monogamous sexual partnerships at constant rate

This model system exhibits classical “threshold” behaviour. For instance, for sufficiently low values of the transmission probability per sex act,

In order to examine how inclusion or exclusion of partnerships affects projected prevalence, the dynamics of the pair model can be explored for a range of possible values of the separation rate

The impact of the partnership turnover rate in the “

The analyses presented in

The model variables are _{i} (the average (mean) number of single females of infection status _{j} (the average (mean) number of single males of infection status _{ij} (the average (mean) number of pairs of infection status ^{*}, ^{*}, and ^{*} are the equilibrium number of single females, single males and pairs, respectively (see

The quantities _{R}, _{R} and _{RR} wherever they appear according to _{0} analytically is difficult due to the large dimensionality of the system of equations. Parameters are given in

The first conclusion of this research is that models of STI transmission where partnerships of nonzero duration are explicitly included in the model yield projections that will generally differ from models where partnerships are not explicitly included (

However, if transmission rates are first calibrated to match observed prevalence and then used to predict the impact of vaccination, the predictions of these two types of models will be very similar, which may be surprising. Homogeneous (non-pair) models often use this type of calibration approach. This suggests that homogeneous mixing models may suffice for modelling STI transmission if they are calibrated to prevalence data and only being used to predict the impact of vaccination.

In practical terms, this means that the predicted impact of vaccination policies according to homogeneous (non-pair) mixing models cannot necessarily be discarded on grounds that sexual partnerships have not been accounted for. However, we note that there are important aspects of real-world sexual contact networks that we did not include in this analysis, such as concurrency (overlapping partnerships), stochasticity (random effects), sexual risk groups, and age structure. These elements have been shown to influence disease dynamics [

Finally, a more exhaustive exploration of parameter space may reveal parameter regimes where pair model projections of the impact of vaccination diverge from homogeneous mixing model projections, even when both models are calibrated to pre-vaccine prevalence. However, the biological plausibility of these parameter regimes would have to be considered. Rigorous analysis of the model equations, permitting a derivation of the basic reproduction number _{0} for example, would facilitate understanding the relationship between model predictions and model parameters. However, derivation of _{0} is difficult due to the large dimensionality of the system. Moreover, an expression for _{0} would not provide much information about the transient nature of the solutions, and it is the transient solutions that are relevant to public health since transient solutions describe prevalence in the first few years or decades after a vaccination program is implemented.

Additionally, this model only considers infection from HPV types 16 and 18 since the vaccine is preventative for these types. However, there are 30–40 HPV types that are transmitted sexually and infection with certain types may inhibit or activate infection of other types. The interaction between HPV types, or the impact of potential highly multivalent vaccines, could also be studied in future work. Many of these limitations suggest areas for future research on this topic.

In summary, if homogeneous mixing models that neglect partnerships are used to assess the impact of vaccination programs or other interventions for sexually transmitted infections, it is important to understand how and whether the non-inclusion of partnership dynamics influences the projections of these models. This is particularly important if these models are used to inform public health policy.

This research was supported by grants from the Canadian Institutes of Health Research, the Ministry of Research and Innovation of Ontario, the University of Guelph, and the Natural Sciences and Engineering Research Council of Canada. The funders had no role in the study.

Schematic of pair dynamics. Single females, _{i}, and single males, _{j}, form partnerships _{ij} at a rate

Schematic of infection dynamics. Females only are vaccinated, at a rate

Transmission dynamics (equilibrium prevalence and time series) for various parameter values. Baseline parameters used are, μ = 1/15/yr, κ = mu*N/2, σ = 1/6/month, ρ = 33.66, γ = 1/yr, ω = 0/yr, h = 130/yr unless otherwise stated. Figure

The impact of changes in the separation rate, σ, on percent infected for “ρ dependent” and “ρ fixed” cases. Baseline parameters to achieve 3% prevalence include μ = 1/15/yr, κ = μ*N/2, σ = 2/yr = 1/6/month, γ = 1/yr, h = 130/yr, β = 0.0737/act, ρ = 33.66/yr, and ω = 0. For the “ρ dependent” case, the equilibrium number of pairs was held constant at 8,875 according to

Impact of transmission rate, β, on prevalence for various turnover rates, σ for the “ρ dependent” case; baseline parameters μ = 1/15/yr, κ = μ*N/2, γ = 1/yr, h = 130/yr, ω = 0/yr were used. The equilibrium number of pairs was held constant at 8,875 according to

Time series of prevalence for various turnover rates, with vaccination introduced at year 300 at a rate ω = 0.05/yr. Baseline parameter values are as in the 3% prevalence case, μ = 1/15/yr, κ = μ*N/2, γ = 1/yr, h = 130/yr. The transmission rate per sex act, β, is calibrated for each turnover rate, σ to achieve 3% prevalence using the “ρ dependent” approach. The equilibrium number of pairs was held constant at 8,875 according to

Model parameters, values and sources.

Symbol | Definition | 3% prevalence scenario | Source |
---|---|---|---|

ɛ | Vaccine efficacy | 95% | [ |

μ | Rate at which individuals leave the age group of peak sexual activity /yr | 1/15/yr | [ |

κ | Rate at which individuals are recruited into the age group of peak sexual activity /yr | μN/2 | Derived (see Methods) |

σ | Pair break-up rate /yr | 2/yr | [ |

ρ | Pair formation rate /yr | 33.66/yr | Derived using [ |

h | Number of sex acts /yr | 130/yr | [ |

β | Transmission rate per sex act | 0.073/act | [ |

ω | Rate at which females are vaccinated | 0.05/yr | [ |

γ | Infection clearance rate/yr | 1/yr | [ |

δ | Natural immunity waning rate/yr | 1/10/yr | [ |