# When Do Sexual Partnerships Need to Be Accounted for in Transmission Models of Human Papillomavirus?

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Human Papillomavirus

#### 1.2. HPV Vaccination

#### 1.3. Mathematical Models of HPV

#### 1.4. Pair Models for Sexually Transmitted Infections

_{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, i.e., the average number of secondary infections produced by an infectious person in an otherwise susceptible population.) Therefore, R

_{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.

_{0}to determine the minimum intervention efforts required to eradicate an infection.

#### 1.5. Outline of Paper

## 2. Results and Discussion

## 3. Methods

_{i}(the average (mean) number of single females of infection status i = S, I, R), y

_{j}(the average (mean) number of single males of infection status j = S, I, R), and P

_{ij}(the average (mean) number of pairs of infection status i = S, I, R and j = S, I, R). An individual is defined as single if they do not have a partner at a given time. The model equations resulting from the assumptions described at the beginning of Section 2 are given by

^{*}, Y

^{*}, and P

^{*}are the equilibrium number of single females, single males and pairs, respectively (see equation (14)). The formation of new partnerships is described by the function:

_{R}, y

_{R}and P

_{RR}wherever they appear according to equation (14) and the following relations:

_{0}analytically is difficult due to the large dimensionality of the system of equations. Parameters are given in Table 1.

## 4. Conclusions

_{0}for example, would facilitate understanding the relationship between model predictions and model parameters. However, derivation of R

_{0}is difficult due to the large dimensionality of the system. Moreover, an expression for R

_{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.

## Acknowledgments

## References and Notes

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**Figure 1.**Schematic of pair dynamics. Single females, X

_{i}, and single males, Y

_{j}, form partnerships P

_{ij}at a rate ρ, where i,j = S,I,R. These partnerships are destroyed either when the relationship breaks up (which occurs at a rate ⌠) or when one partner dies (which occurs at a rate μ).

**Figure 2.**Schematic of infection dynamics. Females only are vaccinated, at a rate ⌉. Infection can only occur within a partnership at a rate h®. Infected individuals recover at a rate γ and both natural and vaccine-derived immunity wane at a rate δ. Single females and males are recruited to the sexually active population at a rate /.

**Figure 3.**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

**3a**shows the impact of transmission rate, β on prevalence. Figure

**3b**shows impact of average number of sex acts per year, h on prevalence, for and β = 0.4/act. Figure

**3c**shows the impact of transmission rate, β, on prevalence for various vaccination rates ω, at baseline parameters. Vaccination rate include ω = 0/yr, ω = 0.05/yr, ω = 0.1/yr and ω = 0.2/yr. Figure

**3d**shows a time series of prevalence for various vaccination rates introduced at year 300, with baseline parameters and β = 0.3/act.

**Figure 4.**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 equation (14) (see methods).

**Figure 5.**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 equation (14) (see methods).

**Figure 6.**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 equation (14) (see methods).

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

ɛ | Vaccine efficacy | 95% | [11] |

μ | Rate at which individuals leave the age group of peak sexual activity /yr | 1/15/yr | [2,3] |

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 | [27] |

ρ | Pair formation rate /yr | 33.66/yr | Derived using [27] (see Methods) |

h | Number of sex acts /yr | 130/yr | [31] |

β | Transmission rate per sex act | 0.073/act | [41,42], calibrated (see Methods) |

ω | Rate at which females are vaccinated | 0.05/yr | [42], calibration (see Methods) |

γ | Infection clearance rate/yr | 1/yr | [14,16,42] |

δ | Natural immunity waning rate/yr | 1/10/yr | [13,14,16] |

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Muller, H.; Bauch, C.
When Do Sexual Partnerships Need to Be Accounted for in Transmission Models of Human Papillomavirus? *Int. J. Environ. Res. Public Health* **2010**, *7*, 635-650.
https://doi.org/10.3390/ijerph7020635

**AMA Style**

Muller H, Bauch C.
When Do Sexual Partnerships Need to Be Accounted for in Transmission Models of Human Papillomavirus? *International Journal of Environmental Research and Public Health*. 2010; 7(2):635-650.
https://doi.org/10.3390/ijerph7020635

**Chicago/Turabian Style**

Muller, Heidi, and Chris Bauch.
2010. "When Do Sexual Partnerships Need to Be Accounted for in Transmission Models of Human Papillomavirus?" *International Journal of Environmental Research and Public Health* 7, no. 2: 635-650.
https://doi.org/10.3390/ijerph7020635