Article Text
Abstract
Objectives Reinfection after treatment for Chlamydia trachomatis or Neisseria gonorrhoeae reduces the effect of control interventions. We explored the impact of delays in treatment of current partners on the expected probability of reinfection of index cases using a mathematical model.
Methods We used previously reported parameter distributions to calculate the probability that index cases would be reinfected by their untreated partners. We then assumed different delays between index case and partner treatment to calculate the probabilities of reinfection.
Results In the absence of partner treatment, the medians of the expected reinfection probabilities are 19.4% (IQR 9.2–31.6%) for C trachomatis and 12.5% (IQR 5.6–22.2%) for N gonorrhoeae. If all current partners receive treatment 3 days after the index case, the expected reinfection probabilities are 4.2% (IQR 2.1–6.9%) for C trachomatis and 5.5% (IQR 2.6–9.5%) for N gonorrhoeae.
Conclusions Quicker partner referral and treatment can substantially reduce reinfection rates for C trachomatis and N gonorrhoeae by untreated partners. The formula we used to calculate reinfection rates can be used to inform the design of randomised controlled trials of novel partner notification technologies like accelerated partner therapy.
 MATHEMATICAL MODEL
 CHLAMYDIA TRACHOMATIS
 PARTNER NOTIFICATION
 NEISSERIA GONORRHOEA
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Introduction
If sexual partners are not treated concurrently for Chlamydia trachomatis or Neisseria gonorrhoeae, the index case may be reinfected.1 ,2 Our previous work has shown that the impact of screening interventions for C trachomatis infection is limited by reinfection from current partners that remain untreated.3 Repeated infection might also increase the risk of adverse sequelae, particularly in women.
Partner notification (PN) limits reinfection in people who have been treated for C trachomatis or N gonorrhoeae. Expedited partner therapy and accelerated partner therapy are PN technologies that reduce the time between diagnosis and treatment of the index patient to partner referral and treatment.1 ,4 In randomised controlled trials (RCTs), expedited partner therapy reduced the incidence of repeated detection of either C trachomatis or N gonorrhoeae infection.1 It was not possible to determine the proportion of repeated infections that resulted from reinfection by an untreated current partner, infection from a new partner or antibiotic treatment failure. The incremental benefits of reducing the delay to partner treatment on reinfection of index cases by partners who have not yet been treated are not well understood.
The effects of PN can be explored with mathematical models. Dynamic transmission models often assume that index cases and their current partners are treated at the same time so the risk of reinfection with different delays to partner treatment cannot be studied.3 Calculating the probability of reinfection requires a model that takes into account competing risks of reinfection, partner treatment, clearance of infection in the partner and dissolution of the sexual partnership. Such a model could help us better understand the differences between C trachomatis and N gonorrhoeae reinfection after treatment.
The objective of this study was to develop a model to calculate the expected probability with which a treated index case will be reinfected with C trachomatis and N gonorrhoeae by their untreated current partner. We explore the effect that varying the time delay to partner treatment has on reinfection of index cases.
Methods
Probability of reinfection
We devised a formula to calculate the probability of reinfection of index cases by their as yet untreated partners:
p_{p} is the probability that the index case is in an ongoing sexual partnership and p_{i} is the probability that the partner is infected. The first term in the brackets represents the probability that the index case is reinfected before the partner is treated, the partner's infection is cleared or the sexual partnership is dissolved. The successfully treated index case is assumed to engage in new sex acts with the partner at a frequency f, where the infection can transmit at a per sex act transmission probability β. Transmission can occur until the partner is either treated at rate δ (1/δ is the average delay in the time to partner treatment), the infection is spontaneously cleared in the partner at rate γ, or the sexual partnership dissolves at rate σ. The second term in the brackets corresponds to the probability that reinfection occurs in cases in which the partner has not been successfully treated, where ɛ is the treatment efficacy in the partner. We assume that all probabilities and events are independent of each other.
Parameter values
To calculate the expected range of reinfection probabilities, we generated 10^{6} parameter sets by randomly drawing from uniform (U(a, b) from a to b) or binomial (B(n, p) with sample size n and probability p) distributions. The probability that index cases are in an ongoing partnership was assumed to be in the interval U(0, 1). C trachomatis or N gonorrhoeae positivity in current partners were calculated from the studies by Quinn et al5 and Lycke et al6 and drawn from B(77, 0.69) and B(188, 0.80), respectively. We assumed an average duration of sexual partnerships in C trachomatis infected individuals from U(1 week, 6 months).7 For N gonorrhoeae infected individuals, we assumed that sexual partnerships are shorter (U(1 day, 2 weeks)). For both infections, the frequency of sex acts was drawn from U(1, 7) per week, reflecting the range reported in sexual behaviour studies.8 The per sex act transmission probabilities for C trachomatis and N gonorrhoeae were drawn from U(0.06, 0.167)7 and U(0.19, 0.53).9 The average duration of C trachomatis and N gonorrhoeae infection in the current partner is drawn from U(6, 12)7 and U(1, 6)10 months. We assume that the current partner of each index case receives treatment, that is, 100% PN. Efficacy of antibiotic use for a notified and treated partner was 90%2 for C trachomatis and 100% for nonresistant N gonorrhoeae.11
Results
We compare reinfection with C trachomatis and N gonorrhoeae in the absence of PN to expected reinfection in the presence of PN and used the following average delays to treatment of the current partner: 14 days, 3 days and 1 day (figure 1). Without PN, median reinfection probabilities from the 10^{6} parameter combinations are 19.4% (IQR 9.2–31.6%) for C trachomatis and 12.5% (IQR 5.6–22.2%) for N gonorrhoeae. The values for C trachomatis reinfection support previous observations in empirical and mathematical modelling studies.2 ,12 Less is known about N gonorrhoeae reinfection in the absence of PN. Although the per sex act transmission probability for N gonorrhoeae is higher than for C trachomatis, we find a lower probability of reinfection because sexual partner change is more frequent and partnerships do not last as long in N gonorrhoeae infected individuals.
Reducing the delay between index case and partner treatment from 14 days13 to 1 day or 3 days4 substantially reduces the risk of reinfection (figure 1). For C trachomatis, median probabilities of reinfection are 8.8% (IQR 4.3–14.6%), 4.2% (IQR 2.1–6.9%) and 2.8% (IQR 1.4–4.6%) for decreasing delays of partner treatment. For N gonorrhoeae, the respective probabilities are 9.7% (IQR 4.5–17.1%), 5.5% (IQR 2.6–9.5%) and 2.6% (IQR 1.3–4.6%). Thus, the probability of reinfection after treatment for C trachomatis and N gonorrhoeae become more alike as the time to partner treatment grows shorter.
Discussion
Our formula provides a general framework within which one can assess the risk that index cases treated for C trachomatis or N gonorrhoeae will be reinfected by their, as yet, untreated partners. We found that reducing the time to partner treatment to 1 day to 3 days can reduce reinfection of treated index cases from untreated partners substantially. The formula is intuitive and flexible, and can be easily adapted to model variations in the mechanisms of transmission and PN. For example, it can easily accommodate reduced frequency of sexual intercourse or condom use.
Because we do not know all the parameters that influence transmission of C trachomatis and N gonorrhoeae, the generalisability of our findings is limited. There are more studies with prospectively collected data about C trachomatis reinfection than for N gonorrhoeae, at least for women.2 Reinfection rates might differ between women and men or by age. Our study applies to young adults in whom the majority of reinfections with C trachomatis occur.14 Because of limited information about sexspecific parameter differences, we did not consider them in this study. We had information about the average length of sexual partnerships of C trachomatis infected7 but not N gonorrhoeae infected individuals. We assumed that treated index cases can only be reinfected by a single current partner but acknowledge that concurrent sexual partnerships also exist. Finally, our study concentrates only on reinfection of index cases and does not investigate the effects that different periods of delay to partner treatment might have on transmission in the population. Without dynamic transmission models we could not address the latter question, but we have shown recently that preventing C trachomatis reinfection of the treated index is the most effective way of increasing the populationlevel effect of PN.3
The results from our formula can be compared with findings from empirical studies. For example, recurrent or persistent C trachomatis and N gonorrhoeae infections were measured in an RCT of the effects of expedited partner therapy.1 Our calculated risk of reinfection for a PN delay of 14 days is consistent with the RCT findings for standard referral of partners. Golden et al1 further found that expedited partner therapy reduced persistent or recurrent infections more for N gonorrhoeae than for C trachomatis. Our study did not replicate this finding but this might be because one could not distinguish between treatment failure in index cases and repeated infection from current or new partners in the RCT.
Our model can help to understand the potential impact on C trachomatis and N gonorrhoeae reinfection of reducing delays to partner treatment. Its value will increase as we improve our estimates of transmission and sexual behaviour, particularly for individuals infected with N gonorrhoeae. We conclude that PN technologies that shorten delays to partner treatment, such as expedited and accelerated partner therapy, substantially reduce index case reinfection by untreated partners, when compared with standard patient referral.
Key messages

Our formula calculates the probability of reinfection of index cases with Chlamydia trachomatis and Neisseria gonorrhoeae by their untreated partners.

Quicker treatment of current partners can substantially reduce reinfection rates for C trachomatis and N gonorrhoeae.

The intuitive and flexible formula can be easily adapted to inform the design of studies such as randomised controlled trials.
References
Footnotes

Handling editor David A Lewis

Contributors NL and CLA designed the study and wrote the manuscript. CLA developed the mathematical model. All authors contributed to the analysis and interpretation of the results and commented on the manuscript.

Funding This study is part of a project funded by the NIHR Health Technology Assessment programme (project number 07/42/02). The project will be published in full in the Health Technology Assessment journal series. Visit the HTA programme website for more details (http://www.hta.ac.uk/1722). The views and opinions expressed therein are those of the authors and do not necessarily reflect those of the Department of Health. CLA, JCMH and SAH acknowledge financial support by the Swiss National Science Foundation (project numbers PZ00P3_136737, 320030_135654, PDFMP3_124952).

Competing interests In 2010, JCMH and NL received fees from GlaxoSmithKline for attending a meeting about Chlamydia trachomatis vaccines.

Provenance and peer review Not commissioned; externally peer reviewed.