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Abstract
Objectives To determine the effects of using discrete versus continuous quantities of people in a compartmental model examining the contribution of antimicrobial resistance (AMR) to rebound in the prevalence of gonorrhoea.
Methods A previously published transmission model was reconfigured to represent the occurrence of gonorrhoea in discrete persons, rather than allowing fractions of infected individuals during simulations.
Results In the revised model, prevalence only rebounded under scenarios reproduced from the original paper when AMR occurrence was increased by 105 times. In such situations, treatment of high-risk individuals yielded outcomes very similar to those resulting from treatment of low-risk and intermediate-risk individuals. Otherwise, in contrast with the original model, prevalence was the lowest when the high-risk group was treated, supporting the current policy of targeting treatment to high-risk groups.
Conclusions Simulation models can be highly sensitive to structural features. Small differences in structure and parameters can substantially influence predicted outcomes and policy prescriptions, and must be carefully considered.
- GONORRHOEA
- NEISSERIA GONORRHOEA
- MATHEMATICAL MODEL
- ANTIMICROBIAL RESISTANCE
- ANTIBIOTIC RESISTANCE
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Introduction
The worldwide incidence of infection caused by Neisseria gonorrhoeae is currently the highest of any bacterial sexually transmitted infection (STI),1 and rates of antimicrobial resistance (AMR) are on the rise.2 Because maintenance of gonorrhoea in a population is reliant upon high-risk individuals—those with multiple sex partners, or high rates of concurrency3 ,4—control methods typically focus on treating these ‘core groups’ responsible for perpetuating transmission of infection. A mathematical model published in 20115 predicted that this strategy was counter-productive in the presence of AMR, and could, in fact, lead to a rebound in and persistence of prevalence of AMR-gonorrhoea above that predicted for treating the low-risk or intermediate-risk groups. Our objective was to reproduce the original model and explore the effects of allowing only whole numbers of infected individuals to transmit infection in the model.
Methods
Model
We used Vensim DSS6—a popular numerical software for ordinary differential equations—to reconstruct a previously published5 risk-stratified susceptible infectious-susceptible model to simulate the transmission dynamics and equilibrium prevalence of gonorrhoea in a hypothetical population. The reconstructed model was numerically integrated using the Euler method with a timestep of 0.0078125. Briefly, the compartmental model allows for individuals to be either infected or susceptible. Infected individuals are captured by a variety of compartments representing infection susceptible to treatment, resistant to drug A, resistant to drug B or resistant to both A and B. The model is risk-stratified, with 97.4% of the population classified as low risk, 2.3% as intermediate risk and the rest as high risk, to reproduce the proportions used in the original model. The duration of infectiousness was 42 days, and the chance of developing resistance upon treatment was 10−6. Proportional mixing, with homogeneous mixing within risk groups, was represented using a mixing matrix (see online supplementary appendix).5 This reconstructed model accurately reproduced all the scenarios described in the paper.
Modifications
In the original version, the simulations allow for arbitrarily small (real-number) counts of individuals to be infectious, with or without AMR, at any given timestep. This means that infection, or resistance, can gain ground in the model by transmission between ‘parts of’ individuals. Given that an individual can only be infectious, or not infectious, requiring whole number-based calculations is more representative of the way in which gonorrhoea infections are transmitted. To examine the effect of conditions that more accurately reflect infection transmission in finite at-risk populations, we altered the code in our model to incorporate rounding up or down, so that only whole numbers were used in prevalence calculations and, therefore, in the force of infection. This means that only whole individuals could transmit the infection in our model. Specifically, any fraction below 0.5 was rounded down to the next integer, and those with values of 0.5 and above were rounded up.
Additionally, we used an effective contact rate of 10.27 partnerships per year for the high-risk group, which was derived from the originally published equations and confirmed by the authors. In the balance of the paper, we use the term ‘original model’ to refer to this Chan et al model using this derived contact rate.
We used our modified model to re-create several scenarios from the original study, including the baseline scenario (no treatment given) and scenarios under which the low-risk, intermediate-risk or high-risk group is treated with one drug, as well as where one-third of each risk group is treated with one drug, all in the absence of AMR. We then ran these same scenarios in the presence of AMR.
Results
In the absence of AMR
Our model accurately reproduced the baseline prevalence, in the absence of treatment, of 69 per 100 000 in the overall population, reported by Chan et al (figure 1A). Our model also produced results similar to the original model for the scenario under which the low-risk group is treated (figure 1A). Our model predicted slightly higher equilibrium prevalence in scenarios where the intermediate-risk group is treated, and where one-third of each group is treated. In the original model, when the high-risk group is treated, prevalence of gonorrhoea decreased over time until it essentially disappeared within 50 years. Our version differed slightly; rather than disappearing, after 6.5 years, infection levels decreased to an equilibrium prevalence of 6.6 per 100 000 (figure 1A).
(A–C) Substantial differences exist in predictions obtained using the original model and the version using whole numbers, specifically for scenarios where one-third of all groups, or the high-risk group, is the focus of treatment, in the presence of antimicrobial resistance (AMR).
In the presence of AMR
When the model parameters are adjusted to allow for development of AMR, our results accurately reflect those of the original model for the scenario in which the low-risk group alone is treated, and were only slightly different for the intermediate-risk group treatment scenario (figure 1B). However, the whole number-based model showed no rebound in gonorrhoea prevalence when the high-risk group is treated or when one-third of each group is treated (figure 1B). Rather, we obtained the same results for these scenarios as when there was no AMR development (figure 1A). This is markedly different than the results of the original model, which showed a modest response to treating low-risk and intermediate-risk groups and an initial response followed by rebound to baseline prevalence when treating the high-risk group and one-third of each group.
Sensitivity analysis
To explore the sensitivity of our model to changes in selected parameters, we examined the effects of increasing population size, increasing the contact rate among high-risk individuals in the model and increasing the chance of de novo development of resistance upon treatment. These analyses were performed using the scenario where the high-risk group is treated with one drug, which is the scenario that resulted in the greatest rebound in the original model. For each scenario, we examined the resulting steady-state prevalence; by way of comparison for the results reported below, the initial prevalence of the model was 69 per 100 000. We found that even when the population is increased from 1 000 000 by a factor of 1000, no rebound in prevalence occurred. The 10-year steady-state prevalence, if the initial population size was increased by 1000 times was 66, 65 and 3 per 100 000, if the low-risk, intermediate-risk and high-risk groups were treated, respectively.
The model seemed to be most sensitive to contact rate among the high-risk individuals; increasing the contact rate in this group from 10.27 to 12 partnerships per year was enough to provoke an increase in prevalence (with no initial decline) under the whole numbers-based framework. The 10-year steady-state prevalence, if the contact rate for the high-risk group was increased, was 152, 150 and 90 per 100 000, if the low-risk, intermediate-risk and high-risk groups were treated, respectively.
Only if the frequency of de novo development of resistance was increased by 105 times, did we see an initial decline followed by rebound when the high-risk group was treated (figure 1C). However, the steady-state prevalence following treatment of the high-risk group was not substantially different (68 per 100 000) than that seen if the low-risk or intermediate-risk group was treated, instead (67 per 100 000).
Discussion
The results from our model illustrate the sensitivity of compartmental models to assumptions and details of model structure. The differences in predictions obtained when whole numbers, rather than fractions, are used in the simulation are substantial and have important implications for effective policy development on gonococcal AMR. In contrast with the original model, our results predict that gonorrhoea infection may, in fact, be drastically reduced by treatment of core groups, even in the presence of AMR. Because AMR never gains a foothold in the whole number-based model, results of scenarios in the presence of AMR are identical to those without AMR. By contrast, a simulation that allows for fractional levels of infected individuals—akin to ‘partial’ individuals being infected—permits a small reservoir of AMR infection to be maintained and transmitted, leading inevitably to an eventual rebound.
The policy implications advanced from the original model differed from those arising from the whole-person model even after sensitivity analysis of the parameter estimates. While there is indisputable empirical evidence attesting to an observed rebound in gonorrhoea prevalence over the last several decades, such evidence does not undercut the alternative ‘whole person’ assumptions examined here. Alternative explanations could lead to an observed rebound even under the whole-person assumptions we have introduced. However, the only scenario where we could reproduce a rebound in infection rates resulted from a very high occurrence of AMR. Like for the original model, there was minimal response to treatment of the low-risk or intermediate-risk groups and an initial response in the high-risk group followed by a rebound. However, the prevalence after rebound in the high-risk group was only marginally above that seen when the other groups were treated. For all other scenarios examined, treatment of the high-risk group resulted in the lowest prevalence. While the original model yielded findings that provocatively challenged the wisdom of established policy prescriptions, the slight variant introduced here does the opposite.
Given the stylised nature of both models examined here, neither is likely to be highly representative of real-world dynamics. However, the marked qualitative differences in AMR dynamics—and, more importantly, policy implications—between these two nearly identical models illustrates the degree to which model findings of great policy import may ride on simple but delicate model assumptions, and caution against arriving at premature policy conclusions based on a single model.
Supplementary materials
Supplementary Data
This web only file has been produced by the BMJ Publishing Group from an electronic file supplied by the author(s) and has not been edited for content.
Files in this Data Supplement:
- Data supplement 1 - Online supplement
Footnotes
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Handling editor Jackie A Cassell
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Contributors JRD, NDO: conceived the idea; MAT, DJH, CLW, NDO: performed the analysis; MAT, NDO, CLW: drafted the manuscript; JRD, CLW, NDO: edits and final approval.
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Funding MAT is supported by a CIHR Frederick Banting and Charles Best Canada Graduate Scholarship and funding to JRD from the University of Saskatchewan in part supported this research. This research was carried out in partial fulfilment towards PhD requirements for MAT.
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Competing interests None.
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Provenance and peer review Not commissioned; externally peer reviewed.