We developed a mathematical model that describes transmission of antibiotic-sensitive and -resistant gonorrhoea, clinical pathways for diagnostic testing with culture, NAAT, or POC, and treatment with first- and second-line antibiotics (Additional file 1: Section Model). Here we describe the model, focusing on testing and treatment of gonorrhoea (Fig. 1 and Table 1).
Table 1 Gonorrhoea testing and treatment parameters and their default values
Basic model structure
The model is based on our previously published compartmental model of gonorrhoea transmission and resistance spread [19]. The model describes a population with two sexual activity classes i∈C, where C={L, H} indicates that there are two sexual activity classes L and H with low and high partner change rates. The model incorporates sexual mixing between the sexual activity classes, sexual behaviour change, migration in and out of the population, and gonorrhoea transmission. Individuals in the population can be susceptible to infection, S
i
, infected with antibiotic-sensitive gonorrhoea, \(\phantom {\dot {i}\!}I_{\text {Sen}_{i}}\), infected with gonorrhoea resistant to the first-line antibiotic, \(\phantom {\dot {i}\!}I_{\text {Res}_{i}}\), or infected with gonorrhoea resistant to the first-line antibiotic and waiting for re-treatment, W
i
. Depending on the parameters for sexual behaviour, transmission, and gonorrhoea natural history (Additional file 1: Table S2), the model describes a population of MSM or HMW.
Gonorrhoea testing and treatment
Antibiotic-sensitive gonorrhoea
Individuals infected with antibiotic-sensitive gonorrhoea, \(\phantom {\dot {i}\!}I_{\text {Sen}_{i}}\), can recover spontaneously at rate ν or seek care (Fig. 1, left). Symptomatic care-seekers receive treatment on the same day at rate τ
S
. Asymptomatic care-seekers, i.e., those who are screened for gonorrhoea or were notified through an infected partner, are tested at rate τ
A
. Gonorrhoea is detected with sensitivity ξ
G
. On average, a fraction λ
A
of asymptomatic individuals returns for treatment after δ days. The treatment rate for asymptomatic individuals is approximated by 1/(1/τ
A
+δ), the inverse of the average time until individuals are tested, 1/τ
A
, and the time until they return for treatment, δ. Both symptomatic and asymptomatic individuals are treated with a first-line antibiotic that has treatment efficacy η
1. We assumed that individuals whose treatment was inefficacious remain infected and do not seek care again immediately. This assumption reflects the notion that treatment failure of antibiotic-sensitive gonorrhoea is most likely to occur in pharyngeal infections, which are usually asymptomatic [21].
Antibiotic-resistant gonorrhoea
Individuals infected with gonorrhoea resistant to the first-line antibiotic, \(\phantom {\dot {i}\!}I_{\text {Res}_{i}}\), can also recover spontaneously at rate ν (Fig. 1, right). Asymptomatic care-seekers that return for treatment (fraction λ
A
) receive treatment with the second-line antibiotic at rate 1/(1/τ
A
+δ) if both gonorrhoea (sensitivity ξ
G
) and resistance (sensitivity ξ
R
) are detected. Symptomatic care-seekers receive the first-line antibiotic as treatment on the same day, but remain infected due to resistance and return for treatment after δ days. At their second visit, symptomatic care-seekers receive the second-line antibiotic if both gonorrhoea (sensitivity ξ
G
) and resistance (sensitivity ξ
R
) are detected. If either test fails, they do not receive the second-line antibiotic. If they remain symptomatic (fraction λ
S
), they wait for re-treatment in compartment W
i
, where they either receive re-treatment with the second-line antibiotic at rate ω or recover spontaneously at rate ν. The assumption that re-treatment occurs with the second-line antibiotic follows recommendations from the World Health Organization (WHO) [20] and the Centers for Disease Control (CDC) [22] to obtain a specimen for culture-based antibiotic resistance testing at a patient’s second visit. The second-line antibiotic has efficacy η
2; individuals whose treatment is inefficacious remain infected and can recover spontaneously or seek care at a later point. De novo resistance to the first-line antibiotic or resistance to the second-line antibiotic are not considered in the model.
Testing scenarios
We simulated clinical pathways of gonorrhoea patients who are tested with culture, NAAT, or POC test at their first visit by adapting the parameters δ, λ
A
, and ξ
R
(Table 2). For culture, test results are not available immediately (δ
culture>0), resistance can be detected (ξ
R,culture>0), and asymptomatic infected individuals might not return for treatment (λ
A,culture<1). For NAAT, test results are not available immediately (δ
NAAT>0), resistance cannot be detected (ξ
R,NAAT=0), and asymptomatic infected individuals might not return for treatment (λ
A,NAAT<1). For POC, test results are available immediately (δ
POC=0), all individuals are followed up (λ
A,POC=1), and thus all individuals are treated at the first visit. We explore the impact of a POC test with (ξ
R,POC>0, POC+R) and without resistance detection (ξ
R,POC=0, POC−R); we use the term “POC” alone when ξ
R,POC is variable.
Table 2 Culture, NAAT, and POC testing scenarios are determined by the values of δ, λ
A
, and ξ
R
Impact measures
We evaluated the impact of a testing scenario by calculating the proportion of resistant infections among all infections, observed cases averted, and the rate at which resistance spreads, compared with another testing scenario. We measured the proportion of resistant infections up to 30 years after the introduction of resistance into the resistance-free baseline scenario. If applicable, we also calculated the time until resistance levels reached 5%, the level above which an antibiotic should not be used for empirical gonorrhoea treatment [23]. We defined observed cases averted as the difference between the cumulative incidence of observed cases (i.e., cases diagnosed and successfully treated at baseline; fraction ϕ [19]) using NAAT and the cumulative incidence of observed cases using culture or POC tests. We calculated the observed cases averted 5 years after the introduction of resistance. The rate at which resistance spreads describes how fast resistant infections replace sensitive infections in a human population [19]. We calculated the ratio of the rate of resistance spread between POC with different test sensitivities to detect resistance (ξ
R,POC) and culture or NAAT scenarios (Additional file 1: Section Rate of resistance spread). If the ratio of the rate of resistance spread is >1, resistance spreads faster when using POC tests compared with other tests. If the ratio is <1, resistance spreads slower when using POC tests compared with other tests.
Parameters
We used the parameters describing sexual behaviour, gonorrhoea transmission, natural history, and treatment from our previous model [19]. There, we estimated sexual behaviour parameters from the second British National Survey of Sexual Attitudes and Lifestyles (Natsal-2), which is a nationally representative population-based survey [24]. We calibrated all other parameters to yield prevalence and incidence rates within empirically observed ranges (Tables 3 and 4). We assumed that the empirically observed values refer to a period during which treatment was mostly effective and thus, used model simulations without resistance for the calibration. For this study, we used subsets of 1000 calibrated parameter sets from the previous study to simulate MSM and HMW populations (prevalence and incidence rates in Additional file 1: Figures S2 and S3). For each calibrated parameter set, we derived the care-seeking rate of asymptomatic (τ
A
) and symptomatic (τ
S
) individuals using the fraction of successfully treated individuals who were symptomatic at baseline ϕ (Additional file 1: Section Derivation of τ
A and τ
S). We set default values for the testing and treatment parameters (ψ, ξ
G
, ξ
R
, η
1, η
2, δ, ω, λ
A
, and λ
S
) guided by the literature (Table 1).
Table 3 Gonorrhoea prevalence and incidence in baseline scenario (before resistance introduced) for men who have sex with men
Table 4 Gonorrhoea prevalence and incidence in baseline scenario (before resistance introduced) for heterosexual men and women
Sensitivity Analyses
We performed sensitivity analyses to confirm that our model results are robust in scenarios with different properties of tests (ξ
G
and ξ
R
), antibiotics (η
1 and η
2), and populations and clinics (δ, ω, λ
A
, and λ
S
). First, we performed sensitivity analyses of the observed cases averted with regard to changes in both the fraction of asymptomatic individuals who return for treatment at baseline (λ
A,baseline) and the fraction of successfully treated individuals who were symptomatic at baseline (ψ), as well as to changes in single testing and treatment parameters (ξ
G
, ξ
R
, λ
A,baseline, λ
S
, ψ, δ
baseline, and ω). Second, we evaluated the sensitivity of the ratio of resistance spread with regard to changes in the test sensitivity to detect resistance against the first-line antibiotic when using POC (ξ
R,POC), the fraction of asymptomatic individuals who return for treatment at baseline (λ
A,baseline), and the fraction of successfully treated individuals who were symptomatic at baseline (ψ). Third, we tested the sensitivity of our model results to the assumption that the test sensitivity to detect N. gonorrhoeae is 99% for culture testing. For this, we simulated an alternative scenario where culture has a lower test sensitivity to detect N. gonorrhoeae and only culture is used at baseline (ξ
G,baseline=ξ
G,culture=90%, all other parameters as in Table 1).
Simulation
For each parameter set, we first simulated a resistance-free baseline scenario where either culture or NAAT is used (δ>0, λ
A
<1). We simulated the baseline scenario until it reached equilibrium using the function runsteady in the package rootSolve [25] from the R language and software environment for statistical computing [26]. Next, we introduced resistant strains by converting 0.1% of all sensitive infections into resistant infections. We then set the parameter ξ
R
to reflect the different testing scenarios (culture, NAAT, POC+R, or POC−R). For POC tests, we additionally set δ=0 and λ
A
=1. Finally, we simulated the model using the function lsoda from the R package deSolve [27].