A novel Bayesian approach to predicting reductions in HIV incidence following increased testing interventions among gay, bisexual and other men who have sex with men in Vancouver, Canada

Abstract

Increasing HIV testing rates among high-risk groups should lead to increased numbers of cases being detected. Coupled with effective treatment and behavioural change among individuals with detected infection, increased testing should also reduce onward incidence of HIV in the population. However, it can be difficult to predict the strengths of these effects and thus the overall impact of testing. We construct a mathematical model of an ongoing HIV epidemic in a population of gay, bisexual and other men who have sex with men. The model incorporates different levels of infection risk, testing habits and awareness of HIV status among members of the population. We introduce a novel Bayesian analysis that is able to incorporate potentially unreliable sexual health survey data along with firm clinical diagnosis data. We parameterize the model using survey and diagnostic data drawn from a population of men in Vancouver, Canada. We predict that increasing testing frequency will yield a small-scale but long-term impact on the epidemic in terms of new infections averted, as well as a large short-term impact on numbers of detected cases. These effects are predicted to occur even when a testing intervention is short-lived. We show that a short-lived but intensive testing campaign can potentially produce many of the same benefits as a campaign that is less intensive but of longer duration.

1. Introduction

An estimated 75 500 (63 400–87 600 95% confidence interval) people were living with HIV in Canada at the end of 2014 [1]. Although prevalence in the overall Canadian population is low (approx. 0.2%), it is far higher in certain high-risk populations. Of all people living with HIV in Canada, an estimated 49.3% are gay, bisexual and other men who have sex with men (GBMSM) [1]. In British Columbia (BC), the current rate of HIV diagnosis is similar to the national average across Canada (0.4–0.5 per 100 000 general population per year [1]). However, despite a declining trend of diagnoses across the whole population in BC over the last decade, the diagnosis rate has remained essentially constant among GBMSM, who now account for 59% of newly diagnosed infections in 2015 (136 diagnoses among GBMSM province-wide). The prevalence among GBMSM in the greater Vancouver area (the largest city in BC) was estimated to be 14% (10%, 18%) in 2009 [2,3].

With sustained high incidence and prevalence of HIV among GBMSM in Vancouver, a comprehensive programme of HIV prevention in this population remains critical, with frequent HIV testing as a central component [4]. Interventions to increase testing have potential as a useful strategy to decrease incidence of HIV, as individuals change their sexual behaviour to reduce transmission risk after receiving a HIV diagnosis and, with treatment, achieve undetectable HIV viral loads at an earlier point during infection, further reducing transmission risk [5]. In BC, recent efforts to improve HIV testing and earlier diagnosis among GBMSM include: (i) implementation of pooled nucleic acid amplification testing to detect acute infections; (ii) expanded availability of point of care testing; (iii) routine offer of HIV testing in healthcare settings; (iv) development and expansion of an online testing service for sexually transmitted and blood-borne infections called GetCheckedOnline [4]. However, little is known about the impact of these various testing initiatives on testing behaviours of GBMSM and earlier diagnosis of HIV.

One opportunity for resource-rich countries in reducing HIV incidence is increasing the availability and frequency of testing. A modelling study found that over a 20-year period a test-and-treat strategy had reduced the cumulative number of new HIV infections by up to 69.1% in the USA [6], with the largest contribution coming from doubling the annual testing rate from 24% to 48%. Another modelling study considered the outcome of replacing all clinic-based testing with more accessible in-home testing that had a ‘window period’ of three months between HIV infection and its detection by the test. It was found that even with a threefold increase in testing frequency using tests with a three-month window period, the overall HIV prevalence would increase when compared with testing in a clinic setting, due to poor sensitivity of the in-home test [7]. However, if the window period of the test was reduced to two months, then a 2.6-fold increase in testing frequency would be sufficient to reduce HIV prevalence.

The likely impact of increased testing can be great and the impact of testing increases over time as the number of secondary cases averted increases [8]. More frequent testing combined with the initiation of ART at diagnosis can be cost-effective, as long as adherence to ART remains high and patterns of condom-less sex do not change [9]. Importantly, it is not clear how an individual's testing pattern impacts the overall epidemic and what impact a population of non-testers or infrequent testers has on the overall epidemic. To our knowledge, differing testing patterns in the GBMSM population have not been modelled before in the context of the HIV epidemic.

Fitting model parameters to a specific population is challenging in HIV due to the sparsity of the data and the relatively slow dynamics of the epidemic. Often, model parameters will need to be directly estimated from data [10,11], through partial fitting of parameters [12], maximizing likelihood of a sub-sample of parameters or through iterated sampling [13,14], or trial-and-error [7,15], thus making it difficult to reconcile different data sources and to understand the uncertainty in the model fit. Methods such as sensitivity analysis have been used to explore model fits empirically, but this typically does not involve uncertainty derived from model fitting as well as uncertainty in parameter estimates [13,14,16]. In the context of this study, we would like to use survey data obtained from the Vancouver GBMSM population to estimate essential parameters related to testing behaviour, rates of change of testing behaviour, frequency of sexual contact and type of sexual contact. Our novel approach here is to apply a Bayesian methodology using well-informed priors to systematically combine these disparate datasets and account for differing uncertainties in the parameter estimates themselves given the data and our transmission model. This approach allows us to incorporate the greater uncertainty inherent in behavioural survey data, but still constrain the model using definitive laboratory results. We achieve a coherent set of model parameters, explicitly capturing both model and parameter uncertainty. Using our calibrated model, we can investigate different plausible scenarios of increased testing and make predictions about the impact on the incidence and prevalence of HIV in the future. We further compare how the parameter distributions are updated by the incidence data and what consequences this has on the predicted impact of testing. These predictions can then be used to estimate the likely impact of GetCheckedOnline or similar testing initiatives.

2. Material and methods

2.1. Model description

We implemented a compartmental model of HIV infection in the GBMSM community in Vancouver, Canada, stratifying individuals according to HIV infection status, testing pattern and risk group (a summary of the model is given in figure 1). Individuals are assumed to adopt either high- or low-risk sexual behaviour (indexed by i). Furthermore, each individual's testing pattern is assumed to be one of infrequent/no testing, regular testing or frequent testing (indexed by j). Initially, individuals are susceptible (S_ij), but may become infected at a rate determined by their risk group. Once infected, they move to an unaware group (U_ij). They then become aware (A_ij) of their positive status at rate τ_j determined by their current testing pattern. As there are three infection status categories, three testing patterns and two risk groups, the total number of categories is 18 (3 × 3 × 2). There is also turnover in the population at rate e. This turnover reflects the length of time individuals are sexually active members of the at-risk population. The parameter also captures effects of emigration and immigration and any other processes that lead to population turnover. We impose a constant population size constraint by requiring that the rates of entry and exit of individuals to/from the population are equal. It has previously been shown that episodic risk, where individuals transition between high and low risk, can greatly increase the risk of transmission [17]. We, therefore, also allow that individuals can change their testing patterns or risk group. The rates of these changes form a linear transfer operator T (see the electronic supplementary material for derivation). To evaluate different scenarios, we track the cumulative numbers of new infections I and diagnoses D over time. The cumulative incidence I and cumulative diagnoses D are connected through the number of individuals unaware of their HIV status in the MSM population (U_ij). Altogether, the model is specified by the following set of ordinary differential equations:

2.1a

2.1b

2.1c

2.1d

2.1e

The transmission model is specified by the vector function β(U, A), which is set up as follows. We define a mixing parameter ρ that defines how much mixing occurs between the low-risk group (size N₁) and high-risk group (size N₂). We define the mixing vector as R = (ρ/2, 1 − ρ/2) so when ρ = 0, there is no mixing between the groups. The transmission is a sum over all risk categories and testing groups of both unaware and aware individuals. The contribution towards ongoing transmission is different in the aware and unaware group (represented as p_A and p_U vectors, respectively, where individuals who are aware they are HIV-positive have different sexual behaviour [5]), as well as if the individual is high or low risk reflected in differing per-encounter risk of transmission. The rate of sexual encounters varies depending on risk group (r₁ for low risk, r₂ for high risk and r = (r₁, r₂)). A proportion of aware individuals are virally suppressed with probability p_ART, effectively removing them from ongoing transmission. Using the Hadamard–Schur product, this can be compactly written as

2.2

Each component in the sum is a contribution from each of the risk categories and aware/unaware infection stages to the total force of infection. To account for the fact that high-risk individuals are more commonly encountered due to their increased rate of sexual encounter r_i, the transmission rate includes a probability of encountering an individual in a given group weighted by that group's encounter rate. A summary of the model parameters is given in table 1 and estimation procedures are summarized below.

Figure 1.

Overview of GBMSM infection model. The GBMSM population is divided according to infection status (from left to right), testing pattern (top to bottom) and risk group (forward and behind). Individuals start in the susceptible class with an initial testing pattern and risk group. Testing pattern and risk group can change over time. If an individual becomes infected, they move to the unaware class, where they can still change testing pattern or risk group. Once individuals are diagnosed, they move to the aware class where they can continue to change risk groups, but their testing pattern is now irrelevant.

Table 1.

Table of parameters used in HIV incidence model.

parameter	symbol	value	SE	CI	prior	reference
1. total size of sexually active GBMSM population	N	20 000		(12 943, 53 471)	normal	[18]
2a. fraction of GBMSM who do not test	π1	16.5%	1.1	(14.3, 18.7)	beta	[19,20]
2b. fraction of GBMSM who test regularly	π2	43.8%	1.5	(40.9, 46.7)	beta	[19,20]
2c. fraction of GBMSM who test frequently	π3	31.3%	1.4	(28.6, 34.0)	beta	[19,20]
3. average time of being sexually active	1/e	20 years		(10, 30)	gamma	(assumption)
4a. average rate of HIV testers (non-testers)	τ1	0 year−1		n.a.	n.a.	[20]
4b. average rate of HIV testers (regular testers)	τ2	1.66 year−1		n.a.	n.a.	[20]
4c. average rate of HIV testers (frequenttesters)	τ3	4 year−1		n.a.	n.a.	[20]
5a. average rate of sexual encounter (low-risk group)	r1	51.7 year−1		(30.5, 83.5)	gamma	[21–23]
5b. average rate of sexual encounter (high-risk group)	r2	205.5 year−1		(122.7, 337.7)	gamma	[21–23]
6a. average per-encounter risk among unaware individual (low-risk group)	p(1)U	0.306%		(0.219, 0.445)	beta	[21–23]
6b. average per-encounter risk among unaware individual (high-risk group)	p(2)U	1.39%		(1.01, 1.91)	beta	[21–23]
6c. average per-encounter risk among aware individual (low-risk group)	p(1)A	0.108%		(0.077, 0.155)	beta	[21–23]
6d. average per-encounter risk among aware individual (high-risk group)	p(2)A	0.286%		(0.209, 0.394)	beta	[21–23]
7. fraction of HIV-diagnosed GBMSM on ART and virally suppressed	pART	71.2%	0.8%	(69.5, 72.5)	beta	[3]
8a. average transition rate from non-tester to regular tester	h1,2	0.08 year−1	0.01	(0.06, 0.10)	gamma	[20]
8b. average transition rate from regular to frequent tester	h2,3	0.29 year−1	0.02	(0.25, 0.31)	gamma	[20]
9a. average transition from low to high-risk group	ν1,2	0.137 year−1	0.01	(0.12, 0.16)	gamma	[24]
9b. average transition from high to low-risk group	ν2,1	2.373 year−1	0.05	(2.28, 2.46)	gamma	[24]
10. proportion of mixing between high- and low-risk groups	ρ	n.a.	n.a.	n.a.	uniform [0, 1]	—

2.2. Model calibration

A Bayesian inference approach was adopted. This approach allows survey data on sexual behaviour to be combined with literature estimates of per-exposure risk and serological measurements of the annual incidence. Prior distributions for key model parameters were constructed from questionnaire studies on testing and sexual behaviour (table 1). Where the parameter is a rate, a gamma distribution was fitted and where a parameter was a proportion, a beta distribution was fitted (see the electronic supplementary material for full details). These distributions helped to regularize the model fitting and were flexible enough to fit the estimated mean and standard deviation of the parameter.

To simplify the analysis and increase computational speed, the model was fitted at endemic equilibrium, where the annual incidence of HIV is assumed constant. This necessarily implies that testing patterns and risk behaviour were non-changing over the fitting period. This was justified by examining the annual number of new HIV cases between 2005 and 2014 using a Bayesian Poisson linear model. The fitting was conducted with year being the independent variable and with wide priors for both intercept and year. Although the 95% credible interval for the effect of year did not include zero (95% credible range: (−0.046, −0.005)), the 99% credible interval did (99% confidence range: (−0.052, 0.002)). There is, therefore, little evidence that cases in Vancouver, Canada have been decreasing over the period considered and justifies the use of this equilibrium assumption.

The likelihood function was constructed from the Poisson distribution, using the annual incidence. Each incidence error was assumed to be independent and identically distributed producing the following likelihood

2.3

Here x_i are the annual incidences and λ is the incidence rate at endemic equilibrium for the model. The posterior for the model P(Θ | x) is, therefore, defined as

2.4

To sample from the posterior and produce refined estimates for the parameters incorporating knowledge of the serological data, we adopted a Markov chain Monte Carlo (MCMC) scheme. Sampling steps occurred using an adaptive Metropolis–Hastings update [25].

We performed analysis on the posterior marginalized over each parameter in order to understand how strongly the likelihood influences the prior for each parameter. This was achieved by performing a two-sample Kolmogorov–Smirnov (KS) test on the empirical cumulative density functions of the posterior and prior marginalized on each parameter [26]. The fitting was performed in Python v. 2.7 using the library PyMC [27,28]. The KS test was implemented using the SciPy library and data processing was carried out using the Pandas library [29,30]. Data visualization was performed using the libraries Seaborn & Matplotlib [31].

2.3. Survey parameter estimation

We used data from a cohort study of HIV negative GBMSM in Vancouver to construct well-informed priors for the model. The study population was recruited from 18 June 2011 to 2 March 2012 at a sexual health centre for GBMSM in Vancouver. Of the 1141 eligible individuals, 194 consented and 166 completed the baseline survey. Details of the survey methods and findings have been previously published [32].

Participants completed four surveys and sexual network interviews at baseline and approximately 30, 180 and 360 days after recruitment. The surveys had questions on demographics, HIV testing patterns, personal sexual history, HIV knowledge and attitudes, as well as other questions pertaining to sexual and mental health. The sexual network interviews collected detailed behavioural information for the last sexual encounter with each of up to five most recent sexual partners. The construction procedures of priors from the survey data are detailed below. In general, priors were constructed to take into account the uncertainty in the distribution of parameters and sample size (approx. 1% of population being modelled). The key idea is to construct the priors in such a way that the uncertainty is carried forward into the model predictions so that the posterior reflects the uncertainty within the estimates.

2.3.1. Transition rates between no, regular and frequent testing

The population is divided into three categories based on their testing behaviour. These testing patterns are defined from the surveys as follows: (i) ‘no testing’ for never or infrequent testing (τ₁ = 0.05 year⁻¹), (ii) ‘regular testing’ for testing every six to 12 months, (τ₂ = 1.66 year⁻¹) and (iii) ‘frequent testing’ for testing every three months or more often, (τ₃ = 4 year⁻¹). From the survey, the proportion in each of these groups is 26%, 44% and 30%, respectively. We also used the survey data to estimate the rates of transitions of individuals between the testing categories by considering the likelihood that individuals reported changes in testing behaviour between the three approximately equidistant time points in the survey (baseline, day 180 and day 360). The raw probability that an individual switched behaviour is then converted into a rate using an exponential waiting time distribution, and the rates are then averaged across time-points (see the electronic supplementary material for full details). The same method was used to estimate the rates of transitions between low- and high-risk groups.

2.3.2. Rate of risky events

We define the rate of risky events r as the number of sex acts where HIV may be transmitted per unit time. We obtain this parameter value based on participants' responses to survey questions about their five most recent sexual partnerships in the past six months. For each partnership, the participants provided the date of the first and last sexual encounter, whether the relationship was ongoing, and the average frequency of sex in the past year. From this information, we estimate the number of days of partnership within the last year and the frequency of sex. As the provided responses to either of these quantities are often vague or approximate, we allow a generous lower and upper bound for each relationship in constructing the prior.

We assume a maximum rate of risky sex acts of once per day on average. Some participants did not report any sex act with their sex partners, so we set the minimum number of sex acts to once per relationship. For each participant and each survey time point, we then sum the total number of all risky events over all their partnerships in the past year, and average this count within a defined risk category category to obtain the final estimate.

2.3.3. Per-encounter risk probability of infection

To calculate the per-encounter risk of infection in the presence of more than one sex act per encounter, we assume that the likelihoods of infection from all risk factors are independent. The probability can then be calculated as

This defines the per-encounter risk of infection for an individual. To evaluate this for the population, we split the population into high- and low-risk groups. This was achieved by defining a force of infection for each participant. The per-partner-associated risk was calculated as above (p^partner_u,i) for each participant i in the survey and each partner they reported. This was then multiplied by the rate of sexual encounter for that partner, r^partner_i. The force of infection for the participant is then defined as . Clustering was performed on the statistic to divide the population into high- and low-risk groups (see the electronic supplementary material). The means for each group were used to derive the rate of sexual encounter r_i, the per-encounter risk of transmission of unaware, p_u,i and aware, p_a,i for the high- and low-risk group (figure 2).

Figure 2.

Distribution of per-contact risk associated with HIV derived from survey data. Two-means clustering was used to determine high- and low-risk groups in order to derive the parameters for each. Note the x-axis is a log scale.

3. Testing scenarios

Limited data are available for how public health interventions can alter individual testing patterns. We, therefore, consider a set of plausible scenarios regarding changes to testing patterns and use the calibrated model to predict the long-term effects in terms of detected incidence and averted infections. For a random sample of the parameters from the posterior distribution, initial conditions were chosen using the equilibrium distribution. The model was then simulated forward for 50 years with and without a given intervention. By comparing the output of these two simulations, we calculate the per-year changes in detected cases and number of new infections.

We consider situations where the effects of the testing intervention are permanent or temporary. To faithfully compare these two scenarios, we consider the number of individuals who move from no testing to regular testing in a fixed number of years. This is modelled as a change over time in the probability that an individual who begins in the no-testing category ends up in the regular-testing category. We consider the following parameterization for the rate of transitioning between testing groups (see the electronic supplementary material for a derivation)

3.1

Here, the equilibrium probability of being in testing group i is π_i (i = 1, 2, 3) and the rates of transitioning between groups are α and β. We fit the parameters α, π₁ and π₂ to a given probability p that an individual who started in the no-testing category is in the regular-testing category after x years (figure 3). The probabilities considered are 30%, 40% and 50%, which are reached after a period of 5 years (as a hypothetical duration of a testing initiative). The intervention can further be modelled where the change in testing rates is permanent (continuous) or the change in testing pattern is over a fixed period (fixed). An example of how fixed period testing is implemented is shown in figure 3.

Figure 3.

Change in testing patterns over time. We consider interventions that move 30%, 40% or 50% of individuals who start in the no testing category into at least the regular testing category after 5 years by altering the transition rate between infrequent and regular testing (shown in blue, green and red, respectively). This is achieved through a combination of increasing the rate of switching between the no testing and regular testing categories, and the long-term proportion of individuals in the regular testing category. (a) Individual trajectories over time are presented for individuals who are in the no testing category at the start of the intervention. Although for the 30% intervention the individual sampled here never moves from the no testing category, their corresponding probability of being in the regular testing category does increase over time. (b) Changes in the probability of being in the regular testing category over time, conditional on starting in the no testing category.

The measured outcomes for the testing scenarios were the change in the number of detected cases and the number of infections averted for each year following the start of the intervention. This was measured by sampling a set of parameters from the posterior and then simulating for 50 years with and without the intervention and calculating the difference in the number of detected cases and number of new infections.

We also explored the estimated impact of the intervention under the prior and posterior distributions in order to gain insight into how the fitting methodology affects the predicted impact of testing. We analysed the same scenarios as above with parameters drawn from the prior distribution and then compared with numbers of detected cases and infections averted when parameters were drawn from the posterior distribution.

4. Results

4.1. Model calibration

We took 1 × 10⁵ samples with a burn-in period of 5000 and thinning at every 10 steps to estimate the parameter posterior distribution (figure 4). All parameters have approximately unimodal distributions and show good convergence.

Figure 4.

Estimated posterior from fitted HIV model using samples drawn from MCMC. Marginal distributions for each separate parameter are shown as histograms along the diagonal. Correlations between two parameters are shown in the off-diagonal plots. Contours indicate the density of points and plots were created using kernel-density estimation. (Online version in colour.)

The KS two sample test was used to determine which parameters differed significantly from their prior to posterior marginal distribution (figure 5). The strongest influence was on the mixing coefficient ρ where the prior with mean 0.5 (0.023–0.98 95% credible interval) was updated to 0.96 (0.88–1.0) in the posterior. Similarly, the posterior generated a tighter bound for the population turnover parameter e, although with a similar mean (0.05 (0.033–0.071) to 0.043 (0.035–0.049)). Other parameters, whose marginal posteriors were more strongly informed by the likelihood, were the rate of sexual encounter in the high-risk group r₂ (210 year⁻¹ implying a sexual encounter rate of 0.6 day⁻¹), the per-contact risk of infection in the low-risk aware group p⁽¹⁾_A and the rate of transitioning from high- to low-risk ν₂₁. Although these parameter marginal distributions changed significantly from the prior to the posterior, this results in tighter bounds on each of them in order for samples from the parameter space to produce numbers of diagnoses consistent with the data (see the electronic supplementary material for full list of parameters).

Figure 5.

Violin plots describing each parameter's marginal distribution for posterior and prior. The violin plots show the median (white point) with the inter-quartile range (solid dark line) with an empirical distribution estimated using kernel density estimation [33]. The posterior (right in green) is broadly within the same support as the prior (left in blue). The parameters that show significant difference between the posterior and prior are: ρ, e, r₂, p⁽¹⁾_A and ν₂₁.

Most parameters do not show strong covariance except for p⁽¹⁾_A and population size e. Although p⁽¹⁾_A is higher than p⁽²⁾_A, due to the prevalence of viral suppression in the aware category being estimated to be 71%, this parameter has relatively little impact on the overall rate of new annual cases.

The estimated number of reported cases per year was sampled from the posterior by drawing a set of parameters, simulating the dynamics at equilibrium and then drawing a Poisson random variable using this rate. The estimated number of reported cases per year was 115 (85–139 95% credible interval). All data points lie within the credible interval (figure 6a).

Figure 6.

(a) Predictive interval for the model-based incidence (95% shown in light blue, 75% in dark blue and 50% as light blue line). GBMSM diagnosed cases in Vancouver (Coastal Health) shown as red dots. (b) Model-based prevalence of HIV divided into low-risk, high-risk, unaware and aware. The total estimated prevalence of 18.9% (17.2–20.5) is divided further into estimates of high risk and low risk (two bars), which are further divided by colour into unaware and aware. The 95% credible intervals are shown as (grey) error bars for each category.

4.2. Prevalence of diagnosed and undiagnosed HIV infection

As a validation step, the overall prevalence of HIV was estimated from the posterior distribution (figure 6b). The resulting distribution estimates prevalence to be 18.9% (17.2%–20.5%) among the GBMSM population. This estimated prevalence is very close to a previously published estimate [2]. We predict that the frequency of individuals who are unaware of their HIV-positive status is 1.3% (1.2%–1.6%) of the total GBMSM population or 6.7% (5.3–8.3) of the HIV-positive population. In terms of population size, the steady-state number of individuals who are unaware of their HIV status and can potentially be targeted in a testing intervention is estimated to be 260 (240–290).

Both the prevalence and number of cases diagnosed per year are strongly informed by the likelihood. Before fitting to the incidence data, the prior distribution predicts the number of detected cases as 0 (0–207) (compared to 115 (85–139) for the posterior) and the prevalence as 0% (0%–30%) (compared to 18.9% (17.2–20.5). Although the prior range does include the range determined by the posterior for both prevalence and detected cases, the posterior provides parameter estimates with sharper bounds on the observed and unobserved epidemiological statistics.

4.3. Five-year testing interventions

We now consider the effectiveness of short-lived campaigns to increase testing where specific targets for increasing the number of regular testers are met within 5 years. Five years was chosen as a typical length of a public health intervention or programme. The rate of change between no testing and regular testing and the long-term proportion of the regular testing category were fitted to achieve specific targets (figure 3).

If a campaign is rolled out continuously and the effects of the intervention are considered to be permanent, there is a sharp, significant increase in the number of detected cases. After 2 years, the median change in the number of detected cases is 11 (11,12), 19 (18,21), 31 (28,33) and 54 (49,60) for 50%, 60%, 70% and 80% regular testing targets, respectively (figure 7a). The change in detected cases then declines to zero (compared to where there is no intervention) after approximately 10 years for each scenario considered. When the intervention lasts for 5 years, the initial change in detected cases is similar (figure 7c). However, once the intervention is halted, there is a significant drop in the number of detected cases for all testing targets. The rate of detection of new infections then gradually returns to its pre-intervention level.

Figure 7.

Five-year intervention target scenarios. Interventions are modelled where the testing targets are set at a 5-year time horizon. The intervention either runs continuously for the 50-year period considered (a,b) or for 5 years (c,d, intervention period shown as an orange coloured background). The changes in detected cases compared to where there was no intervention are shown in (a) and (c) and the number of new infections averted are shown in (b) and (d). (Online version in colour.)

The rate of infections averted has a slower decline in both scenarios (figure 7d,b) where there is a gradual increase in the rates of averted cases for 6 years. Afterwards, there is either a decline due to halting the intervention (figure 7d) or a linear increase in averted cases (figure 7b). The per year number of averted cases increases to 5.7 (2.3, 9.8) for the most ambitious scenario and then linearly increases up to 10.2 (4.5, 16.7) after 50 years. Where the lifetime of the intervention is fixed, the most ambitious scenario increases to 5.5 (3.1, 9.8) and then declines to 1.4 (0.8, 1.9) after 50 years (figure 8)

Figure 8.

The influence of the posterior on estimating the impact of an intervention. Comparison between the posterior and the prior for the intervention scenario where 60% of the GBMSM population become regular testers in 5 years where the intervention does not discontinue (a,b) or is over a fixed time-period (c,d). The prior has not been fitted to the incidence data and as such underestimates the impact of the intervention. (Online version in colour.)

The cumulative number of averted infections was considered for a number of time-horizons (electronic supplementary material, table S2). Where the intervention has a lifetime of 5 years, the cumulative numbers of cases averted at a 30-year time-horizon are 29 (11, 48), 43 (17, 79), 68 (34, 113) and 95 (54, 164) with increases in regular testing of 50%, 60%, 70% and 80%, respectively. Where the intervention lasts indefinitely, the total numbers of averted cases after 30 years are 75 (37, 132), 103 (57, 194), 145 (82, 254) and 200 (83, 331), respectively.

5. Discussion

We have constructed a dynamic model of HIV infection in a GBMSM population, stratified by testing patterns and risk of exposure to HIV. We applied a novel Bayesian approach to model fitting that provides an explicit methodology for combining disparate forms of survey data with confirmed new cases of HIV. After fitting to recent data from Vancouver, Canada, we were able to predict long-term outcomes of testing-related public health interventions, incorporating parameter and model uncertainty into the final estimates.

Previous population HIV transmission models have performed fitting without explicitly taking into account behavioural and clinical data sources and incorporating parameter and model uncertainty. By constructing prior distributions from survey data and combining with a likelihood derived from the incidence data in a Bayesian manner, we have developed a method for performing robust model estimation. By including all parameters in the model fitting, this also gives a way of performing uncertainty analysis alongside estimation from the model. The method could be readily adapted to other forms of data, as the prior distributions may be composed from multiple datasets. The likelihood too, may also be composed of prevalence information, or broken down in greater detail by subpopulation.

The use of informative priors has been shown to increase the power of statistical analysis in the face of limited data [34]. By comparing the posterior and prior distributions both under the marginalized parameter distributions and measured outcomes (figure 8), we find the methodology is capable of refining both the estimated outcomes and the estimates for parameters associated with the HIV diagnoses. We found that the prior, although well-informed, was unable to capture the number of cases per year. This is also reflected in an expected prevalence of 0% in the prior, although its range does capture a region where the prevalence is non-zero. Ultimately, without informing the prior parameters with the incidence data, erroneous results are produced when assessing the impact of increased testing. The prior indicates a negligible impact of testing due to the expected prevalence being zero. On the other hand, the posterior refines the prediction by incorporating different data sources, while maintaining a region of the parameter space that is plausible from the prior distribution. We note that fitting the likelihood alone would be extremely ill-posed. However, by combining the likelihood with priors taken from other data sources, our parameter estimates are refined and robustly produce samples where both prevalence and incidence match current estimates.

The prior and posterior were compared through marginal distributions for each parameter and pair-wise joint distribution (figures 4 and 5). This presentation would also allow comparison between different models in how parameters could be informed by the data through the model.

The overall estimated HIV prevalence was predicted by the model to be 18.9% (17.2, 20.5). This value is in excellent correspondence with previously estimated values of HIV prevalence among GBMSM in BC [2]. Furthermore, the estimated number of individuals who are unaware of their positive HIV status was estimated to be 260, or 1.3% of the GBMSM population. It should be noted that this value reflects a dynamic equilibrium where individuals are constantly infected and then diagnosed. However, this provides an upper-bound on the number of individuals that a testing intervention can target at any given time, and shows how even high-impact testing campaigns may have surprisingly little impact on the whole epidemic in a setting where overall uptake of HIV testing is already high—as in the Vancouver region.

Our model predictions show that testing interventions can be expected to have an immediate impact on the rate of detection of cases, but reductions in the rate of new infections are achieved much more slowly. If an intervention is sustained, then there is a multiplicative effect, as each year fewer cases are observed and so the risk for new infections decreases. However, even when a campaign is short-lived, the impact on the number of infections averted is still sustained for up to 30 years after the intervention has ended.

By comparing short-lived and long-lived testing promotions, we found that an intensive short-lived campaign can outperform a lower-impact continuous campaign, even over a 30-year time horizon. For example, we compared a campaign that moves 80% of infrequent/no testers into regular testing for five years, with a campaign that moves 50% into regular testing over a 30-year time-horizon. We find that the 5-year campaign produces 95 (54, 164) averted cases, compared to 75 (37, 132) for the sustained campaign.

5.1. Directions for further study

We have presented and parameterized a testing-focused model of HIV infection among GBMSM. However, we chose to neglect a number of potentially important effects in our analysis. First, we did not model the progression of HIV within an individual patient. There is evidence that the first few months of infection (the acute stage) may have a disproportionate contribution towards onward transmission, due to either viral load or behaviour [10,17], and importantly, this stage often precedes diagnosis. Estimates of the length of the acute stage and of increased transmission rates during this stage vary greatly between studies, further making this a difficult effect to model. We intend to investigate the linkage between testing frequency and diagnosis of acute infections in future work.

We assumed a constant rate of infection between 2004 and 2014 with a dynamic equilibrium, where the proportion of individuals in each category does not change. We intend to further investigate this assumption, in particular how changes in risk and testing can produce a constant rate of diagnoses.

Second, demographic effects related to immigration/emigration and age of individuals are not explicitly modelled. Instead, all those effects are combined into a single population turnover parameter. Although there is some indication that the dynamics of the epidemic are different between different age groups, we assume there is enough mixing between age classes to justify the assumption of no age classes. We also assumed that all infections come from within the population and there are no importations. BC and in particular Vancouver has a relatively large amount of immigration compared with the rest of Canada, but we do not have information to allow us to break out exogenous infections from the remainder.

Finally, we focused solely on the impact of testing interventions and the impact of testing in combination with other HIV prevention interventions are not explicitly modelled. For example, HIV pre-exposure prophylaxis (PrEP) is likely to have a long-lasting impact on the GBMSM HIV epidemic in BC. However, currently PrEP uptake remains low. As such, this and other ongoing interventions related to treatment and prevention have not been explicitly taken into account, except to the extent that they affect the fitted parameters through the incidence.

6. Conclusion

The use of Bayesian methods to combine disparate datasets into a quantitative framework holds much promise in a model fitting for HIV epidemiology and can provide estimates of uncertainty in key outcomes of interventions, incorporating both parameter and model uncertainty. Stratifying a GBMSM population by testing patterns allows for greater realism in how pathways to testing or testing initiatives will impact the epidemic.

Data accessibility

The corresponding author can be contacted with regard to access to cohort study data. Data on clinical diagnoses of HIV can be found at the following URL: http://www.bccdc.ca/resource-gallery/Documents/Statistics%20and%20Research/Statistics%20and%20Reports/STI/HIV_Annual_Report_2014-FINAL.pdf.

Authors' contributions

M.A.I., B.P.K., M.G. and D.C. conceived the study. B.P.K., M.G., W.M. and D.C. developed the initial model and analysis. M.A.I., M.G., R.B. and D.C. developed the extended model and Bayesian methodology. M.A.I. performed the final model fitting and analysis. M.A.I., M.G. and D.C. wrote the initial draft of the manuscript. All authors revised and approved the final version of the manuscript.

Competing interests

We declare we have no competing interests.

Funding

M.A.I., D.C. and M.G. acknowledge support from the Canadian Institutes of Health Research Partnerships for Health Systems Improvement programme (grant no. 318068). D.C. acknowledges support from the Natural Science and Engineering Research Council of Canada.