Increasing HIV transmission through male homosexual and heterosexual contact in Australia: results from an extended back-projection approach

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1 DOI: /j x r 2010 British HIV Association HIV Medicine (2010), 11, ORIGINAL RESEARCH Increasing HIV transmission through male homosexual and heterosexual contact in Australia: results from an extended back-projection approach H Wand, 1 P Yan, 2 D Wilson, 1 A McDonald, 1 M Middleton, 1 J Kaldor 1 and M Law 1 1 National Centre in HIV Epidemiology and Clinical Research, Sydney, Australia and 2 Center for Infectious Disease Prevention and Control Population and Public Health Branch, Ottawa, Canada Objectives The aim of the study was to reconstruct the HIV epidemic in Australia for selected populations categorized by exposure route; namely, transmission among men who have sex with men (MSM), transmission among injecting drug users (IDUs), and transmission among heterosexual men and women in Australia. Design Statistical back-projection techniques were extended to reconstruct the historical HIV infection curve using surveillance data. Methods We developed and used a novel modified back-projection modelling technique that makes maximal use of all available surveillance data sources in Australia, namely, (1) newly diagnosed HIV infections, (2) newly acquired HIV infections and (3) AIDS diagnoses. Results The analyses suggest a peak HIV incidence in Australian MSM of 2000 new infections per year in the late 1980s, followed by a rapid decline to a low of o500 in the early 1990s. We estimate that, by 2007, cumulatively MSM were infected with HIV, of whom 13% were not diagnosed with HIV infection. Similarly, a total of 1050 and 2600 individuals were infected through sharing needles and heterosexual contact, respectively, and in 12% and 23% of these individuals, respectively, the infection remained undetected. Discussion Male homosexual contact accounts for the majority of new HIV infections in Australia. However, the transmission route distribution of new HIV infections has changed over time. The number of HIV infections is increasing substantially among MSM, increasing moderately in those infected via heterosexual exposure, and decreasing in IDUs. Keywords: Australia, HIV/AIDS, incidence, modified back-projection, transmission route Accepted 29 September 2009 Introduction Estimates of past and current HIV and AIDS incidences and prevalences are important for effective public health prevention strategies. The HIV/AIDS epidemic in Australia has been under surveillance since 1981 through notification of AIDS diagnoses, and since 1985 through notification of cases of newly diagnosed HIV infection. Since 1991, Correspondence: Dr Handan Wand, National Centre in HIV Epidemiology and Clinical Research, Sydney, Australia. Tel: ; Fax: ; hwand@nchecr.unsw.edu.au further surveillance has been supplemented by national notification of HIV diagnoses with evidence of newly acquired HIV infection, defined as new HIV diagnoses with either a previous negative HIV test within 12 months, or evidence of a recent seroconversion illness. Although these data are indicative of trends in the HIV epidemic, they cannot be used directly to estimate the incidence of HIV infection. Accurate estimates of the incidence of HIV infection are required at the national and subgroup levels to determine trends in the epidemic and to evaluate the effectiveness of prevention strategies. 395

2 396 H Wand et al. Methods based on back-projection [1] have historically been used to estimate the incidence of HIV infection (numbers of new infections per year) from AIDS surveillance data, based on the probability distribution of the incubation period, that is, the time from HIV infection to AIDS. The availability of effective antiretroviral therapies from 1997 onwards altered the distribution of the incubation period in ways that are difficult to quantify. The interpretation of trends in both AIDS case reporting and HIV infection reporting must take into account the effect of treatment in slowing disease progression and the effect of test-seeking behaviour on the numbers and characteristics of persons being tested for HIV. When monotherapy was the main treatment option, models of the epidemic were adjusted for estimates of its effect on the incubation period from infection to AIDS, but the complexities and stronger effects of the multiple therapies now available have made treatment adjustment too uncertain for use in the modelling. Methods of estimating HIV incidence rates based on the results of the HIV test and a test for a biomarker have been under investigation [2 4]. Although these methods provide a very up-to-date and revealing snapshot of the epidemic, the technology used to detect recent infections is still quite new and expensive. The methodology that we used in this study does not require a test for a biomarker and makes maximal use of all available HIV/AIDS data sources in Australia s surveillance databases, including newly diagnosed HIV infections, newly acquired HIV infections and AIDS diagnoses, to estimate trends in HIV incidence. As there is no established statistical model to link HIV incidence to HIV diagnosis with respect to HIV testing patterns, the current methodology assumed that, if an individual was infected before, or during, a certain year, it was more likely that this individual sought an HIV diagnostic test at the onset of clinical symptoms. However, as HIV testing became more widely available and promoted, individuals infected in later years tended to be more likely to seek testing independent of the onset of clinical symptoms. Surveillance systems based on the reporting of AIDS cases also do not provide a completely up-to-date picture of the trend of the HIV epidemic, highlighting the need for methods with which to estimate the incidence of HIV infection accurately. In recent years in Australia, there have been increasing numbers of HIV diagnoses in some states and territories, particularly among men who have sex with men (MSM) [5]. In this study, we used modified back-projection modelling to estimate the incidence of HIV infection in an attempt to assess whether increases in HIV notifications in recent years truly reflect increases in the underlying incidence of HIV infection. We investigated the three exposure routes with the highest incidences of HIV infection, namely: (1) transmission in MSM, including those who also reported injecting drug use (IDU), and including men with HIV infections without a reported source of HIV exposure; (2) transmission in people who reported IDU but not MSM; and (3) heterosexual transmission in men and women. These subgroup analyses are of particular importance in assessing the success of prevention programmes, and for the allocation of prevention resources. The coupling of the methodology used in this study with a long-established HIV/AIDS database created a unique opportunity to reconstruct the HIV infection curve. Methods Surveillance data In Australia, HIV transmission is monitored through the notification of cases of newly diagnosed HIV infection, including cases with evidence of newly acquired HIV infection, which is defined as HIV infection with evidence of a prior negative test or a diagnosis of primary HIV infection or an indeterminate western blot within 12 months of HIV diagnosis. Therefore, there are potentially three data sources available in each calendar year: HIV surveillance data (first HIV-positive diagnoses by year of diagnosis), data on newly acquired HIV infections (recent infections among new HIV diagnoses) and AIDS case reporting surveillance data (based on physicians reporting on diagnoses of clinical events subject to the AIDS case definition) [5]. Modified back-projection approach The back-projection method was originally proposed by Brookmeyer and Gail and used in Western countries in the late 1980s and early 1990s to estimate trends in HIV infections based on reported AIDS diagnoses [1]. This methodologyuseddataonreportedaidscases,combinedwithan assumption of the rate at which people progress from HIV infection to AIDS diagnosis (the incubation period), to estimate the most likely pattern of past HIV incidence. The availability of effective antiretroviral therapies from 1997 onwards altered the distribution of the incubation period in ways that are difficult to quantify. As a result, application of the method to current AIDS diagnosis data is unlikely to give reliable estimates of HIV infection rates. Some researchers [6] have modified the incubation function to account for the treatment effect, but this approach has generally been unsuccessful because of the difficulty of capturing the complexity of treatment regimens and their effects. Others [7] have incorporated HIV diagnosis data into the back-projection method to improve the reliability of estimation.

3 Increasing HIV transmission in Australia 397 The back-projection method that we used in this study differs from similar approaches in the literature, in that it does not require data linkage between the HIV and AIDS diagnostic registries. It is based on a parametric formulation of the time between the acquisition of HIV infection and the earliest diagnosis of HIV infection obtained from enhanced HIV surveillance systems or from laboratoryconfirmed testing. For an infected individual, a diagnosis of HIV infection may be made as a consequence of an awareness of recent exposure, the onset of symptoms related to HIV disease progression, random detection or frequent testing. In this study we assumed a simplified model that only involves two testing forces: testing during asymptomatic infection and testing driven by clinical symptoms at a late stage of HIV disease progression. Based on these assumptions, two submodels were constructed that are mathematically connected to form the combined progression rate distribution. The HIV incidence curve was then reconstructed by combining two back-projection estimated HIV incidence curves from AIDS diagnostic data (up to 1994, prior to which effective antiretroviral treatment was not available) and HIV diagnostic data using the combined progression rate distribution. The methodology also used the back-calculated HIV incidence to forecast what the trend of AIDS diagnoses over the years would have been in the absence of treatments. This forecast can be compared with the actual trend of AIDS diagnoses from surveillance data. Details of this methodology are given in the Appendix A. User-friendly software for this methodology, written in the R language, together with other technical and methodological documents, is available upon request (ping_yan@ phac-aspc.gc.ca). Results Following a long-term decline, the annual number of new HIV diagnoses has gradually increased recently, from 763 cases in 2000 to 998 in Among the cases of newly diagnosed HIV infection, an increasing number were in people who had acquired HIV infection within the previous year. Summary figures suggest that, by the end of 2006, diagnoses of HIV infection, diagnoses of AIDS and 6723 deaths following AIDS occurred in Australia [5]. Table 1 shows the distribution of HIV diagnoses for three exposure categories. Estimated HIV incidence curves and their pointwise 95% confidence intervals (CIs), which were calculated by bootstrap [7], are plotted for the three main routes of transmission (MSM, IDU and heterosexual acquired for both men and women) in Fig. 1a d. Model-predicted HIV and AIDS diagnoses (in the absence of therapies) along Table 1 HIV diagnoses in Australia over time and by likely route of exposure Exposure category MSM IDU Male heterosexual contact Source: Australian Surveillance Report, IDU, injecting drug use; MSM, men who have sex with men. Female heterosexual contact with their observed counts are also presented in Figs 2a d and 3a d, respectively. In recent years there has been a noticeable increase in the number of HIV diagnoses in MSM (Fig. 2a). The backprojection analyses suggest a peak HIV incidence in MSM of over 2000 new infections per year in the early 1980s, followed by a rapid decline to a low of a little under 500 new infections per year in the early 1990s (Fig. 1a). It is estimated that the incidence of HIV infection then increased gradually through the early 2000s, to 750 new HIV infections in This is in broad agreement with previous reports and conventional back-projection estimates [8]. Our results also show that, to the end of 2006, a total of men were infected with HIV through male homosexual sex, of whom 13% (95% CI 12%, 14%) are estimated not to have been diagnosed with HIV infection (Table 2). In , approximately 4% of HIV diagnoses in Australia were in people who reported a history of IDU (Annual Surveillance Report, 2007). The prevalence of HIV infection among people attending needle and syringe programmes remained low ( 1% in ). According to the back-projection estimates, there was a peak in the incidence of HIV infection through IDU of 122 new infections in 1985, followed by a rapid decline to a low of 9 new infections in 1994 (Fig. 1b). The back-projections suggest some evidence of an increase in HIV incidence in the late 1990s, but a plateau at around 40 HIV infections per year in the 2000s. The models estimate that a total of 1050 people were infected with HIV solely through IDU in Australia to the end of 2006, of whom 12% (95% CI 9%, 15%) are estimated to have remained undiagnosed (Table 2). The number of new HIV diagnoses for which exposure to HIV was attributed to heterosexual contact increased from 775 in to 914 in , accounting for

4 398 H Wand et al. Estimated HIV infections 3500 (a) Estimated HIV infections (b) Estimated HIV infections 120 (c) Estimated HIV infections 100 (d) Fig. 1 The estimated number of HIV infections ( ) and uncertainty bands (0.975 and quintiles) ( ) by year according to the backprojection model for Australian (a) men who have sex with men, (b) injecting drug users, (c) heterosexual men and (d) heterosexual women. Fig. 2 The estimated number of HIV diagnoses by model ( ) and observed HIV diagnoses ( ) by year for Australian (a) men who have sex with men, (b) injecting drug users, (c) heterosexual men and (d) heterosexual women.

5 Increasing HIV transmission in Australia 399 Fig. 3 The estimated number of AIDS cases by model(assumingnotreatmenteffect)( ) and observed AIDS diagnoses (which may be subject to a treatment effect) ( ) by year for Australian (a) men who have sex with men, (b) injecting drug users, (c) heterosexual men and (d) heterosexual women. Table 2 Back-projected number of HIV infections in men who have sex with men (MSM), injecting drug users (IDU) and those infected via heterosexual transmission, and estimated percentage of infections not diagnosed in the time periods before 1990, and Model-based estimates for back-projected HIV infections Before Total MSM Estimated number of HIV infections Percentage of infections not yet diagnosed (95% CI) 2 (1.0, 3.0) 9 (8.0, 10.0) 43 (41.5, 44.4) 13 (12.0, 14.0) IDU Estimated number of HIV infections Percentage of infections not yet diagnosed (95% CI) 3 (1.6, 4.4) 1 (0.4, 2.4) 36 (29.7, 42.3) 12 (9, 15) Heterosexual male Estimated number of HIV infections Percentage of infections not yet diagnosed (95% CI) 2 (0.8, 3.2) 10 (7, 13) 52 (48, 56) 23 (21, 25) Heterosexual female Estimated number of HIV infections Percentage of infections not yet diagnosed (95% CI) 2 (0.5, 3.5) 9 (6, 12) 52 (47, 57) 22 (19, 25) CI, confidence interval. 20% of the total HIV diagnoses (Annual Surveillance Report, 2007). Consistent with these results, back-projection analyses suggest steady increases in new infections attributed to heterosexual exposure to HIV (men and women) since the mid-1990s (Fig. 1c and d, respectively). The model estimates that a total of 1492 men and 1119

6 400 H Wand et al. women were infected through heterosexual exposure, of whom 23% (95% CI 21%, 25%) and 22% (95% CI 19%, 25%), respectively, are yet to be diagnosed with HIV infection (Table 2). Discussion In the absence of accurate tests for biological markers that can be used to determine the duration of infection in individuals, it is important to use all data available to estimate trends in HIV incidence over time. One of the advantages of our model for estimating HIV incidence is its ability to utilize the long history of HIV and AIDS surveillance data while adjusting for changes in testing behaviours. AIDS surveillance data were only used in the analysis for HIV incidence until 1987, just prior to the first antiretroviral drug becoming available. Overall, our results suggest that recent increases in HIV diagnoses in MSM in Australia do reflect an increasing trend in underlying HIV incidence over recent years. Similar increases in HIV diagnoses have been seen in MSM in virtually all developed countries [9]. Deterministic mathematical models suggest that reported increases in unprotected anal intercourse in MSM, and importantly increases in other sexually transmissible infections acting as co-factors for HIV transmission, can explain increases in HIV incidence in Australia [10]. According to our results, the rate of HIV transmission through IDU is currently relatively flat in Australia after an increase in incidence during the late 1990s. The increase in HIV incidence in the late 1990s coincided with an increase in the number of injecting drug users, and with an increase in the incidence of hepatitis C virus (HCV) infection [11]. It is widely acknowledge that, since 2001 in Australia, there has been a reduction in the heroin supply, resulting in some reduction in IDU, and also an estimated decline in HCV incidence [12]. The plateau in HIV incidence during the 2000s may reflect these patterns of reduced IDU, and also raises the possibility that HIV transmission among injecting drug users could re-emerge if the number of injecting drug users increased. An important finding from our analyses is a consistent pattern of increasing estimated HIV incidence in men and women with heterosexual exposure (Fig. 1c and d, respectively), despite relatively inconclusive trends in HIV diagnoses (Fig. 2c and d, respectively). As far as can be ascertained using national surveillance data, the majority of reported diagnoses are either in people from a high HIV prevalence country, or in people with a partner from a high HIV prevalence country. However, a relatively large proportion of HIV infections among heterosexuals are estimated to be undiagnosed. Although these estimates are still much lower than those in other developed countries, combined with increases in reported sexually transmissible infections in the general population [5], these increases in estimated HIV incidence are a real concern. This raises the possibility of an accelerating heterosexually transmitted HIV epidemic in Australia, which has to date largely been avoided. This study is the first to use a modified back-projection method to reconstruct the HIV infection curves for selected populations by linking three data sources in the Australian surveillance database. Previously we investigated the Australian HIV epidemic through the development and analysis of a mathematical transmission model [10] which uses a mechanistic framework to combine epidemiological, behavioural, biological and clinical data, and assess how factors interact and together contribute to the HIV incidence in Australian MSM. One advantage of the backprojection analyses used in this study is that they provide a completely independent statistical method for estimating HIV incidence, the results of which can be compared with those obtained using mathematical transmission models. Both the statistical back-projection models and the epidemic mathematical models are based on a number of uncertain, but different, assumptions. The extent to which these very different approaches agree provides some corroboration of the results. The back-projection analyses do have limitations, chiefly in the assumptions required to generate a rate of progression from HIV infection to diagnosis. Although this rate of progression was allowed to vary over time, this was assumed to be in a fairly strictly increasing manner. This assumption is consistent with testing data for MSM in Australia, where the proportion tested each year has increased over time; in the absence of similar data for heterosexuals, this assumption is not unreasonable. Furthermore, although the relationship among newly acquired HIV infection, HIV diagnosis and AIDS diagnosis (until 1987) is to some extent exploited in generating the progression rate distribution, it is not possible for external information, for example rates of HIV testing, to be built into the models using the current formulation. It is also a well-known limitation of back-projection analyses that estimates of recent HIV incidence, the period that is of most interest, are highly uncertain. The methodology used in this study has several advantages over the original back-projection method which was based purely on AIDS data [5]. First, this method utilizes data available from an established national surveillance system and maximizes the available information to estimate the HIV incidence. Secondly, this approach was able to reproduce the historical trend in HIV infection and the results were broadly consistent with the observed pattern of HIV diagnoses in all exposure groups. Publicly available user-friendly software written in the R language and a user manual describing the

7 Increasing HIV transmission in Australia 401 method used in this study are available upon request from the second author. Conclusion In conclusion, these analyses may help to improve understanding of the dynamics of the HIV epidemic, based on high-quality surveillance data, and provide reasonably reliable estimates of the incidence of HIV infection. Our analyses suggest some increase in HIV transmission through male homosexual and heterosexual contact in Australia in the early 2000s, although not through IDU. This suggests that educational messages around safe sex need to be reinforced. Acknowledgements The National Centre in HIV Epidemiology and Clinical Research (NCHECR) is funded by the Australian Government Department of Health and Ageing, and is affiliated with the Faculty of Medicine, University of New South Wales, Sydney, NSW. Its work is overseen by the Ministerial Advisory Committee on AIDS, Sexual Health and Hepatitis. The NCHECR Surveillance Programme is a collaborating unit of the Australian Institute of Health and Welfare. Competing interests The authors have no conflict of interest. Authors contributions Study concept and design: HW and ML. Analysis and interpretation of data: HW, ML and DW. Data extraction: HW, AM and MM. Drafting of the manuscript: HW and ML. Critical revision of the manuscript for important intellectual content: all authors. NSW: National Centre in HIV Epidemiology and Clinical Research (NCHECR), The University of New South Wales, Munoz A, Hoover DR. Role of cohort studies for evaluating AIDS therapies. In: Finkelstein DM, Schoenfeld DA, eds. AIDS Clinical Trials. New York: Wiley, 1995: Cui J, Becker NG. Estimating HIV incidence using dates of both HIV and AIDS diagnoses. Stat Med 2000; 19: Feachem RGA. Valuing the Past... Investing in the Future: Evaluation of the National HIV/AIDS Strategy to , 2nd ed. Downer, ACT: Looking Glass Press for Publications and Design (Public Affairs), Commonwealth Dept. of Human Services and Health, Grulich AE, Kaldor JM. Trends in HIV incidence in homosexual men in developed countries. Sexual Health 2008; 5: Hoare A, Wilson DP, Regan DG, Kaldor J, Law MG. Using mathematical modelling to help explain the differential increase in HIV incidence in New South Wales, Victoria and Queensland: importance of other sexually transmissible infections. Sexual Health 2008; 5: HCVPWG. Estimates and Projections of the Hepatitis C Virus Epidemic in Australia Sydney, NSW: National Centre in HIV Epidemiology and Clinical Research, The University of New South Wales, Razali K, Thein HH, Bell J et al. Modelling the hepatitis C virus epidemic in Australia. Drug Alcohol Depend 2007; 91: Brookmeyer R, Gail MH. Generalized back-calculation: extension to account for nonstationary incubation distributions. In: AIDS Epidemiology: A Quantitative Approach. New York: Oxford University Press, 1994: Becker NG, Watson LF, Carlin JB. A method of non-parametric back-projection and its application to AIDS data. Stat Med 1991; 10: References 1 Brookmeyer R, Gail MH. Minimum size of the acquired immunodeficiency syndrome (AIDS) epidemic in the United States. Lancet 1986; 2: Brookmeyer R, Quinn TC. Estimation of current human immunodeficiency virus incidence rates from a cross-sectional survey using early diagnostic tests. Am J Epidemiol 1995; 141: Janssen RS, Satten GA, Stramer SL et al. New testing strategy to detect early HIV-1 infection for use in incidence estimates and for clinical and prevention purposes. JAMA 1998; 280: Karon JM, Song R, Brookmeyer R, Kaplan EH, Hall HI. Estimating HIV incidence in the United States from HIV/AIDS surveillance data and biomarker HIV test result. Stat Med 2008; 27(23): NCHECR. HIV/AIDS, viral hepatitis and sexually transmissible infections in Australia Annual Surveillance Report. Sydney, Appendix A The approach we used in this study is based on the assumption that all people infected with HIV will eventually be diagnosed with HIV, either close to infection and be reported as having a newly acquired HIV infection, later during chronic HIV infection and be notified as a new HIV diagnosis, or much later during infection at the onset of clinical symptoms (AIDS). This assumption was modelled using the following submodels. Submodel 1: HIV testing during asymptomatic infection It is assumed that a proportion of people infected with HIV will be diagnosed with HIV prior to clinical symptoms or AIDS. A heterogeneous mixed exponential model was used to model the rate at which people in this group are diagnosed with HIV. Each individual in this group was

8 402 H Wand et al. assumed to have a constant testing rate l, corresponding to an exponential model with probability density function (p.d.f.) f a ðxjlþ ¼le lx for a given l. We also assume heterogeneity such that the testing rate l itself varies across individuals. We let l follow a standard exponential distribution with p.d.f., gðlþ ¼e l. This leads to a mixed exponential model R 1 0 f aðxjlþgðlþdl ¼ 1=ðx þ 1Þ 2 which is the p.d.f of the Pareto distribution for the duration X between HIV infection and HIV diagnosis, which essentially steps down over time. Then the corresponding survivor and hazard functions will be: S a ðxþ ¼PrfX > xg ¼ 1 1 þ x and h aðxþ ¼ 1 ð1þ 1 þ x We define the probability of testing x years after infection as follows: f a ðxþ ¼S a ðxþ S a ðx þ 1Þ ¼1=ðx þ 2Þðx þ 1Þ; x ¼ 0; 1; 2;... Submodel 2: HIV testing driven by clinical symptoms at a late stage of HIV disease progression A proportion of HIV diagnoses are assumed to be made at a late stage of HIV infection, essentially as a result of clinical symptoms close to, or at, AIDS diagnosis. For this group, we assumed that the progression from HIV infection to the earliest HIV diagnosis follows a distribution similar to the progression to CD4 counts of o200 cells/ml without any treatment. A Weibull distribution was used, with median time to HIV diagnosis of 6.5 years and shape parameter 2.08 [13] with the following survivor and hazard functions: S b ðxþ ¼expf 0:014x 2:08 g and h b ðxþ ¼0:029x 1:08 ð2þ We define the probability of testing x years after infection as follows: f b ðxþ ¼S b ðxþ S b ðx þ 1Þ ¼ expf 0:014x 2:08 g expf 0:014ðx þ 1Þ 2:08 g; x ¼ 0; 1; 2;... The Weibull distribution has the property that the hazard increases with increasing time from infection, which intuitively would mirror the risk of progression to HIVrelated symptoms in untreated HIV infection. Overall rate of progression to HIV diagnosis The overall rate of progression to HIV diagnosis f ðxjt; jþ was then formulated based on combining the two submodels [i.e. f a (x) and f b (x)] described above by using a mixture distribution model as follows: f ðxjt; jþ ¼Sðxjt; jþ Sðx þ 1jt; jþ where Sðxjt; jþis the survival function: Sðxjt; jþ ¼m t ðjþs a ðxþþ½1 m t ðjþšs b ðxþ ð3þ where m t ðjþ ¼ pedþgðt t 0 Þ ; t t 1þe dþgðt t 0 Þ 0, is a mixing function with j 5 (p,d,g); p represents the proportion of infected individuals who were not tested because of clinical symptoms; d determines the overall shape of the curves and g denotes the rate of increase in infection at time t. When HIV testing became available at t 5 t 0, m t0 ðjþ ¼ ped,asm 1þe d t (j) increases with time t at rate l to a saturation level p ¼ lim t!1 m t ðjþ. We assume that there will be a proportion 1 p of infected individuals who are driven by clinical symptoms to be tested (as specified by submodel 2). The parameter d determines the overall shape of the curves. Prior to the availability of HIV testing in 1985, HIV diagnosis was only made on the basis of AIDS symptoms. This information was incorporated into our model by allowing the model to vary over time, so that the proportion of diagnoses resulting from clinical symptoms would decrease after Therefore, the mixture distribution, f ðxjt; jþ, results in an overall bath-tub shaped hazard, with a relatively high rate of HIV diagnosis in the first year following HIV infection, which then decreases over time, before increasing again as clinical symptoms appear. The two submodels given by (1) and (2) are then mathematically connected based on HIV diagnostic data. For this purpose, we first define the following distribution functions by using (3): Fðtjt; jþ ¼1 Sðtjt; jþ ¼ PrfHIV diagnosed during ðt; t þ tšj HIV infection at time t; jg and rðtjtþ ¼Prfinfection in ½t t; tþjdiagnosis at time tg where Fðtjt; jþ is the distribution of time to testing and r(t t) is the distribution of time since infection. For small t, one has the approximation rðtjtþi 2 ðtþ i 1 ðtþfðtjtþ, implying that the number of individuals who are infected and diagnosed during a short duration of time t around the infection time t. Therefore, the back-calculation predicts ~rðtjt; jþ ¼ ~ i 1 ðt; jþ Fðtjt; jþ: ~i 2 ðt; jþ The data on recent infections (k t ) among newly diagnosed individuals (n t ) were used to identify the parameters in j. As the pair (k t, n t ) follows a binomial distribution, the likelihood function for j can be written as Lðjj~i 1 ðt; jþþ ¼ ~rðtjt; jþ kt nt kt ½1 ~rðtjt; jþš ð4þ The expectation-maximization-smoothing (EMS) algorithm [14] is used to back-calculate the HIV incidence from HIV diagnostic data and determine the final estimate for the

9 Increasing HIV transmission in Australia 403 HIV incidence. For observed values of (k t, n t ), the methodology searches all possible values in the parameter space for j 5 (p, d, g) togeneratethe~rðtjt; jþ that most closely agrees with the observed proportion kt n t. This also implies that, for each combination of (p, d, g) in the parameter space, the following back-calculations are made in order to determine the final incidence, ^iðtþ ¼~i 1 ðt; ^jþ, (i) based on AIDS diagnostic data and (ii) based on HIV diagnostic data by searching the set of parameters ^j until (3) is maximized (details are omitted).