# Modelling Based on Australian HIV Notifications Data Suggests Homosexual Age Mixing Is Primarily Assortative

Objective: To understand age-specific sexual mixing patterns of men who have sex with men in Australia, for behavioral insight, and to determine which age groups are at greatest risk of acquiring HIV in the future.

Methods: Diagnoses data based on Australia's HIV surveillance system are analyzed with a mathematical sexual transmission model to estimate the age-mixing patterns of sexual partnerships amongst men who have sex with men. The degree of assortativity and the relative rate of partner acquisition across different age groups are estimated.

Results: Although there are some differences between age groups, sexual mixing is primarily assortative for men of all ages, between completely random partner acquisition and fully assortative mixing, with greatest assortativity among middle-aged men. Men aged 25-34 years were shown to have the greatest level of sexual partner acquisition. There is little evidence to support disassortative mixing on average across the population.

Conclusions: Although there is a tendency for partners of a similar age to be chosen and the average age of HIV-infected people in Australia is relatively old, suggesting that the epidemic may eventually “age out” depending on death rates, there seems to be sufficient cross-age mixing and high sexual mixing in younger age groups to sustain the HIV epidemic over the medium to long term.

From the National Centre in HIV Epidemiology and Clinical Research, University of New South Wales, Sydney, Australia.

Received for publication April 16, 2008; accepted January 29, 2009.

Supported by ARC grant DP0771620. The National Centre in HIV Epidemiology and Clinical Research is funded by the Australian Government Department of Health and Aging and is affiliated with the Faculty of Medicine at the University of New South Wales.

Correspondence to: David P. Wilson, PhD, Level 2, 376 Victoria Street, Darlinghurst, Sydney, New South Wales 2010, Australia (e-mail: dwilson@nchecr.unsw.edu.au).

## INTRODUCTION

New sexual partner acquisition is relatively frequent in the male homosexual population in Australia.^{1} Given the moderately high prevalence of HIV in this population (∼10%)^{2} and that transmission of the virus through unprotected anal intercourse is relatively high,^{3-8} it is important to gain further understanding of the age-specific sexual mixing patterns of this population. Theoretical studies have indicated the epidemiological importance of such mixing for the future course of epidemics.^{9,10} Mixing patterns provide some insight into the age groups at risk of acquiring HIV. In this study, the age-specific contact matrix of partnerships in the Australian homosexual population will be estimated. This will be estimated purely from data of new diagnoses and analysis with a simplified mathematical disease transmission model. This simple methodology may provide easily obtained insight into actual mixing patterns.

Although estimates of the overall rate of sexual partner acquisition are easy to derive from surveys or model calculations, the range and heterogeneity of sexual mixing patterns is less apparent. Various patterns of mixing between groups are possible: from fully assortative [like with like (ie, partners are chosen only from the same age group)] to fully disassortative [like with unlike (ie, partners are chosen only from different age groups)] through random age-independent mixing. Of course there are also other types of mixing patterns, based on other population stratifications, but age-related sexual mixing is addressed in this study.

Since the early 1980s, relatively detailed HIV epidemiological data has been collected in Australia. This data includes when each newly diagnosed case occurred, the age of the patient, the likely route of transmission, CD4 count at diagnosis, and the number of AIDS-related deaths. The HIV diagnosis data, incorporating the age of newly diagnosed patients, by year of diagnosis, will be used in this study^{2} (Fig. 1A for this data since 1998) to estimate the mixing patterns in the male homosexual population over the last 10 years.

## METHODS

### Validity of Diagnoses Data

The average time from infection to diagnosis is very important and it can vary between settings, between groups within the same setting, and it can vary over time within the same group. The assumed time from infection to diagnosis may also greatly affect the estimated transmission rate.^{11} This is particularly important if diagnoses occur in late disease stages.^{12} Therefore, trends in testing rates and time of diagnosis were investigated. In Figure 1B, estimates of the proportion of Australian men who have sex with men (MSM) who tested for HIV in the last 12 months is shown from 1999 to 2007, based on annual reports of behavior among gay men.^{13} The testing rate has not substantially changed over this period. In addition, there is no significant change in the distribution of CD4^{+} T-cell levels of patients at the time of diagnosis over this period (Fig. 1C); similarly, no trends were observed within any age groups (results not shown). This suggests that the time from infection to diagnosis has not changed over time. It is thought that high HIV testing rates were attained by the mid-1990s, and these rates have been maintained over the last decade. Therefore, the trend in the number of diagnosed HIV infections is thought to be relatively well representative of incidence, and changes in testing patterns do not need to be taken into consideration for the purposes of investigating sexual mixing.

### Mathematical Model

In this study, the number of newly diagnosed HIV infections in the year “*t*” of age group “*a*” is denoted symbolically by Λ(*a*, *t*). This variable definition has direct data correspondence for every age and year (Fig. 1A). By matching this dataset to its equivalent mathematical expression from a standard transmission model, the mathematical system will be solved to estimate the age-dependent contact mixing that likely led to this observed data. The number of new infections per year, Λ(*a*, *t*), is referred to as the force of infection in disease transmission modeling. The force of infection, per susceptible person, is typically represented in sexually transmitted models as the product of the number of sexual partnerships per unit time (represented by the symbol, *c*), the probability that a new partner will be infected with the disease (the ratio of number of infected people, *I*, to the total number of people, N), and the probability of disease transmission per serodiscordant partnership per unit time (*β*). In mathematical symbols, the force of infection is

and the total number of new infections per unit time is

where *S* is the total number of susceptible people. This expression can be translated into the discrete year and age-specific number of new infections, namely:

where *A* is the number of age groups (taken to be 10; Fig. 1A), *c*(*a*,*a*′) is a matrix of *A* × *A* parameters representing the average number of sexual partners people in age group *a* have with people in age group *a*′ per year, and *S*(*a*,*t*), *I*(*a*,*t*), and N(*a*,*t*) are the number of susceptible, infected, and total number of people in age group *a* at the end of year *t*, respectively; N(*a*,*t*) is defined to be the sum of the susceptible and the infected people, N(*a*,*t*) = *S*(*a*,*t*) + *I*(*a*,*t*). This model has provision for each age group to have a different average rate of sexual partner acquisition; that is, younger men may have different numbers of sexual partners per year compared with older men. But heterogeneity in sexual behavior within a single age group is not considered; like most compartmental ordinary differential equation models, here an average partner change is assumed within each age group. Similarly, although some MSM do not have sex with other men until they are relatively old, in this model, the simplifying assumption is made that that size of the homosexual population in each age group is approximately constant, N(*a*,*t*) = N_{0}. Then, the modeled number of new infections can be expressed as:

For conservation of partnerships, the total number of partnerships that people of age *a* have with people of age *a′* must be equivalent to the total number of partnerships that people of age *a′* have with people of age *a*. That is, *c*(*a*,*a*′) N(*a*,*t* − 1) = *c*(*a*′,*a*) N(*a*′,*t* − 1) or *c*(*a*,*a*′) = *c*(*a*′,*a*). This reduces the number of unknown contact partnership variables, from *A*^{2} to *A* (*A* + 1)/2 (ie, from 100 to 55 for *A* = 10 age groups); the unknown variables are:

These variables denote the different average rates of partnership acquisition for men in each age group as they mix with men of the same or different age groups. The total number of HIV-infected men in age group *a*, at time *t*, *I*(*a*,*t*), is estimated from the diagnoses data by summing the number of new diagnoses, aging 20% of the population of each 5-year age into the next age group, and removing the number of age-specific HIV-related deaths reported each year. That is:

where Deaths (*a*,*t*) is the reported data of the number of deaths in year “*t*” and age “*a*”. It must be noted that this is an underestimated approximation to the actual number of infections because not all infections are diagnosed soon after infection. However, Australian homosexual men are known to test regularly for HIV and other sexually transmitted infections, and it is thought that ∼90% of all HIV infections are diagnosed.^{14} Therefore, the estimated number of HIV infections used in this model, based on the diagnosis data, should adequately reflect the true number of HIV-infected men for this simplified analysis.

### Determining the Age-Mixing Partnership Matrix

To determine the partnership-mixing matrix, *c*(*a*,*a*′), for the average rate of sexual partner acquisition, the system of equations described by (3) must be solved simultaneously for the unknown variables. Because mixing of partnerships are being calculated over a 10-year period and for each year we have 10 age groups, there are 100 equations and 55 unknowns, an overdetermined system. This constrained [*c*(*a*,*a*′) ≥ 0] optimization problem was solved numerically with the “fmincon” command in Matlab (version 7.4.0; Mathworks, Natick, MA). The parameters *β* and *N*_{0} were estimated from available literature; uncertainty analyses of these parameters on influencing the resultant contact matrix was performed with Latin Hypercube Sampling^{15} (1000 simulations in total) over reasonable ranges [0.001 < *β <* 0.3^{3,4,6-8}; 7500 < *N*_{0} < 15,000 per 5-year age group, and there are 10 age groups (this specifies ∼1%-2% of the male adult population as homosexual^{16})]. The solution for the partnership-mixing matrix, *c*(*a*,*a*′), estimated the average number of new partners that men of age group *a* form with men who are in age group *a*′. This solution is used to estimate the different relative sexual partner acquisition rates between age groups and to estimate the degree of assortative, random, and disassortative mixing in the population.

## RESULTS

Fully assortative mixing is represented by nonzero values on the diagonal of the mixing matrix [*c*(*a*,*a*) > 0 and *c*(*a*,*a*′) = 0 for all *a* ≠ *a*′], and in contrast, fully disassortative mixing must have zeros on all diagonal elements of the mixing matrix [*c*(*a*,*a*) = 0 for all *a*]. To compare and illustrate mixing patterns, the mixing matrices should be normalized to represent the probability of partnerships forming in each group. It should be noted that disassortative mixing is not unique and any combination of row elements that sum to 1, but where the diagonal elements are zero, represents disassortative mixing, whereas assortative mixing would result in the normalized mixing matrix equal to the identity matrix. In Figure 2A, the average (over all 1000 simulations) proportion of partners that people from a given age group have with people from any other age group is illustrated. It can be seen that the greatest proportion of partnerships are with men who are of the same age group (darkest boxes in Fig. 2A appear on the diagonal). For most age groups, the majority of partnerships are with men of the same age group, and then other partnerships are formed decreasingly with men who are of older ages (Fig. 2A). However, the exception to this seems to be for older men (age greater than 60 years); the model-based results suggest that older men tend to form partnerships with men of all ages. But the greatest assortativity is seen for middle-aged men (in their 30s and 40s) (Fig. 2A). Thus, the degree of assortative, random, and disassortative mixing seems to change with age.

The overall degree of mixing in the population can be evaluated by various methods; here we use both the Garnett and Anderson^{17-19} and the Gupta et al^{9} methods. Based on the Australian diagnoses data and the algorithm described above, the partnership mixing matrix was determined for each simulation (1000 simulations in total) and the corresponding Garnett and Anderson^{18,19} and Gupta^{9} measures of the degree of assortative-disassortative mixing were calculated. These measures estimate an overall degree of mixing across all age groups. The results from these simulations are shown in the inset of Figure 2A; the 2 measures provided very similar values, of 0.27 [interquartile range (IQR) 0.22-0.35] for the Garnett and Anderson index and 0.29 (IQR 0.21-0.37) for the Gupta index. It can be seen from the uncertainty analysis over the 1000 simulations that the range of uncertainty is relatively small and mixing is between random and assortative (Fig. 2A inset). Mixing is much more inclined to be assortative than disassortative, indicating that partnership formation is typically within the same age group. Degree of assortative mixing was estimated as ∼0.3. In this special case, where there are 10 age groups and each contributes the same number of partnerships, this could be interpreted as approximately 30% of new partners, for each individual are chosen from the same age group. Although considerable random cross-age mixing occurs in the Australian homosexual population, the tendency is for partners to be chosen from the same age group. The results are also robust to model parameters as there was not a significant trend in the association between the measured indices and the parameter values (small partial rank correlation coefficients of 0.09 for *β* and −0.05 for *N*_{0}). This suggests that although simplifying assumptions used in this model are extreme caricatures of reality, the calculated values are an appropriate and reliable first-order approximation and relatively independent of other detailed factors that could influence the system.

In addition to estimating the degree of assortativity, the model solution was used to compare the average rates of sexual partner acquisition between each age group. A summary (from the 1000 model simulations) of the relative average number of new partners that men in each age group have per year, compared with men of age 15-19 years as a reference, is shown in Figure 2B. The rate of partner change is calculated to increase with age up to a peak for the 30- to 34-year old age group at 6.5 (median, 2.2-22.9 IQR) times the number of partners per year compared with the number of partners per year for men aged 15-19 years (Fig. 2B). The rate of partner change then decreases with age for men older than ∼35 years (Fig. 2B). This age profile of the rate of sexual partner acquisition is highly consistent with mixing of MSM in other settings around the world (eg, ^{20-22}). Although there are some differences in assortative mixing patterns (Fig. 2A) and the overall rate of partner exchange (Fig. 2B) between age groups, these results suggest robustness in the conclusion that age mixing generally tends to be assortative with respect to age.

The HIV incidence in the future, based on current age-mixing patterns, was forecasted over the next 20 years using the current model to simulate HIV transmission among Australian MSM (Fig. 2C). The death rate among the HIV-infected population in Australia, and around the world, has changed substantially since the introduction of highly effective antiretroviral therapy. It could also be assumed that the death rate will continue to decrease in the future. To forecast epidemic trajectories, 3 different death rates were simulated: 1%, 2%, and 5% per year. For each of these scenarios, the median, IQR, and 95% limits of the assortativity indices (Fig. 2A inset) were used (Fig. 2C). It was determined that the variability in the degree of assortativity calculated by the model did not highly influence the variability in the projected incidence, but the death rates were important (Fig. 2C). However, even with a very high death rate (5%), the epidemic is still sustained at relatively high levels over 20 years, albeit with decreasing incidence.

## DISCUSSION

The degree of assortative mixing has important implications on the HIV epidemic. Understanding age-specific sexual mixing patterns provides insight into behavior and assists in determining which age groups are at greatest risk of acquiring HIV. It is particularly relevant to the aging nature of Australia's HIV epidemic (which is similar to other settings, including the United Kingdom^{23}). The average age of HIV-infected individuals in the homosexual population in Australia is increasing at a rate of approximately 0.6 years of age per year (generated from the current model) and the average age of persons living with HIV infection is currently approximately 47 years. The current study estimated that MSM in Australia primarily choose partners who are of similar age, and this is especially true for men in their 30s and 40s. This may explain why the average age at diagnosis is increasing and the aging nature of the epidemic. However, it was also estimated that there is sufficient mixing across age groups to sustain significant HIV incidence in all age groups. Because the mixing is not purely assortative, older HIV-infected men will transmit infection to younger men (on average), and thus the “wave” of infections aging out of the population will leave a trail behind of HIV infections in the younger age groups. Furthermore, 20- to 44-year olds (and particularly 25- to 35-year olds) were found to have the greatest sexual mixing rates (Fig. 2B). This is consistent with what is intuitively expected and similar to the age profile of sexual mixing of MSM seen in other settings. If the age profile of sexual partner acquisition rates remains unchanged over time, then the aging nature of the epidemic in Australia could be expected to saturate in the future such that the average age at diagnosis will remain in this highest activity age group.

Currently there are no detailed data on the degree of assortative mixing in the Australian MSM population. It is recommended that empirical studies be performed to confirm or correct the estimates provided here of the degree of assortative sexual mixing patterns in Australian MSM. It has been estimated that overall mixing is primarily assortative but it is between random and fully assortative. However, the degree of assortativity changes with age with more random mixing in the older age groups and also in younger age groups, with greater assortativity in the middle-aged groups. Confirming these findings would be important social research to provide greater understanding of sexual mixing and to inform public health campaigns to prevent HIV infections in the future.

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**Keywords:**

assortativity; Australia; HIV; men who have sex with men; sexual mixing