The effect of changing sexual activity on HIV prevalence

Abstract

In a one-sex preferred mixing model, reductions in the rate of partner change by those with low sexual activity increase the average probability of HIV infection in the remaining pool of available partners. This increases prevalence among people with high activity, and since high activity people disproportionately influence the spread of HIV, may increase long-run prevalence in the population as a whole. Calculations using the model and survey data on sexual activity indicate that in low prevalence populations, many people have low enough activity that reductions in their activity might increase the endemic steady-state prevalence. If these results prove robust in more realistic models, they would support the case for targeting public health messages urging reduced sexual activity to high activity people.

Introduction

This paper shows that, in a simple single-sex preferred mixing model, increases in the frequency of partner change by low-activity people may reduce long-run HIV prevalence. To see the intuition, suppose that 9 partners per year were required for HIV to be endemic in a homogeneous population. Consider a population in which a small minority had 10 partners a year, and the majority had no partners at all. In this population, the disease would persist among the active minority. Suppose that the inactive majority decided to have 1 partner each over their lifetimes, that the groups mixed randomly, and that the proportions of the two groups were such that on average the high-activity people might have 5 partnerships a year with the low-activity people and 5 a year with other high-activity people. In this case, the disease would die out, because half the new infections would occur among low-activity people who would not infect others.

To see the intuition slightly more mathematically, recall that, under a simple random mixing model, the disease dies out or is endemic according to whether the threshold function, R₀, is greater or less than 1, where

$$\text{R}{}_0\text{=}\text{β}\text{δ}\text{μ+}\text{σ}{}^2\text{μ}\text{;}$$

μ and σ are the mean and standard deviation of the rate of partner change, and β and δ are the birth and death rates [1].

This can be rewritten as

$$\text{R}{}_0\text{=}\text{β}\text{δ}\text{Σα}{}_{\mathrm{k}}\text{i}{}_{\mathrm{k}}{}^2\text{Σα}{}_{\mathrm{k}}\text{i}{}_{\mathrm{k}}\text{,}$$

where there are N groups in the population classified by the number of partners per year, i_k, and each group represents a portion α_k of the population. Differentiating with respect to i_k shows that small increases in activity by individuals with activity less than $\text{1}\text{2}\text{μ+σ}{}^2\text{/μ}$ will reduce R₀. Consider a population on a knife-edge between endemic HIV and the disease dying out, so that R₀=1. If members of the population with activity less than $\text{1}\text{2}\text{μ+σ}{}^2\text{/μ}$ increase activity, R₀ will decrease, and the disease will die out. In fact, a uniform small increase in activity by the entire population will reduce R₀ if σ>μ [2].

It turns out that these effects on the stability of the endemic steady-state are far too sensitive to the assumption of homogeneous mixing among sexual partners to be of empirical importance. As we show in this paper, however, similar effects may apply to the level of steady-state prevalence, even if the disease remains endemic: namely, an increase in activity by low-activity members of the population may decrease prevalence in the long run. These effects on prevalence are robust enough with respect to preferred mixing among sexual partners that they may be of some practical significance.

In particular, crude calibration of a simple single-sex model using survey data on sexual activity suggests that these counter-intuitive effects may be more than just a theoretical curiosity. We develop expressions for a cut-off level of sexual activity such that increases in activity in the groups below this level will reduce steady-state prevalence. More than 80% of the population in a comprehensive study of sexual activity in the UK had low enough activity that reductions in activity would increase steady-state prevalence in a standard single-sex susceptible-infected (SI) epidemiological model with preferred mixing. Simulations using this simple model suggest that if everybody who had 1 partner every 5 years reduced their frequency of partner change by 5%, steady-state prevalence would increase by 7% under random mixing. However, the simulations also indicate that for reasonable parameter values, increases in activity would not lead to the eradication of the disease.

There are several important caveats. First, reductions in activity by low-activity people could only increase steady-state prevalence in populations that have low steady-state prevalence given current activity levels and transmission probabilities. The counterintuitive effects discussed in this paper thus may be relevant for heterosexuals in developed countries, but not for homosexuals, heterosexuals in the highest-prevalence areas of Africa, or IV drug users. Second, increases in activity by low-activity people will increase prevalence temporarily. Third, we consider the effects of changes in activity by one group while keeping the activity levels of all other groups constant. The conclusions in this paper may be weakened if reductions in activity by low-activity people make it harder for high-activity people to find partners, and they consequently reduce their own activity. More generally, since the model abstracts from important features of the epidemic (for example, our model does not allow concurrent partnerships or different sexes), the results should be considered provisional.

To the extent that the results of this paper prove robust in more realistic models, though, they reinforce arguments for targeting public health messages urging reductions in the frequency of partner change to high-activity people. Because anyone increasing activity will, at least in the short term, increase his or her risk, we would absolutely never recommend public health messages designed to increase activity by lower activity groups: people have a reasonable right to expect that public health policy will not act directly to increase their individual risk, even for the sake of a long-term reduction in prevalence in the population as a whole.

A large literature examines the dynamics of sexually transmitted diseases under a variety of mixing patterns, and considers the effect of changes in activity 1, 3, 4, 5, 6, 7, 8, 9, 10. Both the present study and that of Kremer 2, 11 follow independent work by Whitaker and Rentin [12]. They show that in a two group example with random mixing, an increase in activity by the low-activity group may reduce steady state prevalence. They, however, explicitly disclaim empirical relevance of its model for AIDS. Our analysis differs from Whitaker and Rentin [12] in that it extends the results to a preferred mixing model with an arbitrary distribution of activity, and uses empirical data to show that these effects may be important in low-prevalence populations, but not in high-prevalence populations.

Kremer [11] analyzes the externalities from sexual activity using the economic concept of asymmetric information. This paper presents the results in purely epidemiological terms. It also further develops the theory and mathematical methods of Ref. [11] using a different approach to determine the cut-off levels of activity, and extending the theory to models with preferred mixing. Kremer [2] considers how the argument in this paper is modified if high-activity people change their activity in response to changes in activity by low-activity people.

The paper is organized as follows: In Section 2, we define the model we use, and cite some results concerning its stability and the endemic steady state. In Section 3, we solve for the cut-off levels of activity below which increasing activity reduces steady-state prevalence among other members of the population, or prevalence in the population as a whole. In Section 4, we examine the special case of random mixing. In Section 5, we calculate the cut-off values of activity under a preferred mixing model using data on the frequency of partner change among heterosexuals in the UK. Section 6discusses the time-path of prevalence in response to reductions in the rate of partner change. In Section 7, we discuss directions for future research and possible policy implications.