Modelling sexually transmitted infections: The effect of partnership activity and number of partners on $R_0$

Abstract

We model a sexually transmitted infection in a network population where individuals have different numbers of partners, separated into steady and casual partnerships, where the risk of transmission is higher in steady partnerships. An individual's number of partners of the two types defines its degree, and the degrees in the community specify the degree distribution. For this structured network population a simple model for disease transmission is defined and the basic reproduction number $R_0$ is derived, $R_0$ being a size-biased (i.e. biasing individuals with many partners) average number of new infections caused by individuals during the early stages of the epidemic. First a homosexual population is considered and then a heterosexual population. The heterosexual model is fitted to data from a census survey on sexual activity from the Swedish island of Gotland. The main empirical finding is that, for relevant transmission rates, the effect that so-called superspreaders have on $R_0$ is over-estimated when not admitting for different types of partnerships.

Introduction

In the present paper we investigate, by means of mathematical modelling, the potential spread of a sexually transmitted infection (STI) in a community of interest, with particular focus on heterogeneities in terms of number of sex partners and sexual activity within partnerships.

Similar questions have been addressed elsewhere in the literature using various approaches. A complete survey is beyond the scope of the present paper, but here are some examples. Inspired by Hethcote and Yorke (1984), May and Anderson (1987) acknowledge variations in sexual activity by dividing the community into subgroups, where the subgroup an individual belongs to is defined by the average number of new sex partners per unit of time s/he has. By approximating the initial stages of the outbreak by a set of differential equations they derive an expression for the basic reproduction number, an expression containing the variance of the number of new sex partners. In Diekmann et al. (1991) the forming and separation of partnerships is modelled dynamically in time—partnerships break up and new partnerships are formed with specified intensities. This leads to a Markov-type model and the number of partners during the infectious period will typically follow the Poisson distribution (or a sum of Poisson variables when there is more than one disease state). The initial phase of the disease outbreak is approximated by a suitable branching process assuming a large population and an expression for the basic reproduction number $R_0$ is obtained. A similar model, but where variation in sexual activity within partnerships is acknowledged by distinguishing between “steady” and “casual” partnerships, is analysed by Kretzschmar et al. (1994) and $R_0$ is derived. A somewhat different approach is taken by Diekmann et al. (1998) where each individual has k fixed “acquaintances” chosen uniformly within the community, and contacts between each pair of acquaintances occur independently and randomly in time. For such a model they derive expressions for $R_0$ but also the final size of the epidemic in case a major outbreak occurs. Eames and Keeling (2004) study a model in which each individual has at most one active partnership among a set of fixed potential partnerships but these active partnerships may change over time. The number of sexual partners over a given period can thus be fitted to empirical data. The spread of the disease in the dynamic network of active partnerships is approximated by differential equations relying on a large population and the moment closure method. Newman (2002) studies a general model for infectious diseases where the focus is in modelling social relationships by means of static network models (an approach used also in the present paper). A model for a heterosexually transmitted disease is also presented and briefly analysed. Nordvik and Liljeros (2006) look at an STI-model where the transmission probability depends on the number of sex acts, fits the model to the data reanalysed in the present paper, and derives an expression for the expected number of new infections caused by a randomly selected individual in the community.

In the present paper we study a model which takes into account both individual variation of number of sex partners and variation in sexual activity between partnerships. Empirical evidence shows that the distribution of the number of sexual partners, the degree distribution, is usually heavy-tailed (Colgate et al., 1989, Liljeros et al., 2001). For this reason our model allows an arbitrary degree distribution. Further, in the model we incorporate two types of partnerships, “steady” and “casual” (cf. Kretzschmar et al., 1994), having distinct transmission probabilities. This distinction refers to the number of sex acts over a period corresponding to the length of the infectious period. We distinguish between steady and casual (or high and low transmission risk) partnerships for two reasons. First, the number of sex acts affects the probability of disease transmission in such a way that the more sex acts the higher over-all probability of disease transmission (e.g. Rottingen and Garnett, 2002). Secondly, individuals having many sex partners tend to have fewer sex acts per partner compared to individuals with one or few partners (e.g. Giesecke et al., 1992, Blower and Boe, 1993, Nordvik and Liljeros, 2006).

Assuming a large population we approximate the initial phase of the epidemic by a suitable multitype branching process and derive an expression for the basic reproduction number $R_0$ which determines whether a major outbreak is possible or not. We then neglect the fact that there are steady and casual partnerships, simply treating all partnerships identically, and derive $R_0$ under this assumption. By calibrating parameters in the two models we can compare $R_0$ under the different assumptions. These questions are first addressed for a homosexual community (Section 2) and thereafter for a heterosexual community in which the degree distributions as well as transmission probabilities may differ between sexes (Section 3).

We model the sexual partnerships in the community by a static random network. It would of course be more realistic to allow new partnerships to be formed and old to break up, as is done in some of the papers cited above. However, if in reality new partnerships are formed and old break up in a time-stationary way, then the fixed set of partners of a given individual in the present model can be interpreted as the partners of that individual during a time period corresponding to the typical length of the infectious period in a dynamic network model. With this interpretation the present model can approximate a time-dynamic network model.

In Section 4 the heterosexual model is applied to data on sexual activity collected in the island of Gotland, Sweden (Giesecke et al., 1992, recently reanalysed in Nordvik and Liljeros, 2006). The data are from a representative community sample and contain the number of sex partners of the individuals and also information about the sexual activity in each partnership. $R_0$ is then computed numerically in terms of transmission parameters for two distinctions between steady and casual partnerships, as well as for the case when not distinguishing between partnerships. A comparison is made between the three partnership definitions. The comparison is “fair” because the transmission probabilities of the different partnership definitions are calibrated by computing them as means, within the specified partnership, of a simple per-sex-act transmission model. The main empirical finding from the analysis is that, for relevant transmission probabilities, $R_0$ is higher (and hence over-estimated) when neglecting differences in partnerships as compared to the case when partnerships are separated into steady and casual. The paper is concluded with a discussion on limitations of the present work and important future problems.