Modelling disease spread through random and regular contacts in clustered populations

Abstract

An epidemic spreading through a network of regular, repeated, contacts behaves differently from one that is spread by random interactions: regular contacts serve to reduce the speed and eventual size of an epidemic. This paper uses a mathematical model to explore the difference between regular and random contacts, considering particularly the effect of clustering within the contact network. In a clustered population random contacts have a much greater impact, allowing infection to reach parts of the network that would otherwise be inaccessible. When all contacts are regular, clustering greatly reduces the spread of infection; this effect is negated by a small number of random contacts.

Introduction

When trying to understand the spread of an infectious disease through a population, many different factors are important. Pathogen genetics, host immune response, host behaviour, and public health interventions all have a role. Here, we focus on the part played by the social behaviour of the host population, specifically considering a human population.

Infections are often transmitted through close contact. Sexually transmitted infections are spread by intimate physical contact whereas common airborne infections such as influenza require less close proximity (Hethcote and Yorke, 1984, Anderson and May, 1992, Wallinga et al., 1999). Vector or water-borne diseases do not generally rely on an interaction between infector and infected but instead tend to be spatially localised (Anderson and May, 1992). In this paper we will consider infections that are transmitted through direct interaction between individuals, and observe that the relevant definition of “interaction” is pathogen-specific.

Many contacts are long lasting. Those who share houses, workspaces, or those who are long-term sexual partners interact repeatedly and over a long period of time (Edmunds et al., 1997, Martin and Yeung, 2006). It is no surprise that much transmission of infectious diseases takes place within the close household, since these individuals interact with high frequency (Eichner, 2003, Ferguson et al., 2006).

Some contacts can be considered as random events: interactions that take place once and only once. Those who sit next to each other on a train, or bump into each other in the street, for example, would fall into this category (Eichner, 2003, Ferguson et al., 2006).

Other contacts are harder to categorise: if an infection lasts for a fortnight then a barber visited each month or an aunt seen every Christmas is a contact that will not be repeated during the infectious period, and is, by this measure, indistinguishable from a random interaction; however, such contacts may have a place in the wider social network—linked through mutual friends, for instance—that gives them more significance than a random event.

For the purposes of this paper, we will categorise contacts as regular or random: regular contacts are those that are seen repeatedly during the course of an infection, random contacts are those that will not be encountered multiple times by an infectious individual. This categorisation of contacts as regular (repeated) or random (one-off) interactions simplifies greatly the reality of human social behaviour, but it allows the tractable exploration of the effect of different types of interaction without the need for large scale data collection to parameterise detailed simulations.

When modelling the spread of an infection, the two distinct types of interaction require different treatment. Regular interactions can be viewed as a contact network (Watts and Strogatz, 1998, Bearman et al., 2004, De et al., 2004, Eames and Keeling, 2004, Eubank et al., 2004, Andre et al., 2006), with individuals appearing as nodes and those individuals who have regular contact with each other being joined by lines. This network representation illustrates very clearly that each individual has, in general, regular contact with only a small subset of the population.

Random, one-off, contacts can be well captured with more familiar mean-field models (Anderson and May, 1992). These models, which have been applied to a huge range of infection scenarios, assume that all individuals in a population are capable of interacting, in contrast to the network paradigm that limits the number of interactions. Mean-field models adhere nicely to the idea of brief, unrepeated, contacts, resulting in the force of infection experienced by a susceptible individual being proportional to the level of infection present in the whole population.

If mean-field and network models with the same contact intensity are compared we would expect infection to spread more slowly within a network. This is because of a local “burning out” of susceptible individuals and is manifested by correlations in infection status emerging between connected individuals—infected individuals tend to be linked to other infected individuals, by whom they were infected or to whom they have transmitted infection; therefore the number of contacts between infected and susceptible individuals is reduced and the spread of infection slowed (Diekmann et al., 1998, Keeling, 1999).

It has been shown that the structure of a contact network has important implications for the spread of infection. Individuals with particularly high numbers of contacts can dominate the dynamics and are key targets for interventions (Kretzschmar et al., 1996, Rothenberg et al., 1998, De et al., 2004, Andre et al., 2006), an effect also shown by high-mixing subgroups in mean-field models (Hethcote and Yorke, 1984, Anderson and May, 1992). It has also been shown that highly clustered networks, those in which two linked individuals are particularly likely to have other contacts in common, result in reduced epidemic spread; in clustered networks infection tends to be confined to highly interconnected cliques and is less likely to emerge into the wider population (Rothenberg et al., 1996; Keeling, 1999; Eubank et al., 2004) although it may be more likely to infect all of a well-connected cluster (Newman, 2003). Clustering makes contact tracing a more effective control measure since it allows multiple routes for intervention efforts to follow to reach infected individuals (Eames and Keeling, 2003, Tsimring and Huerta, 2003, Kiss et al., 2005). Further, by increasing the chance that two infected individuals will compete to infect the same susceptible individual, clustering alters the evolutionary pressures that a pathogen experiences (Boots and Sasaki, 1999, Read and Keeling, 2003, Boots et al., 2004).

In this paper we will examine the impact of regular and random contacts on disease spread, looking particularly at the differences between clustered and unclustered social networks. We will see that disease spreads further and more rapidly when contacts are random, and that the difference between regular and random contacts is most apparent in highly clustered populations.