Network models of epidemics

Level 2 module

Basic problem

Simple mathematical models of epidemics assume random mixing between individuals, which is clearly not the case in reality. Populations have structure, and individuals tend to interact with a very limited subset of the population. Contacts or links between the individuals constitute a network, and infectious diseases that propagate by direct contact can only spread along the paths of these networks. In this module, we implement explicit models of contact and transmission networks to explore the effects of network structure on epidemic spreading.

This is one of the more challenging modules.

General approach

We will build simulation models to track the network of contacts within a population and the spread of a disease over the network. We will explore alternative network structures (random, clustered, small-world, scale-free), dynamic networks with changing links, and how epidemics spread or can be prevented over the various types of networks.

What can be learned

Concepts:

• Individual-based models
• Simulation and analysis of networks (graphs)

Methods:

• Stochastic individual-based simulation with discrete time steps
• Construction and analysis of graphs

Starting point
Download the reader and the starting script for the model. You are also advised to check out the related modules SIR models of epidemics and Stochastic simulation of epidemics for some comparison and background information. Install the igraph package in R.

Interesting questions that you can investigate

• What is the effect of network structure on the spreading of epidemics, compared with random mixing? Are there differences according to network type?
• Can we define a basic reproductive ratio (cf. SIR models of epidemics) in a network model?
• How "determined" or stochastic is the outcome of a simulated epidemic? Does it depend on the structure of the network?

Advanced questions:

• How can we implement a dynamic network (with removal and entry of new nodes and edges)?
• How efficient can targeted intervention (vaccination or treatment) be over the various network types?
• What are (if any) the special properties of sexual contact networks?
• What factors influence the evolution of virulence? Does the structure of the network matter?

Glossary

• Individual-based model (also known as 'agent-based model'): a simulation model which tracks distinct individuals or "agents" explicitly. Such models often exhibit emergent complexity, which means that the interaction of individuals (that follow simple rules) results in complex behaviour at the level of the whole system.
• Graph or network: an abstract representation of a set of objects where some pairs of the objects are connected by links. In graph theory, the objects can be called nodes or vertices; the links can also be called edges. In epidemiology, you can simply refer to individuals and their contacts. Further graph-related terms are defined in the reader.

Literature & Weblinks

• Keeling MJ & Eames KT (2005). Networks and epidemic models. J. R. Soc. Interface 2, 295-307. (This is a very nice review of the field.)
• Liljeros F et al. (2001). The web of human sexual contacts. Nature 411, 907-908. (A brief introduction to sexual networks.)
• Gómez-Gardenes J et al. (2008). Spreading of heterosexually transmitted diseases in heterosexual populations. Proc. Natl. Acad. Sci. USA 105, 1399-1404. (With a lot of data on sexual networks.)
• Szendrői B & Csányi G (2004). Polynomial epidemics and clustering in contact networks. Proc. R. Soc. Lond. B 271, S364-S366. (On the general effect of network structure on epidemic spreading.)
• Eubank S et al. (2004). Modelling disease outbreaks in realistic urban social networks. Nature 429, 180-184. (A tour de force of predicting epidemic spreading in huge populations.)
• Halloran ME et al. (2002). Containing bioterrorist smallpox. Science 298, 1428-1432. (To show that such models can advise policy.)
• Barabási A-L & Albert R. (1999). Emergence of scaling in random networks. Science 286, 509-512. (A seminal paper on scale-free networks.)
• de Sousa JD et al. (2010). High GUD incidence in the early 20th century created a particularly permissive time window for the origin and initial spread of epidemic HIV strains. PLoS ONE 5(4), e9936. (with an R model and parameters on HIV epidemics in Africa)
• Graphs and Graph Theory on Wikipedia.
• Homepage of the igraph package.