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Modelling of Contact Structure and the Spread of Infectious Diseases

A report by Mirjam Kretzschmar (RIVM). Organizers: Dr. M. Kretzschmar (RIVM), Dr. M.C.M. de Jong (ID-DLO).

On September 24, 1998, a workshop took place in Bilthoven, The Netherlands, about "Modelling of Contact Structure and the Spread of Infectious Diseases". The aim of the workshop was to bring together researchers who have worked on different approaches in modelling contact patterns, epidemiologists who have studied actual contact patterns in specific populations, and modellers who are applying epidemic models to public health and veterinary questions. The speakers were Lisa Sattenspiel (University of Missouri, Columbia, USA), Minus van Baalen (University of Amsterdam & Universite M. et P. Curie, Paris), Gianpaolo Scalia-Tomba (Universite La Sapienza, Rome, Italy), John Edmunds (University of Warwick, United Kingdom), Mart de Jong (ID-DLO, Lelystad), Nico Nagelkerke (RIVM, Bilthoven & KNCV, The Hague), and Tom Snijders (University of Groningen). In the first lecture Lisa Sattenspiel gave an account of the spread of influenza between settlements in Canada in 1918, and showed the importance of understanding the social structure and contact patterns between those settlements for explaining the observed mortality patterns. Next, Minus van Baalen talked about the application of the pair approximation approach in modelling the invasion of new strains of a pathogen depending on the contact structure of the population. His simulation results showed that in a population where every individual only contacts a small number of neighbors, it is possible that a less virulent strain of a pathogen may invade and establish itself locally, whereas globally a more virulent strain is dominant. Gianpaolo Scalia-Tomba succeeded in intuitively explaining some mathematically intricate results about the spread on an infectious disease in a population with two levels of contacts: local contacts that take place in small groups such as households, and global contacts with other parts of the population. John Edmunds then gave us some insight into the contact patterns of the Warwick student population with special reference to his own research group. He showed results of an empirical study about contact patterns pertaining to contacts of varying intensity, ranging from "having a two-way conversation" to "having sex". Then Mart de Jong talked about the interpretation of different threshold quantities and showed how the basic reproduction ratio R0 depends on the average number of aquaintances in a contact network. Nico Nagelkerke demonstrated how molecular epidemiology can contribute to the study of tuberculosis transmission. From the clustering of DNA-fingerprints of TB and their relationship to different ethnic groups in the Netherlands he drew conclusions about the contact patterns among and between those groups. Finally, Tom Snijders gave an introduction into the main concepts of social networks and discussed the possible applicability of those concepts in modelling transmission dynamics of infectious diseases in networks. He also talked about some statistical approaches in connection with the sampling of networks.

The workshop was continued on September 25, 1998, in a small group consisting mainly of the speakers of the day before. In an open discussion several of the problems that had been mentioned in their talks on the day before, were discussed in more detail. Although the background of the participants of this discussion group was very diverse, the discussion was fruitful for all. In total, the workshop was perceived as stimulating and as an inspiration for future work.


Lisa Sattenspiel University of Missouri, Columbia (USA)
Furs, flus, and feather beds: or why Canadian fur-trappers should stay in bed when they have the flu
In recent years, many researchers in mathematical epidemiology have developed models that incorporate realistic contact structures to allow for nonrandom interactions among populations separated in time, space, or social behaviors. I describe here such a model for the geographic spread of infectious diseases, and, more importantly, describe how the model is actually being used to study the impact of the 1918-19 influenza epidemic among three communities of native Canadian fur trappers. This application is made possible because of an unusually rich data base of information on both the mortality rates from this epidemic and on the daily mobility patterns of the fur trappers throughout the region during the time of the epidemic. Computer simulations in combination with ethnohistoric analysis of parish and Hudson's Bay Company post records have been used to assess the importance of trapper mobility on disease spread, the possible impact of quarantine practices during the epidemic, and the potential effects of seasonality on the spread of influenza.

Minus van Baalen Universiteit van Amsterdam & Universite Pierre et Marie Curie, Paris (F).
Contact networks and the evolution of virulence: Implications for virulence management
Some parasites require close contact between their hosts for transmission. If the host population is `well-mixed' standard epidemiological methods can be used to study their dynamics and evolution. If, however, the host population is socially structured other methods are necessary. When parasites can transmit themselves to their host's immediate neighbours, the parasite population becomes `viscous': even if there is no large-scale subdivision in the host population, the parasite cannot spread immediately throughout the entire population. To what extent are parasites adapted to their host's social structure? How are parasites expected to respond when this contact structure changes? To address these questions, I will assume that the host population has a fixed contact network in which every host is socially connected to a fixed number of hosts. Using cellular automaton models and the correlation dynamics approach I will show that in a simple SIR model, less virulent parasites may be favoured in `viscous' than in `well-mixed' host populations. When a mutant parasite arises, it will form a local focus of infection (cluster). A cluster of less virulent parasites can invade and replace more virulent strain if it spreads locally more effectively, i.e. if the cluster can grow. In the particular model that I studied, based on the assumption that more infectious (virulent) parasites induced a longer period of immunity, virulent parasites tend to be surrounded by immune hosts, whereas avirulent parasites benefit from having more susceptibles in their neighbourhood. Effectively, this is an instance of kin selection among parasites across hosts. Any change in the host population that disrupts the pattern of relatedness among the parasites is therefore expected to favour more virulent strains.

Gianpaolo Scalia-Tomba Universita La Sapienza, Rome (IT)
Final size of epidemics in structured populations
Recently, many theoretical efforts to understand the effects of mixing heterogeneity on epidemic spread in a population have been made (see e.g. Andersson 1998). The relation between heterogeneity caused by two-level mixing, i.e. high infection probability within a small group (e.g. a family) and small infection probabilities with other individuals in the population, and the probability and size of a large epidemic has been studied by Ball, Mollison & Scalia Tomba (1997). While the mathematical machinery used in the proofs of results is slightly complicated and heavy, the main intuitions can still be recovered by heuristic considerations about the outgoing and incoming contact graphs (cf. Barbour & Mollison 1990) and about the initial imbedded renewal structure (cf. Heesterbeek 1992). However, these methods yield little information about the real time behaviour of the epidemic. Furthermore, the usual deterministic approach to the problem runs into difficulties because of the essentially discrete nature of the small groups composing the population. Further work will be needed to clarify these issues.

Andersson H. (1998) Epidemic models on graphs and lattices: a short survey. Research report no. 204, Inst. Actuarial math. and math. stat., Stockholm University, Sweden.
Ball F., Mollison D., Scalia Tomba G. (1997) Epidemics with two levels of mixing. AAP 7, 46-89.
Barbour A.D., Mollison D. (1990) Epidemics and random graphs, in Stochastic processes in epidemic theory, Gabriel J.P, Lefevre C., Picard P., Eds., Lecture Notes In Biomathematics 86, 86-89.
Heesterbeek J.A.P (1992) R0, Ph.D. thesis, CWI, Amsterdam.

W.J. Edmunds Department of Biological Sciences University of Warwick Coventry, UK.
Who mixes with whom? Measuring contact patterns for air-borne infections
The rates and patterns of host mixing are critical determinants of the transmission dynamics of infectious diseases. For microparasitic human diseases spread by close contact (e.g. measles, rubella and influenza) there have been remarkably few attempts to directly quantify the relevant patterns of mixing. Instead, analysts tend to make (strong) assumptions regarding mixing patterns then indirectly estimate the mixing parameter(s) from epidemiological data. In contrast, the existence of clearly defined and well understood at-risk events for sexually transmitted diseases (a sex act and/or a new sexual partnership) have allowed researchers in this field to obtain estimates of relevant mixing patterns directly from sexual behaviour surveys or from contact tracing. We have applied these same survey techniques to estimate mixing patterns for close- contact infections by defining simple at-risk events for these infections and asking individuals to record these events in diaries. For example, we assume one-to-one conversations to be sufficient for the transmission of some aerosol infections, while skin-to-skin physical contact is required for others. Thus, at-risk events are often hierarchically arranged. For instance, conversation is not an at-risk event for an STD but sexual contact is an at-risk event for close contact infections. The data provided by these surveys have allowed us to tentatively estimate the mean and variability in the numbers of contacts that individuals make for a range of possible infections. Since we record information on the study participants and ask them to note the characteristics of their contacts (age, sex, where the contact occurred etc) we have been able to estimate how individuals mix within and between age groups and across different social settings (home versus work etc). In addition, by asking individuals to identify their contacts (by name) and sampling individuals through time we are beginning to be able to identify the structure and dynamics of social networks through which close-contact infections spread. Though still in the pilot stage, we believe these techniques have wide applicability and may lead to significant improvements in parameter estimation, model formulation and epidemiological prediction.

Nico Nagelkerke National Institute of Public Health and the Environment, Bilthoven
Martien Borgdorff Royal Netherlands Tuberculosis Association, The Hague
The role of the population structure of The Netherlands in TB transmission
Population structure, in terms of age, gender, ethnicity and behaviour plays an important role in the transmission dynamics of many infectious diseases. Unfortunately, empirical epidemiology is complicated by the difficulty or impossibility to establish who infects whom. Molecular fingerprinting now provides new opportunities to explore transmission, by identifying "clusters" of individuals linked by (recent) transmission. Since 1993 molecular fingerprinting, using IS6110, has been carried out on all diagnosed cased of tuberculosis in The Netherlands. We will show two applications of this data: 1. Clusters of two patients of Dutch nationality are analysed to establish the role of (similarity in) age in TB transmission. 2. Using incidence on unclustered patients, a probabilistic approach to identifying "sources" of "clusters" was developed. Using this approach, we estimated the contribution of different nationalities to the Dutch TB burden.

Tom Snijders Department of Statistics, Measurement Theory, and Information Technology, University of Groningen
An introduction to social networks and their relevance to diffusion of disease
Patterns of contacts in groups of individuals have received much attention in sociology. This has led to the development of the field of "Social Networks" since the 1970s. Social networks appear to be an important bridge between individual behavior and macro-level consequences. In this presentation some concepts and methodological approaches of social network analysis will be presented, focusing on the role of social networks in research on diffusion.

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