DIMACS Center, CoRE Building, Rutgers University

**Organizers:****Valerie Isham**, University College, London, valerie@stats.ucl.ac.uk**Alun Lloyd**, Institute for Advanced Study, alun@alunlloyd.com**Bryan Grenfell**, Cambridge University**Matthew Keeling**, Cambridge**Denis Mollison**, Heriot-Watt University

This working group will study issues involved in mathematical modeling of the spread of infectious diseases among human, animal, and plant populations, using models with explicit time and space parameters and models based on networks of individuals. We will investigate the use of novel computational methods, emphasizing the use of mathematical models to gain understanding of transmission mechanisms and to determine and evaluate control strategies. The group will explore a variety of computer-intensive stochastic spatio-temporal models, including compartment models that divide the population into locally well-mixed populations, continuum models that assume individuals are located at points in a continuous plane, and network models that locate individuals at the vertices of a graph (see, e.g., [Dieckmann, Law and Metz, (eds.) (2000), McGlade, (ed.) (1999)]). We will review connections with current research on interacting particle systems and probabilistic cellular automata (see, e.g., [Levin and Durrett (1997), Mollison (ed.) (1995), Rand Keeling and Wilson (1995)]) and examine statistical modeling issues arising in the collection and analysis of spatial data, such as in the use of Markov chain Monte Carlo methods to fit models to data (see, e.g., [Gilks Richardson and Spiegelhalter (1995), O'Neill, Balding, Becker, Eerola and Mollison (2000)]). We will develop computational models of spread of disease on networks (e.g., work on STD's and AIDS in [Altmann, M. (1995), Altmann, Wee, Willard, Peterson and Gatewood (1994), Peterson, Willard, Altmann, Gatewood and Davidson (1990)]) and analytical models for epidemics on networks, which are in their infancy (see, e.g., [Altmann (1995), Ball, Mollison, and Scalia-Tomba (1997)]). These models provide major computational challenges in dealing with the complexity of networks [Kretzschmar and Morris (1996), Levin, Grenfell, Hastings and Perelson (1997), Morris and Dean (1994)]. As computers become more powerful, we are able to simulate more and more complex biological interactions. This is especially true of stochastic processes in spatial environments or on large networks. Recently, techniques such as pairwise approximations and moment closure techniques have been used to mimic the effects of space and stochasticity with remarkably good results [Keeling (2000)]. Many of these techniques have been developed in isolation, and essentially none with the aid of computer scientists. One of the aims of the working group will be to connect the many researchers in this field with computer scientists, with the hope of producing a coherent framework. The group will also investigate a suite of models for spatially-extended dynamics that goes across heterogeneous scales, includes periodic forcing (due to seasonal change), involves temporally disturbed environments (due to abrupt changes in the presence of favorable periods for transmission), and deals with changes in host characteristics with age. Among the specific research challenges facing this working group are: synthesizing available epidemiological data and using it to validate and parametrize models; learning how models, particularly stochastic models, scale from small populations to large ones; and applying models to policy questions about endemic and emerging diseases. As noted in the discussion of the Data Mining working group, data about important diseases usually comes in many different forms, from spatially extensive data with few details, such as mortality records or livestock destruction reports, to detailed information from specific places and times. Combining widely varying sources of information is a challenging area requiring both theoretical and computational expertise. Examples of diseases for which large amounts of spatio-temporal data of varying kinds are available include tuberculosis, malaria and HIV, as well as newly emerging or re-emerging diseases illustrated by the recent outbreak of foot and mouth disease in Great Britain, the Four Corners outbreak of Hantavirus, and many other human, animal and plant diseases (see, e.g., [Keeling and Gilligan (2000)]). We will explore analytic and simulation techniques for integrating such data from different sources and these should be valuable in assessing the effects of past interventions and thus in refining models used to answer policy questions. We will consider theoretical questions about which details are important in scaling models up to large populations and these should be critical in developing workable, large-scale models. The agenda of this working group is sufficiently broad that it will probably make sense to break it into several groups after awhile.

Altmann, M., Wee, B., Willard, K., Peterson, D., and Gatewood, L. (1994),
"Network analytic methods for epidemiological risk assessment,"
*Stat. in Med.*, **13**, 53-60.

Ball, F., Mollison, D., and Scalia-Tomba, G. (1997),
"Epidemics with mixing at two levels,"
*Ann. Appl. Prob.*, **7**, 45-89.

Dieckmann, U., Law, R., and Metz, J.A.J. (eds.) (2000),
*The Geometry of Spatial Interactions*, Cambridge University Press.

Gilks, W., Richardson, S., and Spiegelhalter, D. (1995),
"Markov chain Monte Carlo," *Practice*, Chapman and Hall.

Keeling, M., (2000), "Multiplicative moments and measures of persistence in ecology,"
*J. Theo. Biol.*, **205**, 269-281.

Keeling, M., and Gilligan, C.A. (2000), "Metapopulation dynamics of bubonic plague,"
*Nature*, **407**, 903-906.

Kretzschmar, M., and Morris, M. (1996),
"Measures of concurrency in networks and the spread of infectious disease,"
*Mathematical Biosciences*, **133**, 165-95.

Levin, S.A., and Durrett, R. (1997), "From individuals to epidemics,"
*Phil. Trans. Roy. Soc. Lond.*, **B351**, 1615-1621.

Levin, S.A., Grenfell, B., Hastings, A., and Perelson, A.S. (1997),
"Mathematical and computational challenges in population biology and ecosystems science,"
*Science*, **257**, 334-343.

McGlade, J. (ed). (1999), *Advanced Ecological Theory*, Blackwell.

Mollison, D. (ed.) (1995), *Epidemic Models: Their Structure and Relation to Data*,
Cambridge University Press.

Morris, M., and Dean, L. (1994),
"Effects of sexual behavior change on long-term human immunodeficiency virus prevalence among homosexual men,"
*Amer. J. Epidemiol.*, **140**, 217-231.

O'Neill, P.D., Balding, D.J., Becker, N.G., Eerola, M., and Mollison, D. (2000),
"Analyses of infectious disease data from household outbreaks by Markov chain Monte Carlo methods,"
*J. Roy. Statist. Soc. C*, **49**, 517-542.

Peterson, D., Willard, K., Altmann, M., Gatewood, L., and Davidson, G. (1990),
"Monte Carlo simulation of HIV infection in an intravenous-drug-user community,"
*J. AIDS*, 1086-1095.

Rand, D.A., Keeling, M., and Wilson, H.B. (1995),
"Invasion, stability and evolution to criticality in spatially extended, artificial host-pathogen ecologies,"
*Plroc. R. Soc. Lond.*, **259**, 55-63.

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Document last modified on March 31, 2003.