Abstract
The process of infection during an epidemic can be envisaged as being transmitted via a network of routes represented by a contact network. Most differential equation models of epidemics are mean-field models. These contain none of the underlying spatial structure of the contact network. By extending the mean-field models to pair-level, some of the spatial structure can be contained in the model. Some networks of transmission such as river or transportation networks are clearly asymmetric, whereas others such as airborne infection can be regarded as symmetric. Pair-level models have been developed to describe symmetric contact networks. Here we report on work to develop a pair-level model that is also applicable to asymmetric contact networks. The procedure for closing the model at the level of pairs is discussed in detail. The model is compared against stochastic simulations of epidemics on asymmetric contact networks and against the predictions of the symmetric model on the same networks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, R.M., May, R.M.: Infectious diseases of humans. Oxford University Press, 1992
Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: The Legacy of Kermack and McKendrick. In: Epidemic Models, their Structure and Relation to Data. ed. Mollison, D., Cambridge Univ. Press, pp. 95–115 (1995)
Ferguson, N.M., Donnelly, C.A., Anderson, R.M.: The foot-and-mouth epidemic in Great Britain: Pattern of spread and impact of interventions. Science 292, 1155–1160 (2001)
Grenfell, B.T., Dobson, A.P.: Ecology of infectious diseases in natural populations. Cambridge University Press, 1995
Keeling, M.J.: The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B266, 859–867 (1999)
Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics, 1. Proc. R. Soc. Edinb. A115, 700–721 (1927)
Matsuda, H.N., Ogita A., Sasaki, A., Sato K.: Statistical mechanics of population: The lattice Lotka-Volterra model. Progress in Theoretical Physics 88, 1035–1049 (1992)
Mollison, D.: Spatial contact models for ecological and epidemic spread. Journal of the Royal Statistical Society. B 39 (3), 283–326 (1977)
Morris, A.J.: Representing spatial interactions in simple ecological models. PhD thesis, Warwick University 1997
Murray, J.D.: Mathematical Biology. Springer, 2003
Rand, D.A.: Correlation equations for spatial ecologies. In: Advanced ecological theory ed. McGlade, J, Blackwell Scientific Publishing, pp. 99–143 (1999)
Renshaw, E.: Modelling biological populations in space and time. Cambridge University Press, (1991)
van Baalen, M.: Pair Approximations for Different Spatial Geometries. In: The Geometry of Ecological Interactions: Simplifying Complexity, (eds.) Dieckmann, U., Law, R., Metz, J.A.J., pp. 359-387 (2000)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ``small-world'' networks. Nature 393, 440-442 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
DEFRA funded project FC1153
Rights and permissions
About this article
Cite this article
Sharkey, K., Fernandez, C., Morgan, K. et al. Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks. J. Math. Biol. 53, 61–85 (2006). https://doi.org/10.1007/s00285-006-0377-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-006-0377-3