The spread of infectious diseases in spatially structured populations: an invasory pair approximation.

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.

[1]  Boots,et al.  The evolutionary dynamics of local infection and global reproduction in host–parasite interactions , 2000 .

[2]  J. Filipe,et al.  Hybrid closure-approximation to epidemic models , 1999 .

[3]  Jerram L. Brown,et al.  Animal Societies: Theories and Facts , 1987 .

[4]  T. E. Harris.  Contact Interactions on a Lattice , 1974 .

[5]  Benjamin M Bolker,et al.  The effects of disease dispersal and host clustering on the epidemic threshold in plants , 2004, Bulletin of mathematical biology.

[6]  Space: the final frontier for ecological theory : Spatial theory , 1994 .

[7]  C T Bauch,et al.  A versatile ODE approximation to a network model for the spread of sexually transmitted diseases , 2002, Journal of mathematical biology.

[8]  A. Morris,et al.  Representing spatial interactions in simple ecological models , 1997 .

[9]  V Isham,et al.  Assessing the variability of stochastic epidemics. , 1991, Mathematical biosciences.

[10]  J. Dushoff,et al.  Host heterogeneity and disease endemicity: a moment-based approach. , 1999, Theoretical population biology.

[11]  Ulf Dieckmann,et al.  The Geometry of Ecological Interactions: Simplifying Spatial Complexity , 2000 .

[12]  M. Keeling,et al.  The effects of local spatial structure on epidemiological invasions , 1999, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[13]  S. Levin,et al.  Mathematical and Computational Challenges in Population Biology and Ecosystems Science , 1997, Science.

[14]  B. Bolker,et al.  Spatial Moment Equations for Plant Competition: Understanding Spatial Strategies and the Advantages of Short Dispersal , 1999, The American Naturalist.

[15]  J. Mcglade,et al.  Advanced ecological theory : principles and applications , 1999 .

[16]  C. Bauch,et al.  Using network models to approximate spatial point-process models. , 2003, Mathematical biosciences.

[17]  R. Durrett,et al.  The Importance of Being Discrete (and Spatial) , 1994 .

[18]  Akira Sasaki,et al.  Pathogen invasion and host extinction in lattice structured populations , 1994, Journal of mathematical biology.

[19]  Kei-ichi Tainaka,et al.  Paradoxical effect in a three-candidate voter model , 1993 .

[20]  Stephen P. Ellner,et al.  Speed of invasion in lattice population models: pair-edge approximation , 1998 .

[21]  Minus van Baalen,et al.  The Geometry of Ecological Interactions: Pair Approximations for Different Spatial Geometries , 2000 .

[22]  B M Bolker,et al.  Analytic models for the patchy spread of plant disease , 1999, Bulletin of mathematical biology.

[23]  B. T. Grenfell,et al.  Disease Extinction and Community Size: Modeling the Persistence of Measles , 1997, Science.

[24]  Christopher Thomas Bauch.  Moment closure approximations in epidemiology , 2000 .

[25]  M. Baalen,et al.  The Unit of Selection in Viscous Populations and the Evolution of Altruism. , 1998, Journal of theoretical biology.

[26]  Peter Kareiva,et al.  Special Feature: Space: The Final Frontier for Ecological Theory , 1994 .

[27]  D A Rand,et al.  A moment closure model for sexually transmitted disease transmission through a concurrent partnership network , 2000, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[28]  Ulf Dieckmann,et al.  A DYNAMICAL SYSTEM FOR NEIGHBORHOODS INPLANT COMMUNITIES , 2000 .

[29]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[30]  David A. Rand,et al.  Correlation Equations and Pair Approximations for Spatial Ecologies , 1999 .

[31]  H. Matsuda.  Conditions for evolution of altruism , 1987 .

[32]  Christl A. Donnelly,et al.  The Foot-and-Mouth Epidemic in Great Britain: Pattern of Spread and Impact of Interventions , 2001, Science.

[33]  J A N Filipe,et al.  Inferring the dynamics of a spatial epidemic from time-series data , 2004, Bulletin of mathematical biology.

[34]  Jacqueline McGlade,et al.  Advanced Ecological Theory , 1999 .

[35]  Akira Sasaki,et al.  Statistical Mechanics of Population: The Lattice Lotka-Volterra Model , 1992 .

[36]  David A. Rand,et al.  Invasion, stability and evolution to criticality in spatially extended, artificial host—pathogen ecologies , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[37]  G. Macdonald,et al.  The analysis of equilibrium in malaria. , 1952, Tropical diseases bulletin.