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The effects of disease dispersal and host clustering on the epidemic threshold in plants

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Abstract

For an epidemic to occur in a closed population, the transmission rate must be above a threshold level. In plant populations, the threshold depends not only on host density, but on the distribution of hosts in space. This paper presents an alternative analysis of a previously presented stochastic model for an epidemic in continuous space (Bolker, 1999, Bull. Math. Biol. 61, 849–874). A variety of moment closures are investigated to determine the dependence of the epidemic threshold on host spatial distribution and pathogen dispersal. Local correlations that arise during the early phase of the outbreak determine whether a true global epidemic will occur.

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References

  • Ackermann, R. V. and R. W. Schlische (1997). Anticlustering of small normal faults around larger faults. Geology 25, 1127–1130.

    Article  Google Scholar 

  • Alexander, H. M. (1989). Spatial heterogeneity and disease in natural plant populations, in Spatial Components of Plant Disease Epidemics, M. J. Jeger (Ed.), Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Anderson, R. M. (1991). Discussion: The Kermack-McKendrick epidemic threshold theorem. Bull. Math. Biol. 53, 3–32.

    Article  Google Scholar 

  • Augspurger, C. K. (1984). Seedling survival of tropical tree species: interactions of dispersal distance, light gaps, and pathogens. Ecology 65, 1705–1712.

    Article  Google Scholar 

  • Aylor, D. E. (1989). Aerial spore dispersal close to a focus of disease. Agr. Forest Meteorol. 47, 109–122.

    Article  Google Scholar 

  • Bergelson, J., J. A. Newman and E. M. Floresroux (1993). Rates of weed spread in spatially heterogeneous environments. Ecology 74, 999–1011.

    Article  Google Scholar 

  • Bohan, D. A., A. C. Bohan, D. M. Glen, W. O. C. Symondson, C. W. Wiltshire and L. Hughes (2000). Spatial dynamics of predation by carabid beetles on slugs. J. Anim. Ecol. 69, 267–379.

    Google Scholar 

  • Bolker, B. M. (1999). Analytic models for the patchy spread of plant disease. Bull. Math. Biol. 61, 849–874.

    Article  Google Scholar 

  • Bolker, B. M. and S. W. Pacala (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52, 179–197.

    Article  MATH  Google Scholar 

  • Bolker, B. M. and S. W. Pacala (1999). Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am. Nat. 153, 575–602.

    Article  Google Scholar 

  • Brophy, L. S. and C. C. Mundt (1991). Influence of plant spatial patterns on disease dynamics, plant competition and grain yield in genetically diverse wheat populations. Agr. Ecosyst. Environ. 35, 1–12.

    Article  Google Scholar 

  • Burdon, J. J. and G. A. Chilvers (1976). The effect of clumped planting patterns on epidemics of damping-off disease in cress seedlings. Oecologia 23, 17–29.

    Google Scholar 

  • Carlsson, U. and T. Elmqvist (1992). Epidemiology of anther-smut disease Microbotryum violaceum and numeric regulation of population of Silene dioica. Oecologia 90, 509–517.

    Article  Google Scholar 

  • Carlsson, U., T. Elmqvist, A. Wennstrom and L. Ericson (1990). Infection by pathogens and population age of host plants. J. Ecol. 78, 1094–1105.

    Google Scholar 

  • Cole, B. J., K. Haight and D. C. Wiernasz (2001). Distribution of Myrmecocystus mexicanus (Hymenoptera: Formicidae): Association with Pogonomyrmex occidentalis (Hymenoptera: Formicidae). Ann. Entomol. Soc. Am. 94, 59–63.

    Google Scholar 

  • Cressie, N. A. C. (1991). Statistics for Spatial Data, New York: Wiley.

    MATH  Google Scholar 

  • Dale, M. R. T., P. Dixon, M.-J. Fortin, P. Legendre, D. E. Myers and M. S. Rosenberg (2002). Conceptual and mathematical relationships among methods for spatial analysis. Ecography 25, 558–577.

    Article  Google Scholar 

  • Diekmann, O. (1978). Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130.

    MATH  MathSciNet  Google Scholar 

  • Diekmann, O., J. A. P. Heesterbeek and J. A. J. Metz (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382.

    Article  MathSciNet  MATH  Google Scholar 

  • Dieckmann, U. and R. Law (2000). Relaxation projections and the method of moments, in The Geometry of Ecological Interactions: Simplifying Spatial Complexity, U. Dieckmann, R. Law and J. A. J. Metz (Eds), Cambridge: Cambridge University Press.

    Google Scholar 

  • Diggle, P. (1983). Statistical Analysis of Spatial Point Patterns, New York: Academic Press.

    MATH  Google Scholar 

  • Durrett, R. (1995). Spatial epidemic models, in Epidemic Models: Their Structure and Relation to Data, D. Mollison (Ed.), Cambridge: Cambridge University Press.

    Google Scholar 

  • Durrett, R. and S. A. Levin (1994). The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394.

    Article  MATH  Google Scholar 

  • Filipe, J. A. N. and G. J. Gibson (1998). Studying and approximating spatio-temporal models for epidemic spread and control. Philos. Trans. R. Soc. Lond. B 353, 2153–2162.

    Article  Google Scholar 

  • Filipe, J. A. N. and G. J. Gibson (2001). Comparing approximations to spatio-temporal models for epidemics with local spread. Bull. Math. Biol. 63, 603–624.

    Article  Google Scholar 

  • Fry, N. (1979). Random point distributions and strain measurement in rocks. Tectonophysics 60, 89–105.

    Article  Google Scholar 

  • Gubbins, S., C. A. Gilligan and A. Kleczkowski (2000). Population dynamics of plant-parasite interactions: Thresholds for invasion. Theor. Popul. Biol. 57, 219–233.

    Article  MATH  Google Scholar 

  • Guttorp, P., D. R. Brillinger and F. P. Schoenberg (2002). Point processes, spatial, in Encyclopedia of Environmetrics, Vol. 3, A. H. El-Shaarawi and W. W. Piegorsch (Eds), Chichester: Wiley and Sons.

    Google Scholar 

  • Holmes, E. (1997). Basic epidemiological concepts in a spatial context, in Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, D. Tilman and P. Kareiva (Eds), Princeton: Princeton University Press.

    Google Scholar 

  • Jeger, M. J. (1986). Asymptotic behaviour and threshold criteria in model plant disease epidemics. Plant Pathol. 35, 355–361.

    Google Scholar 

  • Jeger, M. J. (Ed.), (1989). Spatial Components of Plant Disease Epidemics, Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Jeger, M. J. and F. van den Bosch (1994). Threshold criteria for model plant disease epidemics. I. Asymptotic results. Phytopathology 84, 24–27.

    Google Scholar 

  • Keeling, M. J. (1999). The effects of local spatial structure on epidemiological invasions. Philos. R. Soc. Lond. B 266, 859–867.

    Article  Google Scholar 

  • Keeling, M. J. and B. T. Grenfell (2000). Individual-based perspectives on R 0. J. Theor. Biol. 203, 51–61.

    Article  Google Scholar 

  • Kendall, D. G. (1957). Discussion of ‘Measles periodicity and community size’ by M. S. Bartlett. J. R. Stat. Soc. A 120, 48–70.

    Article  Google Scholar 

  • Kermack, W. O. and A. G. McKendrick (1927). A contribution to the mathematical theory of epidemics. Philos. R. Soc. Lond. A 115, 700–721.

    Google Scholar 

  • Kleczkowski, A. and B. T. Grenfell (1999). Mean-field-type equations for spread of epidemic: the ’small world’ model. Physica A 274, 355–360.

    Article  Google Scholar 

  • Kot, M., M. A. Lewis and P. van den Driessche (1996). Dispersal data and the spread of invading organisms. Ecology 77, 2027–2042.

    Article  Google Scholar 

  • Law, R. and U. Dieckmann (2000). Moment approximations of individual-based models, in The Geometry of Ecological Interactions: Simplifying Spatial Complexity, U. Dieckmann, R. Law and J. A. J. Metz (Eds), Cambridge: Cambridge University Press.

    Google Scholar 

  • Levin, S. A. and R. Durrett (1996). From individuals to epidemics. Philos. Trans. R. Soc. Lond. B 351, 1615–1621.

    Google Scholar 

  • Lewis, M. A. and S. Pacala (2000). Modeling and analysis of stochastic invasion processes. J. Math. Biol. 41, 387–429.

    Article  MathSciNet  MATH  Google Scholar 

  • Madden, L. V., M. J. Jeger and F. van den Bosch (2000). A theoretical assessment of the effects of vector-virus transmission on plant virus disease epidemics. Phytopathology 90, 576–594.

    Google Scholar 

  • Matsuda, H., N. Ogita, A. Sasaki and K. Sato (1992). Statistical mechanics of population. Prog. Theor. Phys. 88, 1035–1049.

    Article  Google Scholar 

  • May, R. M. (1990). Population biology and population genetics of plant-pathogen associations, in Pests, Pathogens and Plant Communities, J. J. Burdon and S. R. Leather (Eds), Oxford: Blackwell Scientific.

    Google Scholar 

  • May, R. M. and R. M. Anderson (1989). The transmission dynamics of human immunodeficiency virus (HIV), in Applied Mathematical Ecology, S. A. Levin, T. G. Hallam and L. J. Gross (Eds), Berlin: Springer.

    Google Scholar 

  • McCallum, H., N. Barlow and J. Hone (2001). How should pathogen transmission be modelled? TREE 16, 295–300.

    Google Scholar 

  • McCartney, H. A. and B. D. L. Fitt (1987). Spore dispersal gradients and disease development, in Populations of Plant Pathogens: Their Dynamics and Genetics, M. S. Wolfe and C. E. Caten (Eds), Oxford: Blackwell Scientific.

    Google Scholar 

  • McElhany, P., L. Real and A. Power (1995). Vector preference and disease dynamics: A study of barley yellow dwarf virus. Ecology 76, 444–457.

    Article  Google Scholar 

  • Metz, J. A. J. and F. van den Bosch (1995). Velocities of epidemic spread, in Epidemic Models: Their Structure and Relation to Data, D. Mollison (Ed.), Cambridge: Cambridge University Press.

    Google Scholar 

  • Minogue, K. P. (1989). Diffusion and spatial probability models for disease spread, in Spatial Components of Plant Disease Epidemics, M. J. Jeger (Ed.), Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Mundt, C. C. (1989). Modeling disease increase in host mixtures, in Plant Disease Epidemiology, Vol. II, K. J. Leonard and W. Fry (Eds), New York: McGraw-Hill.

    Google Scholar 

  • Mundt, C. C. and J. A. Browning (1985). Development of crown rust epidemics in genetically diverse host populations: Effect of genotype unit area. Phytopathology 75, 607–610.

    Article  Google Scholar 

  • Nasell, I. (1995). The threshold concept in deterministic and stochastic models, in Epidemic Models: Their Structure and Relation to Data, D. Mollison (Ed.), Cambridge: Cambridge University Press.

    Google Scholar 

  • Onstad, D. W. (1992). Evaluation of epidemiological thresholds and asymptotes with variable plant densities. Phytopathology 82, 1028–1032.

    Google Scholar 

  • Ramsay, J. G. and M. I. Huber (1983). The Techniques of Modern Structural Analysis, vol. 1, London: Academic Press.

    Google Scholar 

  • Real, L. A. and P. McElhany (1996). Spatial pattern and process in plant-pathogen interactions. Ecology 77, 1011–1025.

    Article  Google Scholar 

  • San Jose, J. J., M. R. Farinas and J. Rosales (1991). Spatial patterns of trees and structuring factors in a Trachypogon savanna of the Orinoco Llanos. Biotropica 23, 114–123.

    Article  Google Scholar 

  • Sato, K., H. Matsuda and A. Sasaki (1994). Pathogen invasion and host extinction in lattice structured populations. J. Math. Biol. 32, 251–268.

    Article  MATH  Google Scholar 

  • Thieme, H. R. (1977). A model for the spatial spread of an epidemic. J. Math. Biol. 4, 337–351.

    Article  MATH  MathSciNet  Google Scholar 

  • Thrall, P. H. and J. J. Burdon (1997). Host-pathogen dynamics in a metapopulation context: The ecological and evolutionary consequences of being spatial. J. Ecology 85, 743–753.

    Google Scholar 

  • Thrall, P. H. and J. J. Burdon (1999). The spatial scale of pathogen dispersal: Consequences for disease dynamics and persistence. Evol. Ecol. Res. 1, 681–701.

    Google Scholar 

  • Watve, M. G. and M. M. Jog (1997). Epidemic diseases and host clustering: An optimum cluster size ensures maximum survival. J. Theor. Biol. 184, 165–169.

    Article  Google Scholar 

  • Wright, S. (1946). Isolation by distance under diverse systems of mating. Genetics 31, 39–59.

    Google Scholar 

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Authors and Affiliations

  1. Department of Agronomy and Range Science, University of California, Davis, CA, 95616-8633, USA

    David H. Brown

  2. Department of Zoology, University of Florida, USA

    Benjamin M. Bolker

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  1. David H. Brown
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  2. Benjamin M. Bolker
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Correspondence to David H. Brown.

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Brown, D.H., Bolker, B.M. The effects of disease dispersal and host clustering on the epidemic threshold in plants. Bull. Math. Biol. 66, 341–371 (2004). https://doi.org/10.1016/j.bulm.2003.08.006

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  • DOI: https://doi.org/10.1016/j.bulm.2003.08.006

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