Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T17:29:59.583Z Has data issue: false hasContentIssue false
  • English
  • Français

Markovian contact processes

Published online by Cambridge University Press:  01 July 2016

Denis Mollison
Denis Mollison*
Affiliation:
Heriot-Watt University, Edinburgh
Get access Rights & Permissions [Opens in a new window]

Abstract

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St(θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists.

An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St/t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes.

Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

Keywords

BRANCHING RANDOM WALKCONTACT DISTRIBUTIONECOLOGICAL DIFFUSIONEPIDEMIC DIFFUSIONGEOGRAPHICAL SPREADMARKOVIAN CONTACT PROCESSMIGRATIONNON-LINEAR DIFFUSIONPERCOLATION PROCESSPOPULATION PROCESSSPACE-TIME MODELLINGSPATIAL CONTACT PROCESSSPREAD OF EPIDEMICSSTEPPING-STONE MODELSVELOCITY OF SPREAD
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 1 , March 1978 , pp. 85 - 108
Copyright
Copyright © Applied Probability Trust 1978 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ammerman, A. and Cavalli-Sforza, L. L. (1973) A population model for the diffusion of early farming in Europe. In Institute of Archaeology Research Seminar, ed. Renfrew, Colin. Duckworth, London, 343–357.Google Scholar
Atkinson, C. and Reuter, G. E. H. (1976) Deterministic epidemic waves. Math. Proc. Camb. Phil. Soc. 80, 315–330.Google Scholar
Bailey, N. T. J. (1968) Stochastic birth, death and migration processes for spatially distributed populations. Biometrika 50, 189–198.Google Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
Biggins, J. D. (1976) Asymptotic Properties of the Branching Random Walk. Oxford University .Google Scholar
Biggins, J. D. (1978) The asymptotic shape of the branching random walk. Adv. Appl. Prob. 10, 62–84.Google Scholar
Daniels, H. E. (1975) The deterministic spread of a simple epidemic. In Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett. ed. Gani, J. Distributed for the Applied Probability Trust by Academic Press, London, 373–386.Google Scholar
Daniels, H. E. (1977) The advancing wave in a spatial birth process. J. Appl. Prob. 14, 689–701.CrossRefGoogle Scholar
Ellis, R. S. (1977) Review of McKean (1975) Math. Rev. (to appear).Google Scholar
Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369.Google Scholar
Halliday, , (1929) in discussion of Soper, H. E. J. R. Statist. Soc. 92, 67–69.Google Scholar
Hammersley, J. M. (1966) First-passage percolation. J. R. Statist. Soc. B 28, 491–496.Google Scholar
Hammersley, J. M. (1974) Postulates for subadditive processes. Ann. Prob. 2, 652–680.Google Scholar
Howell, R. S. (1975) Dutch elm disease data (abstract). Adv. Appl. Prob. 7, 463–464.Google Scholar
Jardine, N. and McKenzie, D. (1972) Continental drift and the dispersal and evolution of organisms. Nature (London) 235, 20–24.Google Scholar
Kendall, D. G. (1965) Mathematical models of the spread of infection. In Mathematics and Computer Science in Biology and Medicine. Medical Research Council, London, 213–225.Google Scholar
Kesten, H. (1973) in discussion of Kingman, J. F. C., Subadditive ergodic theory. Ann. Prob. 1, 903.Google Scholar
Kingman, J. F. C. (1975) The first birth problem for an age-dependent branching process. Ann. Prob. 3, 790–801.CrossRefGoogle Scholar
Kolmogoroff, A. N., Petrovsky, I. G. and Piscounoff, N. S. (1937), Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. de l'Univ. d'État Moscou (sér. intern.) A 1(6), 1–25.Google Scholar
McKean, H. P. (1975) Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piscunov. Comm. Pure Appl. Maths. 28, 323–331.Google Scholar
Mollison, D. (1972a) Possible velocities for a simple epidemic. Adv, Appl. Prob. 4, 233–258.Google Scholar
Mollison, D. (1972b) The rate of spatial propagation of simple epidemics. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 579–614.Google Scholar
Mollison, D. (1977a) The velocity of stochastic processes. Proc. 1974 E.M.S. (Prague).Google Scholar
Mollison, D. (1977b) Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.Google Scholar
Mollison, D. and Daniels, H. E. (1978) The deterministic simple epidemic unmasked. Submitted for publication.Google Scholar
Morgan, R. W. and Welsh, D. J. A. (1965) A two-dimensional Poisson growth process. J. R. Statist. Soc. B 27, 497–504.Google Scholar
Neyman, J. and Scott, E. L. (1964) A stochastic model of epidemics. In Stochastic Models in Medicine and Biology, University of Wisconsin Press, Madison, 45–83.Google Scholar
Radcliffe, J. (1973) The initial geographical spread of host-vector and carrier-borne epidemics. J. Appl. Prob. 10, 703–717.Google Scholar
Richardson, D. (1973) Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar
Skellam, J. G. (1951) Random dispersal in theoretical populations. Biometrika 38, 196–218.Google Scholar
Stakman, E. C. and Harrar, J. G. (1957) Principles of Plant Pathology. Ronald Press, New York.Google Scholar
Stirzaker, D. (1977) On a birth process. In Recent Developments in Statistics, Proc. 1976 E.M.S. (Grenoble). North-Holland, Amsterdam.Google Scholar
Wierman, J. C. (1978) The front velocity of the simple epidemic.Google Scholar
Williams, T. and Bjerknes, R. (1972) Stochastic model for abnormal clone spread through epithelial basal layer. Nature (London) 236, 19–21.Google Scholar
Wright, P. B. (1969) Effects of wind and precipitation on the spread of foot-and-mouth disease. Weather 24, 204–213.Google Scholar
Ziegler, P. (1969) The Black Death. Penguin Books, Harmondsworth.Google Scholar